Determining Melt-Crystallization Mechanisms and Structures of Disordered Crystalline Solids

ABSTRACT

DILL, ERIC DELAMARTER. Determining Melt-Crystallization Mechanisms and Structures of Disordered Crystalline Solids. (Under the direction of James D. Martin).

The crystal growth kinetics of the halozeotype CZX-1 were explored with a series of isothermal crystallization experiments performed with Differential Scanning Calorimetry (DSC) and 2-D Temperature and time Resolved X-ray Diffraction (2-D TtXRD). Fitting the transformations to the well-known Kolmogorov-Johnson-Mehl-Avrami (KJMA) reaction model resulted in rate constants that appeared to be strongly dependent upon the sample size. Crystal growth simulations were designed to elucidate the relationship between the observed crystal growth rate and the sample size. These simulations unequivocally demonstrated that the KJMA rate constant is highly dependent on the sample size and the sample shape. Under the simulated conditions of slow nucleation relative to crystal growth, sample volume and sample anisotropy correction affords a means to eliminate the experimental condition dependence of the KJMA rate constant, producing the material specific parameter, the velocity of the phase boundary. Additionally, the crystallization simulations allow us to clearly define the KJMA parameters: t0 is nominally the induction time of the first crystallite in the system; n is the dimensionality of the transformation process; k, when volume and anisotropy is accounted for, is equivalent to the crystallization phase boundary velocity within a factor of two. The nucleation process in CZX-1 was explored from the “bulk” perspective with DSC and radially integrated TtXRD. No evidence for nucleation according to the mechanism proposed by Classical Nucleation Theory was found. However, a maximum in the initial nucleation “rate” was observed, defined as the inverse of the initial nucleation time from the -1 KJMA model: knuc=t0 . A method for direct analysis of the 2D TtXRD diffraction images is presented where the intensity of individual diffraction spots are fit to the KJMA model to determine their t0 and knuc values. The direct analysis of 2D diffraction images demonstrates the capability to quantitatively estimate the number of crystallites that appeared during an isothermal crystallization experiment. The structure of the plastic crystalline carbon tetrabromide, -CBr4 was investigated as a potential model of the structure of the phase boundary between liquid and crystalline phases. Two-dimensional single-crystal synchrotron X-ray diffraction of the high-temperature plastic phase of carbon tetrabromide (α-CBr4, 퐹푚3̅푚, a≈8.82 Å) was collected which reveals a twinned plastic crystal and twinned structured diffuse scattering with sharp 푮 ± {110}∗ sheets of diffuse intensity (where G represents the set of 퐹푚3̅푚 Bragg reflections). The real- space manifestation of the intense and highly structured diffuse scattering is demonstrated through 2D Patterson Function analysis. The structural understanding is augmented by a series of simulations. The analysis of the diffraction images and the simulations suggest that the CBr4 units reside in D2d site symmetry on average, consistent with previous reports. All these structural analyses find that the structure of α-CBr4 is well-described by static disorder, in clear contrast to the commonly accepted notion of the plastic phase possessing dynamic reorientational freedom.

by Eric Delamarter Dill

A dissertation submitted to the Graduate Faculty of North Carolina State University in partial fulfillment of the requirements for the degree of Doctor of Philosophy

Chemistry

Raleigh, North Carolina

2014

APPROVED BY:

______James D. Martin David A. Shultz Committee Chair

______Mike H. Whangbo Paul A. Maggard

DEDICATION This work is dedicated to my incredible parents and my amazing wife, without whom none of this would have been possible.

BIOGRAPHY

Eric was born August 9th, 1895 in Memphis, Tennessee to parents Robert and Kristen Dill. Eric grew up in Raleigh, North Carolina and went through the Magnet program of the Wake County Public School System. He attended the John W. Ligon GT Magnet Middle school and William G. Enloe Magnet High School. He went to North Carolina State University where he pursued degrees in Chemistry and Physics. Eric began working with Professor Jim Martin as an undergraduate researcher in the 2006 when they crossed paths in Jim's Inorganic Chemistry 1 course. Eric remained at North Carolina State University to pursue a graduate degree in Inorganic Chemistry with Jim after completing his undergraduate studies. Good thing he decided to stay at NC State for graduate school because he met his amazing wife while in graduate school.

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ACKNOWLEDGEMENTS

First and foremost, this work is a testament to the dedication to teaching and mentoring of my advisor, Jim. Your open door policy provides so much support to your graduate students (though I suppose you know this). I always felt that I could come ask you questions and knew that you wouldn’t feel bothered or put out by having us come ask you questions. Every day I knew I would learn something new from you, be it in the classroom while I was your TA, at group meeting, or during the course of one of our countless rounds of debate over crystallization and disorder. You taught me what it means to be a scientist by proving guidance when I needed it most and giving me the intellectual freedom of self-discovery. Jaap, my dissertation is also a testament to your open door policy and willingness to (almost always) drop everything to teach me about things that can be done with statistics that seem more like voodoo than math, to teach me about the black magic of fourier transforms and sometimes just to debate the finer points of esoteric phase transition theory (Landau, anyone?) Your ability to play devil’s advocate is unparalleled and therefore, simply put, maddening, because you’re so freaking good at it. Every time I thought I had come to a reasonable solution, you would find a clever way to poke a hole in it. I knew that I could always come to you with questions and you would always be excited to dive into the weeds. Thank you Jim and Jaap. Without these two brilliant men, the research reported in this dissertation would most certainly not have been possible. Both of you have taught me what it means to be a great teacher. Your passion for teaching and your desire to watch students succeed is inspiring. Thank you. Thank you to the Martin Group members, past and present, especially Feier, whose love of all things cute always provided a much needed break from disorder and mechanisms. Your herculean effort to beat the hydra (i.e., DSC crystallization kinetics) into reliable results proved to be a cornerstone of the crystallization mechanism research. Every time you thought you were done with that analysis, two more heads sprang up. Thank you to Amanda for being a wonderful mentor and friend and for paving the way with the crystallization mechanism studies. Thank you to Brad for becoming our resident expert in all things computationally quantum and keeping me honest about my appreciation for the dark magic of computational

quantum mechanics! Best of luck with that zinc chloride hydrate structure and remember, if it’s yellow, let it mellow. Thank you to the wacky and wild cast of undergraduate researchers that I’ve worked with over the years. Eric R., you’re one of the finest people I’ve ever met and it was wonderful to be able to work with and mentor you over the years. Elijah, I wish we had a chance to work together more, but best of luck in graduate school! I know you guys will do incredible things. To Mick the pirate; keep doing your thing because you’re awesome at it. I will most certainly miss those days when I walk into the lab and there’s a pirate carefully tinkering with the microbalance. James, keep on doing your thing. You’ve got the incredible gift of creating seemingly magical devices from everyday things. That dip-probe spectrometer will probably amaze me until the day I die. Daniel, even though your time with us at NC State was short and you’ll likely return to Brown to finish your studies, you’re the most dedicated undergraduate student I’ve ever met and your work ethic is unparalleled. I know you’ll be successful at whatever you set your mind to. Perhaps you’ve got enough baseball gloves though? Troy, though we only had limited interaction, I feel comfortable advising you to watch out for exploding (or was it an implosion?) drybox gloves. It’s been an absolute treat to work with all of you over the years. Please consider me a resource to pull from whenever you need and let me know if I can ever be of assistance! (Brad and Feier, that goes for you too!) Thank you to Dave B. and Rob K., who were always willing to listen to me babble on about crystallization mechanisms and disordered materials, even when you would rather be doing (literally) almost anything else. Thank you to Justin K. for being a great friend and overnight-work-companion when we ended up on the same weird schedule while I was writing my prelim and you were writing your dissertation. Thank you to those of you who participated in those weekly morning meetings and the biweekly writing workshops in the spring semester of 2013 (Michael, Meghan, Alecia, Ryan, Sandra, Suzanne, Timia, Rhonda and Mike). Those meetings took quite a lot of the sting out of the dissertation writing process. Finally, thank you to my committee members: Dr. Mike Whangbo, Dr. David Shultz, Dr. Paul Maggard and Dr. Mike Carter; for taking the time to read this document and for taking

part in my final defense. Thank you to the Department of Chemistry at North Carolina State University. Thank you to the scientists at Brookhaven National Laboratory and Argonne National Laboratory who have allowed our group to perform experiments even though our rate of publication was not what you wanted. Without the support of those national facilities, none of this research would have been possible. I would like to thank especially Dr. Elaine DiMasi at Brookhaven National Lab (National Synchrotron Light Source) and Dr. Peter Chupas and Dr. Karena Chapman at Argonne National Lab (Advanced Photon Source).

TABLE OF CONTENTS Chapter 1: Introduction ...... 1 1.1. Motivation ...... 2 1.2. Prior Work/Framing Questions ...... 4 1.3. The Experimental Systems...... 7

1.3.1. The Halozeolite, CZX-1: [HNMe3][CuZn5Cl12]...... 8 1.3.2. Plastic Crystalline Carbon Tetrabromide...... 9 1.4. Summary of Research Presented ...... 9 Chapter 2: Experimental Determination of the Crystallization Phase Boundary Velocity in the Halozeotype CZX-1 ...... 11 2.1. Introduction ...... 12 2.2. Experimental Methods ...... 14 2.2.1. Isothermal Crystallization: DSC ...... 14 2.2.2. Isothermal Crystallization: TtXRD ...... 15 2.2.3. Sample Volume and Density ...... 17 2.3. Results and Discussion ...... 19 2.3.1. Isothermal Crystallization: DSC ...... 19 2.3.2. Isothermal Crystallization: 2D TtXRD ...... 20 2.3.3. KJMA Parameter Correlations ...... 26 2.3.4. Isothermal Crystallization Rate...... 29 2.4. Conclusions ...... 33 Chapter 3: Crystal Growth Simulations to Establish Physically Relevant Kinetic Parameters from the Empirical KJMA Model ...... 35 3.1. Introduction ...... 36 3.2. Background ...... 36 3.3. Theory ...... 40 3.4. Simulation ...... 44 3.4.1. Simulation Methods ...... 44 3.4.2. Simulated Sample Geometry ...... 45

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3.4.3. Nucleation and Growth...... 46 3.4.4. KJMA Analysis of Simulations ...... 47 3.5. Results and Discussion ...... 49 3.5.1. Isotropic Geometries ...... 52 3.5.2. Anisotropic Geometries ...... 55 3.6. Conclusion ...... 59 Chapter 4: Probing the Mechanisms of Nucleation in Melt-Crystal Reactions ...... 63 4.1. Introduction ...... 64 4.2. Classical Nucleation Theory ...... 66 4.3. Experimental Methods ...... 68 4.3.1. Determination of the quench time ...... 69 4.3.2. 2D TtXRD measurement and analysis of individual crystallites ...... 69 4.3.3. Spotpicking Algorithm ...... 70 4.3.4. Individual Spot Kinetics ...... 75 4.4. Results & Discussion ...... 76 4.4.1. Bulk Nucleation Kinetics: DSC and 1-D TtXRD...... 76 4.4.2. Nucleation Kinetics: Direct Analysis of 2D Diffraction Images...... 81 4.4.3. Structural Evolution during the Induction Time ...... 88 4.5. General Discussion & Conclusion ...... 95 Chapter 5: Plastic Crystalline Carbon Tetrabromide ...... 98 5.1. Introduction ...... 99

5.2. Structural Modifications of CBr4 ...... 101 5.3. Experimental Methods and Results ...... 108 5.3.1. Experimental Methods ...... 108 5.3.2. 2D Diffraction ...... 108 5.3.3. 2D Patterson Function ...... 114 5.3.4. Bragg and Diffuse Scattering in the Patterson Function ...... 120 5.4. Simulation Methods & Results ...... 122 5.4.1. Simulation Methods ...... 123

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5.4.2. Simulation Results ...... 132 5.5. Discussion ...... 167 5.5.1. High-Q diffuse scattering...... 168

5.5.2. Tetrahedral distortion in α-CBr4 ...... 169 5.5.3. Low-Q Diffuse Scattering ...... 170

5.5.4. Twinning in α-CBr4 ...... 173 5.6. Conclusion ...... 175

LIST OF TABLES Table 2.1. Experimental sample volumes (cm3) used in 2-D TtXRD and DSC experiments...... 18

Table 2.2. CZX-1 lattice constant extrapolated to 0 °C and thermal expansion as determined from variable temperature TtXRD...... 19

Table 2.3. KJMA parameters for fits of the data given in Figure 2.7 demonstrating strong correlation between parameters ...... 27

Table 3.1. Tabulation of metrical parameters for the simulated sample volumes...... 46

Table 3.2. Correlation between the model parameters averaged across all simulations...... 48

Table 3.3. Average phase boundary velocity (unitstime-1) from fitting the crystallization of individual crystallites and the bulk sample with and without the sample container anisotropy correction...... 54

Table 5.1. Bonds lengths for the bromines surrounding the four unique tetrahedra...... 124

Table 5.2. Br-C-Br bond angles for the four unique tetrahedra...... 124

Table 5.3. Interaction parameters for CBr4...... 124

Table 5.4. Some examples of the coordinate relationships between image (pixel) space and reciprocal space. ΔQ=0.1, Qmax=20...... 130

Table 5.5. Symmetry operations to generate the twelve cubic 110 directions from a single direction. To obtain the new direction, perform the operation listed in the row labels then the operation in the column labels...... 148

Table 5.6. Relative axes to compute the C-Br bond vector projection...... 164

LIST OF FIGURES Figure 2.1. Schematic diagram of the forced air furnace used for isothermal crystallization experiments at the synchrotrons located at Argonne and Brookhaven National Labs. T1, T2, and T3 are the thermocouple locations used to monitor and control the temperatures. The sample in a capillary is affixed to a goniometer head with epoxy. (a) Pneumatic actuator for switching the air flow coming out the top between the two furnaces. As currently sketched, the air from Furnace 1 is flowing out the top and across the sample, while the air from Furnace 2 is being vented away from the sample...... 17

Figure 2.2. DSC heat flow curves resulting from a quench of a 20.2 mg CZX-1 sample to isotherms between 145 and 160 °C, as labeled. Five repetitions are shown for crystallization at each isotherm. The instrumental quench response is shown as the hashed region. Inset plotted with a compressed time scale to highlight the slow crystallization at the 160 C isotherm...... 20

Figure 2.3. TtXRD diffraction frames from the Tiso = 135 °C crystallization of CZX- 1. (Top left) supercooled liquid 610 seconds after quenching to the isotherm, (top right) fully crystallized sample, (bottom) time series (0.5 Hz) of diffraction frames...... 21

Figure 2.4. SVD analysis of a TTXRD crystallization reaction quenched from a 230 C melt to Tiso = 133C. (a) Basis vectors u1 and u2 and (b) corresponding time traces v1 and v2...... 22

Figure 2.5. Modified basis vectors u1′ and u2′, from rotating basis functions u1 and u2 by θ = 205°, which represent the experimental liquid and crystalline diffraction patterns in (a) and (b), respectively. (c) Corresponding rotated basis vectors v1′ and v2′ demonstrate the simultaneous disappearance of liquid diffraction (red) and the appearance of crystalline diffraction (blue)...... 24

Figure 2.6. Rotated basis vectors v1′′ and v2′′ (rotation of v1 and v2 by θ = 6°) to maximize variance in v1′′ and minimize variance in v2′′. The variance, v1′′ is fit to the KJMA model (solid line)...... 25

Figure 2.7. TtXRD data for a crystallization reaction at Tiso = 139 °C (black circles). Data are fit to the KJMA model with n = 3 (red) or with t0 fixed to 0 s (blue), 30 s (cyan) or 60 s (yellow). Insets are an expansion of the onset and termination portions of the transformation...... 27

Figure 2.8. A single crystal of CZX-1 (2 x 2 x 2 mm) demonstrating cubic morphology of crystal growth...... 28

Figure 2.9. (a) KJMA rate constants as a function of temperature extracted from isothermal DSC (red) and TTXRD (blue/cyan) experiments. Samples X1 and X2 were collected at the NSLS and X3 and X4 were collected at the APS synchrotron sources, respectively. (b) Phase boundary velocities as a function of temperature...... 29

Figure 2.10. Plot of vpb as a function of temperature with the DSC experiments corrected for their anisotropic sample shape...... 32

Figure 3.1. (a) Normalized transformation ( vs. t) of a spherical crystal nucleated in the center of a cubic volume (yellow circles), compared to unbounded cubic growth (red) and the KJMA model (blue). (b) Residuals, of cubic and KJMA models compared to actual growth. White circles indicate tnuc and the time at which termination begins...... 42

Figure 3.2. Plot of individual terms of the series expansion of Equation 3.5 in increasing order from left to right (narrow black lines) with the first term highlighted (blue), and the sum of the series expansion (thick black line). Simulated transformation of a spherical crystal growing in a cubic volume (circles) and the corresponding unbounded cubic model (red) are also shown...... 44

Figure 3.3. Condensed and exploded views of simulated crystallites within (a,b) X4 and (c,d) D1 geometries...... 49

Figure 3.4. Histogram contour plot of the natural log of KJMA rate constants versus the natural log of sample volume for all simulations. The color scale contour axis is the frequency of occurrence. The black line indicates the theoretical maximum rate constant for cubic growth (Equation 3.7). Individual crystallite simulations for (a) all XRD (b) D1 and (c) D2 geometries. (d) Bulk crystallization simulations. Roman numerals I, II and III differentiate (DSC-I) small crystallites terminated by crystal-crystal impingement from (DSC-III) the large crystallites terminated by sample container impingement and (DSC-II) exhibiting both types of termination...... 50

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(white) D1-I and D2-I crystallites with >500 volume units and (black) D1-III and D2-III crystallites. (f) (gray) D2-II and (black) D1-II...... 52

Figure 3.6. Apparent bulk (a) dimensionality, n’, and (b) velocity of the phase boundary, vpb', plotted against the natural log of the sample volume aspect ratio, ln(d/h). Blue to white contours represent the frequency of observance, circles are the average value for each simulation geometry. Green lines in b are the anisotropy correction functions, Equation 3.9...... 56

Figure 3.7. Histogram contours of the natural log of the KJMA rate constants vs. the natural log of the sample volume using data from Figure 3.4 corrected for sample anisotropy. (a) Individual crystallites and (b) bulk crystallization. The contour axis is the frequency of occurrence. The black line is the theoretical maximum rate constant for cubic growth (Equation 3.7)...... 58

Figure 3.8. Difference between t0 and tnuc after applying the anisotropy corrections to DSC-II & III. Individual crystallites corresponding to (white) DSC-I and (black) DSC-III for simulation volumes (a) D1 and (b) D2. (c) Bulk transformations for (black) D1 and (gray) D2. (d) Individual crystallites corresponding to DSC-II for simulations volumes (black) D1 and (gray) D2...... 59

st nd Figure 4.1. A Gaussian peak (A=1, x0=3, σ=0.3) and its 1 and 2 derivatives. Vertical lines indicate the position along x where the 2nd derivative is zero...... 71

Figure 4.2. Isothermal crystallization of CZX-1 (145 °C). Black cross, calibrated image center. (a) Raw image. (b) d2/dy2. (c) d2/dx2. (d) d2/dy2+d2/dx2. (e) Black, spots found by the algorithm described in the text in black. (f) Purple, spots found by the algorithm on the original image...... 74

Figure 4.3. Temperature dependent (a-b) initial nucleation time and (c) initial nucleation rate from bulk crystallization measurements. The error in data outlined in orange in c are not significantly different from zero...... 78

Figure 4.4. Normalized nucleation rate ρnuc plotted on (a) normal y-scale and (b) logarithmic y-scale. (c) Logarithmic plot of knuc for comparison...... 80

Figure 4.5. Direct analysis of in-situ isothermal crystallization experiments that resulted in (a,d) oligo-, (b,e) single and (c,f) polycrystalline samples...... 83

Figure 4.6. Isothermal crystallization reactions at (a,d) 145 °C, (b,e,g) 150 °C and (c,f) 96 °C. Black, fraction transformed; Green, fit t0 histogram; Blue,

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cumulative fraction; ×###, total number of fit diffraction spots; Broken red, bulk tnuc; t=0 on the x-axis is tiso...... 87

Figure 4.7. Azimuthally averaged 2D TtXRD experiment, Tiso=145°C. (a) S(Q). (b) G(r). (c) Final crystalline diffraction image...... 89

Figure 4.8. SVD of induction time. (a) Primary two components in the PDF (left singular vectors) and a 94 point moving average of the second LSV. (b) Time-dependence (right singular vector) of the first two LSV’s...... 92

Figure 4.9. (a) Normalized peak intensity of specific pair-pair correlations. Black, Zn- Cl; Olive, Zn-Zn/Cl-Cl; Blue, 2nd nearest Zn-Zn. (b) Zn-Cl peak. (c) Zn-Zn/Cl-Cl peak. (d) 2nd Zn-Zn peak...... 92

Figure 4.10. (a) Liquid CZX-1 temperature-dependent diffraction pattern. (b) Liquid CZX-1 difference pattern. (c) Unit cell volume versus temperature for (red) liquid and (blue) crystal...... 94

Figure 5.1. Comparison between the three condensed phases of CBr4: Liquid in blue at 373 K, α in red at 333 K and β in dashed black at 303 K...... 102

Figure 5.2. (a) Diagram of the α-CBr4-type unit cells superimposed on the β-CBr4 unit cell. The correspondence to the FCC α unit cell (projected along its [110]) is apparent by considering the figure enclosed by six solid lines (with the upper edge of the FCC cell represented by the dashed line). Atoms from one FCC unit cell are located at the black circles at fractions of b (Recreated from Figure 2 in reference165). (b) Crystallographically unique molecules in β-CBr4 highlighted in four colors. The monoclinic cell is oriented so as to emphasize the [100] projection of a pseudo FCC molecular arrangement. (Solid) Monoclinic axes. (Dashed) Pseudo-FCC packing outline...... 103

Figure 5.3. (a) The β-CBr4 cell rotated to demonstrate that the tetrahedral 4 axes are aligned with the pseudo-cubic 110 real-space directions. (b) The six alignments of the tetrahedral 4 axes with the cubic 100 real-space directions. The two alignments within the orange box indicate the preferential orientations of half of the tetrahedra (blue/green) in the monoclinic cell. The remaining four alignments reflect the preferential orientation of the other half of the tetrahedra (red/yellow). (c) The “average molecular construct” which is created by randomly distributing the six orientations in b onto the FCC lattice sites...... 104

Figure 5.4. (a) Structure factor plot of α-CBr4 obtained at the APS and the Bragg diffraction calculated from a perfectly ordered Frenkel model. Orange

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box corresponds to the diffuse scattering that could not be fit by many of the early investigations...... 105

Figure 5.5. Experimentally observed diffraction of α-CBr4 with the beam stop and its holding arm visible. Capillary rotation axis is the vertical direction in the images. (a) 10° from the 111,002, (b) 20° from the 111,111, (c) near 211,022. (d) Schematic representation of the unit cell orientation of each of the twinned crystals. The green line represents the [110] twinning axis and the white line is the capillary rotation axis, close to the 021 of the red crystal. Bragg reflections labeled in white correspond to the red crystal in (d) and those labeled black correspond to the blue crystal...... 110

Figure 5.6. Experimentally observed diffraction of α-CBr4 with the beam stop and its holder visible showing highly structured diffuse scattering along a twinned 111,111 high-symmetry projection. The 111’s of the twinned crystals are separated by 9.25°. (a) Experimental image Qmax = 20Å-1 (b) Reprint of the diffuse scattering previously observed with single crystal synchrotron diffraction. Reprinted from Phys. Rev. B.38 (c) Cartoon of the diffuse scattering. (d) Experimental image Qmax = 6Å-1...... 111

Figure 5.7. Projection along the 111 (white text) and 100 (black text). (a) Twinned Bragg reflections. White and black text correspond to the red and blue crystallite’s 111 in Figure 5.6d. (b) Expanded view to highlight the twinned diffuse scattering. (c) Twinned diffuse pattern highlighted in red/blue as corresponding to the twins described in Figure 5.6d...... 113

Figure 5.8. High symmetry 111,111 diffraction image manipulated with GIMP to remove the beam stop to (a) Q=21 Å-1 and (b) Q=7 Å-1. 2D Patterson functions of (a) to (c) r=60 Å and (d) r=11 Å. 1 C-Br: 1.91Å. 2 Intramolecular: Br-Br 3.2Å. 3 Intermolecular Br-Br: 3.8Å. 4 1st nearest neighbor: 6.2Å. 5 2nd nearest neighbor: 10.4 Å. 6 3rd nearest neighbor: 12.3 Å...... 116

Figure 5.9. An expanded portion of the 2D Patterson function from Figure 5.8c overlaid with the structure of a 111 slice of the average molecular construct to illustrate the location of various pair-pair correlations. (a), (c) Molecule centered overlay. (b), (d) Bromine centered overlay. (e) (black) Plot of intensity along a cross section of the PF along the white line in (b). (red) Electron-density weighted pair-pair correlations calculated from the average molecular construct...... 119

Figure 5.10. Experimentally obtained 2D diffraction images (letters) and corresponding Patterson functions (primed letters). Each set is separated by 1° of rotation around the capillary axis...... 122

Figure 5.11. Pseudocode random walk algorithm...... 128

Figure 5.12. Cartoon of the method for projecting C-Br bonds onto the 001 planar surface to visualize the alignment of the C-Br bonds with the 001 direction. The black outline with blue ribs corresponds to the spherical surface where the bromines are located. The green outline corresponds to the 100 projections shown in Figure 5.29 and Figure 5.30...... 131

Figure 5.13. Diffraction patterns and Patterson functions computed along the 100 directions of an fcc lattice constructed with random molecular orientations. (a-c) Initial lattice construction with no relaxation. (d-f) 1 cycle of random walk relaxation. (g-i) 5 cycles of random walk -1 -1 relaxation. (a,d,g) Qmax=20 Å . (b,e,h) Qmax≈6 Å . Certain Bragg reflections are labeled. (c,f,i) 2D Patterson functions to rmax ≈ 13 Å. Numbers correspond to atomic positions. For each diffraction image, the left half of the image is the raw computed image and right half is the image with a 5×5 median filter mask to remove the Bragg peaks...... 134

Figure 5.14. Diffraction patterns and Patterson functions computed along the 111 directions of an fcc lattice constructed with random molecular orientations. (a-c) Initial lattice construction with no relaxation. (d-f) 1 cycle of random walk relaxation with a walk length of 256. (g-i) 5 cycles of random walk relaxation with a walk length of 256. (a,d,g) -1 -1 Qmax=20 Å . (b,e,h) Qmax≈6 Å . Certain Bragg reflections are labeled. (c,f,i) 2D Patterson functions to rmax ≈ 13 Å. Numbers correspond to atomic positions of intra and intermolecular correlations. For each diffraction image, the left half of the image is the raw computed image and right half is the image with a 5×5 median filter mask to remove the Bragg peaks...... 135

Figure 5.15. Larger version of the 2D Patterson function in Figure 5.14i, with the maxima labeled in angstroms. The average molecular construct centered at carbon and a bromine, respectively, are shown in the upper right and lower left quadrants. Compare to Figure 5.8e...... 140

Figure 5.16. Diffraction patterns and Patterson functions computed along the 100 directions of an fcc lattice constructed by randomly placing one of six molecular orientations on each lattice site. (a-c) Initial lattice construction with no relaxation. (d-f) 1 cycle of random walk relaxation with a walk length of 256. (g-i) 5 cycles of random walk

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-1 relaxation with a walk length of 256. (a,d,g) Qmax=20 Å . (b,e,h) -1 Qmax=9 Å . (c,f,i) 2D Patterson functions to rmax ≈ 13 Å. For each diffraction image, the left half of the image is the raw computed image and right half is the image with a 5×5 median filter mask to remove the Bragg peaks...... 143

Figure 5.17. Diffraction patterns and Patterson functions computed along the 111 directions of an fcc lattice constructed by randomly placing one of six molecular orientations on each lattice site. (a-c) Initial lattice construction with no relaxation. (d-f) 1 cycle of random walk relaxation with a walk length of 256. (g-i) 5 cycles of random walk -1 relaxation with a walk length of 256. (a,d,g) Qmax=20 Å . (b,e,h) -1 Qmax=9 Å . (c,f,i) 2D Patterson functions to rmax ≈ 13 Å. For each diffraction image, the left half of the image is the raw computed image and right half is the image with a 5×5 median filter mask to remove the Bragg peaks...... 144

Figure 5.18. Patterson Function of the 1×256 random walk simulation starting from random D2d orientations (Figure 5.17f) with labeled distances. The average molecular construct centered at carbon and a bromine, respectively, are shown in the upper right and lower left quadrants. Inset: Experimental Patterson Function (Figure 5.8d, rotated by 30°)...... 145

Figure 5.19. Side-by-side comparison between the 001 diffraction images calculated by starting with the lattices constructed with (a) D2d orientations and (b) random orientations and performing 5×256 random walks...... 146

Figure 5.20. Diffraction patterns and Patterson functions computed along the 100 directions of the first shell monoclinic fcc lattice (see text for details). (a-c) Initial lattice construction with no relaxation. (d-f) 1 cycle of random walk relaxation. (g-i) 5 cycles of random walk relaxation. -1 -1 (a,d,g) Qmax=20 Å . (b,e,h) Qmax=9 Å . (c,f,i) 2D Patterson functions to rmax ≈ 13 Å. For each diffraction image, the left half of the image is the raw computed image and right half is the image with a 5×5 median filter mask to remove the Bragg peaks...... 150

Figure 5.21. Diffraction patterns and Patterson functions computed along the 111 directions of the first shell monoclinic fcc lattice (see text for details). (a-c) Initial lattice construction with no relaxation. (d-f) 1 cycle of random walk relaxation with a walk length of 256. (g-i) 5 cycles of - random walk relaxation with a walk length of 256. (a,d,g) Qmax=20 Å 1 -1 . (b,e,h) Qmax=9 Å . (c,f,i) 2D Patterson functions to rmax ≈ 13 Å. For each diffraction image, the left half of the image is the raw

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computed image and right half is the image with a 5×5 median filter mask to remove the Bragg peaks...... 151

Figure 5.22. Large version of Figure 5.17f with labeled distances. Compare to Figure 5.8e and Figure 5.15...... 153

Figure 5.23. Comparing the (a) diffraction patterns and Patterson Functions (b) for the randomly distributed D2d orientations and monoclinic first shell constructs after 1×256 random walks...... 154

Figure 5.24. Diffraction patterns and Patterson functions computed along the 100 directions of the first shell monoclinic fcc lattice (see text for details). (a-c) Initial lattice construction with no relaxation. (d-f) 1 cycle of random walk relaxation with a walk length of 256. (g-i) 5 cycles of - random walk relaxation with a walk length of 256. (a,d,g) Qmax=20 Å 1 -1 . (b,e,h) Qmax=9 Å . (c,f,i) 2D Patterson functions to rmax ≈ 13 Å. For each diffraction image, the left half of the image is the raw computed image and right half is the image with a 5×5 median filter mask to remove the Bragg peaks...... 158

Figure 5.25. Diffraction patterns and Patterson functions computed along the 111 directions of the first shell monoclinic fcc lattice (see text for details). (a-c) Initial lattice construction with no relaxation. (d-f) 1 cycle of random walk relaxation with a walk length of 256. (g-i) 5 cycles of - random walk relaxation with a walk length of 256. (a,d,g) Qmax=20 Å 1 -1 . (b,e,h) Qmax=9 Å . (c,f,i) 2D Patterson functions to rmax ≈ 13 Å. For each diffraction image, the left half of the image is the raw computed image and right half is the image with a 5×5 median filter mask to remove the Bragg peaks...... 159

Figure 5.26. Expanded view of Figure 5.25f with labeled distances...... 160

Figure 5.27. (a) Condensed view of α-CBr4 from RMC fit to powder neutron diffraction data from Temleitner et al.146. (b)-(d) Condensed view of 20 10×10×10 simulation cells (40000 total molecules) from the monoclinic Second Shell simulation for the (b) initial build, (c) 1×256 random walk relaxation and (d) 5×256 random walk relaxation. Blue spheres represent carbon positions and rose spheres represent bromine positions. (b-d) were visualized with VMD.184 ...... 162

Figure 5.28. One 103 simulation supercell for a single random walk monoclinic- Second Shell, shown as a stick drawing (a) full simulation cell and (b) expanded view to emphasize the orientational disorder and (c) full simulation cell showing only the carbon centers...... 162

xviii

Figure 5.29. Projections of the orientation of the C-Br bond vectors from 50 simulations of a 30×30×30 monoclinic-Second Shell construct after 1×256 random walk relaxation. The reference frame for each set of projections is given to the left of each row. Columns represent all, and the 1st to 4th most aligned C-Br bond vectors. All panels are plotted on the same (x,y) scale. The z-scale for each panel is unique and automatically scaled for maximum contrast...... 165

Figure 5.30. Projections of the orientation of the C-Br bond vectors from 50 simulations of a 30×30×30 monoclinic-Second Shell construct after 1×256 random walk relaxations. The reference frame for each set of projections is given to the left of each row. All panels are plotted on the same (x,y) scale. The z-scale for each panel is unique and automatically determined for maximum contrast...... 167

Figure 5.31. Average D2d molecular constructs oriented to emphasize the expected diffuse scattering along the (left) [211], (mid) [111] and (right) [100]...... 169

Figure 5.32. Removal of certain aspects of the diffraction pattern before calculating the Patterson function. Scale bars for the diffraction images are available in (a) and for the Patterson functions in (aʹ). The indexing in (aʹ) corresponds to 1 the C-Br bond 2 intramolecular Br-Br 3 intermolecular Br-Br and 4-6 1st, 2nd and 3rd intermolecular nearest neighbors. The removed diffraction feature is labeled on the diffraction image and Patterson function. The color scale for the diffraction patterns go from white to black as the intensity increases, plotted on a linear scale. The color scale for the Patterson functions go from black to purple to yellow as intensity increases, plotted on a linear scale. The images were manipulated and created with FTL-SE...... 172

Figure 5.33. Remarkable similarity between the “streaky” Bragg scattering in β-CBr4 and the diffuse maxima in α-CBr4...... 174

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Chapter 1: Introduction

1.1. Motivation Often, validating or refuting a scientific theory requires technology that does not yet exist. As such, established theories may require reevaluation when more advanced technology becomes available. We have reached this point with respect to mechanistic understanding of crystallization from homogeneous liquids and amorphous materials. Modern synchrotron diffraction and area detector technology allows for crystallization to be probed at the atomic level with rapid sampling (<1 Hz) and a single workstation provides the computational power needed for advanced data analysis techniques and simulations. Together, these tools afford the capability to evaluate the fundamental assumptions and mechanisms of the seminal theories that form the basis of our current understanding of the mechanisms and kinetics of crystal formation. Crystallization, as a process, is a tool that has been used since our ancestors obtained salt by evaporating seawater. In the intervening tens of thousands of years, crystallization has been used for innumerable applications in the physical sciences from purification and synthesis to energy and information storage. Additionally, crystallization is at the heart of many naturally occurring phenomena. The history of crystal growth and crystallography is documented by Dr. Bohm,1 whose fascinating manuscript is worth reading in its entirety, with a brief excerpt given below which notes the foundational work of current theories of crystal nucleation and growth. “Foremost there is the masterly theoretical work of Gibbs2-3 on heterogeneous equilibria, but the value of this work was generally recognized only with great delay. Gibbs determined the energy needed to generate a nucleus and derived the equilibrium form of a crystal that fulfils the condition of minimum total free surface energy. But in a footnote he pointed out that the equilibrium form may determine the nature of small crystals only whereas the larger ones will be confined finally by such faces onto which the attachment of material proceeds most slowly.

These concepts, developed by Gibbs, were used by Volmer and Weber,4-5 Farkas6 and Becker and Döring7 to produce the theoretical framework generally recognized as Classical Nucleation Theory (CNT) which accurately describes the nucleation phenomenon in the vapor-

to-liquid, vapor-to-crystal and dilute solution-to-crystal transformations. Turnbull, Fisher and Hollomon’s extension of CNT into condensed systems8-11 in combination with the Kolmogorov12-Johnson and Mehl13-Avrami14-16 (KJMA) condensed-phase reaction model forms the basis of our current understanding of the kinetics of phase changes in condensed systems. The collection of these works and their extensions encompass nearly all theoretical approaches to the current understanding of the manner in which phase changes occur in the condensed phases. Since these seminal works, little progress has been made in understanding the phenomena of nucleation and growth of crystals from the melt. The KJMA model is commonly expressed as 푛 1.1 훼(푡) = 1 − exp(−(푘(푡 − 푡0)) ). with rate constant k, nucleation time t0 and dimensionality n. These kinetic parameters, particularly the dimensionality, are often used to infer mechanistic detail12-17 in spite of the growing body of literature demonstrating they are empirical,18-22 significantly affected by experimental factors23-26 and do not necessarily reflect the true reaction mechanism. 27-30 Understanding nucleation remains a challenge because it is experimentally difficult to separate from growth. Such separation would allow for the independent measurement of the two phenomena to either validate or refute these classical theories. This dissertation addresses the aforementioned challenges by combining simulations with experimental measurements to establish the physical basis of the KJMA rate expression and to extend the KJMA formalism such that an intrinsic rate parameter, the velocity of the phase boundary, vpb, can be obtained directly from experimental measurements. The modified KJMA expression, along with use of 2-D Temperature and time-resolved X-ray Diffraction (2- D TtXRD) affords an unprecedented ability to decouple the measurement of crystal growth from that of nucleation, which challenge accepted ideas from classical nucleation theory. Additionally, to understand the intermediate structural organization across the liquid-to-crystal phase boundary, plastic crystalline structure is explored through a combination of experimental single crystal diffraction, simulation and 2-D Patterson Function techniques.

1.2. Prior Work/Framing Questions Past work in the Martin Group has focused on developing materials and methods to achieve such an experimental separation of nucleation and growth. In this spirit, Dr. Josey 31 explored the CZX-1 system; a templated ZnCl2 network with the empirical formula 32 [HN(CH3)3][CuZn5Cl12]. CZX-1 crystallizes as a sodalite-type zeolite below its melting temperature of 173 °C with the space group 퐼4̅3푚, a = 10.5887(3) Å and forms a glass when rapidly quenched to below ≈ 30 °C. As a result of the cubic symmetry, diffraction studies are simplified since the Bragg peaks are few in number, high in intensity and evenly spaced through reciprocal space. Dr. Josey demonstrated that the combination of high-flux synchrotron diffraction sources (2nd and 3rd generation) and rapid-acquisition area detectors could be used to both spatially and temporally resolve the nucleation events of individual crystallites and track the growth of these crystallites as a function of time. The research presented in her dissertation focused on understanding the crystal growth kinetics of crystalline CZX-1 from its melt, probing the liquid structure of CZX-1 with a combination of neutron and X-ray diffraction and presented the critical experimental observation that the KJMA rate constant appeared to be dependent upon the number of mols present in the measured sample. While this result was initially quite surprising, a straightforward geometric argument (see §3.3) demonstrates that under most experimental conditions, the KJMA rate constant should indeed be related to the size of the experimental sample. Such a relationship suggests that the KJMA rate constant does not represent an intrinsic parameter to the experimental system and therefore requires modification before mechanistic information can be obtained. To develop an appropriate modification it is important to first determine the appropriate physical significance for the KJMA rate constant. In molecular kinetics, the rate constant generally represents the concentration- dependence of the reaction rate. However, there is no condensed phase analog for the concentration-dependence of a congruently melting system because there is no concentration difference between the liquid and crystalline phases. Consider instead the physical process that occurs when a system crystallizes. Crystallization starts from some point in the sample, be it a single defect that propagates or a larger collective domain of hundreds or even thousands

of atoms. This is termed the “nucleation event.” Once the crystal is formed, its size is increased by the advancement of the crystal faces, called “crystal growth,” until those faces encounter another crystal or the edges of the sample container, at which point growth ceases along their interface, known as “termination.” When nucleation does not dominate the rate of the crystallization transformation, the advancement of the crystal faces (i.e., the phase boundary) do. It is therefore reasonable to suggest that the KJMA rate constant should be related to the velocity of the phase boundary, vpb. Historically, this simple concept of the phase boundary velocity being related to the crystallization rate constant seems to have been lost. Instead, solid-state reaction modeling has relied upon a different concept: that of the ‘fraction transformed,’ which has been a way to side-step the lack of a concentration-dependence to the reaction. The fraction transformed is given the parameter α (see Equation 2.1) and is simply the crystallized volume divided by the total volume (i.e., normalization such that 0 ≤ α ≤ 1). It is a fairly obvious statement that when α is used, the amount of material under investigation is ignored because the total sample volume is explicitly removed from consideration. Normalizing out the sample volume from the experimental signal is perfectly acceptable as long as the parameters to be studied are independent of the sample volume. The initial nucleation time is one such parameter whose physical interpretation is immune to volume normalization but the KJMA rate constant is not, as is demonstrated in §3.3. By asserting that the KJMA rate constant must be related to a physical parameter of the system, the concept of a phase boundary velocity can be reintroduced into the KJMA formalism, thus maintaining the critical effects of sample size in the experimental measurement of crystallization. The KJMA rate constant then recovers its status as a physically significant model parameter that is intrinsic to the bonds being formed, broken and rearranged at the growing interface.

Within the KJMA model, there seems to be a general consensus that the t0 parameter is physically significant and related to the beginnings of the crystallization process either as a representation of the time when the growing crystalline phase first exceeds the detection limit33 or as an actual temporal measure of the initial nucleation event(s). As such, we assert that the t0 parameter can be used as an experimental probe to begin to evaluate the assumptions of

condensed-phase CNT. The nucleation literature is heavily skewed towards studying what is termed the “steady-state nucleation rate” which essentially ignores all time before the system reaches a point where nuclei are produced at a consistent rate. As such, less effort has been expended to study the ‘transient nucleation’ regime which encompasses the time when the crystalline phase became the thermodynamically favored state (i.e., the system was brought below its melting temperature) and when the ‘steady-state’ regime is reached. It is in this transient nucleation regime that the crystalline phase first appears and so the question becomes not how fast are nuclei appearing, but simply: How are nuclei appearing? By using the t0 parameter from the KJMA model, we can begin to understand the temperature dependence to the initial nucleation rate. Classical Nucleation Theory argues that nuclei appear by the stepwise addition or loss of fundamental units. Physically, the energy change of the system that results from the formation of this cluster is related to the difference between the negative interaction energy with its surroundings and the positive interaction energy within the cluster. Assuming that the rate of addition is greater than the loss rate, the cluster will increase in size. As the cluster increases in size, the surface area to volume ratio of the cluster decreases and the free energy change of the system goes from being positive to negative (i.e., spontaneous). Once the free energy change of the system becomes negative, the cluster is considered to be stable and regularly-shaped crystals proceed to grow. This mechanism arises from collision theory where particles change size by inelastic collisions with other particles and is a completely valid description when there is a significant volume (gas-to-liquid and gas-to-crystal) or concentration (dilute solution-to-crystal) difference. However, in the liquid-to-crystal transformation for congruently melting materials, the concentration of the two phases is identical and there is only a minimal volume difference (generally less than 15%). It is further interesting to note that the liquid, glassy and crystalline phases of many covalently networked systems (ZnCl2 and SiO2, for example) possess little difference in local bonding (< 5 Å).34-35 Given that the local bonding environments are virtually identical between the condensed phases, why is the condensed phase nucleation mechanism described by a gas-phase collision theory? Furthermore, given the minimal

volume difference between the two phases, isn’t the nucleation event more appropriately described not by a region increasing in size, but by a region becoming ordered? The mechanism of nucleation, according to condensed phase CNT, is built upon the conceptualization that the free energy change of a growing cluster can be defined by the difference between the surface energy and the bulk energy. Such an assumption is completely valid for nucleation in the vapor-to-liquid, vapor-to-solid and dilute solution-to-crystal transformations, for which CNT was originally developed. In each case, there is a significant volume (or concentration) difference between the reactants and the products and so it is reasonable to assume a sharp interface and therefore significant interfacial energy between the two phases. However, return these concepts to the liquid-to-crystal transformation and consider the results of a recent neutron diffraction/isotopic labeling study of vitreous ZnCl2, a corner-shared tetrahedral covalently networked material. This study demonstrated that atomic correlation in this system persists to the 22nd nearest molecular neighbor (approximately 62 Å).35 By combining the observations that the local bonding environment is virtually identical between all condensed phases and that amorphous materials exhibit correlations to the 22nd nearest neighbor, we must therefore ask a series of questions: What is the meaning of a “surface” between the amorphous and crystalline phases? Is truly a surface where one atomic layer is considered “bulk” crystal and the next “bulk” liquid? If not infinitely sharp, is the interface one molecular layer? Ten? Fifty? It is illogical to suggest that the region between an ordered domain (crystal) and disordered domain (liquid) can have a shorter correlation length than the disordered domain. Therefore, at minimum, the “surface” between the liquid and crystalline phases must be at least the correlation length that is present in the more disordered phase.

1.3. The Experimental Systems. In this dissertation, two chemical systems are used as probes for crystallization kinetics and structural disorder. CZX-1 was the system used to study the rates of crystal growth and 32 nucleation. Carbon tetrabromide (CBr4) was the system used to understand intermediate structural organization across the liquid to crystal phase boundary.

1.3.1. The Halozeolite, CZX-1: [HNMe3][CuZn5Cl12]. The CZX-1 system was used to study the rates of crystal growth and crystal nucleation because it has shown previous success in the experimental separation of nucleation from 31 growth. CZX-1 is comprised of a 5:1:1 mixture of ZnCl2:CuCl:[HN(CH3)3]Cl and crystallizes isostructurally to sodalite in the cubic space group 퐼4̅3푚 with a=10.5887(3) Å.

The crystal structure is described as a templated network of corner shared MCl4/2 tetrahedra (M=5:1 Zn:Cu) CZX-1 is a glass former and congruently melts.32 The β-cage structure of crystalline CZX-1 was shown to persist well into the liquid state,36 persisting to 50 Å at 230 °C, 57° above the melting temperature. The correlation length of 50 Å indicates a strong structural similarity between the crystalline, glassy and liquid phases. As such it is reasonable to suppose that crystallization follows an A→B type mechanism which, along with the system’s cubic symmetry, simplifies crystallization studies. As previously noted, many aspects of the CZX-1 system were characterized by Dr. Josey.31 Dr. Josey determined that molten CZX-1, studied with neutron diffraction, possessed atomic pair correlations to at least 50 Å, where the signal was lost to the noise. The phase boundary width of CZX-1 must therefore be at least this correlation length of 50 Å. The CZX-1 experimental system, as studied with time-resolved synchrotron diffraction, provides a unique window into the proposed serial aggregation mechanism of CNT because there is frequently a long (>5 minutes) ‘induction time’ before crystallization is observed. After these long induction times, crystallization is frequently complete within tens of seconds. This induction time is observed after CZX-1 has been quenched from well over its melting point (Tm+100°) to isotherms between Tg and Tm. The induction time is shortest at intermediate temperatures and lengthens as Tg and Tm are approached. Interestingly, there appears to be an upper limit to crystallization, as no crystallization has been observed 162 °C, notably 11° below Tm. The crystallization kinetics of CZX-1 were studied at all temperatures between 40 °C

(Tg+10°) and 162 (Tm-11°). DSC measurements were not possible below 145 °C because the CZX-1 would begin crystallizing during the quench and be almost completely crystallized before the desired isotherm was reached. Crystallization kinetics studies with 2D TtXRD were

not performed above 153 °C because slow initial nucleation times and limited synchrotron beam time conspired to render these studies resource-inefficient. Instead, 2D TtXRD was used primarily to explore the rest of the temperature window that DSC could not access: all temperatures between 40 °C and 153°C. It is important to note that while there is only a narrow overlap between the two experimental techniques used, there is no indication that the results should be technique dependent.

1.3.2. Plastic Crystalline Carbon Tetrabromide. As an additional probe into the mechanism of the liquid-to-crystal transformation, consider the structural relationship between the liquid and crystalline phases, as these provide initial and final states which bound the transformation. Determining the structure of a liquid is a formidable task since a true crystallographic ‘unit cell’ can only be defined as containing an infinitely large number of atoms, given that there is no long-range periodicity present in the liquid state.37 Therefore, states of matter that exhibit properties between that of a crystal and a liquid can be exploited to more fully understand the relationship between liquid and crystalline structure. A few common states of matter exhibit such properties: liquid crystals comprised of long, anisotropic molecules which exhibit many fluid crystalline phases, the glassy state with little periodicity and high mechanical strength and the plastic crystalline state with significant periodicity like that of a crystal but low mechanical strength reminiscent of a liquid. The plastic crystalline phase of carbon tetrabromide (α-CBr4) is used as the model system for these investigations because of the Martin Group’s prior experience in understanding the structure 38 of α-CBr4. Additionally relevant to choosing α-CBr4 as a model system is the wealth of literature covering a wide range of experimental and theoretical investigations into its multiple solid and liquid phases.

1.4. Summary of Research Presented This dissertation utilizes simulation and computational methods to augment experimental measurements. This approach is leveraged to provide critical analyses of our

current understanding of crystallization kinetics and nucleation as well as to structurally probe the poorly understood plastic crystalline phase. In Chapter 2, I evaluate the results of melt-crystallization experiments of our model system, CZX-1, measured with differential scanning calorimetry (DSC) and 2-D Temperature and time-resolved X-Ray Diffraction (TtXRD) to demonstrate that the intrinsic materials property, the phase boundary velocity, can be extracted directly from the widely used but empirical Kolmogorov-Johnson-Mehl-Avrami (KJMA) condensed phase reaction model. In Chapter 3, I create and use crystallization simulations to evaluate specific aspects of the crystallization process and demonstrate that the parameters of the KJMA reaction model are highly correlated and impacted by the geometry of the sample container. A series of expressions are provided to remove the empiricism from the KJMA parameters rendering them accurate descriptors of intrinsic materials properties. In Chapter 4, I expand upon the techniques pioneered by Dr. Josey to probe the assumptions of CNT by re-analyzing CZX-1 isothermal crystallization data in light of the results presented in Chapters 2 and 3. I present software and analysis techniques to directly analyze 2D diffraction images to determine the nucleation time of all diffraction spots that appear during time-resolved 2D diffraction experiments. In Chapter 5, simulations of the plastic crystalline phase are constructed whose calculated diffraction patterns are in good agreement with those obtained experimentally. Analysis of the simulated structures provide evidence that the highly structured diffuse scattering results from strongly correlated intermolecular orientations. Real-space analysis of the experimental single crystal diffraction images via 2D Patterson Function Analysis provides compelling evidence of intermolecular correlations to at least the fifth molecular coordination shell; conceptually antithetical to the commonly held belief that plastic crystals have dynamic reorientational freedom. Finally, the strong intermolecular correlations in α-CBr4 suggest that the nucleation mechanism must be concerted; it is impossible to reorient a single molecule (i.e., CNT nucleation mechanism) without reorienting at least some of its neighbors.

Chapter 2: Experimental Determination of the Crystallization Phase Boundary Velocity in the Halozeotype CZX-1

2.1. Introduction Fundamental understanding of the mechanism(s) by which homogenous liquids and amorphous systems crystallize is critical to diverse areas of science and technology ranging from environmental science to advanced information technologies. Despite this importance, understanding of crystallization mechanisms is limited. Seminal works from 1876-1953 remain the basis for much current understanding of nucleation and crystal growth.18, 39-41 In 1876 Gibbs determined the energy required to generate a nucleus and derived the equilibrium form of a crystal based on minimization of the surface energy.2-3 This work was applied to describe the nucleation phenomenon in the vapor-to-liquid and vapor-to-crystal transformations in the works of Volmer and Weber, 5 Farkas, 6 and Becker and Döring; 7 the collection of which is frequently referred to as Classical Nucleation Theory (CNT). Turnbull, Fisher and Holloman extended CNT into condensed systems8, 10-11 which, along with the Kolmogorov12-Johnson and Mehl13-Avrami14-16 (KJMA) model, forms the basis for modern understanding of crystallization kinetics in condensed systems. The KJMA crystalline growth model is used to extract kinetic parameters for isothermal crystallization processes from supercooled melts and superheated glasses. It describes crystallization as a phase boundary controlled process, and recognizes the dimensionality of the growth process impacts the observed rate of bulk crystallization. The fraction of material crystallized, α, per time is directly proportional to the progress of the phase boundary for a 1-D growth process and increases with the square or cube of the progress of the phase boundary for 2-D and 3-D growth, respectively. The rates thus appear to accelerate until growth impinges on domain boundaries and termination ensues, resulting in a sigmoidal shape to observed rate curves. The KJMA model is commonly expressed as 푛 2.1 훼(푡) = 1 − exp(−(푘(푡 − 푡0)) ). with rate constant k, nucleation time t0 and dimensionality n. These kinetic parameters, particularly the dimensionality, are often used to infer mechanistic detail12-17 in spite of the growing body of literature demonstrating they are empirical,18-22 significantly affected by experimental factors23-26 and do not necessarily reflect the true reaction mechanism. 27-30

Given the broad use of the KJMA model, much effort has been expended to verify the models’ assumptions42-46 and to demonstrate situations in which the model does not hold.47-48 Additionally, numerous extensions to the KJMA model have been investigated.40, 49-51 When identified, attempts to correct experimentally dependent kinetic parameters often result in non- physical values such as rate constants extrapolated to zero sample thickness25 or zero mass.52 To obtain mechanistic information from kinetic parameters there is a need to remove the empiricism and establish physical significance. It is the goal of this work, and that of a corresponding simulation manuscript,53 to decipher the physical significance of kinetic parameters. Starting with Equation 2.1 we find that with sample specific corrections, the transformation kinetics of our experimental system can be effectively described exclusively with physically significant parameters. In this manuscript, investigations into the mechanism of melt-crystal growth of the halozeotype CZX-1, [HNMe3][CuZn5Cl12] by differential scanning calorimetery (DSC) and temperature- and time-resolved synchrotron X-ray diffraction (TtXRD) are described. CZX- 1 crystallizes in the cubic space group 퐼4̅3푚 with a=10.5887(3) Å, and is isostructural to sodalite. CZX-1 is a glass former and congruently melts.32 The β-cage structure of crystalline CZX-1 was shown to persist into the glassy and liquid states,36 indicating a strong structural similarity between the crystalline, glassy and liquid phases. As such it is reasonable to suppose that crystallization follows an A→B type mechanism which, along with the system’s cubic symmetry, simplifies crystallization studies. As described in the initial part of this manuscript, use of multiple techniques to investigate the rate of crystallization reveals that depending on the technique employed, the KJMA rate constant varied by about 5× at common isotherms. To understand this method dependence, crystallization simulations were performed and are described in a corresponding manuscript.53 In the simulations, all information with respect to the crystallization process is precisely defined or known (nucleation location, orientation, time and frequency; crystallite growth geometry, time dependent shape and size; velocity of the phase boundary). Fitting the crystallization simulations to the KJMA model and comparing the fitted kinetic parameters

with the defined simulation parameters established a sample volume and sample container shape correction. As described in the latter part of this manuscript, application of the volume and shape correction removes the experimental dependence and yields the intrinsic, material specific rate parameter, the velocity of the phase boundary, vpb. These demonstrate sample volume and shape are the major contributors to the observed method dependence of KJMA kinetic parameters in our experimental system.

2.2. Experimental Methods

2.2.1. Isothermal Crystallization: DSC For the DSC isothermal crystallization experiments, three sample masses of CZX-1 were used: 12.2, 20.2, and 26.5 mg. All samples were sealed in high-pressure stainless steel DSC pans with gold foil seals (Perkin-Elmer product # B0182901). The samples were melt- crystal cycled five times between 40 and 230 °C at a constant rate of 5 °Cmin-1 before isothermal crystallization experiments were performed to remove heterogeneous nucleation sites and to ensure uniform thermal contact between pan and sample. The samples were again heated to 230 °C, held at the melt isotherm for 5 min and quenched to a crystallization isotherm -1 (Tiso) of between 145 and 160 °C at maximum instrumental cooling rate of ~40 °C·min . Heat flow data were measured at the isotherm until crystallization was complete, typically between

1 and 100 min. Crystallization began before Tiso was reached for temperatures below 155 °C, but the crystallization heat flow was distinguishable from the instrumental quench response until the target isotherm was below 145 °C. The time scale of the raw heat flow data was adjusted such that the quench initiation for each experiment occurred at t = 0 min. Isothermal

crystallization experiments were repeated 3-5 times for each sample, at each isotherm, for a total of 49 isothermal DSC experiments. The instrumental quench response (IQR), which must be subtracted from the raw signal to obtain the crystallization heat flow, is primarily dependent upon the sample mass, but instrumental response as a function of Tiso is also observed. For each sample, the IQR of an isotherm for which crystallization was well separated in time from the quench, e.g. Tiso ≥ 160 °C, was fit with a combination of two Gaussian and one Lorentzian functions. These Gaussian and Lorentzian parameters were then used as the starting parameters to fit the data using SOLVER55 to obtain the IQR for subsequent reactions of the same sample at different isotherms.

2.2.2. Isothermal Crystallization: TtXRD Synchrotron diffraction data were obtained on beam lines 11-ID-B (90 KeV, λ = 0.13702 Å, collimated beam 1.0 × 1.0 mm) at the Advanced Photon Source (APS), Argonne National Laboratory, and X6b (19.1 KeV, λ = 0.646 Å, collimated beam 0.3 × 0.3 mm) at the National Synchrotron Light Source (NSLS), Brookhaven National Laboratory. Data were collected in a Debye-Scherer geometry at sampling rates of 0.05 to 8 Hz, each diffractogram hereafter referred to as a frame, using 2048 × 2048 GE Silicon (APS)56 or 2084 × 2084 SMART CCD detectors (NSLS). Samples were sealed into fused silica capillaries (Charles Supper Co. Natick, MA) and affixed to a single-axis goniometer head with epoxy. During crystallization experiments the samples were oscillated 10° in synchronization with the duration of X-ray exposure to illuminate a larger region of reciprocal space and minimize thermal gradients in the sample. The wavelength and detector alignment were calibrated to LaB6 or CeO2 standards using fit2d57 to correct all experimental data. A melted and recrystallized ingot of CZX-1 in a fused silica capillary was centered in the synchrotron beam. The sample was heated to 230 °C, noting the temperature at which the sample melted, which when compared to the CZX-1 melting point (Tm) of 173 °C provided an internal temperature calibrant. To ensure melt isotropy, the sample was held at 230 °C, well above Tm, for 5 min. Data collection was initiated for at least 1-3 frames prior to quenching to

a Tiso of between 40 and 160 °C. Diffraction data was recorded at the isotherm until crystallization was complete. When a sample did not nucleate within 2 h, the reaction was aborted because of limited synchrotron time; which was the case for Tiso >155 °C. The melt- quench-crystallization cycle was then repeated. Data from a total of 55 crystallization experiments are reported here. Additional data were collected for samples whose diffraction patterns were found to exhibit an excess of ZnCl2. These are excluded from the kinetic analysis to remove as many extraneous effects as possible. No sample degradation was observed after multiple (>15) cycles, indicating that CZX-1 is stable with respect to decomposition under both rapid temperature shifts and multiple hours of high-energy X-Ray exposure. Temperature control of the crystallization reaction was afforded by a pneumatically switchable manifold that directs airflow through one of two ¾” in-line air heaters (Omega), schematically depicted in Supporting Information (SI) Figure SI-1. One furnace was set to the temperature of the high temperature melt isotherm (230 °C) and the other to the temperature of the desired Tiso (40-155 °C). The temperature of each heating element was controlled using a Eurotherm 91p temperature controller to a precision of ±0.5 °C. Using the phase transitions of elemental sulfur and the melting temperature of CZX-1 as calibrants, temperature accuracy of ±2.5 °C was obtained; the relatively large range is a result of sample environment conditions including temperature gradients, variations in air flow, and sample/thermocouple placement within the air stream. The temperature of the air stream stabilized to the quenched isotherm at the controlling thermocouple T3 (Figure 2.1) within 15 sec after the air streams were switched affording rapid quenching to Tiso, irrespective of melt and quench temperatures.

Sample

Goniometer a

T T1 2

Furnace 1 Furnace 2

Figure 2.1. Schematic diagram of the forced air furnace used for isothermal crystallization experiments at the synchrotrons located at Argonne and Brookhaven National Labs. T1, T2, and T3 are the thermocouple locations used to monitor and control the temperatures. The sample in a capillary is affixed to a goniometer head with epoxy. (a) Pneumatic actuator for switching the air flow coming out the top between the two furnaces. As currently sketched, the air from Furnace 1 is flowing out the top and across the sample, while the air from Furnace 2 is being vented away from the sample.

2.2.3. Sample Volume and Density

In order to obtain the material specific vpb from the KJMA rate constant k it is necessary to obtain an accurate estimate of the sample volume.53 In the synchrotron experiments, for which the sample is contained within a sealed capillary, an ingot longer than the diameter of the synchrotron beam was utilized. However, the observed signal results only from crystal growth within the irradiated volume, equivalent to the volume of the intersecting rectangular prism of the synchrotron beam and the capillary. Two sizes of capillaries were used, 0.7 and

0.5 mm outer diameter (OD) with 0.01 mm wall thickness. The four XRD sample volumes are presented in Table 2.1.

Table 2.1. Experimental sample volumes (cm3) used in 2-D TtXRD and DSC experiments. Simulation Geometry53 Volume (cm3) Synchrotron Source Capillary diameter (cm) X1 3.80 × 10-5 NSLS 0.05 X2 5.84 × 10-5 NSLS 0.07 X3 1.81 × 10-4 APS 0.05 X4 3.63 × 10-4 APS 0.07 Sample mass (mg) D1 5.02 × 10-3 12.2 -- 8.31 × 10-3 20.2 D2 1.09 × 10-2 26.5

The volume of the DSC samples was determined knowing the pan diameter of d = 0.50 cm and with the assumption of a 90º contact angle between CZX-1 and the sample pan. With such a cylindrical approximation the sample height is approximated by Equation 2.2. (푚⁄휌) ℎ = 2.2 휋(푑⁄2)2 Sample density was determined based on the crystallographic density. The crystallographic thermal expansion was computed by a least squares fit of the lattice constant for six independent TtXRD measurements detailed in Table 2.2. The intercept and slope represent the lattice constant at 0 °C and the thermal expansion, respectively. The average crystallographic density of CZX-1 (at 273 K) is 2.478(1) gcm-3 with an average thermal expansion resulting in a density decrease of 2.96(7) × 10-4 gcm-3 K-1. Averaged DSC sample volumes (Table 2.1) are computed using the density at Tiso.

Table 2.2. CZX-1 lattice constant extrapolated to 0 °C and thermal expansion as determined from variable temperature TtXRD.

Heating Rate Tstart Tfinish Lattice Constant Thermal Expansion

(°C min-1) (°C) (°C) (Å) (Å K-1) × 104 5 35 181 10.5474(3) 4.12(2) 5 100 168 10.5447(10) 3.99(8) 10 31 170 10.5590(3) 3.75(3) 10 120 166 10.564(3) 4.3(2) 10 125 168 10.552(2) 5.09(15) 20 107 167 10.543(2) 4.47(15)

2.3. Results and Discussion

2.3.1. Isothermal Crystallization: DSC Measurement of liquid-to-crystal transformations via DSC ideally exhibits two well- separated signals corresponding to the instrumental quench response (IQR) and the heat evolved from crystallization. As shown in Figure 2.2 for the 20.2 mg CZX-1 samples, these are well separated for Tiso ≥ 155 C. Similar plots for the 12.2 and 26.5 mg samples are given in Appendix A. The IQR was subtracted and the heat flow integrated as a function of time, and normalized to the fraction transformed, . The  vs. t data were then fit to Equation 2.1 over the range 0 ≤  ≤ 0.5 to extract KJMA parameters. For a given isotherm the sample-to-sample variation in the observed rate of crystal growth is less than a factor of 2, albeit the greatest being faster rates for the smaller sample and slower rates for the larger sample. The most significant variation in crystal growth rates is the marked deceleration with increasing Tiso. Such behavior is clearly non-Arrhenius as has commonly been observed in the literature for crystallization isotherms proximate to the melting point. The amount of heat evolved at a given isotherm exhibited no change over more than 20 cycles indicating no significant sample degradation. No crystallization is observed for Tiso ≥ 163 C even after 48 h; notably 11 C below the melting point of CZX-1.

2.3.2. Isothermal Crystallization: 2D TtXRD A representative series of 2-D TtXRD images collected at the APS for the isothermal crystallization of a sample of CZX-1 at Tiso = 135°C is given in Figure 2.3. In the experiment shown, the first evidence of crystallization is observed approximately 604 s after quenching, with crystal growth occurring over the next 30 seconds. The area detector utilized for the TtXRD experiments is invaluable to visualize and differentiate nucleation and crystal growth. The final crystalline diffraction pattern shown in Figure 2.3 provides clear evidence that in this reaction the bulk of crystallization is accounted for by a single crystallite, with a few satellite reflections from subsequent nucleation. Such single (or near single) crystal growth occurs under conditions where nucleation is slow with respect to the rate of crystal growth. By contrast, at both lower and higher temperature when the rate of growth is slowed with respect to the rate of nucleation, polycrystalline samples are observed.

Figure 2.3. TtXRD diffraction frames from the Tiso = 135 °C crystallization of CZX-1. (Top left) supercooled liquid 610 seconds after quenching to the isotherm, (top right) fully crystallized sample, (bottom) time series (0.5 Hz) of diffraction frames.

The final frame from each of the TtXRD crystallization experiments in this study is given in Appendix B. While no consistent trend has yet been established with respect to nucleation, single crystal growth was most frequently observed for the largest X4 sample size for isotherms between 120 and 150 °C. Approximately 10 or fewer crystallites are observed at Tiso > 90 °C while ten to hundreds of crystallites are observed for Tiso < 90°C, generally increasing as isotherm temperature is decreased. Evaluation of the diffraction frames for polycrystalline samples as a function of time clearly demonstrates that nucleation proceeds in a continuous fashion, albeit with subsequent nucleation occurring at a much faster rate than the initial nucleation. Direct analysis of individual crystallite data by 2-D TtXRD across a time series is currently under development. For this manuscript the 2-D diffraction patterns are azimuthally averaged to create 1-D diffraction patterns and analyzed by singular value decomposition (SVD). The 1-D patterns are aggregated into an m × n data matrix A, where m is the number of reciprocal lattice vectors in each pattern and n is the number of diffraction patterns collected. These time dependent crystallization data are deconvoluted into two sets of orthonormal basis functions according to Equation 2.3, where U and VT are the left singular vectors (LSV) and

right singular vectors (RSV) corresponding to time-independent diffraction patterns and their time- dependence, respectively, and Σ is composed of singular values (i.e. weighting factors) that describe the contribution of the corresponding singular vectors to the data and are sorted in decreasing order.58 Detailed descriptions of the general theory behind SVD and applications are available.58-62 An example of this SVD analysis of a crystallization reaction is shown in

Figure 2.4 for a reaction quenched from T = 230 to Tiso = 133 C. 퐴 = 푈Σ푉푇 2.3

u 1 u 2

Intensity (arbitrary units) (arbitrary Intensity

0 2 4 6 Q (Å-1) b

v 1 v 2

Signal (arbitrary units) (arbitrary Signal

0 100 200 Time (s) Figure 2.4. SVD analysis of a TTXRD crystallization reaction quenched from a 230 C melt to Tiso = 133C. (a) Basis vectors u1 and u2 and (b) corresponding time traces v1 and v2.

The first two basis vectors of the resultant U matrix, u1 and u2 in Figure 2.4a, contain elements of the broad amorphous scattering from the liquid phase and the sharp diffraction

from the crystalline phase. These basis vectors describe approximately 80% of the variance of the original A matrix. The third component (not shown) represents 1.2% of the data set and corresponds to a slight shift in peak positions during the transformation and the remaining 18.8% of the data set is spread across 263 additional basis vectors which have no physical interpretation, i.e. noise. The first two basis vectors of the resultant V matrix, time traces v1 and v2, demonstrate u1 and u2 correspond to simultaneous changes that occur between 65 and 100 s, Figure 2.4b. Given that all columns of U and V are orthogonal, physical meaning can be ascribed to the basis vectors u1 and u2 by taking the linear combination of basis vectors u1 and u2, producing modified basis vectors, u1′ and u2′. Using the rotation matrix shown in Equation 2.4, where θ is the angle over which the original basis vectors are rotated, a rotation of θ = 205 was found to minimize the difference between u1′ and the experimental liquid diffraction pattern (a), and correspondingly the difference between u2′ and the crystalline pattern (Figure 2.5b).

′ ′ cos(휃) − sin(휃) [푢 푢 ] = [푢1 푢2] [ ] 2.4 1 2 sin(휃) cos(휃)

Basis vectors v1 and v2 must also be rotated over the angle of θ = 205°, producing modified basis vectors v1′ and v2′. Since u1′ and u2′ represent the time-independent experimental liquid and crystalline diffraction patterns, v1′ and v2′ correspond to the time- dependent loss of liquid scattering and appearance of crystalline diffraction, respectively. The normalized basis vectors v1′ and v2′, along with their fit to Equation 2.1 are shown in Figure 2.5c. Note specifically the loss of the liquid scattering and the emergence of the crystalline scattering cross at 50%, providing strong support for a mechanistic conclusion that CZX-1 crystallization proceeds via an 퐴 → 퐵 type mechanism, i.e. with no intermediates.

An 퐴 → 퐵 crystallization mechanism indicates the crystalline phase can only grow at the expense of the liquid phase. As a result, the time-dependence of crystallization is spread across v1′ and v2′, which serves to decrease the signal-to-noise (S/N) ratio in each. However, taking further advantage of the orthogonality of the U and V matrices, it is possible to transfer essentially all of the time-dependence of crystallization into one of the basis vectors by rotating v1′ and v2′ (or v1 and v2) to produce v1′′ and v2′′. A rotation angle of 6° from the original basis vectors v1 and v2 achieves this, as shown in Figure 2.6. The corresponding u1′′ and u2′′ (not

shown), which contain a mixture of both liquid and crystalline scattering, are physically meaningless. However rotating all time-dependence into one singular vector maximizes the S/N for these TtXRD kinetic experiments.

The superior quenching capability in the TtXRD experiments provide access to a greater range of temperatures over which the crystallization rate can be measured than was accessible by DSC. As was observed in the DSC measurements, reactions conducted in the high temperature region within about 40° of the melting point, exhibited a slowing in the rate of crystallization with increasing Tiso. However, for isotherms above Tg but below 135 °C, the rate of crystallization is observed to increase with increasing Tiso. Notably, the single, or near single crystal pattern observed in the final frame for a range of higher temperature experiments (Appendix B) demonstrates the slowed rate of crystallization in the higher temperature region does not correspond to a slowed rate of nucleation as suggested by Classical Nucleation

Theory.5-8, 10-11 The observed variation in crystallization rates is observed both for experiments resulting in single- and polycrystalline products. Details of the crystallization rate as a function of temperature are given below in the presentation of measured rate constants.

2.3.3. KJMA Parameter Correlations To obtain significant mechanistic information from rate measurements, a minimum requirement is that any physical parameters be intrinsic to the material. However, substantial variation between all KJMA parameters extracted from the DSC and TtXRD experiments was observed, even for experiments conducted at common isotherms. Nucleation is presumed to be a random event, but n and k are expected to be material, not technique dependent. The KJMA exponent n has previously been described to include the dimensionality of growth, λ = 1, 2, or 3, as well as the probability of nucleation β such that n = λ + β. β is considered to be 0 for fixed nucleation and 1 for continuous nucleation. The many single, or near single crystal reactions observed (Figures SI-4 - SI-7) should result in β ≈ 0, yet values of n as low as 2 and well above 4 were obtained. Similarly, widely variant values of k were observed for reactions measured at a common isotherm but with different techniques. Initial insight into the KJMA parameter problem is obtained by understanding the 53 extent to which the KJMA parameters t0, n and k are correlated. Consider, for example, the TtXRD isothermal crystallization experiment shown in Figure 2.7 where the sample reached the Tiso = 139 °C at 0 s. Fixing n = 3, a reasonable assumption given the cubic structure of

CZX-1 and the observed single crystal growth, yields a very good fit to the data when t0 = 91 -2 -1 s and k = 2.28(6) × 10 s . However, as shown in Table 2.3 and Figure 2.7, the first half of the transformation is equally fit when t0 is fixed to earlier time points and n is allowed to vary.

When t0 is set to 0 s, the time at which Tiso was achieved, the data are well fit by n = 12.9(3) and k = 7.56(1) × 10-3 s-1. These data demonstrate the strong, positive correlation between k and t0, and the negative correlation between both and n. While each of the four cases considered numerically fit the KJMA model with statistical significance, three produce physically unreasonable KJMA parameters for t0 and n. The above analysis of parameter correlations demonstrates that care must be taken before ascribing physical significance to

systems described with large KJMA dimensionalities. It is likely that cases in which large

KJMA exponents are reported, are in fact ones in which t0 was not accurately determined.

Table 2.3. KJMA parameters for fits of the data given in Figure 2.7 demonstrating strong correlation between parameters

k n t0 Fisher’s Z 0.0228(6) 3 91(8) 2.60 0.0134(6) 6.39(15) 60 2.76 0.00976(3) 9.6(2) 30 2.76 0.007561(15) 12.9(3) 0 2.70

Correlation problems,63 such as noted above, can be avoided with prior knowledge of, or by independent determination of t0 or n. In the CZX-1 crystallization exper iments it is difficult to pre-determine t0 because nucleation has been observed to occur both as quickly as a few seconds after the system has reached Tiso and as slow as hours after quenching to the growth isotherm. Nevertheless, it is clear that t0 is not equivalent to the time at which the system reaches the quench isotherm. In fact, one can only determine an upper bound to t0 since nucleation must have occurred prior to observation of the first experimental signal. Assuming

a nucleation event is the result of organization of a few hundred unit cells out of ~1017 unit cells in a TtXRD sample it is unlikely that any signal resulting from nucleation will be stronger than the noise inherent to the measurement, further exacerbating the determination of t0. Several features of this CZX-1 system present a strong case for the KJMA exponent to be fixed at 3, indicative of three-dimensional crystal growth with little or no contribution from continuous nucleation. The cubic space group of CZX-1 (퐼4̅3푚) suggests the system should exhibit isotropic growth in three dimensions, unless constrained into an anisotropic sample geometry.53 As shown in Figure 2.8, crystals of CZX-1 with a cubic morphology have been observed to grow from a sample sealed within a fused silica tube that was quenched from the melt to a glass, and then held at room temperature (23-25 C) for several weeks. Furthermore, several of the 2-D TtXRD experiments, Figure 2.3 and Appendix B, demonstrate single crystal growth, and nearly all those with Tiso greater than 90 C isotherms exhibit fewer than 10 crystallites. For these systems a continuous nucleation probability term β is likely at or near zero. Thus, to mitigate correlation effects, all XRD experimental data in were fit with n fixed at 3.

Figure 2.8. A single crystal of CZX-1 (2 x 2 x 2 mm) demonstrating cubic morphology of crystal growth.

2.3.4. Isothermal Crystallization Rate Crystallization rate constants determined by fitting isothermal DSC and TtXRD crystallization data to Equation 2.1 with n fixed at 3 are presented in Figure 2.9a. As described in the results above, the distribution of rate constants as a function of temperature clearly demonstrate that the rate of crystallization increases as the isothermal reaction temperature increases from above the Tg to some maximum Tiso after which point the rate decreases with increasing reaction temperature. However, on more careful inspection, it becomes apparent that the rate constants are distributed with respect to the sample size used in the respective experiments. At common isotherms, the rate constants extracted from TtXRD data collected at the NSLS synchrotron are distinctly greater than those obtained for data collected at the APS synchrotron, which are significantly greater than those obtained from DSC experiments.

Given that condensed phase crystallization is a phase boundary controlled process, i.e. atomic rearrangement is required at the phase boundary between the melt and crystal, it is implausible that the physical and chemical processes controlling the reaction have changed when measured with different volumes. That the volume of a sample must be considered with respect to the rate at which the sample crystallizes is intuitive. Starting from an assumption that the velocity of the crystallization phase boundary should be a material specific and constant value, albeit temperature dependent, a large sample will take longer to crystallize than a small sample. As the sample is transformed, the crystal volume must grow at a rate proportional to the velocity of the phase boundary, vpb, raised to the power of dimensionality of crystal growth. Conversely, in the simplest system of a cube growing in an unrestricted volume, not impacted by termination effects, vpb should be directly proportional to the cube root of the transformed volume. The normalization of the crystallization transformation to  in the KJMA model implicitly exchanges the volume dependence of the experimental signal for a volume dependent rate constant k. We thus propose to reintroduce the sample volume into the KJMA model according to Equation 2.5 to produce the chemically meaningful phase

-1 boundary velocity where vpb has units of distance  s . A similar relation was previously introduced to describe sample mass effects on kinetics of thermal decomposition reactions.52, 64 As shown in Figure 2.9b, application of Equation 2.5 to the crystallization rate constants obtained from crystallization experiments successfully accounts for the majority of the technique-dependence from the rate constant.

3 푣푝푏 ∝ 푘√푉 2.5 To establish the quantitative validity of the empirical relationship proposed in Equation 2.5, an extensive set of crystal growth simulations were performed.53 The simulations allowed precise definition of vpb, nucleation time/nucleation rate, crystallite shape, sample size and shape, nucleus location and orientation. The simulated data were then fit in the same manner as the experimental data to compare the fit KJMA parameters t0, n, k, and vpb to the defined simulation parameters. The simulations validate Equation 2.5 with the addition of a sample container geometric shape term, g, and when crystallization is performed in reasonably isotropic sample containers, defined by the minimum and maximum sample axes being within

a factor of about 2.67. It is also shown that the observed rate of crystallization is strongly dependent upon the location and orientation of the growing crystal within the sample volume. Random nucleation location and orientation results in an intrinsic variation in the measured rate constants of about a factor of two. The simulations also demonstrate that the relative anisotropy of the sample volume in which the crystal grows can impact the measured rate. The KJMA model with the volume correction is appropriate for reasonably isotropic sample volumes but outside of this range, impingement of crystal growth will begin earlier as the sample geometry approaches the limiting cases of two-dimensional (disc) and one-dimensional (capillary). Such progressively earlier termination blocks one (disc) or two (capillary) available growth dimensions producing a lower apparent dimensionality, n′, a lower apparent k′ and therefore a lower apparent phase boundary velocity vpb′.

To obtain the most accurate vpb it is preferable to conduct experiments in sample geometries that are as isotropic as possible. The XRD geometries reported in this work are sufficiently isotropic (0.4 ≤ d/h ≤ 0.7) that no anisotropy correction is required. However, the experimental DSC geometries result in aspect ratios of 19.5, 11.8 and 9.0, respectively, for the 12.2, 20.2 and 26.5 mg samples under the cylindrical approximation. As demonstrated by the series of constant volume simulations53 a decreasing apparent dimensionality n′ and apparent rate constant k′ is observed with increasing sample anisotropy. However, this effect can be removed with an empirical expression whose numerical terms result from a fit to Figure 3.6 shown in Equation 2.6.

푎 = exp (−0.29 푙푛 (푑) + 0.24) , 푑 > 2.67 2.6 푐 ℎ ℎ The function to describe the variation of the dimensionality and rate constant with respect to the aspect ratio of the sample must be scaled to the relative number of crystallites in that sample (thus a function of nucleation rate, growth rate and sample size), because for any given sample aspect ratio, additional nuclei further subdivide the sample causing each crystallite to be less anisotropic. To account for the experimentally observed sample container shape effect, we simulated the relative volume of each DSC sample (1.5 × 106 to 3.2 × 106 volume elements) with otherwise identical simulation parameters.53 The DSC simulations

were fit using the KJMA model with t0 fixed at the known simulated tnuc to obtain an apparent dimensionality, n′, which was then averaged over all simulations performed. The aspect ratio corresponding to that fit value of n′ in the simulation shown in Figure 6a of Chapter 3 was then used to fit the experimental data for each of the DSC sample geometries. Specifically the modified exponents used to correct the DSC data are n′ = 2.5, 2.62 and 2.7 for the 12.2, 20.2 and 26.5 mg DSC samples, respectively, corresponding to apparent aspect ratios of 5.17, 4.75 and 3.95. As applied to the DSC data reported in Figure 2.9, the sample anisotropy correction results in an increase from the values obtained for vpb when n was fixed to 3 of 75%, 54% and 35% for the 12.2, 20.2 and 26.5 mg samples, respectively.

Figure 2.10. Plot of vpb as a function of temperature with the DSC experiments corrected for their anisotropic sample shape.

The sample volume and anisotropy corrections, Equation 2.5 and Equation 2.6, along with a term to describe the intrinsic geometry of crystal growth, g (g = 1 for cubic growth), are combined to give Equation 2.7. This correction affords the intrinsic, material specific kinetic parameter for a condensed phase reaction, the velocity of the phase boundary, to be determined from the simple form of the KJMA expression, Equation 2.1. Application to the experimental data of this work yields the vpb reported in Figure 2.10, demonstrating complete consistency between the TtXRD and DSC measurements.

3 푘푔√푉 2.7 푣푝푏 = 푎푐

2.4. Conclusions Utilization of multiple experimental techniques to measure the rate of crystallization revealed the need for a critical sample volume correction that, when applied to the KJMA model for phase boundary controlled reactions, provides the physically meaningful and material specific kinetic parameter, the velocity of the phase boundary, vpb. This is in clear contrast to the method-dependent and therefore not material specific KJMA rate constant, k.23- 26 The set of experimental measurements, and corresponding simulations53 demonstrate that to obtain the most accurate kinetic information, it is necessary to independently determine at least one of the KJMA parameters, most reasonably the growth dimensionality n, to reduce parameter correlation effects. Furthermore, it is demonstrated that kinetic measurements should be performed in isotropic sample volumes to maximize unrestricted crystallite growth, thereby mitigating sample container effects. Because nucleation location and orientation can cause the rate of crystallization to vary by about a factor of two, a large number of kinetic measurements should be made to ensure appropriate statistical sampling.40 Specifically with respect to the rate of crystallization of the halozeotype CZX-1, the combination of DSC and TtXRD techniques provides access to kinetic measurements over the entire temperature range between Tg and Tm. The DSC measurements provide invaluable access to measurement of the slow crystallization rates at temperatures near the melting point. By contrast, the TtXRD measurements, which clearly differentiate between liquid and crystalline phases, afford a rapid quenching such that kinetic measurements in deeply supercooled melts can be achieved. Additionally, DSC experiments measure the entire sample, in contrast to TtXRD, which can only probe the portion of the sample which intersects the synchrotron beam. However, using an area detector for the TtXRD experiments affords insight into the number of nucleation events that occurred during the sample’s transformation, in contrast to DSC measurements which cannot differentiate between single and polycrystalline transformations.

Our sample volume and anisotropy correction to the KJMA model affords an unprecedented measurement of the velocity of the phase boundary from bulk crystallization

-1 kinetic measurements. These range from 0.02 μms at 37 °C near Tg to a maximum of 32

-1 -1 μms at 135 °C (Tmax) and 1.5 μms at our highest observed crystallization temperature about 11° below the melting temperature. Given the unit cell length for CZX-1 is 10.85 Å, the maximal phase boundary velocity indicates the crystalline axes increase by approximately 29 unit cell lengths per millisecond.

Finally we recognize that vpb at temperatures between Tg and Tmax seem to exhibit Arrhenius-type kinetics, though it has been noted in the literature that the theoretical justification for application of the Arrhenius model to reactions in the condensed phase is unclear.19, 65 The observation that the rate of crystal growth exhibits a maximum is not unique to this system. To explain this effect in an Arrhenius context, CNT presumes the reduction in rate is due to a reduction in nucleation. However, the observation here that the crystallization rate is slowed even for the growth of single crystals cannot be explained by classical models. Additionally, these models cannot account for the observation that crystallization is not observed until ~10° below Tm. While an energetic model that accounts for the observed crystallization behavior is not yet developed, replacing the empirical kinetic parameters by those with clear intrinsic physical significance is a necessary starting point.

Chapter 3: Crystal Growth Simulations to Establish Physically Relevant Kinetic Parameters from the Empirical KJMA Model

3.1. Introduction Understanding the mechanism(s) of condensed phase crystallization is of broad interest to numerous scientific fields. Crystallization kinetics usually follow a sigmoidal-shaped transformation curve, first described with a phenomenological model for the decomposition of austenite steel.66 A theoretical framework accounting for this sigmoidal shape was developed through the combined works of Kolmogorov,12 Johnson and Mehl13 and Avrami,14-16 commonly referred to as the Kolmogorov-Johnson-Mehl-Avrami (KJMA) model. Much work has been done to verify29, 42-46, 67 or refute47-48 the validity of the KJMA assumptions and to develop extensions of the KJMA model.18-19, 40, 49-51, 63, 67-72 However, a growing body of literature demonstrates these parameters are empirical18-22 and/or significantly affected by experimental parameters.23-26 As reported in our corresponding manuscript describing experimental measurement of the rate of crystallization of the halozeotype CZX-1, the KJMA rate constant systematically varies with sample size and shape53 indicating an incompleteness to the KJMA model.18 However, for any mechanistic understanding a material-specific, not method-dependent, rate constant is a minimum requirement. Furthermore, that work demonstrated strong statistical correlations between the KJMA parameters, which must be addressed before rational mechanistic interpretations can be made. Reported here are the results of simulations of crystal growth, which address the correlations between KJMA parameters and demonstrate necessary modifications to obtain a material- specific rate constant, the velocity of the phase boundary vpb, from experimentally determined crystal growth rate constants using the KJMA model.

3.2. Background The KJMA model provides the integrated rate expression given in Equation 3.1 where α(t) is the fraction of the material transformed as a function of time, t, a crystallization rate -1 constant, k, with units of time , the induction time, to, and the KJMA kinetic exponent, n, which is understood to contain contributions from the growth dimensionality and the nucleation probability. Before using simulations to address how the KJMA model parameters

(n, k, t0) are physically manifest, it is useful to discuss their current literature usage.

푛 3.1 훼(푡) = 1 − exp(−(푘(푡 − 푡0)) )

The t0 parameter corresponds to the time at which crystallization begins and therefore is representative of the first nucleation event(s), tnuc. However, as a practical experimental matter, nucleation is not precisely defined, being limited by the sensitivity of the measurement. 73-74 While effort has been made to analytically describe t0, in practice it is generally determined from fitting the transformation with linear or non-linear methods or defined by various approximations including visual selection,74 the time when the testing temperature reaches 1 K of the specified isothermal crystallization temperature,75 as the intersection of the tangent taken at the inflection point of the sigmoid with the x-axis,76 or when the signal of the growing crystalline phase first emerges above the measurement detection limit.33 It is also not uncommon for the t0 parameter to be excluded from application of the KJMA expression; a 25 problem that has been recognized. When t0 is excluded it is unclear whether t = 0 corresponds to the initiation of recording data, the time at which the crystallization isotherm is achieved, or whether t has been adjusted such that t = 0 corresponds to tnuc. A numerically reasonable fit to the KJMA expression can be obtained from any of these starting assumptions, but the resultant parameters will not accurately describe the crystallization transformation due to correlation between t0 and n. Furthermore, exclusion of t0 hides significant information with respect to chemistry involved in crystal nucleation. The KJMA exponent, n, is defined to contain contributions from the growth dimensionality, λ, and the nucleation type, β, with limiting cases of 0 for instantaneous nucleation (pre-existing nuclei) and 1 for continuous nucleation.12-17, 41 Thus n = λ + β where 0 ≤ β ≤ 1 and λ = 1, 2 or 3 for 1-, 2- or 3-D growth, respectively. Given such a specific definition for the KJMA exponent, numerous publications detail reaction mechanisms for crystallization transformations based solely on the KJMA exponent; though many reports caution that kinetic exponents may not reflect the reaction mechanism.19, 27-30, 45 Numerous experimental manuscripts demonstrate variation of the KJMA exponent with isothermal crystallization temperature,25, 77 time,30, 40, 45, 77-78 heating/cooling rate48, 72 and pre-annealing temperature.79-81 These often are used as evidence that the crystallization mechanism changes over the course of a reaction. While change of mechanism during crystallization is unlikely for

congruently melting systems,23, 29-30, 50 such may be possible for non-congruently melting systems where the composition at the crystallization front changes with time.79, 82 Consensus has not been reached as to the physical significance of, or the consistent conditions under which the KJMA exponent can be correctly interpreted. Additional complications in determining n arise due to its high correlation with t0 and k. Any deficiency in the determination of one of these parameters dramatically impacts the value obtained for the others.53 Therefore, special care must be taken when interpreting the value of n in a mechanistic framework to avoid drawing inappropriate conclusions. The rate constant, k, provides a time constant in KJMA theory, with no further physical or chemical significance implied. Several reports attempt to describe the temperature dependence of k from an Arrhenius context18, 40-41, 50, 52, 64, 73, 75, 77, 79-82 (or various nonisothermal methods63, 78, 81-83) to determine the activation energy of crystallization and/or nucleation according to Equation 3.2. 퐸 푘(푇) = 푍 ∗ exp (− 푎 ) 3.2 푅푇 While the Arrhenius relationship presumes a Boltzmann distribution of atomic or molecular collisions in a homogeneous system, in the condensed phase one might presume a Boltzmann-type distribution of phonons22, 84 that result in a jump frequency whereby some unit of matter crosses the phase boundary from non-crystalline to crystalline. Thus, Z is a pre- -1 exponential factor related to a collisional or jump frequency (time ), Ea is an activation energy for growth (J mol-1), T is absolute temperature and R is the gas constant. Consistent with this approximation many crystallization systems measured for isotherms well below the melting point yield linear plots of ln(k) vs. 1/T. At higher temperatures, crystallization kinetics exhibit anti-Arrhenius-type dependence of k(T) (i.e. the reaction slows with increasing T 25, 75, 85-86). To date, most explanations of the anti-Arrhenius behavior presume competitive influences of nucleation and crystal growth. However, as demonstrated in our corresponding experimental manuscript, the rate of crystal growth is slowed for isotherms that approach the melting temperature, even for single crystals.85 Thus, contrary to Classical Nucleation Theory (CNT),5-8, 10-11 the rate of nucleation cannot be the

reason for the slowed rate of crystallization at higher temperatures. While attractive to access information about the activation energy of crystallization processes, the applicability of the Arrhenius equation to the crystallization transformation is questionable.22, 65, 84, 87-88 Interestingly experimentalists and theoreticians both seem to believe that the other has established its validity.19 Adding further complication, our experimental measurements exhibited dramatically different rate constants from KJMA fits of differential scanning calorimetry (DSC) and synchrotron temperature and time resolved X-ray diffraction (TtRXRD) data, collected for samples at the same crystallization isotherm.85 Before it is possible to extract valid mechanistic information from kinetic measurements it is critical to obtain material-specific, not method- dependent, rate constants. The observed method dependence of k appears to be a result of the size and shape of the sample container; a matter not adequately addressed with the KJMA model. Theoretical, computational and experimental efforts in polymer science have addressed certain aspects of the complex relationship between the exponent, rate constant, sample thickness and nucleation rate.23-26 For example, crystallization simulations have shown an inverse relationship between the KJMA exponent and the sample thickness for instantaneous nucleation and constant nucleation density23 and that the sample shape impacts the transformation curve of single- crystal growth. 26 Continuous nucleation simulations have also shown an inverse relationship between the KJMA exponent and both the sample thickness and nucleation density. 24 An inverse relationship between k and both the sample thickness23 and the square root of the sample mass25 has been experimentally observed. Similarly a relationship between the sample mass, geometry and rate constant was articulated for thermal decomposition of solids.52, 64 No sample size or shape dependence has been reported for t0. There is clearly dissonance within the condensed matter kinetics community regarding the use and significance of the KJMA parameters under various experimental conditions. At the most fundamental level, it would seem the chemistry and physics of bond formation/reorganization that must take place during a crystallization reaction should be intrinsic to a given material and consistent throughout the course of the crystallization reaction,

with possible exceptions during termination and heterogeneous nucleation. To address the challenge of parameter interpretation, a series of simulations was performed, for which explicit crystallization parameters were compared with parameters extracted from fitting the simulations to the KJMA model. From these analyses, expressions are derived to extract physically meaningful information and intrinsic parameters from experimentally measured rates of phase transformations in real systems.

3.3. Theory The original Kolmogorov derivation and a derivation of general KJMA theory have been recently summarized30, 44 and so will not be repeated here. We simply note the KJMA formalism is based on the geometrical relationship between the fractional volume of material crystallized, α(t) = V(t)/Vtot, a rate constant of crystallization, k, and the dimensionality of that transformation, n. To demonstrate the application of the KJMA model to crystal growth, consider the idealized case of growth of a spherical crystal. Assuming unbounded growth, the volume of the sphere as a function of time is given by Equation 3.3 where vpb is the velocity of the phase 4 boundary in units of distance per time, g is a shape factor equal to 휋 for a sphere, and t0 is the 3 difference between the initiation of the reaction time (e.g. time at which a melt is quenched to a crystallization isotherm) and when the crystallite actually begins to grow at t = t0 = tnuc. 3 3.3 (푡) = 푔[푣푝푏(푡 − 푡0)] In a real system, sample container walls bound crystal growth. If the spherical crystallite starts growing from the center of a cubic sample volume of dimension a3 then Equation 3.3 gives the exact volume of the growing crystallite until the radius of the sphere reaches the faces of the cubic volume, i.e. when vpb(t - t0) = a/2. After this point, even though the phase boundary velocity remains unchanged, the rate of crystal growth (i.e. V/t) is reduced since the crystal cannot grow outside the bounds of the cubic sample volume. Growth will not fully terminate until the radius of the growing, albeit increasingly truncated, sphere has traversed the distance between the center and corners of the cube, vpb(t - t0) = a2√3. These result in the observed sigmoid shaped growth curve, seen in Figure 3.1a, with the volume of

the crystallite increasing proportionally to the cube of the progress of the phase boundary until impingement of growth by the walls of the sample container slows the transformation rate. Notably, termination of the sigmoid is extremely dependent upon the sample geometry and location of the nucleus within that sample volume. For example, if the spherical crystallite nucleated in the corner of a cubic sample volume, then the shape factor would be 푔 = 1 × 8 4 휋 휋 = , the unbounded growth model would exactly model the transformation until vpb(t - t0) 3 6

= a, and full termination of growth would occur when the radius equals the body diagonal, vpb(t

- t0) = a√3. Thus, without knowledge of the exact location and orientation of the nuclei within the sample volume, measurement of crystallization rates will have an intrinsic uncertainty. Few experimental techniques provide a direct measure of actual crystallite volume or the location and orientation of nuclei.33, 89-90 Three common techniques record the total amount of heat (DSC), scattered intensity (XRD, TEM) or spectroscopic signal (NMR) that evolves throughout the course of the crystallization reaction. To apply the KJMA model, the signal is normalized to construct the fractional volume transformed, α. Though not recognized in the KJMA model, this normalization of the transformation introduces a relative rate constant k, related to vpb by Equation 3.4 which appears in a similar context in thermal decomposition literature to describe reactions proceeding by shrinking the reaction interface. 52, 64

푣푝푏 푘 ≈ 3.4 3√푉 As demonstrated experimentally,85 and by the simulations presented below, failure to account for the volume normalization in the rate constant is the reason for the observed sample- size dependence of the KJMA rate constant. Replacing V(t) by α and vpb with k, the unbounded growth model, Equation 3.3, exactly fits the data of the sphere growing in the cubic volume over the range 0 ≤ k(t - t0) ≤ a/2, (i.e. when α=π/6). This is shown in Figure 3.1 for which the data were generated by nucleating a sphere at t = 60 (arbitrary time units) in the center of a

3 -1 cubic sample volume, V = 30,000 units , at a rate of vpb = 1 unittime . Fitting the normalized data to the cubic growth model with the KJMA model gives the relative rate constant k = 0.0321(3) s-1, clearly in agreement with the prediction of Equation 3.4, 1 × 30000-1/3 = 0.0322.

Thus far the discussion has focused on the idealized case of spherical growth in a cubic volume, which is a simple problem to solve analytically since the termination time (and therefore the deviation from unbounded 3-D growth) depends entirely on the nucleation location relative to the nearest sample wall. However, real systems with non-spherical and often multiple crystallites are far more complex rendering this problem intractable via exact analytical solutions. The KJMA formalism attempts to account for this complexity through the concepts of “extended volume,” which is the sum of the volumes of all growing crystallites as though their growth is unimpeded by other growing crystallites, and the “overlap of extended volumes,” which includes the fraction of unimpeded volume that cannot exist in actuality because of impingement by neighboring crystallites. The contrasting influence of these extended and overlapping volumes is represented in the KJMA model as the Taylor series of the exponential in Equation 3.1 with the addition of the crystallite shape factor g, given in Equation 3.5.

∞ (−푔[푘(푡 − 푡 )]푛)푚 훼(푡) = − ∑ 0 3.5 푚! 푚=1

푣푝푏 푘 ≈ 3.6 3√푉 The first term in the series expansion is given in Equation 3.6 and is identical to the extended volume, Vext, and an overestimation of the total volume transformed since mutual crystallite impingement is not accounted for. The second term in the series expansion, which begins to become significant (i.e. has an absolute value > 0.01) at about 13% of the crystalline transformation, subtracts the volume of singly overlapped regions. The sum of the first and second terms results in an underestimation of the transformed volume. The third term adds the volume of doubly overlapped regions, overestimating the transformed volume. This procedure is repeated to produce the Taylor series expansion of the KJMA model. Notably, if the rate constant is the relative rate constant k, i.e. with vpb normalized with respect to V according to Equation 3.4, then the first term in the Taylor series expansion, Equation 3.6, is identical to the unbounded cubic model, Equation 3.3. When the test case data for the spherical crystal grown inside of a cube, Figure 3.1a, is fit by the KJMA model over the known unbounded range 60 ≤ t ≤ 85 (arbitrary time), a KJMA rate constant of 0.0343(10) time-1 is obtained; a 7% error from the exact unbounded cubic model rate constant. Using the KJMA parameters from the fit of the data in Figure 3.1a, the first 11 terms of this series expansion are plotted in Figure 3.2. Given that this test case has only one crystallite, there are no extended or overlapping volumes requiring additional terms. Thus the KJMA model’s expectation of additional crystallites explains the observed 7% overestimation of the actual rate constant for single crystalline systems. The accuracy of the KJMA model improves with additional crystallites in the sample as expected given its mathematical derivation. By accounting for the volume normalization as described in Equation 3.4 a reasonably accurate determination of the velocity of the phase boundary can be obtained from the measured KJMA rate constant for the growth of single crystallites. By extension, the growth of individual crystallites is reasonably fit with the KJMA model.

3.4. Simulation The above theoretical analysis using a simple but precisely defined model system clearly demonstrates thesample volume relates vpb and k. However, real systems are significantly more complex, with significant likelihood of multiple crystallites and generally unpredictable location of nucleation events; both of which impact the determined values of k. Furthermore, crystals are rarely spherical. And it has been shown that both growth anisotropy and crystallite orientation within the sample volume can impact the kinetic parameters.29-30, 33, 48, 50, 91-92 In addition, nucleation occurs when experimental measurements have the worst signal to noise ratio thus t0 is difficult to precisely determine. To address these complexities, and to establish experimental utility of this sample size corrected condensed phase kinetic model, a series of simulations were performed to replicate salient features of the experimental system utilized in the measurement of the crystallization of CZX-1 described in our corresponding experimental manuscript.85

3.4.1. Simulation Methods

Simulations were parameterized with respect to the velocity of the phase boundary, vpb, nucleation probability, Pnuc, nucleation location, and crystallite orientation. Because cubic crystallites of CZX-1 were observed, crystallization was simulated by nucleating cubic

crystallites at random locations and with random orientations, which were then grown along the {100} growth planes. Once nucleated, the crystallites were grown in sample volumes of specific size and shape. Simulations and automated fitting algorithms were written in Java (build 1.7.0-b147) in the Eclipse IDE (Build id: 200110615-0604) and the JAMA matrix package93 was used for fitting algorithms and error determination. Simulation source code is available at http://www.github.com/ericdill/CrystalSim and this program is described in Appendix C. Further documentation is available at the github website in the ‘doc\’ folder.

3.4.2. Simulated Sample Geometry A majority of the simulations were constructed to model the experimental CZX-1 crystallization system.85 Sample volumes were configured to approximate six unique experimental sample geometries of four synchrotron TtXRD experiments (X1 to X4) and two DSC experiments (D1 and D2). TtXRD geometries were approximated by cylinders with dimensions related to the width of the square cross section of the synchrotron beam (0.3 mm at X6B, National Synchrotron Light Source (NSLS), Brookhaven National Lab, and 1.0 mm at 11-ID-B, Advanced Photon Source (APS), Argonne National Lab) and the diameter of the sample capillary (ID = 0.48 and 0.68 mm). DSC geometries were approximated as cylinders where d = 5 mm, the diameter of the DSC pan, with the height determined from the sample mass and density.85 Each sample volume was simulated by a grid of discrete volume elements, organized as a simple cubic lattice. A consistent grid was used for all sample volumes, and scaled such that the smallest experimental dimension (r = 0.15 mm) is divided into 10 units. While this provides a unit length of 15 μm, it should be noted that there are no metric restrictions relating the simulation unit length to a physical distance (e.g. mm, μm, nm), as long as the unit length is greater than the width of the physical phase boundary, presumably a few unit cells wide.94 The unit parameters chosen result in simulations with between 104 and 3.20×106 volume elements. The relative sample volumes and aspect ratios (diameter divided by height) are given in Table 3.1.

Table 3.1. Tabulation of metrical parameters for the simulated sample volumes.

X1 X2 X3 X4 D1 D2 Avg. 2.9 3.4 4.3 5.1 14.5 14.9

>500) Min. 1 1 1 1 6 5

xtal

(V Max. 7 8 9 12 26 27

Num. cryst. cryst. Num. R 10 10 16 23 166 166 h 32 45 67 67 17 37 Aspect ratio 0.63 0.44 0.48 0.69 19.53 8.97 Volume (×104) 1 1.4 5.4 11 147 320 Number of 600 600 600 600 522 537 simulations

To investigate the impact of the sample container shape on the crystallization parameters, a set of constant volume simulations were conducted with the shape varying from capillary to disc. The simulation volume was fixed to 104 volume elements and the aspect ratio varied from 1-D (capillary, r = 4, h = V / 16 / π), to 3-D (isotropic cylinder, d ≈ h), to 2-D (disc, h = 1, r = √(V/π)).

3.4.3. Nucleation and Growth 2-D TtXRD data from the CZX-1 model system85 demonstrate variations in the nucleation density, ranging from single to polycrystalline reactions. Polycrystalline reactions further demonstrate nucleation occurs throughout the course of the reaction. With the exception of the lowest temperature isotherms, the TtXRD experiments revealed a relatively small number (< 10) of distinct crystallites, i.e., nucleation events. Thus simulations were -1 constructed using a continuous nucleation model with Pnuc = 0.05 time such that less than ten crystallites (Vxtal > 500) nucleated for all but the largest X-ray sample volume. Simulations -1 were performed for combinations of Pnuc and vpb values ranging from 0.001 time and 0.1 units∙time-1 to 0.7 time-1 and 4 units∙time-1. These, generally support the inclusion of a β term in the KJMA kinetic exponent17 as the nucleation rate, relative to growth, increases. However,

under the conditions observed for CZX-1 crystallization where nucleation is slow with respect to the rate of growth, experiments are appropriately modeled with β = 0. Nuclei were generated in random locations and orientations using the pseudorandom number generator in Java 1.7. At every time step, the possibility of nucleation is considered by generating a pseudorandom number 0 ≤ r ≤ 1. If r < Pnuc, then a new crystallite is nucleated at a location inside the not yet crystallized sample volume, whose position and orientation are defined by six additional pseudorandom numbers to define its x, y, z coordinates and the angles (between 0 and π) to rotate around each of the global x- y- and z- axes. A series of simulations were also performed using a constant nucleation density assumption, allowing comparison to a previous report which considered the impact of sample thickness on the KJMA parameters.23 A constant nucleation density assumption requires either instantaneous nucleation of all crystallites or a mechanism to terminate crystal growth at a certain volume such that all crystalline domains are equivalent. A density of one random nucleus per 103 volume elements was considered a reference nucleation density. Multiple simulations were performed for systems with one to twenty-eight nucleation events corresponding to model sample volumes between 103 and 2.8×104 volume units. For all simulations, after determining the nucleus location and orientation the crystallite was grown by increasing each crystallite {100} growth face by vpb = 1 unit length/unit time. The number of simulation volume elements that are engulfed by the growing cubic crystal during each time step determines crystallite growth.

3.4.4. KJMA Analysis of Simulations

For each simulation, nucleation time t0, growth dimensionality n and KJMA rate constant k of individual crystallites and the bulk sample were fit to Equation 3.1. Fits were performed for the first 50% of the crystallization transformation to minimize the impact of termination effects due to growth impingement on the sample container walls or other crystallites. Some reports recommend fitting  over a smaller range.95-96 However, the data sampling rate of our TtXRD system relative to the fastest growth rate requires a larger range to obtain sufficient data points for curve fitting. Additionally, it critical to include the onset of

the sigmoidal curve if n is to be a fitting parameter. Thus, for consistency, analysis of all experimental and simulated data were fit over 0 ≤  ≤ 0.5. Bulk parameters represent the transformation of the entire sample volume, which may include single crystals or numerous crystallites growing simultaneously. By contrast, because very small individual crystallites often exhibit irregular transformation curves precluding meaningful analysis with respect to any model, data for individual crystallites is only considered for crystallites that are larger than 500 simulated volume elements. A total of 20489 individual crystallites spread across the six geometries met the above criteria and were evaluated. As demonstrated in the corresponding experimental manuscript, there is a strong correlation between kinetic parameters.85 The correlation problem is quantified by consideration of the off-diagonal elements of the correlation matrices for the KJMA parameters averaged over all simulations given in Table 3.2. Such strong correlations between variables can be addressed by fixing one variable to a reasonable value,40, 63 preferably determined using an orthogonal method. Having experimentally established that CZX-1 grows as cubes, and defining the simulation for explicit cubic growth, crystallization parameters for the simulations were evaluated by fixing n = 3 and fitting k and t0. Because t0 can be explicitly determined in the simulation, simulation data were also evaluated with fixed t0, fitting k and n, to quantify the relationship between crystallization parameters and the KJMA dimensionality.

Table 3.2. Correlation between the model parameters averaged across all simulations.

k/ t0 k/n t0/n

Fit all Bulk 0.932 -0.846 -0.971 parameters Individual 0.866 -0.738 -0.958 Bulk 0.979 - - Fit k and t0 and fix n=3 Individual 0.979 - -

3.5. Results and Discussion To evaluate the effects of sample size and geometry on the KJMA parameters, as well as the proposed relationship between k and vpb, between 500 and 600 simulations of crystal growth were performed for each of six sample geometries, representative of the corresponding experimental TtXRD and the largest and smallest of the DSC sample geometries.85 Examples of the crystallites grown in the most and least isotropic simulation geometries, X4 and D1, respectively, are given in Figure 3.3. Immediately apparent are the significantly plate-like crystallites from D1, constrained by the shape of the sample container, whereas the crystallites from X4 are reasonably isotropic, yet surprisingly non-cubic. In both simulations, impingement by other crystallites and the container walls impacts the final shape.

Figure 3.3. Condensed and exploded views of simulated crystallites within (a,b) X4 and (c,d) D1 geometries.

Crystallization rate constants, extracted from fits of the bulk and individual transformation curves with n fixed to 3, are plotted in Figure 3.4 as a function of the total sample or individual crystallite volume, respectively. Individual crystallites (Figure 3.4a-c) demonstrate an essentially continuous variation in final crystallite size as a result of continuous nucleation. The maximum possible crystallite size is equivalent to the total volume of the sample container. However, as a direct consequence of continuous nucleation and the chosen

Pnuc, a single crystallite accounting for the entire transformation is rare, occurring in less than 12% and 0.5% of the smallest and largest XRD simulations, X1 and X4, respectively. No single crystal transformations were observed for the D1 or D2 simulations.

Comparison of the rate constants from the simulations and the proposed theoretical volume dependence is readily visualized using Equation 3.7, the linearized form of Equation 3.4. Plotted in all panels of Figure 3.4, Equation 3.7 represents the maximum theoretical rate constant as a function of volume, for unimpeded cubic growth. The slope and y-intercept correspond respectively to the growth dimensionality and natural logarithm of vpb.

ln(푘) = −3 ln(푉) + ln(푣푝푏) 3.7 The rate constants extracted from both the individual and bulk analyses of the simulations demonstrate the expected volume dependence. However, the contours in the ln(k) vs. ln(V) plot are distinct for the XRD and DSC geometries. Bulk analyses demonstrate the average KJMA rate constants for XRD geometries reasonably replicate the value defined in the simulation, whereas the rate constants are significantly underestimated for the DSC

simulation geometries. Analysis of the individual crystallites within the simulated volumes demonstrate it is the largest crystallites, D1-III and D2-III, that exhibit the most reduced rate constants. This likely is primarily a result of the disk sample geometry causing premature termination. These DSC-III crystallites individually are greater than 4% and 5%, for D1 and D2, respectively, of the total simulation volumes. By contrast, the smallest DSC-I crystallites, individually with volumes less than 1% and 1.5% of the total D1 and D2 simulation volumes, respectively, are reasonably isotropic, similar to the majority of individual crystallites from the XRD geometries. For the latter DSC-I crystallites, growth is significantly impeded by other crystallites. The effects of crystal growth constraints are also apparent in frequency histograms showing the difference between the fit values of t0 and the actual simulated tnuc, shown in

Figure 3.5. The tnuc is well estimated by t0 in the XRD geometry (Equation 3.4a) and for the small DSC-I crystallites (Equation 3.4b & c, open bars). However, for the larger DSC-III crystallites (Equation 3.4b & c, filled bars) the fit value of t0 is generally shifted to times significantly earlier than the simulated tnuc with D1 crystallites experiencing a greater shift to earlier times as a result of its more anisotropic simulation volume. The influence of the sample container shape on the simulated crystallization transformation is the primary cause of the differences between the defined and fit values of k and t0 in the XRD and DSC geometries. The geometric contrast is readily apparent in Figure 3.3 and the aspect ratios in Table 3.1. The disk shape of the DSC geometries exhibit aspect ratios of 20 and 9 for D1 and D2, respectively, in contrast to the more isotropic XRD geometries with limiting aspect ratios of 0.4 and 0.7. Irrespective of the location of nucleation within a disk-like geometry, it is far more likely that termination of unbounded cubic growth will ensue in the direction of the cylindrical axis than due to other crystallites for the larger DSC-III crystallites. This gives significantly smaller rate constants than that expected for unbounded cubic growth since the geometry of the sample container imposes an effectively lower dimensionality for crystal growth.

Figure 3.5. Difference between t0 and tnuc for (a-c,f) individual crystallites and (d-e) bulk transformation. (a,d) Aggregate of all XRD geometries. (a) (white) XRD crystallites with >500 volume units and <5% of total simulation volume and (black) >5% of total simulation volume. (b-c) (white) D1-I and D2-I crystallites with >500 volume units and (black) D1-III and D2-III crystallites. (f) (gray) D2-II and (black) D1-II.

3.5.1. Isotropic Geometries As shown in Figure 3.4a-c, the most frequently observed crystallization rate constants for relatively isotropic individual crystallites (XRD and DSC-I), exhibit nearly the maximum expected for unbounded cubic growth, Equation 3.7. The fraction of rate constants observed slightly above that line correspond to crystallites growing unbounded for more than 13% of the transformation; i.e. past the point where growth impingement becomes significant in the KJMA model. There is a significantly larger population of the XRD and DSC-I crystallites whose rate constants fall below this expected line. For those crystallites, impingement of growth likely began at a point earlier than assumed by the KJMA model, as would be the case for a crystallite nucleating near the edge or corner of a free growth volume (i.e. container wall or neighboring crystallite). Nevertheless, all observed rate constants fall within the range of uncertainty expected; a factor of about 2-3, based on the position and orientation of the

crystallite relative to other crystallites and the sample volume boundaries, both of which determine the distance though which the phase boundary can propagate. Furthermore, because of the strong correlation between k and t0 (Table 3.2), the smaller rate constants resulting from increased impingement of growth also result in an apparently early t0, Figure 3.5. Analysis of the bulk data by fitting the entire transformation provides a slightly different perspective. As seen in Figure 3.4d for the XRD-geometries, the most frequently observed rate constants correspond closely to the theoretical maximum predicted for unbounded cubic single crystal growth while the DSC rate constants are again significantly reduced. However, unlike the individual crystallites, the most frequent occurrence for bulk simulations are centered about the maximum theoretical rate constant. Importantly, this does not imply that rate constants (or by extension vpb) for crystal growth can be faster than the theoretical maximum. Instead it must be recognized that the bulk rate constants provide a single value to describe the growth of crystallites of all sizes within the sample. The differences between the individual crystallite and bulk analyses are similarly observed in the fitted values of t0, shown in Figure 3.5. The correlation between k and t0 is apparent with the later values of t0 corresponding to the larger values of k. In addition the spread in t0 - tnuc for the bulk XRD analyses is twice that of the individual crystallites. Like the bulk analysis of rate constants for which all different sized crystallites are described by a single rate constant, the KJMA model provides only a single value for t0 to describe the nucleation of all crystallites. However, continuous nucleation results in actual nucleation times for individual crystallites throughout the entire transformation. Nevertheless, the average fit value of t0 for the reasonably isotropic

XRD and DSC-I geometries accurately reproduces the known simulation tnuc. Importantly, while variation in the analyzed value of the rate constants can be understood with the complexities described above, the average observed rate constants for these simulations, from both individual crystallite and bulk sample analysis, closely reproduce the expected size-dependent rate constant for cubic growth. Using the volume of the individual crystallites and the total sample volume for the individual and bulk analyses, respectively, it is possible to convert the rate constant to the velocity of the phase boundary according to Equation 3.4. These values, collected in Table 3.3, demonstrate that for the isotropic XRD

geometries, the sample size correction to the KJMA model very reasonably reproduces the input simulation parameter, vpb=1.

Individual vpb Bulk Simulation Actual (Increasing Uncorrected Corrected Uncorrected Corrected Effective aspect n’ d/h) v v v v aspect ratio pb pb pb pb ratio X2 0.856(4) - 0.856(8) 0.987(10) 0.44 2.61(2) 0.45 X3 0.858(4) - 0.957(9) 0.996(10) 0.48 2.80(2) 0.62 X1 0.876(5) - 0.928(9) 0.965(9) 0.63 2.79(2) 0.61 X4 0.840(4) - 1.055(9) 1.055(9) 0.69 3.01(1) 1.0 D2 0.663(8) - 0.853(6) 1.038(7) 8.97 2.64(1) 3.62 I/bulk D1 0.652(10) - 0.648(4) 0.893(6) 19.53 2.49(1) 5.10 D2 0.588(13) 0.742(15) II D1 0.538(12) 0.765(14) D2 0.450(5) 0.915(9) III D1 0.602(7) 0.945(9)

In contrast to simulations using the continuous nucleation model described above, there is no such variation in the KJMA rate constant as a function of sample size in simulations for which a constant nucleation density was employed, consistent with the findings in a previous report.23 Constant nucleation density, when combined with instantaneous nucleation, creates a scenario in which the crystallite size distribution is independent of the total sample volume. Simulations constructed with a constant nucleation density of one nuclei per 103 volume units across all sample volumes exhibited a rate constant k = 0.1 time-1. When corrected for the

3 -1 average grain size of 10 , instead of total sample volume, the simulation vpb = 1.0 unittime is correctly reproduced. Thus the applicability of the volume correction is highly dependent upon the experimental system under investigation, requiring correction that range from total

sample volume to average grain size for systems exhibiting relatively slow but continuous nucleation to those with significant “instantaneous” nucleation, respectively.

3.5.2. Anisotropic Geometries

Whereas Equation 3.4 effectively recovers the vpb for the simulations of isotropic geometries, this sample size correction poorly recovers vpb for the large DSC-III crystallites.

Data shown in Figure 3.4b-c and Figure 3.5b-c indicate that k is smaller and t0 is earlier for the large, anisotropic DSC crystallites than for their isotropic counterparts. Constant volume simulations were utilized to directly assess the impact of crystal growth in anisotropically constrained sample geometries, for which the simulated sample geometry was varied from a disc (d >> h), to an approximately isotropic cylinder (d = h), to a capillary (d << h). Because the apparent dimensionality, n', is strongly influenced by anisotropic sample containers, the bulk data were fit to Equation 3.1 with t0 fixed to tnuc to evaluate the magnitude of the sample container effects on both n' and k. A summary of the results of these constant volume simulations is shown in Figure 3.6, with the anisotropy of the sample plotted as the natural log of the sample aspect ratio such that the disk and capillary geometries exhibit positive and negative anisotropy values, respectively. For cubic crystal growth in disk shaped sample volumes, the apparent dimensionality (Figure 3.6a) begins to decrease from n' = 3 at an aspect ratio of about 2.5 (ln(d/h) = 0.9) to n' ≈ 2.4 for anisotropy greater than about 10. In the capillary direction the apparent dimensionality decreases to an apparent dimensionality of n' ≈ 1.6 for aspect ratios less than 0.04. Modeling the growth of a single crystal in these geometries results in the apparent n' dropping to 2 and 1 for the disk and capillary limiting geometries, respectively. The difference in n' between the single crystal and bulk is consistent with concept of β in the KJMA exponent. As 1-D and 2-D geometric limits are approached, the distance over which the phase boundary must traverse goes to infinity, thus the KJMA model’s normalization of the transformation to  results in the apparent rate constant k' approaching zero. Correspondingly, this yields an apparent phase boundary velocity, vpb', which also approaches zero (Figure 3.6b).

The decrease in the apparent phase boundary velocity, vpb', as a function of sample aspect ratio can be fit with the empirical function given in Equation 3.8 푑 푣′ = exp (푚 ∗ ln ( ) + 푏) 3.8 푝푏 ℎ 푠 Because the phase boundary velocity for the aspect ratio variation simulation, 푣푝푏, was set to a value of 1 unittime-1, this function provides a normalized sample anisotropy correction, ac, to the KJMA model. The regression parameters (m & b) in Equation 3.8 are fit to the regions d/h < 0.46 and d/h > 2.67, respectively, generating Equation 3.9. 푑 푑 ′ exp (0.44 × ln ( ) + 0.31) < 0.46 vpb 푎 = = { ℎ ℎ 3.9 푐 푣푠 푑 푑 푝푏 exp −0.29 × ln ( ) + 0.24 > 2.67 ℎ ℎ

The corrective term ac can be used to obtain the actual vpb from the apparent bulk KJMA rate constant according to Equation 3.10.

3 푘′ × √푉 3.10 푣푝푏 = 푎푐 The exact shape of the curves in Figure 3.6 is dependent on the range of  over which the data are fit (here 0 <  < 0.5), and the relative number of nuclei in the sample volume. Fitting to a smaller extent of  increases the width of the “isotropic plateau.”

For a given sample aspect ratio, an increased number of crystallites, obtained from a larger sample size or faster nucleation rate, divides the anisotropic volume into smaller, more isotropic domains, decreasing the apparent sample anisotropy and increasing both n’ and vpb’. Rather than regenerating the curves of Figure 3.6 and Equation 3.9 for every sample size or nucleation probability, it is possible to achieve an equivalent anisotropy correction by fitting the D1 and D2 simulations with fixed t0 to obtain n'. (Note that fixing t0 = tnuc is not a valid approach for fitting experimental data since there is no known experimental technique to obtain the exact value for tnuc. Additionally, small variations in the fixed value of t0 dramatically impact n'.) Based on the fit values of n' (2.49(1) and 2.64(1) for D1 and D2, respectively) effective aspect ratios can be extrapolated from Figure 3.6a (5.10 and 3.62 for D1 and D2, respectively). This procedure was also applied to XRD geometries, for which the most anisotropic

X2 exhibits a capillary-retarded bulk vpbʹ of similar magnitude to that observed for the D2 disk geometry. Applying this anisotropy correction to the simulated data accurately reproduces the simulated vpb (Table 3). The effect of the anisotropy correction can be visualized by extrapolating its impact to the rate constant k, such that Figure 3.4d is corrected to Figure 3.7b. A similar anisotropy correction accounts for the individual anisotropic D1-III and D2- III crystallites, transforming Figure 3.4c-d to that shown in Figure 3.7a. To remove the sample container shape effect from the DSC-III crystallites a measure of their anisotropies is required. In the simulations, the exact crystallite shape as a function of time is known thus an exact correction could be used. However, such an exact correction is irrelevant because to experimentally obtain such crystallite-shape time-dependence, vpb must already be known, removing the need to use the KJMA model.

To be more experimentally relevant, an estimate of individual crystallite anisotropy was developed using knowledge of only the final crystallite volume and the sample container shape. With growth primarily limited by the sample height for a disk geometry, the final crystallite shape is approximated as a cylinder with height equal to that of the sample, hsample, and corresponding diameter derived from the crystallite volume, 푑푥푡푎푙 ≈ 2√푉푥푡푎푙 /휋ℎ푠푎푚푝푙푒. Evaluation of the individual crystallites in the disk-shaped geometries of the constant volume simulations indicate no significant anisotropy effect for volumes with an aspect ratio of less than 2. With greater anisotropy nʹ drops from 3 to 2, and vpbʹ drops from 1 to 0 as the anisotropic limit is approached. As demonstrated for the bulk transformation, accounting for sample-shape effects requires fitting the transformation with an appropriate nʹ and subsequently correcting kʹ based on the effective aspect ratio. Similarly, individual crystallite transformations are fit with an estimated apparent KJMA dimensionality given in Equation 3.11 and kʹ is corrected with the empirical function given in Equation 3.12 which was obtained from the fit to the average vpbʹ of the crystallites from the constant volume simulations with 2 ≤ d/h ≤ 40. As shown in Table 3.3 and Figure 3.7b, this anisotropy correction based on the cylindrical crystallite approximation effectively corrects the DSC-III crystallites, but is inadequate for DSC-II crystallites.

(2푑 + ℎ ) 푛′ = 2 [ 푥푡푎푙 푠푎푚푝푙푒 ] − 2 3.11 푑푥푡푎푙 푑 푑 3.12 푎 = exp (−0.291 × ln ( )) > 2 푐 ℎ ℎ Frequency histograms for the sample anisotropy corrected D1 and D2 simulations are presented in Figure 3.8, showing the difference between the fit values of t0 and the actual simulated tnuc for both bulk and individual crystallite analyses when k and t0 are fit with n fixed to nʹ. Bulk analysis of the anisotropic samples fit with nʹ (Figure 3.8c) reasonably reproduces the actual tnuc. For individual crystallites, the difference between regions I and III fit with n = 3 (Figure 3.8b-c) largely vanished when fit with the simple cylindrical crystal anisotropy approximation of nʹ for both D1 and D2 simulations, Figure 3.8a-b. Thus, by using a reasonable geometric approximation for the dimensionality parameter, the physical significance of t0 is largely recovered.

3.6. Conclusion The crystal growth simulations described in this work provide clear definition and physical interpretation of the parameters t0, n and k of the KJMA kinetic model for isothermal

phase boundary controlled transformations. These parameters are highly correlated, and are substantially impacted by the sample geometry in which the phase transformation occurs.

The parameter t0 is equivalent to the time of initial nucleation. However, it can only reasonably be extracted from rate measurements if the dimensionality parameter is fixed to an appropriate value determined by independent means. The dimensionality parameter n is an intrinsic value determined by the morphology of the growing crystal itself. However, if the transformation occurs in an anisotropically constrained sample volume, an apparent dimensionality, n', must be utilized when fitting actual kinetic data, which may be significantly reduced from that of its intrinsic value. Such external constraints are least impactful at the early stages of a transformation, prior to the phase boundary being impinged by other crystallites or sample containment boundaries. The normalized KJMA rate constant k, when corrected with respect to the sample volume, V, and sample container anisotropy, ac, as needed, provides the material specific parameter, the velocity of the phase boundary, vpb. Because the location and orientation of nucleation events within the sample volume determines the distance over which the phase boundary must travel and thus the time required to complete the phase transformation, without direct knowledge of these, measured vpb values are expected to vary by about a factor of 2 for a system exhibiting random nucleation throughout the sample. However, with sufficient sampling, the average values from bulk analyses appear to accurately reproduce the actual vpb. Together, these results yield the more comprehensive kinetic expression of Equation

3.13, where ac is the anisotropy correction and g is a geometric factor. Experimentally, it is highly recommended that measurements be conducted in as isotropic a sample volume as possible such that ac = 1 and nʹ = n.

푛′ 푣푝푏푎푐 훼(푡) = 1 − exp (− ( (푡 − 푡0)) ) 3.13 푔3√푉

The crystallization of the halozeotype CZX-185 proved to be an effective system upon which to base the above simulations. The specific simulations were designed to approximate the number of crystallites and the continuous nucleation experimentally observed by TtXRD

in the CZX-1 system. However, simulations conducted using parameters with the vpb: nucleation probability ratio spanning four orders of magnitude indicate that the conclusions of Equation 3.13 are generally applicable to systems with reduced nucleation frequency, and to those for which the vpb : nucleation probability ratio is about an order of magnitude greater than the system modeled here. With greater nucleation in a given sample the concept of adding the β term to the kinetic exponent appears to become relevant. Furthermore, with a significantly higher number of nucleation events, correction of the KJMA rate constant with respect to the cube root of the total measured sample volume increasingly over estimates vpb. Under such conditions, a correction related to the grain size distribution is likely necessary.42, 67, 97 The cubic morphology of CZX-1 further simplified the measurement and interpretation of the crystallization kinetic parameters. If the crystallite growth habit of a system deviates from cubic, but growth is still isotropic, then additional inclusion of only the appropriate g factor is required. However, for a system in which distinct particle faces grow at different rates, then vpb,(hkl) for each growth face must be considered; limited by the slowest growing faces. Previously, general investigations into particle growth habit indicated the rate of overall crystallization transformation decreases with increasing particle anisotropy. 29-30, 33, 48, 91-92 It is almost certain that a correction for the effect of particle anisotropy will be similar to the sample-anisotropy correction discussed in this work. Deconstructing the method dependent kinetic parameters of the KJMA model into parameters that are intrinsic to the material opens great opportunity for further mechanistic studies. Importantly, the identification of parameter correlations and identification of sample shape effects, suggests extreme caution should be exercised with respect to mechanistic interpretations that imply variation in the kinetic exponent and/or apparent activation energy throughout a phase transformation.17, 40, 77-78, 81-82, 98 Most likely any such changes during the course of a reaction are evidence of a transition between unbounded and restricted growth. Furthermore, determination of the velocity of the phase boundary, independent of the rate of nucleation, will allow careful investigation of the non-Arrhenius behavior of vpb, which

experimentally is shown to be depressed with increasing temperature between some vpb(Tmax) and the melting temperature. 25, 75, 85

Chapter 4: Probing the Mechanisms of Nucleation in Melt-Crystal Reactions

4.1. Introduction Understanding crystallization mechanisms is of broad interest. In Chapters 2 and 3, the growth of crystals was explored. Here, we work on understanding how those crystals first appear. Bulk crystallization measurements are an effective tool for extracting the nucleation time of the first crystallite.53 To obtain a nucleation rate, one might consider simply taking the -1 inverse of the t0 parameter in the M-KJMA model, knuc=t0 . Such an approach necessitates * -1 having a physically significant zero point (t=0) on the measurement time-axis or t0 is not an intrinsic parameter but simply the inverse of the time between the observation of crystallization and some arbitrary time in the past. Therefore, defining a physically significant zero point is a critical aspect of nucleation rate measurements. There are two reasonable definitions for a physically significant t=0 in melt- crystallization: Define t=0 to be the initiation of the quench (tq) or define t=0 to be the end of the quench when the system reaches the isotherm (tiso). One attractive concept of the tq definition is that the nucleation time can never be negative as this would seem to imply that nucleation occurred while the system was above the melt temperature. However, the time it takes the system to reach the desired crystallization isotherm significantly impacts the measurement of the nucleation rate as shown by recent melt-crystallization kinetics85 and preliminary work by Feier Hou in the Martin Group.99 In these works, the 40-100 K∙min-1 DSC quench rate resulted in nucleation occurring during quenches in melt-crystallization from 230 °C to isotherms below 150 °C and in glass-crystallization from -40 °C to above 80 °C. However, the second-generation dual-barrel forced-air furnace Figure 2.1 used in many of the synchrotron diffraction measurements can quench to all isotherms between 40 and 160 °C in under 20 s, frequently reaching the isotherm before nucleation is observed. Recognizing that the quench rate can have a significant impact on the nucleation kinetics, we choose to define t=0 as tiso and not tq, thereby avoiding the incorporation of extrinsic factors such as the quenching rate into what is nominally an intrinsic materials property. The nucleation rate is then defined by Equation 4.1. By using tiso as the zero time, the nucleation time, tnuc=t0-tiso,

* Note that t=0 is different from t0.

becomes negative when nucleation occurs during the quench, giving clear indication when crystallization experiments are non-isothermal; something that the tq formalism does not immediately provide.

−1 −1 푘푛푢푐 = 푡푛푢푐 = (푡0 − 푡𝑖푠표) 4.1 The nucleation rate is only an intrinsic materials property when some factor external to the system does not initiate the formation of the new phase. External factors can include dust and container walls, among others, and are problematic because they lower the activation barrier to nucleation. We call this “extrinsic” nucleation. In the absence of these complicating factors, the nucleation mechanism “intrinsic” to the sample is in operation which is usually a slower process as it lacks the lowered activation barrier of extrinsic nucleation. As it is nearly impossible to remove all external nucleating agents from a sample, intrinsic nucleation can be challenging to study. However, upon recognizing that intrinsic nucleation is the slower of the two processes, a reasonable effort to remove external nucleating agents can be combined with numerous crystallization measurements to arrive at a distribution of initial nucleation times. The longest of these initial nucleation times (or smallest initial nucleation rates) is then a good measure of the intrinsic nucleation rate of the system under investigation. The nucleation rates, as determined with DSC and azimuthally averaged 2D diffraction images are presented in §4.4.1. As the area detector utilized in these investigations is invaluable for visualizing and differentiating crystal nucleation from crystal growth, individual diffraction spots are analyzed to provide further insight into the nucleation process in CZX-1 in §4.4.2. Most nucleation theories make the assumption that there is a significant surface energy between the melt and crystalline phases which is a considerable driving force in the formation of nuclei. Classical Nucleation Theory proposes that actual nucleation event is precipitated by the serial aggregation of fundamental “units” into “embryos” until a sufficiently large embryo is formed that can support crystal growth to a macroscopic size. This validity of these assumptions is evaluated in the context of the local structure in CZX-1 in its molten and crystalline phases in §4.4.3.

4.2. Classical Nucleation Theory The basic thermodynamic treatment and the kinetic analysis of the process of nucleation has changed very little since seminal works of 1876-1953.2-3, 5-8, 10-11 The collection of these works are frequently referred to as Classical Nucleation Theory (CNT). For a thorough treatment of CNT, the reader is directed to recent reviews.18, 39-41, 100 Briefly summarized here are some of the more critical assumptions that were made in the application of CNT to the condensed phases. Classical Nucleation Theory (CNT) was initially developed to describe spontaneous droplet formation from the vapor phase and proposes that the formation of critical-sized nuclei required for irreversible growth of the new phase is governed by the formation of aggregated of the vapor phase molecules, termed “embryos.” An embryo, Ai, contains i molecules of phase

A and by capturing or losing a single molecule becomes an Ai+1 or Ai-1 embryo, respectively. The fluctuations of embryo size in the vapor phase therefore involves only this process of aggregation and all other processes by which embryos can form are ignored as they are claimed to be too infrequent. Examples of such processes include the formation of a large embryo by the collision of two or more smaller embryos and its inverse, the dissolution of a large embryo into two or more smaller embryos. Embryo formation is then described as 퐴𝑖 + 퐴 ⇌ 퐴𝑖+1. Under conditions in which the forward process is faster than the reverse, the size distribution of embryos steadily increases with time. With sufficient condensation, a point is reached at which a critical-sized embryo, Ac, is formed where the attachment of a single additional molecule a single additional molecule attaching A to an embryo of size Ac+1 is termed the

“nucleation event” and the Ac+1 embryo becomes the “nucleus” which supports irreversible growth to macroscopic size. This is the vapor-to-liquid phase change mechanism according to CNT. The “classical” rate of nucleation (Equations 5.4-5.5) was derived by Becker, Döring 7, 101-102 and Zeldovich which relates the nucleation barrier (ΔFc) to the surface energy (σ), molar volume of the new phase (vII) and the ratio between the droplet (pI) and planar surface (pI∞) vapor pressure.

퐽 = 푍훽푐푛퐼 exp(−훥퐹푐/푘푇) 4.2

16휋휎3푣2 퐼퐼 4.3 훥퐹푐 = 2 2 2 푘 푇 log (푝퐼/푝퐼∞) This measure of the rate of nucleation of spontaneous droplet formation from a supersaturated vapor has been shown to be a reasonably accurate descriptor of the nucleation process primarily because there is a significant difference between the vapor and liquid phases. In fact, the difference between the volume occupied by a mole of a liquid or crystalline phase and its corresponding vapor phase is often three orders of magnitude which makes the aggregation of a sufficient number of molecules likely a rate determining step. This formalism has also been shown to sufficiently describe the nucleation of a crystalline phase from dilute media, such as crystallization from a solution. The application of CNT to describe melt-crystallization requires the assumption that there is a significant surface energy between the liquid and crystalline phases (σcm) which modifies the nucleation barrier (ΔFc) for the dilute-condensed (Equation 4.3) to that given in

Equation 4.5 where vII is the volume of a molecule in the crystal, ΔHm is the enthalpy of fusion 101 per molecule and T∞ is the bulk melting point. Becker assumed that an activation energy

EA was required to transport a molecule across the interface from the melt to the nucleus; the magnitude of which will be close to the activation energy for self-diffusion, given in Equation 4.5 16휋 휎3 푣2 푇2 푐푚 퐼퐼 ∞ 4.4 훥퐹푐 ≅ 2 2 3 훥퐻푚훥푇

퐽 = 퐶 exp(−퐸퐴/푘푇) exp(−훥퐹푐/푘푇) 4.5 Further derivations of nucleation theory have been developed and discussed in numerous reviews,39, 86, 92, 100 building on the original concepts of CNT and simplifying the expression for the steady state nucleation rate to the single exponential function given in Equation 4.6 where W* is the activation energy for nucleation and A* is a dynamical factor describing the cluster growth rate.

푆 ∗ ∗ 퐼 = 퐴 exp[−푊 /푘퐵푇] 4.6 These thus suggest the nucleation rate to be considered proportional to the thermodynamic probability of having a fluctuation that leads to the formation of a critical

cluster through serial molecular aggregation. Condensed phase nucleation rates predicted by these extensions of CNT have a qualitatively similar temperature dependence but the predicted magnitude of the theoretical nucleation rates are frequently in poor agreement with experimental observations, especially for networked materials where the disparity between observed and calculated nucleation rates has been observed to be upwards of 1020 [cite]. Such gross disparity between experiment and theory is raises significant question as to the validity of applying CNT to describe crystal nucleation from liquids and glasses.103-112 With such inadequate predictive ability, one must question whether the theoretical framework developed for nucleation in the vapor to liquid, vapor to crystal and solution to crystal transformations is truly applicable to the melt to crystal transformation.

4.3. Experimental Methods

All manipulations were performed under an inert N2 atmosphere in a glove box or using vacuum lines. ZnCl2 and HNMe3Cl were purchased from Aldrich and purified via double sublimation prior to use, as previously reported.85 CZX-136 and CuCl54 were prepared according to previously reported procedures. The purity of all starting materials was confirmed by powder X-ray diffraction (XRD, INEL CPS-120) and differential scanning calorimetery (DSC, TA Instruments Q100). Data collection with DSC and Temperature and time resolved X-ray Diffraction (TtXRD) was performed as described previously in Chapter 2.85 DSC samples were sealed in high-pressure stainless steel pans with gold foil seals with three sample masses: 12.2, 20.2 and 26.5 mg. Temperature and time resolved X-ray Diffraction (TtXRD) data were obtained on beam lines 11-ID-B (90 keV, λ=0.13702 Å) at the Advanced Photon Source (APS), Argonne National Laboratory (ANL) and X6B (19.1 keV, λ=0.646 Å) at the National Synchrotron Light Source (NSLS), Brookhaven National Laboratory (BNL). Samples of CZX-1 in Lindemann capillaries (0.48 or 0.68 mm inner diameter), centered in the synchrotron beam, were equilibrated in the melt at 230 °C then quenched to a crystallization isotherm, Tiso, using a forced-air quenching furnace () with ±2.5° temperature control, according to previously

reported methods.85 Nucleation and growth data for individual crystallites was obtained using the “spot-picking” software I designed and wrote, described below. Source-code for this project is available upon request. Pair distribution function analysis (PDF) of melt and bulk crystalline samples was performed with PDFGetX3.113

4.3.1. Determination of the quench time The nucleation time is defined here as the time between when the system reached the desired crystallization isotherm, tiso, and the fit value of t0 from the M-KJMA expression in

Equation 3.13. For the 2D TtXRD measurements, tiso was manually determined by recording the frame number where the system reached the isotherm which imposes systematic error of up to the collection time of the measurement of 2 and 6 seconds at the APS and NSLS, respectively. During many of the DSC experiments, the system nucleated before the system reached the desired crystallization isotherm as shown by the lack of separation from the quenching peak (hatched) and the crystallization peak in the DSC measurements in Figure 2.2 (p20).53 Multiple Gaussians were fit to the quench peak (q(t)) so that it could be subtracted from the raw signal to obtain the crystallization heat flow. The system was assumed to be very close to the desired crystallization at the time when 99% of the quench peak had been

푡 ∞ −1 [ 푖푠표 ( ) ] [ ( ) ] accounted for, i.e., ∫0 푞 푡 푑푡 × ∫0 푞 푡 푑푡 = 0.99.

4.3.2. 2D TtXRD measurement and analysis of individual crystallites Time-dependent synchrotron diffraction data were collected in a Debye-Scherrer geometry with a 2048×2048 pixel GE Silicon detector (APS)56 or a 2084×2084 SMART CCD detector (NSLS). The wavelength and detector alignment were calibrated to LaB6 or CeO2 standards using fit2d57 to correct all experimental data. A melted and recrystallized ingot of CZX-1 in a fused silica capillary was centered in the synchrotron beam. The sample was heated to 230 °C, noting the temperature at which the sample melted, which when compared to the CZX-1 melting point (Tm) of 173 °C provided an internal temperature calibrant. To ensure melt isotropy, the sample was held at 230 °C, well above Tm, for 5 min. Diffraction images were collected at 0.5 Hz (APS) and 0.05 Hz (NSLS),

initiated for at least 1-3 frames prior to quenching to a crystallization isotherm (Tiso) of between 40 and 160 °C. Diffraction data was recorded at the isotherm until crystallization was complete. When a sample did not nucleate within 2 h, the reaction was aborted because of limited synchrotron time; which was the case for Tiso >155 °C. The melt-quench-crystallization cycle was then repeated. Data from a total of 55 crystallization experiments are reported here. During crystallization experiments the samples were oscillated 10° in synchronization with the duration of X-ray exposure to illuminate a larger region of reciprocal space and minimize thermal gradients in the sample.85 Temperature control of the crystallization reaction was afforded by a pneumatically switchable manifold that directs airflow through one of two ¾” in-line air heaters (Omega), schematically depicted in Figure 2.1. One furnace was set to the temperature of the high temperature melt isotherm (230 °C) and the other to the temperature of the desired Tiso (40- 155 °C). The temperature of each heating element was controlled using a Eurotherm 91p temperature controller to a precision of ±0.5 °C. Using the phase transitions of elemental sulfur and the melting temperature of CZX-1 as calibrants, temperature accuracy of ±2.5 °C was obtained; the relatively large range is a result of sample environment conditions including temperature gradients, variations in air flow, and sample/thermocouple placement within the air stream. The temperature of the air stream stabilized to the quenched isotherm at the controlling thermocouple T3 within 15 sec after the air streams were switched affording rapid quenching to Tiso, irrespective of melt and quench temperatures. Evaluation of the appearance and growth of individual diffraction spots should directly correspond to the nucleation and growth of individual crystallites, thus affording single crystal data even for polycrystalline reactions. The time-dependent intensity of all Bragg diffraction spots was determined and fit with the KJMA expression as described below, thereby 53, 85 independently determining kinetic parameters, t0 and vpb, of individual crystallites.

4.3.3. Spotpicking Algorithm To analyze the nucleation and growth of individual crystallites from the 2D TtXRD data, the “active” pixels that correspond to diffraction spots must be located and extracted from

each of the images in the experimental time series. It was necessary to develop an algorithm to automatically find diffraction spots in 2D images and correlate them through time as it would be prohibitively time consuming to analyze each set of diffraction images by hand. However, differentiating the signal of a crystallite from the amorphous background is challenging. To locate diffraction spots, we recognized that the first and second derivatives can be used to locate various points of a peak. The first derivative corresponds to the slope and curvature of the original peak which are zero at local maxima and inflection points, respectively, as shown in Figure 4.1 for a Gaussian peak (A=1, x0=3, σ=0.3). The position where the first derivative is zero could theoretically be used to determine the peak positions but due to the presence of noise in the collected data, detecting zero is often extremely cumbersome in practice. The problem is further exacerbated because each derivative taken reduces the signal to noise ratio. However, as the 2nd derivative is negative between the inflections points and strongly so at the center of the peak in the original function, as shown in Figure 4.1, using the 2nd derivative as opposed to the 1st derivative can circumvent the detection-of-zero problem as detecting signal is considerably simpler.

st nd Figure 4.1. A Gaussian peak (A=1, x0=3, σ=0.3) and its 1 and 2 derivatives. Vertical lines indicate the position along x where the 2nd derivative is zero.

Derivatives can be analytically calculated only if the original function is known, i.e.

2 푓ʺ(푥) = 푑 푓(푥). If the original function is not known then derivatives must be calculated 푑푥 numerically, for which many methods exist. We have chosen to use the method of Savitsky and Golay,114 generalized to two dimensions, where local linear regression is used to determine the coefficients for an nth order polynomial. This method works by taking the inner product of the data and a “mask” which corresponds to a specific numerical manipulation like the first derivative, second derivative, mixed derivative, etc. The mechanism by which masks are constructed is not necessarily intuitive, necessitating a short description. To compute the second derivative at each pixel in the 2D diffraction images, such as the one shown in Figure 4.2a, a 4th order polynomial in x and y was used (Equation 4.7) with nd a 7×7 mask, where the 2 derivative is computed for the pixel at the center of the mask (xc,yc). The derivation of our 2nd derivative masks will therefore be presented with those parameters. 2 2 3 2 푓(푥, 푦) = 풂ퟎ + 풃ퟎ푥 + 풃ퟏ푦 + 풄ퟎ푥 + 풄ퟏ푥푦 + 풄ퟐ푦 + 풅ퟎ푥 + 풅ퟏ푥 푦 2 3 4 3 2 2 + 풅ퟐ푥푦 + 풅ퟑ푦 + 풆ퟎ푥 + 풆ퟏ푥 푦 + 풆ퟐ푥 푦 4.7 3 4 + 풆ퟑ푥푦 + 풆ퟒ푦

푥 푦 푥2 푥푦 푦2 푥3 푥2푦 ⋯ 푦4

푎0 푏0 푏1 푐0 푐1 푐2 푑1 푑2 ⋯ 푒4 1 −3 −3 9 9 9 −27 −27 ⋯ 81 1 −2 −3 4 6 9 −16 −18 ⋯ 81 1 −1 −3 1 3 9 −1 −9 ⋯ 81 1 0 −3 0 0 9 0 0 ⋯ 81 푴풙풚 = 4.8 1 1 −3 1 −3 9 1 9 ⋯ 81 1 2 −3 4 −6 9 16 18 ⋯ 81 1 3 −3 9 −9 9 27 27 ⋯ 81 1 −3 −2 9 6 4 −27 −12 ⋯ 16 ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋱ [ 1 3 3 9 9 9 27 27 81]

푻 −ퟏ 푻 4.9 푨풙풚 = (푴풙풚푴풙풚) 푴풙풚 Determining the “mask” for the 2nd derivatives in x and y requires constructing the matrix Mxy, as shown in Equation 4.8 and transforming it into the linear regression coefficients,

Axy, with Equation 4.9. Each row in the resultant linear regression coefficient matrix is

associated with a specific operation. Given the specifics of the construction of Mxy, the first six rows in Axy correspond to the smoothing operation, the first derivative in x, the first

2 derivative in y, the second derivative in x, the mixed derivative 푑 and the second derivative 푑푥푑푦 in y, respectively. The numerical values of these six masks are presented in Appendix D. A 7×7 subsection of the 2D diffraction image, centered on the pixel which the 2nd derivatives in x and y are to be calculated (xc,yc), is then extracted and converted to a vector such that the (x,y) coordinates of the pixel and mask are in correspondence (see the columns nd labeled “x” and “y” in Equation 4.8). Finally, the 2 derivatives in x and y at (xc,yc) can be determined by computing the inner product between rows 4 and 6 of the Axy 4 and 6 4 and 6 4 and 6 matrix and the data vector. The results of computing the 2nd derivatives in y and x are shown in Figure 4.2b-c, respectively and further discussed below. To visually present our algorithm for finding pixels that comprise Bragg spots, consider the oligocrystalline diffraction image that resulted from an isothermal crystallization experiment carried out at 145 °C, shown in Figure 4.1a. In this image, the intensity increases from white to black such that Bragg diffraction is black. The cross in the center of the image is the location of the beam center, as determined by CeO2 and LaB6 calibrants. Shown in Figure 4.1b-c are the 2nd derivatives in the y- and x-directions, respectively, colored such that blue is negative, white is zero and red is positive. It is immediately apparent that the Bragg diffraction spots appear as red-blue-red sequences relative to the directionality of the 2nd derivative. The sum of the 2nd derivatives is shown in Figure 4.1d where the location of the diffraction spots is appear as blue circles surrounded by a red ring. Pixels that are less than an empirical “threshold” value in both directions of the 2nd derivatives are marked as “active” and therefore potentially part of a diffraction spot. The empirical intensity threshold value depends on the signal-to-noise ratio and is therefore dependent upon the synchrotron flux, exposure duration, detector characteristics, chemical composition of the sample, sample thickness, etc. It was found that a threshold value of -5 was effective at excluding much of the noise at the expense of very little of the diffraction spots, though the exact value is fairly flexible.

Our criterion for determining if active pixels become part of a “spot” is that more than five “active” pixels must be in direct contact along the x- or y-direction. The spot size parameter serves to exclude random active pixels, differentiating them from actual crystallites. Collections of five or more neighboring “active” pixels was determined to be an effective minimum size threshold for a diffraction spot. The actual size of the diffraction spot on the detector generally depends on the crystallinity of the sample, sample-to-detector distance and the pixel density. The image shown in Figure 4.1e has been overlaid with the spots found by this method with the threshold value set to -5 and the number of active pixels per spot set to 5.

It is apparent that the majority of experimentally observed diffraction spots have a corresponding algorithmically determined diffraction spot. Finally, the algorithmically determined Bragg spots are overlaid onto the experimentally obtained diffraction image in Figure 4.1f, demonstrating that this approach, based on a set of rudimentary criteria, successfully finds the majority of the diffraction spots. This process is called “spotpicking.”

The intensity of the algorithmic diffraction spots, Itot, can be obtained by summing the corresponding pixels. As shown in Figure 4.1f, many of the experimentally observed diffraction spots are still visible in black behind the algorithmically determined spots, indicating that the algorithm, while successfully locating the majority of the Bragg diffraction, is not as effective at encompassing all diffraction from all spots. Additionally, the efficacy seems to be related to the size of the experimental spot as the smaller spots appear to be better

“hidden” by the purple overlay. As such, it is likely that Itot for the small spots will provide a better quantitative agreement with the experimental intensity. The center of each diffraction spot can be determined by taking an intensity-weighted average of the spot’s pixels, as described in Equation 4.10, which is an effective measure of the (x,y) position of the spot since the discrepancy between algorithm and experiment is greatest at the lower intensity edge of the spot. 푎푙푙 푝𝑖푥푒푙푠 퐼 (푥 , 푦 ) ∑ 푝 푝 푝 4.10 퐼푡표푡 푝=1

4.3.4. Individual Spot Kinetics For each isothermal crystallization experiment, a spotpicking routine is performed on the final crystalline 2D diffraction image. The time-dependent evolution of the scattered intensity at each diffraction spot in the final crystalline pattern can then be extracted from evaluating its image domain in the previous diffraction images of the time-series. Each diffraction spot was normalized with respect to the initial intensity of those pixels without

Bragg diffraction in the first frame (I0) and the total intensity of that spot (If), as outlined in Equation 4.11, to yield a sigmoidal curve which goes from 0 to 1.

퐼(푡) − 퐼0 퐼푛표푟푚(푡) = 4.11 퐼푓 − 퐼0 These sigmoidal curves for all diffraction spots were fit to the M-KJMA model (Equation 3.13) with the dimensionality of growth fixed to 3 to determine the induction time 53, 85 (t0) and the phase boundary velocity (vpb) of the crystallization front with an automated non-linear fitting algorithm written in Java 1.7 using JAMA.93

4.4. Results & Discussion

4.4.1. Bulk Nucleation Kinetics: DSC and 1-D TtXRD. Measurement of liquid-to-crystal transformations with DSC and 2D TtXRD is discussed at length in Chapter 285 and so will only be briefly summarized here. The heat evolved from crystallization measured with DSC was integrated and normalized to the fraction transformed, α, which was then fit with the M-KJMA expression over the range 0 ≤ α ≤ 0.5 to extract vpb and t0. To directly compare the crystallization experiments measured with diffraction to the DSC experiments, the 2D TtXRD images were azimuthally averaged to create 1D diffraction patterns and analyzed with Singular Value Decomposition (SVD) to extract the time-dependence of the disappearance of the liquid phase and the appearance of the crystalline phase. Those time-dependencies were then normalized and evaluated with the M-KJMA expression to extract vpb and t0.

The initial nucleation times for bulk crystallization, defined as tnuc=t0-tiso, are given in Figure 4.3a-b obtained from DSC (red) and 2D TtXRD at the NSLS (teal) and the APS (blue).

The determination of tiso is discussed in the methods section. The negative tnuc for a few APS and NSLS measurements and many of the DSC measurements indicates that nucleation occurred before the system reached the desired crystallization isotherm. The initial nucleation rates for bulk crystallization are simply defined as the inverse of the initial nucleation times and are given in Figure 4.3c. At low temperatures (T<70 °C) the initial nucleation time is long with a maximum of 29 minutes at 37 °C. At intermediate temperatures (70

to be more consistent (Figure 4.3b) is a result of more rapid quenching at the APS (≈410 K∙min- 1) than the NSLS (≈180 K∙min-1) and a higher 2D image collection frequency at the APS (1 s) than the NSLS (9.28 s or 21.5 s). These combine to improve the kinetic measurement quality at the APS such that the isotherm is reached more quickly and more data points are collected such that better fitting parameters are obtained. Further increase in the temperature to between 120 and 155 °C results in widely scattered initial nucleation times. At the highest temperature (T>155 °C), the initial nucleation time of the DSC samples rapidly increases such that at 158 °C nucleation did not occur in a 20.2 mg sample of CZX-1 for almost an hour, significantly longer than the 30 minutes before nucleation was observed at 37 °C. It is unclear whether the significantly reduced nucleation time in the DSC measurements is a result of the larger sample size (×10-300), the slower cooling rate compared to the TtXRD measurements, extrinsic nucleation due to imperfections at the sample-pan interface or some combination thereof.

Upon careful inspection, it becomes apparent that tnuc and knuc are distributed with respect to the sample size. At common isotherms, the initial nucleation times extracted from the NSLS synchrotron are greater than those from the APS synchrotron which are greater than those from the DSC experiments, with the inverse holding true for knuc. It is not necessarily clear why the sample volume must be considered from the context of Classical Nucleation Theory which describes the nucleation process to result from the serial aggregation of fundamental “units” until some critically sized nucleus is formed which then can support crystal growth. However, under the context of nucleation being a thermodynamic process that occurs through microstate sampling, the importance of volume is clear: As the volume increases, the number of microstates that can be sampled in some unit time increase and so nucleation should occur faster in larger samples simply because the appropriate microstate reached more quickly. There also appears to be a temperature dependence to the minimum observed rate constant which is presumably the intrinsic nucleation rate in CZX-1, though the clear separation of the nucleation rates by measurement technique is troubling.

This apparent sample size trend is extremely reminiscent of that observed in the crystallization kinetics of CZX-185 where the crystal growth rates were argued to be proportional to the cube root of the sample size such the rates obtained at the NSLS appeared significantly faster than those obtained with DSC measurements. In that work, the sample size discrepancy was removed by multiplying the cube root of the sample volume and the KJMA rate constant, recognizing that their product has units of distance per time (velocity) and is equivalent to the phase boundary velocity of crystallization, an intrinsic materials property. The form of a correction to account for the apparent sample-size dependence of the initial nucleation rate in Figure 4.3c is not apparent a priori, though one might propose to simply “normalize” the nucleation rate by the sample volume, i.e., Equation 4.12. In this -1 -3 expression, the product corresponds to a nucleation rate density, ρnuc, with units of s ∙cm .

−1 휌푛푢푐 = 푘푛푢푐 × 푉 4.12

With the sample volumes given in Table 2.1, ρnuc is computed with Equation 4.12 and plotted in Figure 4.4a on a normal y-scale where there appears to be a clear maximum at 100 °C but the values near the glass transition and melt temperatures are not differentiable. Note that pnuc has a maximum at 100 °C while the crystallization phase boundary velocity has a significantly higher maximum, observed at 135 °C.85 It is this difference between maximum nucleation rate and maximum growth rate that allows the growth of single- or near single- crystals from the melt. Plotting the data logarithmically (Figure 4.4b) demonstrates that the sample-volume effect appears to have been successfully removed and a maximum in the nucleation rate at 99 8, ° C is visible. Such a maximum nucleation rate between Tg and Tm is in stark contrast to CNT 10-11 which proposes that the nucleation rate monotonically increases as the temperature is lowered from the melt. -1 -3 -1 -3 The observed limits of ρnuc are 0.027 s ∙cm at 157 °C, 4400 s ∙cm at 99 °C and 15.1 s-1∙cm-3 at 37 °C. The normalized nucleation rate appears to go to zero at high temperatures, consistent with crystallization in CZX-1 never having been observed above 162 °C (notably 11° below the melt temperature).85 A “dead zone” between the highest observed crystallization temperature and the melt temperature is not unique to CZX-1.115

Figure 4.4. Normalized nucleation rate ρnuc plotted on (a) normal y-scale and (b) logarithmic y-scale. (c) Logarithmic plot of knuc for comparison.

Intriguingly, the trajectory of ρnuc as the glass transition temperature is approached (≈30

°C) suggests that either the nucleation rate discontinuously drops to zero at Tg or that the nucleation rate may have a non-zero value for some temperature range below Tg. Given that the glass transition is not a sharply-defined transition, it seems inconsistent to propose that the nucleation rate undergoes a discontinuous change to zero at Tg and therefore likely that nucleation can occur below Tg. In fact, such a non-zero nucleation rate below Tg seems to be consistent with aging in polymers and metals. A few of the reactions reported here resulted in single crystalline (or near single crystalline) transformations, as evidenced by the diffraction patterns shown in Figure 4.4d-l. The data points that correspond to these reactions have been labeled, enlarged and outlined with orange in Figure 4.4b. These single-crystalline transformations generally appear to be at

or below the average nucleation rate suggests that there is a relationship between the initial nucleation rate and the resultant crystalline microstructure.

The initial nucleation rates presented here, from the perspective of either knuc or ρnuc, seem to contradict the widely-used modified Turnbull-Fisher expression which describes the crystallization process in terms of the competing influences of nucleation and growth. In the

Turnbull-Fisher concept, the nucleation rate monotonically decreases from Tg to Tm while the growth rate does the opposite, increasing from Tg to Tm, which is inconsistent with both the temperature-dependent crystallization phase boundary velocity (Figure 2.10)85 and the initial nucleation rates shown in Figure 4.3 and Figure 4.4. In our experimental system, both rates are slow near the glass transition temperature, increase to a maximum at 100 and 135 °C for nucleation and growth, respectively, and slow until crystallization is not observed past 162 °C.

4.4.2. Nucleation Kinetics: Direct Analysis of 2D Diffraction Images. Bulk crystallization measurements are an effective tool for extracting the phase boundary velocity of the crystallization front and the nucleation time of the first crystallite in the sample,53, 85 However, bulk measurements provide no information about nucleation that continues to occur as the sample crystallites, i.e. “continuous nucleation.” By contrast, the area detector utilized for the TtXRD experiments56 affords the capability to visualize and differentiate nucleation and crystal growth such that the nucleation of new crystallites concurrent with the growth of existing crystallites can be detected and analyzed. Furthermore, an estimate of the number of crystallites that nucleated during the crystallization transformation can be obtained from the direct analysis of 2D TtXRD experiments. Such information is a critical aspect of crystallization transformations, as preliminary work by Feier Hou in the Martin Group suggests that nucleation can significantly alter the growth kinetics, an effect that is largely overlooked in KJMA theory.99 Consider the diffraction images given in Figure 4.5d-f which correspond to the final diffraction pattern collected for crystallization reactions of three different samples of CZX-1 which produced oligo-, single- and poly-crystalline samples, respectively. Each crystallization reaction occurred over 320 s with an image collection rate of 1 Hz. The bulk (1D)

transformations for each reaction were determined by azimuthal averaging and SVD analysis -1 -1 on the resultant set of 320 1D diffraction images with kinetic parameters (vpb (μm∙s ), t0 (s )) of (15.6, 89), (22.4, 178) and (3.5, 19) for Figure 4.5a-c, respectively. The normalized bulk transformation is given in Figure 4.5a-c (black) along with the M-KJMA fit (red) and the total number of diffraction spots per image normalized to the number of diffraction spots in the completed reaction (blue) as determined by “spotpicking” each diffraction image with the algorithm outlined in §4.3.3. A visual comparison between the bulk transformation (black) and the number of diffraction spots per image (blue) seems to suggest that nucleation and growth are correlated as they follow almost identical trajectories. Presumably nucleation and growth are coupled through dynamic effects induced from the melt-crystal transformation since these growing crystallites are denser than the surrounding subcooled liquid, the liquid must be expanding to fill the void space that would otherwise be generated. If the liquid cannot expand sufficiently then void space is created by the formation of crystallites which can serve as an extrinsic nucleation site. During our experimentation at the synchrotron facilities it was observed that in capillaries, the crystallization process can cause what appears to be a vortex to be created in the center of the sample as the denser crystalline phase is formed. Given the close agreement between normalized crystalline transformation and normalized number of diffraction spots, an initial interpretation might be that crystallization process is autocatalytic since the growth of crystallites strains the system, inducing the nucleation of additional crystallites. This nucleation mechanism would not appear to significantly alter the growth rate since this type of extrinsic nucleation is only operable in the vicinity of growing crystallites, suggesting that perhaps “autocatalytic” is not an appropriate description. Instead, we call this process “growth-induced nucleation.”

Figure 4.5. Direct analysis of in-situ isothermal crystallization experiments that resulted in (a,d) oligo-, (b,e) single and (c,f) polycrystalline samples.

That growth-induced nucleation is operable near the surface of growing crystallites suggests that there is likely a strong relationship between the orientation of the crystallite that caused the extrinsic nucleation event and the new crystallite that was nucleated as a result. Indeed, such an effect is visible in many of the experimental diffraction patterns that were collected during this investigation (Appendix G). Examination of the diffraction pattern in

Figure 4.5d will show that there are four (110) Bragg diffraction peaks separated by only a few degrees which indicate that there are at least four crystalline domains as the minimum separation for any two (110) Bragg peaks is 60°. Additionally, much of the higher order Bragg diffraction is azimuthally “streaked” in this image, consistent with a “mosaic” of similarly oriented crystalline domains. The number of Bragg diffraction spots that appear during a crystallization experiment is generally dictated by the number and orientation of the crystallites that appear. The number of Bragg diffraction spots should provide a reasonable estimate for the number of crystallites that appeared during a crystallization reaction. Given the location of all Bragg spots on the area detector one could theoretically obtain the orientations of all crystallites as well as a reasonable estimate of their total volume from the Bragg intensities. In practice, this is an inordinately complex problem to solve. We take the simpler, admittedly less accurate, approach of using the number of Bragg diffraction spots on the {110} and {222}, two of the more intense reflections, to estimate the number of crystallites. Given in Figure 4.5a-c are the number of {110} and {222} Bragg spots in orange and green, respectively, that appeared during the crystallization experiment. Up to six {110} Bragg peaks can be simultaneously visible if a crystallite is oriented such that a ⟨111⟩ direction is coincident with the X-ray beam, though no peaks may be visible along lower symmetry directions. In a set of randomly oriented crystallites, each will contribute a different number of {110} peaks. For the purposes of estimation it is assumed that crystallites will have 3 visible {110} peaks on average. Similarly, between zero and four {222} Bragg peaks can be simultaneously visible which suggests an average number of about 2 might be appropriate. Though these estimates are crude, in our study of the geometric aspects of crystallization it was seen that a small variation in the number of nucleation events does not significantly impact the overall crystallization kinetics, as a fivefold increase in the number of nucleation events is required to change the crystallization kinetics by a factor of two.53, 85, 99 Consider the diffraction patterns given in Figure 4.5d-f. In Figure 4.5d, the number of spots determined by the spotpicking algorithm were averaged over the final 50 frames, determining that there are 14.7(2) {110} and 12(1) {222} diffraction spots. Given the

estimated average of 3 {110} and 2 {222} diffraction spots per crystallite suggests that between 5 and 6 crystallites are present in this diffraction image; a not unreasonable estimate given that there are at least four different crystallites, as evidenced by the four (110) Bragg spots that are separated by less than the minimum angle between {110} peaks originating from the same crystallite. The diffraction pattern given in Figure 4.5e presents additional challenges due to the intense diffuse component centered on some of the Bragg peaks. According to our spotpicking algorithm there are 2.2(3) {110} Bragg spots and 13(2) {222} Bragg spots. The {110} Bragg spots suggest that the diffraction pattern in single crystalline while the {222} Bragg spots suggest that there are about 4 crystallites. However, there is no possible arrangement of crystallite orientations that could produce this combination. Visual inspection of the diffraction image indicates that there are only four {222} Bragg spots suggesting that there is a problem with the spotpicking algorithm; a not terribly surprising result as no advanced decision-making capabilities have been incorporated. The polycrystalline diffraction pattern given in Figure 4.5f shows a combination of powder-like rings and discrete Bragg spots. The spotpicking algorithm determined that there are 27.0(1) {110} and 41.4 {222} spots, suggesting that between 9 and 20 crystallites appeared during this experiment. Simply using the number of diffraction spots that our algorithm finds in each diffraction image seems to give a reasonable estimate of the number of crystallites that comprised that crystallization reaction, though the spotpicking algorithm is crude and does not always find the appropriate number of spots. For this reason, we developed an algorithm to correlate the diffraction spots across the diffraction image time series, affording the capability to extract the time-dependent intensity of each diffraction spot and thus kinetic parameters for each spot individually. Ideally, nucleation times and phase boundary velocities could be extracted from these time-dependent spot intensities with the M-KJMA model. As the crystallite volume was shown to be a critical aspect of the M-KJMA model53 and we have not deconvoluted the diffraction patterns into the fundamental single crystalline patterns, we cannot know the volumes of the crystallites that the diffraction spots originated from. We must therefore return to using the KJMA model to fit the spot intensities. Though the KJMA model does not

determine the intrinsic materials property, vpb, it is an effective measure of the nucleation time, 53 t0, when the system is reasonably isotropic. The fit values of the nucleation times of all diffraction spots, given in Figure 4.6, provide a distinctly different view of the relationship between nucleation and growth than the number of diffraction spots per image (Figure 4.5). The initial nucleation times for of all diffraction spots, determined with the KJMA model (Equation 3.13), are given in frequency histograms (green curves), binned to 1 s, and integrated and normalized in blue (the “Cumulative Fraction”), with the total number of fit diffraction spots in blue text and α(t), the measured fraction transformed, in black. In all reactions shown, it seems that the majority of nucleation is complete before any significant fraction of the system has transformed from melt to crystal. The crystallization reaction that resulted in the oligocrystalline diffraction pattern

(Figure 4.6a,d) appears to have four peaks in the t0 histogram which corresponds well to the observation of four diffraction peaks from different crystals in the 2D image, as previously described. Note that about 17% of the diffraction spots have nucleation times (t0) before the bulk tnuc (dashed red line). The single crystal diffraction pattern (Figure 4.6b,e) shows two clearly defined peaks in the t0 histogram where the first peak is coincident with tnuc obtained from the bulk fit (dashed red line). That there is a second peak is initially surprising as the diffraction pattern appears to be single crystalline, but satellite reflections are visible upon closer inspection (Figure 4.6g); consistent with the second peak in the t0 histogram. Here, about 41% of the diffraction spots have fit values of t0 before the bulk tnuc. In the polycrystalline diffraction pattern (Figure 4.6c,f), the majority of nucleation has already occurred before an experimental signal is appreciably detectable. The inflection point of the blue curve occurs at around 37 s where 55% of the nucleation is complete while only 0.76% of the sample has crystallized. Similar to the single crystal reaction, there is a peak in the t0 histogram that is coincident with tnuc obtained from the bulk fit. In this reaction, 16% of the individual diffraction spots appear to have nucleated before the bulk tnuc.

Figure 4.6. Isothermal crystallization reactions at (a,d) 145 °C, (b,e,g) 150 °C and (c,f) 96 °C. Black, fraction transformed; Green, fit t0 histogram; Blue, cumulative fraction; ×###, total number of fit diffraction spots; Broken red, bulk tnuc; t=0 on the x-axis is tiso.

In KJMA theory, only a single parameter is included to account for a delay between the quench and the observation of crystallization (t0). If new crystallites are nucleated concurrent with growth (i.e., “continuous nucleation”) as seems to be the case for CZX-1, then the single-valued t0 parameter in the KJMA model cannot effectively describe the nucleation times for all crystallites. The crystallization kinetics simulations (Chapter 3:) suggest that the bulk t0 fitting parameter is equivalent to the nucleation time of the first crystallite in the sample in the presence of continuous nucleation.53 For the reactions given in Figure 4.6b-c, respectively, this conclusion holds true; the first peak in the t0 frequency histograms (green) are reasonably coincident with bulk tnuc (dashed red). By contrast, the oligocrystalline reaction given in Figure 4.6a has a distinct peak at 137 s, well before the initial nucleation time determined from the bulk fits (147 s), suggesting that tnuc obtained from the bulk may not

always be an accurate description of the nucleation time of the first crystallite. We do not yet fully understand the reaction conditions which result in a significant difference between tnuc determined from individual diffraction spots and tnuc determined from bulk fits. Analysis of the remaining 2D TtXRD melt-crystallization experiments is a necessary first step to determine the frequency with and the conditions under which there is a discrepancy between the the nucleation time of individual diffraction spots and the bulk nucleation times.

4.4.3. Structural Evolution during the Induction Time In many of the CZX-1 melt-crystallization reactions performed at temperatures greater than 140 °C, a significant “induction time” was observed between the quench and the first evidence of crystal growth, but the system rapidly crystallized once nucleation occurred. Such a situation was seen in Figure 4.6a-b where the nucleation occurred at 137 and 178 s, while crystallization was complete (99%) within 101 and 73 s, respectively, after nucleation occurred. This significant induction time presents a unique opportunity to evaluate the nucleation mechanism of CNT which proposes that pre-nuclei “embryos” increase in size through serial aggregation of some fundamental structural unit (e.g., a molecule in the gas phase) until some critical size is reached, precipitating the nucleation event and irreversible crystal growth. It is assumed that this mechanism operates throughout the sample and produces a distribution of domain sizes whose correlation lengths are increasing through time. With the high-energy and high-intensity X-ray diffraction available at the APS, structural evolution of 116-117 RuO2 nanoparticles has been observed with rapid-acquisition Pair Distribution Function (RA-PDF)118 and differential PDF119 analysis techniques. Presuming that the formation of these embryos will be most easily detectable there is a long separation between quench and nucleation, consider the 145 °C 2D TtXRD experiment given in Figure 4.7 where an induction period of 440 s was observed. After this relatively long induction period, crystallization was complete (99%) within 35 s. Given in Figure 4.7a are the structure factor plots (S(Q)) of the azimuthally averaged 2D diffraction images, computed with PDFgetX3,113, 120 with specific frames highlighted: CZX-1 above its melt temperature at 230 °C in red; one image collected during the quench in teal; just after quenching to 145 °C in blue;

the long induction time where no changes are evident in black: just before the nucleation time in purple; the completion of crystallization (99.9%) in green. Each S(Q) represents a collection time of 4 s and only every other frame during the induction time is plotted. There are only two visually discernible changes that occur in S(Q) during this crystallization reaction: The change between hot liquid CZX-1 (230 °C) and subcooled liquid CZX-1 (145 °C) is visible (red to teal to blue) and the appearance of the crystalline diffraction pattern between purple and green.

That there appears to be no change in the diffraction pattern between tiso and tnuc indicates that either the embryo formation mechanism is incorrect or that the structural coherence length of embryos is not sufficient to be visually apparent with traditional diffraction.

102 103 3 103 104

Figure 4.7. Azimuthally averaged 2D TtXRD experiment, Tiso=145°C. (a) S(Q). (b) G(r). (c) Final crystalline diffraction image.

Perhaps these embryos should be considered to be more like nanoparticles given that nanoparticles are known to diffract poorly due to Scherrer broadening. The Pair Distribution Function (PDF) analysis technique is capable of detecting the formation of nanoparticles from

solution.121-123 In the PDF technique, S(Q) is Fourier transformed into real-space number- density weighted atomic pair correlations. Given in Figure 4.7b are the PDFs of the azimuthally averaged 2D TtXRD diffraction patterns with the same color scheme as in Figure 4.7a. The first four peaks in the liquid correspond to the Zn-Cl bond at 2.28 Å, Cl-Cl and 1st Zn-Zn at 3.75 Å, 2nd Zn-Zn at 5.65 Å and 3rd Zn-Zn at 7.34 Å, respectively.31 From the perspective of pair-pair correlations, the major changes that occur during this crystallization reaction are a general increase in the intensity of the peaks in the liquid as a result of the slightly more order in the 145 °C liquid than the 230 °C liquid and the appearance of long-range pair correlations as the system crystallizes. During the induction time (black), there appears to be some minor fluctuations at longer distances (>8 Å) that might be tempting to ascribe to embryo formation. However, given the seemingly random location of the fluctuations and the lack of any coherence through time, these fluctuations are merely a mathematical artifact of the Fourier transform that converts S(Q) to PDF. In nanoparticle formation, PDF peaks appear first at short distances with the longer- distance peaks appearing concurrent with continued nanoparticle growth.121-123 Given the nucleation mechanism proposed by CNT, it was assumed that a similar process would be evident. However, during the time between tiso and tnuc the system does not to appear to be undergoing any structural changes as no additional peaks are observed. Note that the crystalline peaks in the PDF all appear simultaneously and there is no length-scale dependence to their appearance, suggesting that the crystallites are already larger than nanoparticles when they are first detectable by bulk diffraction methods. If embryos are forming during this induction period, it is not clearly evident in either the structure factor or PDF plots. Singular Value Decomposition (SVD) is an analytical technique which is sensitive to small variations that are not visually detectable and can separate the loss of liquid diffraction from the appearance of crystalline diffraction,85 as shown in §2.3.2. The PDFs of the induction time (Figure 4.7b) were processed with SVD, resulting in a series of orthogonal singular vectors; the first two of which are given in Figure 4.8. The x- axis of the left singular vectors (LSV) is the same as the input data set (distance in Å) and the x-axis of the right singular vectors (RSV) is the y-axis of the input data set (time in s). It is

generally true that only a few of the singular vectors have a physical interpretation with the rest being noise. The first two LSVs are given in Figure 4.8a in black and red, respectively, along with a 94 point moving average of LSV2 in blue. LSV1 is equivalent to the PDFs of the induction time (Figure 4.7b) where order can be seen to roughly 40 Å, corresponding to the fourth cage-cage distance in CZX-1, consistent with previous work.31 By contrast, there is little structure evident in LSV2, even at short distances where the signal to noise should be strongest. There is significantly more noise in LSV2 at long distances than LSV1, even after being smoothed by a 94 point moving average; this suggests LSV2 corresponds entirely to noise and therefore of no physical significance.

RSV1, given in Figure 4.8b, initially increases between 12 s and 80 s, then decreases to a minimum at ~300 s and finally increases again until nucleation occurs at 458 s. RSV2 increases to a maximum at ~ 200 s and then decreases until nucleation occurs at 458 s. That both RSV’s are extremely noisy (compare to Figure 2.4-Figure 2.6) suggests that there is little time-dependence to the LSV’s. The initial increase in RSV1 is likely indicating that the system continues to relax after the rapid quench to 145 ˚C, though the subsequent decrease and increase are puzzling. After further analysis is was determined that this decrease-then-increase in RSV1 and the increase-then-decrease in RSV2 did not originate from chemical effects, as the same trend was seen when SVD was performed on the S(Q) plots (Figure 4.7a) at extremely high Q where no scattering was evident (Q > 30 Å-1). Based on these results, we are left but to conclude that the concept of evolving embryos cannot be established through SVD of the PDFs.

Figure 4.9. (a) Normalized peak intensity of specific pair-pair correlations. Black, Zn-Cl; Olive, Zn-Zn/Cl-Cl; Blue, 2nd nearest Zn-Zn. (b) Zn-Cl peak. (c) Zn-Zn/Cl-Cl peak. (d) 2nd Zn-Zn peak.

Perhaps the evidence for embryo formation is not at long distances but instead at short distances, residing in the local order. To search for time-dependent changes to local atomic coordination, each of the first three peaks in the PDF were individually analyzed with SVD. For all peaks, there was a single dominant component that accounted for 98+% of the

transformation. The left singular vectors are indistinguishable from the peaks and so are not shown. The right singular vectors, corresponding to the time-dependence of each peak, are given in Figure 4.9; the Zn-Cl covalent bond distance in black, the first Cl-Cl and Zn-Zn contact in green and the 2nd nearest Zn-Zn contact in blue. In all peaks there is an initial increase as each peak sharpens as a result of the temperature change imposed on the sample by quenching from above the melt temperature at 230 °C to below the melt temperature at 145 °C. As the sample is held at 145 °C, the intensity of each peak is invariant, though somewhat noisy, until nucleation occurs at 458 s where the intensity of each peak increases concurrent with growth. After growth is complete, each peak remains constant until data collection was terminated. That the difference between 145 °C and 230 °C liquids is clearly observable yet the bulk structure does not appreciably change during the induction time would further suggest that the formation of embryos is either not visible with diffraction techniques or that the CNT nucleation mechanism is inaccurate.

It is remarkable that the magnitude of the increase of each peak upon cooling from Tm to Tiso is 25-50% of the subsequent increase when the system crystallizes and suggests that the local coordination environment in the liquid and crystalline phases are quite similar. That a significant change in the liquid between 230 and 145 °C is clearly visible in the PDF suggests that perhaps some insight into the liquid structure can be gained by evaluating the subcooled liquid diffraction patterns that were obtained while waiting for nucleation to occur. The change in liquid structure as a function of subcooling was evaluated by extracting the dominant liquid pattern with SVD, given in Figure 4.10a for a series of repeated quenches to crystallization isotherms between 100 and 150 °C, separated by 5 minutes at 230 °C. There is little visual difference between the liquid diffraction patterns. To more clearly differentiate the temperature-dependence of the liquid structure, the 230 °C liquid was used as a reference point such that the lower temperature isotherms were subtracted from it. These difference patterns are given in Figure 4.10b, demonstrating a peak shift in the first sharp diffraction peak (FSDP), increasing with the amount of subcooling. The change in intensity and amount of peak shift as a function of subcooling was determined by fitting a Gaussian curve to a small range around each of the first four peaks. There was no consistent intensity change of the four peaks as a

function of subcooling. Instead, the intensity of each peak decreased as a function of time due to the slow decay of the synchrotron beam intensity. Each of the four peaks shifted to lower Q with an increasing amount of subcooling with the clearest trend resulting from the third diffraction peak at Q≈2 Å-1.

Figure 4.10. (a) Liquid CZX-1 temperature-dependent diffraction pattern. (b) Liquid CZX-1 difference pattern. (c) Unit cell volume versus temperature for (red) liquid and (blue) crystal.

The third diffraction peak in the liquid phase is dominated by the pseudo-closest packing of the chlorine atoms of the tetrahedral MCl4/2 units (M=Zn,Cu), similar to the (222) of the crystal structure.34, 36 Since the local coordination in CZX-1 is not significantly different between the liquid and crystalline phases31 (Figure 4.9) the third diffraction peak can be used as a reasonable estimate of the liquid phase “unit-cell” volume according to Equation 4.13. With this approximation, the volume of the liquid is 7-13 % greater than the crystallographic density as a function of temperature,85 given in blue in Figure 4.10c, which is not an unreasonable volume increase from the crystalline phase. The “unit-cell” volume of the liquid determined from the liquid diffraction patterns is given in Figure 4.10c in red with an extrapolation to low temperatures where the two intersect at approximately -38 °C, suggesting an experimental estimate of the Kauzmann temperature, TK. 4휋2 × (ℎ2 + 푘2 + 푙2) 3/2 ( ) 4.13 푉푙𝑖푞 푇 = [ 2 ] 푄ℎ푘푙,푙𝑖푞 (푇)

4.5. General Discussion & Conclusion Nucleation of a droplet of liquid from a supersaturated vapor can be described effectively described by a nucleation mechanism requiring mass transport because the difference between the molar volumes of the liquid droplet and vapor is approximately three orders of magnitude. A similar mechanism can be used to describe nucleation of a crystalline phase from solution because a significant concentration difference exists. In both cases, the large volume or concentration differential requires significant molecular transport for atomic or molecular aggregation to form the critical sized nucleus. By contrast, when nucleating a crystal from its corresponding melt in a homogeneous system, neither a concentration nor volume difference exist to any appreciable amount. Here, crystallization requires a change in structural organization. As only minimal differences are observed in the immediate coordination environment of the melt and the crystal in many networked systems,31, 35-36, 124 the atoms and/or molecules in the liquid are already in a nearly close-packed arrangement. Therefore, there is no need for an atom/molecule to be brought to the “surface” of the crystal for it to grow in size; the atoms/molecules are already at the surface. With virtually no difference in the bonding between liquid and crystalline materials, it is the phase boundary between non-crystalline and crystalline material that must move across the system, not the molecules that must move across an embryo-melt interface. The assumption that the driving force of crystallization is dominated by an interfacial energy, surface energy, surface tension, crystal-melt interfacial tension or any other surface- based concept is also problematic as it would imply that nuclei and crystals should frequently appear like suspended liquid droplets (i.e. spherical). However, when nuclei are observed, such as in concentrated colloidal suspensions,125 in recrystallization of deformed metals90 and in simulations of water crystallization,126 only non-spherical nuclei are observed. As such, the interfacial energy terms must be minimal, being outweighed by other factors, such as the lattice stabilization energy. CNT describes the nucleation mechanism as a serial aggregation process of atomic or molecular formula units, 퐴𝑖 + 퐴 ⇌ 퐴𝑖+1. When the rate of the forward reaction is greater than that of the reverse reaction, the size distribution of embryos increases over time and eventually

results in the production of a critically size embryo(s) which precipitates the nucleation event and allows the system to support crystal growth to macroscopic sizes. However, our investigation of the induction period reveals a distinct lack of structural evolution during this time period, raising concern over the CNT nucleation mechanism as applied to the melt- crystallization process. Defining nucleation as an aggregative mechanism requires atomic or molecular mass transport to bring new formula units to the surface of the cluster. It is unclear why nucleation in melt-crystal transformations requires such a mass transport mechanism, given the similarities of the local coordination environments in liquid and crystalline CZX-131, 36 as well as broader congruently melting systems.35, 124 Furthermore, recent work, particularly with respect to the crystallization of polymers, suggests that metastable phase formation, structurally intermediate between that of a liquid and a crystal, precedes crystallization.127-128 As shown in Figure 4.4, the volume-normalized nucleation rate has a clear maximum at 100 °C (Tmax). Such a maximum in the initial nucleation rate is not accounted for in the Turnbull-Fisher formalism which describes the nucleation rate to monotonically increase with decreasing temperature.6, 8, 10-11, 101 The lower temperature volume-normalized nucleation rates exhibit Arrhenius-type behavior suggesting that the temperature dependence between Tg and

Tmax might be understood from an Arrhenius (ln k vs 1/T) or Eyring (ln(k/T) vs 1/T) framework. However, recognizing that the units of the volume-normalized nucleation rate are s-1 cm-3, an additional term is required to remove the volume dependence of the nucleation rate and it is not yet clear what the physical significance of that term is. Nevertheless, Eyring analysis was performed on the low-temperature Arrhenius region, acknowledging that units have not been properly accounted for. The Eyring analysis results in activation parameters of ‡ -1 ‡ -1 -1 ΔnucH =61(8) kJ mol and ΔnucS = -23(3) J mol K . The enthalpy of activation is similar to ‡ -1 that obtained for crystal growth in CZX-1 (ΔgrH 67 kJ mol ) but the entropy of activation for ‡ -1 -1 nucleation is an order of magnitude greater than for crystal growth (ΔgrS = -2.2 J mol K ). It is reasonable that the activation enthalpy should be similar for nucleation and growth, given that a crystalline-like order is being formed in both cases. It is also reasonable that the activation entropy for nucleation should be significantly more negative than for growth as it is not unreasonable for the entropy penalty for creating order from nothing to be immense. By

contrast, the entropy penalty for simply growing the crystallite, while still entropically unfavorable, pales in comparison.

Chapter 5: Plastic Crystalline Carbon Tetrabromide

5.1. Introduction Molecular liquids frequently crystallize on cooling from the melt into a mechanically soft yet crystalline “plastic” phase broadly characterized by significant orientational and positional disorder, high symmetry (usually cubic) and an entropy of fusion less than about 2.5R(≈20 J mol-1 K-1).129 Further cooling eventually results in the system adopting a fully ordered, highly crystalline, mechanically rigid and brittle “nonplastic” crystalline phase. The specifics of the molecular system may dictate additional solid-solid phase transitions between the plastic and nonplastic phases. The plastic crystalline phase is a valuable window into the local order in the liquid state as it can be considered an isolable structural intermediate between an isotropic liquid and a fully ordered crystal. The earliest and simplest picture of the plastic crystalline phase was that molecules are located near average structure lattice sites but rotate largely independently of one another as molecular “spheres.”129 Such a spherical approximation to what is inherently a tetrahedral * 130 molecule may indeed be appropriate for plastic crystalline heavy methane (α-CD4 ). Plastic

CD4 is stable between 27 and 98 K, has an entropy of fusion of 0.55R and crystallizes as 퐹푚3푚 130-131 with a0=5.96 Å. Packing with an fcc-geometry the plastic crystalline lattice implies molecular nearest neighbors are separated by approximately 4.2 Å along the 1 ⟨110⟩ real-space 2 directions . The space required for freely spinning heavy methane molecules can be reasonably approximated as 2.1-2.2 Å from the sum of the C-D bond distance130 and the deuterium van 132 131 der Waals radius. With a mean squared displacement of 0.07 Å in α-CD4, the heavy methane molecules can certainly be considered free rotators. Conversely, the radial extension of the bromine in plastic crystalline carbon 38, 133-135 tetrabromide (α-CBr4) creates sufficient steric hindrance to prevent free rotation.

Plastic CBr4 is observed between 319 K and 363 K, has an entropy of fusion of 1.2R and

136-140 crystallizes as 퐹푚3̅푚 with a0=8.82 Å. The radius of a freely spinning CBr4 tetrahedron

* The highest temperature polymorph is usually referred to as I or α in the literature with lower temperature polymorphs being appropriately enumerated for the ambient pressure polymorphs. High-pressure polymorphs are not as rigidly defined. In this chapter, Greek letters will be used to denote different solid polymorphs.

is roughly equal to the C-Br distance (1.91 Å) plus the bromine van der Waals radius (1.89 Å).

Thus, a free rotor in α-CBr4 would require an intermolecular nearest neighbor distance of 2×3.8

Å. But the experimentally observed intermolecular distance in α-CBr4 is 6.2 Å, with a 0.20 Å mean squared displacement. Thus free rotation would require the average C---C distance to increase by 1-1.4 Å, decreasing the density by about half and doubling the molar volume of the low-temperature non-plastic phase.38, 141 Because the molecules cannot rotate independently, there must be strong nearest-neighbor intermolecular orientational and associated displacive correlations.135, 141 However, there is much debate as to whether the disorder in α-CBr4 manifests as static orientational correlations with dynamical motion limited to vibrations and librations.38, 135, 141-145 or as dynamic reorientational motion.140, 146-158 Recent single crystal synchrotron X-ray diffraction and modeling has suggested that α-

CBr4 is effectively characterized by a set of molecules in well-defined orientations that experience strong static orientational correlations and displacive disorder through nearest neighbor interactions.38 This “static” picture of orientational disorder is supported by earlier single crystal neutron elastic scattering,135, 141 Raman spectroscopy on both sides of the α↔β phase transition142-143 and modeling.144-145 By contrast, recent powder neutron diffraction and modeling was interpreted to suggest dynamic reorientational motions occur, albeit restricted by nearest neighbors, such that the time/ensemble average of the molecular orientations is 146 140, 147-148 149 isotropic. Earlier studies of α-CBr4 with inelastic neutron scattering, IR, thermal analysis,150 Brillouin spectroscopy,151 modeling152-156 and temperature-dependent nuclear resonance methods157-158 argue for the existence of rotational (reorientational) motions in both the plastic and non-plastic crystalline phases of CBr4. Nevertheless, the fundamental principles governing the disorder in plastic crystalline CBr4 remain unresolved. In this chapter, experimental and computational investigations are used to explore the extent to which the structure of α-CBr4 can be described from a static perspective.

Additionally, the role of tetrahedral distortions in α-CBr4 and the structural relationships between α- and β-CBr4 are explored, including possible mechanisms of the α↔β first-order solid-solid phase transformation. The crystal structures of α- and β-CBr4 are presented in §4.2.

In §4.3, experimentally collected 2D synchrotron diffraction images of twinned α-CBr4 show,

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to the best of our knowledge, the first observation of twinned structured diffuse scattering in any system as well as the first observation of a twinned plastic crystal. The 2D images are Fourier transformed (Patterson Function analysis) to obtain 2D real-space pair correlations, weighted by the electron-density. In §4.4, a new approach to simulating disordered materials is presented. Diffraction patterns and 2D Patterson functions along high-symmetry projections are calculated from the resultant simulated structures and compared to the experimental results, elucidating the structural origins of the complex structured diffuse scattering. Additionally, a new technique for visualizing correlations in disordered systems is presented. In §4.5, these experimental and computational results are correlated with data from previous investigations of -CBr4 to advance the ongoing debate regarding dynamic or static disorder in plastic crystals.

5.2. Structural Modifications of CBr4 The orientational and displacive disorder characteristic of the plastic state results in such phases being characterized by average structure Bragg reflections that are few in number and weak in intensity, accompanied by a significant diffuse scattering.159-160 The high temperature plastic phase of carbon tetrabromide (α-CBr4) crystallizes in 퐹푚3̅푚 symmetry 38, 130, 135, 138, 140, 145, 152-153, 161 with a~8.82-8.85 Å and is a prototypical plastic crystal. In α-CBr4, only 16 nonzero sets of Bragg reflections were initially detected,161 though this number was later increased to 22,141 of which only the first two (111 and 002) are relatively intense. From the Martin group’s recent discovery and initial interpretation of the strong structured diffuse

38 scattering from a single crystal of -CBr4, it was clear that additional structured diffuse scattering exists that had not yet been accounted for. To further probe this system, I collected high energy synchrotron diffraction data on this system at beamline 11-ID-B at the Advanced Photon Source (APS) at Argonne National Laboratory (ANL). The structure factor plots of the monoclinic crystalline β-CBr4, the plastic α-CBr4 and liquid CBr4 are shown in Figure 5.1. For a description of the experimental setup see methods in §4.3.

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Figure 5.1. Comparison between the three condensed phases of CBr4: Liquid in blue at 373 K, α in red at 333 K and β in dashed black at 303 K.

Upon cooling, the plastic phase undergoes a first order (nucleated)135, 162-164 solid-solid phase transition at 320 K into a non-plastic monoclinic (퐶2/푐) superstructure (Z=32), with cell parameters a=21.441(10) Å, b=12.116(6) Å, c=21.012(8) Å and β=110.91(3)°, and four 135, 162-164 molecules in the asymmetric unit. Though the unit cell of β-CBr4 is of much lower symmetry and eight times larger, the structure factors obtained from powder synchrotron diffraction for the two solid phases (Figure 5.1) show remarkable similarities. This suggests that much of the FCC molecular site symmetry remains in β-CBr4 but becomes distorted, as shown in Figure 5.2. The correspondence to α-CBr4 is seen in the monoclinic [010] projection shown in Figure 4.2a, which is approximately a [110] projection of FCC cell. The pseudo cubic cell is visualized by the pseudo-rectangles enclosed by six solid lines where the upper edge of the unit cell is the dashed line165 and the black circles correspond to the molecular centers in one α-CBr4-like unit cell. The face-centering vectors in the distorted FCC molecular arrangement in α-CBr4 become (0.057, 0.492, 0.557), (0.57, -0.014, 0.551) and (0.487, -0.017, 0.498).36 Shown in Figure 5.2b is the monoclinic unit cell, aligned to highlight the pseudo- FCC molecular packing (dashed box) and crystallographically unique molecules (colored).

The direct relationship between the monoclinic (am, bm, cm) and cubic (ac, bc, cc) crystal coordinates is given by the transformation matrix in Equation 5.1.38, 161, 165

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−2 1 −1 푎푐 [푎푚 푏푚 푐푚] = [ 0 1 1 ] [푏푐 ] 5.1 2 1 −1 푐푐 Two room-temperature single crystal structure refinements have been performed on β- 161, 165 CBr4 finding the same unit cell parameters within one estimated standard deviation. The initial crystal structure was solved by refining only the bromine positions and assuming the carbon is at their center of mass, finding that there was significant molecular asymmetry with the C-Br bonds between 1.825 and 1.975 Å and the Br-C-Br angles between 105.2° and 114.4°, though the average values are not significantly different from an undistorted tetrahedron (1.912(10) Å and 109.5(5)°).161 When all atomic positions were refined, it was found that the tetrahedra were ideal.165 In §3.4, the impact of distorted versus ideal tetrahedra will be explored with simulations and calculated diffraction images. Raman measurements are more consistent with the tetrahedra being regular at room temperature and reaching a maximum distortion at around 80 K,162, 166 though the irregularity has not been quantified. Raman

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measurements also suggest a higher symmetry for the β phase since less Raman-active modes are observed than predicted based on the monoclinic symmetry162, 167-168 which is consistent with the monoclinic cell being pseudo-cubic.

Figure 5.3. (a) The β-CBr4 cell rotated to demonstrate that the tetrahedral 4̅ axes are aligned with the pseudo-cubic ⟨110⟩ real-space directions. (b) The six alignments of the tetrahedral 4̅ axes with the cubic ⟨100⟩ real-space directions. The two alignments within the orange box indicate the preferential orientations of half of the tetrahedra (blue/green) in the monoclinic cell. The remaining four alignments reflect the preferential orientation of the other half of the tetrahedra (red/yellow). (c) The “average molecular construct” which is created by randomly distributing the six orientations in b onto the FCC lattice sites.

The tetrahedra in β-CBr4 are preferentially aligned such that half of the molecules have their tetrahedral 4̅ axes coincident with the pseudo-cubic [110], [1̅10] and [001] directions, as shown by the blue and green tetrahedra in Figure 5.3a. Those specific 4̅ axes define the

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preferred orientation of the monoclinic supercell as they transform to 푏푚, 푎푚 + 푐푚 and 푎푚 −

푐푚, respectively in the monoclinic coordinates. The other half of the molecules, yellow and red in Figure 5.3a, are evenly distributed across the remaining four orientations shown in Figure 5.3b such that one of the tetrahedral 4̅ axes is coincident with the cubic [010] or [100] directions.

Figure 5.4. (a) Structure factor plot of α-CBr4 obtained at the APS and the Bragg diffraction calculated from a perfectly ordered Frenkel model. Orange box corresponds to the diffuse scattering that could not be fit by many of the early investigations.

The plastic crystalline phase was first characterized by powder neutron diffraction as having 퐹푚3̅푚 symmetry with a=8.82 Å at 325 K at with Z=4.140 In the plastic phase the C- Br bonds were found to be preferentially aligned with all intermolecular nearest neighbor 1 ⟨110⟩ real space directions38, 134-135, 140-141, 147-148, 161, 169 giving the six “average” orientations 2 as shown in Figure 5.3b with equal probability weighting and resulting in the sixfold “Frenkel” model of disorder. A random distribution of these six orientations onto the FCC lattice sites produces an “averaged” molecule Figure 5.3c which has 24 bromines at 1/6 occupancy surrounding the carbon center with pairs of bromines split across the ⟨110⟩ real-space directions. Shown in Figure 5.4 is the structure factor plot of a powdered sample of α-CBr4 collected with synchrotron X-ray diffraction on beamline 11-ID-B at the Advanced Photon

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Source (APS) at Argonne National Laboratory (see §4.3 for methods description) and the Bragg diffraction computed from the sixfold Frenkel model. The Frenkel model is qualitatively similar to the Bragg diffraction observed for α-CBr4, but does not account for any of the diffuse scattering. The Frenkel model could not be refined with traditional structure solution techniques, largely because of the significant diffuse scattering present in α-CBr4. Fits to the total scattering (Bragg + diffuse) observed with elastic neutron diffraction from a powder α-CBr4 sample were obtained by using symmetry-adapted functions140 and spherical harmonics147 to allow for a wider variety of orientations. Both investigations found that the bromine electron density is a maximum along the ⟨110⟩ real-space directions which is consistent with the sixfold Frenkel model, but lack a good fit to the region outlined in orange in Figure 5.4. It was argued that this lack of fit demonstrated the importance of correlations between neighboring molecules;147, 169 an observation confirmed by a later elastic neutron diffraction investigation141 and its corresponding modeling.135 Careful investigation of the peak between 2 and 3 Å-1 showed that it was highly anisotropic with one long axis oriented along certain ⟨111⟩ reciprocal lattice directions (cigar-shaped) and centered at Q=(2.15,2.15,0). By constructing a partition function to account for the compatibility of neighboring orientations based on the Frenkel model, this anisotropic diffuse scattering was effectively fit. Two of these three experimental investigations have concluded that there must be significant reorientational dynamics, a conclusion also drawn from a pair of molecular dynamics (MD) simulations.152-153 The MD studies focused on the collective motions and phonon behavior of α-CBr4, finding that the molecules mostly librate around the equilibrium positions defined by the Frenkel model but occasionally become free rotators if there is a local density fluctuation. Additionally, it was found that librational and translational modes are strongly coupled and the α→β phase transition is a two-step process arising from an instability at the Brilloun zone-boundary in one of the [111] symmetry directions and then half-way along another [111] symmetry direction. However, the dynamic reorientational motions in these studies140, 147, 152-153, 169 can also be explained by static orientational correlations, as shown by recent combined single-crystal X-ray diffraction and Monte Carlo (MC) investigation.38 Folmer et al., found that the major

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components of the structured diffuse scattering in α-CBr4 are well reproduced by statics, accounting for orientational correlations through Monte Carlo (MC) orientational relaxation starting with the six Frenkel orientations, which is in agreement with earlier neutron135, 141 and 144-145 modeling work. The most recent study (2010) on α-CBr4 combined powder elastic neutron diffraction and reverse MC (RMC) and reached the opposite conclusion; their experimental data is argued to show that the intermolecular orientations are correlated, though the orientations are isotropic over a time and ensemble average,146 in agreement with the concept of dynamic disorder. The most often cited experimental evidence for dynamic molecular reorientations in α- 157 13 158 CBr4 arises from bromine nuclear quadrupole resonance (Br-NQR) and C NMR studies.

The Br-NQR study showed a loss of the quadrupole signal between 80 and 220 K in β-CBr4, well below the solid-solid phase transition. They suggested that disappearance of the Br-NQR “probably results from the onset of molecular reorientation or rotation in the solid.” By contrast, temperature-dependent Raman investigations show that the Raman lattice modes are most distinct at 80 K (8-10 visible peaks) while by 250 K only two peaks are still visible.162 Similarly, the tetrahedra reach their most distorted state at 80 K and become significantly less so by 220 K. They are almost ideal by 320 K and appear to be perfectly regular just after the β→α phase transition.162, 166 Recognizing the Br-NQR signal is highly dependent upon the symmetry of the system, taken together the Br-NQR and Raman data are most consistent with a model in which the tetrahedra becoming less distorted between 80 K and 220 K, as opposed to a model in which molecular reorientations appear about 250 K below the solid-solid phase 13 transition. However, the C NMR study found that the T1 relaxation time is continuous across the liquid↔α phase transition but discontinuous across the α↔β phase transition, though they “equivocate on the matter of rotational dynamics.” Nevertheless, Pettitt’s 13C NMR study is cited as supporting dynamic reorientational motion in α-CBr4.

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5.3. Experimental Methods and Results

5.3.1. Experimental Methods Carbon tetrabromide (99%, Aldrich, C11081) was used without further purification. Impurities may serve as nucleation sites in the solid-solid (α↔β) phase transition and lower the transition temperatures,164 but neither of those effects are relevant to our structural investigations. Additionally, less than 2% impurity seems to have no effect on the structure of 157 either solid polymorph. CBr4 samples were sealed in 0.7 mm (0.01 mm wall thickness) borosilicate capillaries (Charles Supper Co. Natick, MA) and affixed to a single-axis goniometer head with epoxy. Synchrotron diffraction data was collected at beamline 11-ID-B (90 KeV, λ=0.13702Å, collimated beam 0.3 x 0.3 mm) at the Advanced Photon Source (APS), Argonne National Laboratory. Temperature control to ±2.5° was afforded by a forced air furnace.85 Data were collected in a Debye-Scherer geometry at sampling rates between 0.1 and 8 Hz with a 2048x2048 GE Silicon detector.56 The wavelength and detector geometry 57, 170 were calibrated to LaB6 and CeO2 standards with fit2d.

Single crystals of -CBr4 were grown in-situ by slow-cooling from the melt (110 °C) at a rate of 0.5 °C/min until diffuse scattering was observed. Upon completion of crystallization, CBr4 was slowly cooled to 60°C to reduce thermal scattering. Plastic crystalline diffuse scattering was evaluated with a series of 180 diffraction images in incremental steps of 1 degree. Since the single plastic crystal of -CBr4 was grown in-situ, it was not possible to pre-determine its crystallographic orientation.

5.3.2. 2D Diffraction

To characterize the highly structured diffuse scattering of α-CBr4 previously seen by single crystal X-ray diffraction and single crystal neutron diffraction, we attempted to grow a single plastic crystal at the APS by slowly cooling from the melt. However, a 50/50 twinned plastic crystal was grown and diffraction images were collected by rotating about the capillary axis as opposed to a high-symmetry crystallographic axis. Two diffraction images have been selected to demonstrate the experimentally observed twinning of Bragg reflections in α-CBr4, shown in Figure 5.5. In Figure 5.5a, two pairs of Bragg reflections can be seen, labeled

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[2̅20], [002] and [22̅0], [002̅], which share a common vector from the image center. These Bragg reflections cannot result from the same crystal as no members of the ⟨220⟩ and ⟨200⟩ families are coincident. Similarly, two pairs of Bragg reflections can be seen in Figure 5.5b, labeled [11̅̅̅̅1], [002̅] and [111̅], [002], which share similar positions in reciprocal space but also cannot result from the same crystal as members of the ⟨111⟩ and ⟨200⟩ are separated by at least 54.7° and thus cannot share similar lattice positions. Similar Bragg pairs are seen in Figure 5.5c.

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Figure 5.5. Experimentally observed diffraction of α-CBr4 with the beam stop and its holding arm visible. Capillary rotation axis is the vertical direction in the images. (a) 10° from the [111̅̅̅̅̅], [002̅], (b) 20° from the [111̅̅̅̅], [1̅11̅], (c) near [211̅̅̅̅], [022̅]. (d) Schematic representation of the unit cell orientation of each of the twinned crystals. The green line represents the [110] twinning axis and the white line is the capillary rotation axis, close to the [02̅1] of the red crystal. Bragg reflections labeled in white correspond to the red crystal in (d) and those labeled black correspond to the blue crystal.

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Figure 5.6. Experimentally observed diffraction of α-CBr4 with the beam stop and its holder visible showing highly structured diffuse scattering along a twinned [1̅11̅̅̅], [1̅11̅] high- symmetry projection. The [111]’s of the twinned crystals are separated by 9.25°. (a) -1 Experimental image Qmax = 20Å (b) Reprint of the diffuse scattering previously observed with single crystal synchrotron diffraction. Reprinted from Phys. Rev. B.38 (c) Cartoon of the -1 diffuse scattering. (d) Experimental image Qmax = 6Å .

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To evaluate the twinning relationship, the coordinates of all observed ⟨111⟩ reflections were extracted and reconstructed in 3-D. By visual inspection it was determined that the ⟨111⟩ reflections could be related by a 90° rotation about the [110]. An idealized version of the crystallographic twin is presented in Figure 5.5d where the green line is the [110] twinning axis, the white line is the capillary rotation axis ([02̅1] relative to the red crystal). The twin law results in two pairs of nearly aligned [111] vectors of the twinned crystals (separated by 9.25°): [11̅1], [1̅11] and [111̅̅̅̅], [1̅11̅]. Fortuitously, the geometry of the capillary rotation axis and the twin law result in commensurate observation of the [11̅1], [1̅11] pair in a few of the experimental images collected. The most nearly aligned image is shown in Figure 5.6. No other high symmetry projections were observed in the collected diffraction images due to the orientation of the twins relative to the capillary rotation axis. The majority of the diffraction visible in the [11̅1], [1̅11] twin pair (Figure 5.6a) is characterized by structured diffuse scattering which connects neighboring Bragg peaks through the {111} reciprocal lattice directions and arises from molecular displacements along the nearest neighbor 1 [110] real space directions.38 There are newly observed structured diffuse 2 scattering features in the high energy synchrotron diffraction images presented here that were not apparent in the previous single crystal synchrotron work38 (Figure 5.6b). Specifically these include the “dog teeth” (red in the schematic of Figure 4c), the structured diffuse hexagon (blue in Fig 4c), the outer structured diffuse (orange in Fig. 4c) and the “inner star” (yellow in Figure 4c) which was hinted at in the previous single crystal synchrotron diffraction, but not accounted for in the simulations.38 In addition to the highly structured diffuse scattering, broad sheets of diffuse scattering observed out to Q≈18 Å-1, seen as hexagons in Figure 4a, are observed with the high energy synchrotron diffraction. The structural origins of these previously unobserved diffuse scattering features will be explored through simulations in §4.4. As was observed previously, the most intense structured diffuse scattering is centered on the {220} reflections and connected to the {111} reflections (vertices of the inner star) and to the {311} reflections by structured diffuse scattering along the ⟨111⟩ directions, as seen by the labeled Bragg reflections in Figure 5.6d. Each {220} is related to the nearest twelve

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reflections by the ⟨111⟩ reciprocal lattice directions and the {311}, {111} account for nine of those possible twelve. The remaining three connect to the {331} reflections which are only just observed in Figure 5.6d, as indicated by the blue arrows. That the structured diffuse scattering connecting to {220} is not equivalent in intensity for all ⟨111⟩ reciprocal lattice directions ({331} < {111} < {311}) indicates that the Laue conditions are not relaxed equally for all ⟨111⟩ reciprocal lattice directions. This is consistent with Folmer et al. (2008) and previous observations from diffuse neutron scattering that the diffuse intensity is “oriented in particular [111] directions”141 if it is assumed that the previous investigations were unable to detect the additional diffuse scattering because it is very weak.

Figure 5.7. Projection along the [̅111̅̅̅̅] (white text) and [1̅00] (black text). (a) Twinned Bragg reflections. White and black text correspond to the red and blue crystallite’s ⟨111⟩ in Figure 5.6d. (b) Expanded view to highlight the twinned diffuse scattering. (c) Twinned diffuse pattern highlighted in red/blue as corresponding to the twins described in Figure 5.6d.

The orientation of the twinned crystals with respect to the capillary rotation axis resulted in many diffraction images being collected where twinned diffuse scattering is evident. Shown in Figure 5.7 is one such image with the Bragg scattering labeled in Figure 5.7a, a magnified image with no labels in Figure 5.7b and a grayscale image with the twinned diffuse scattering highlighted with red and blue in Figure 5.7c. No reference to twinned structured diffuse scattering has been found in a literature search, though its presence is not necessarily

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surprising. Given the observation that the plastic crystal can twin, as shown in this chapter, and the previous observations of structured diffuse scattering in the single crystal, one can reasonably conclude that twinned structured diffuse should be visible in a well-resolved twin.

5.3.3. 2D Patterson Function To obtain a more thorough understanding of the relationship between atomic disorder and structured diffuse scattering, analytical approaches135, 144-145, 154, 171-172 and numerical methods such as molecular dynamics,152-153 traditional173 and reverse Monte Carlo,146 are commonly used. A direct method to analyze the diffuse scattering visible in these experimental 2D diffraction images is Fourier analysis, i.e. the Patterson Function (PF),174-175 which can be used to circumvent the phase problem in crystallography. PF analysis allows real space correlations from diffraction patterns to be obtained without prior assumptions176 which can help to avoid biasing the analysis. The results of PF analysis are electron-density weighted real space atom-atom correlations which differs from the more commonly used Pair Distribution Function (PDF) analysis where the results are number-density weighted.

All diffraction images of the twinned plastic crystal of α-CBr4 collected at the APS were Fourier transformed into Patterson functions with the interactive diffraction image analysis tool, Ramdog.* The computational details of 2D Fourier transforms are rather straightforward. First, all the rows are individually Fourier transformed, then all the columns are individually transformed. As the resultant values are complex, the “power spectrum” is computed to remove the imaginary portion of the complex number. The relationship between real space and reciprocal space is also straightforward: 훥푟 = 휋/푄푚푎푥, 푟푚푎푥 = 휋/훥푄. -1 - The PFs were computed from 2-D diffraction images with ΔQ=0.03Å and Qmax=20Å 1. The high symmetry [1̅11̅̅̅], [1̅11̅] projection shown in Figure 5.6a was manipulated using the open-source image editing software GIMP to remove the beam stop in order to enhance the quality of the resultant Patterson function by removing components that are unrelated to the

* Ramdog is an image analysis tool that I developed. A whitepaper describing its usage and the program itself will be available soon. See http://www.ncsu.edu/chemistry/people/jdm.html or email [email protected] for more information.

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sample diffraction. The manipulated diffraction image is shown in Figure 5.8a-b and its PF is shown in Figure 5.8c-d. The bright red circle at the center of the PF corresponds to the self- correlation function at a distance of 0 Å. Visible in all experimental and simulated PFs presented in this chapter are a horizontal line and a vertical line that pass through the image center. These lines are a mathematical artifact resulting from the image being aperiodic in x and y while the Fourier transform expects it to be periodic, which results in an effect colloquially referred to as “ringing”. In 1D PDF techniques, aperiodicity is significantly more problematic and can completely obfuscate physical interpretation if the data is not appropriately manipulated. As no attempt has been made to enforce translational symmetry on these images, or perform the corrections that are needed for PDF analysis, it is remarkable that the 2D Fourier ringing is so weak and suggests that 2D PF analysis may be significantly more robust than 1D PDF analysis. The remaining features in the PF correspond to real-space interatomic distances. There are only five types of interatomic pair-pair correlations that need to be considered: intramolecular C-Br and Br-Br and intermolecular C-C, C-Br and Br-Br. However, as the features in the Patterson function are weighted by the electron density of the scattering pairs, roughly estimable by the product of their atomic numbers, the Br-Br and C- Br pairs scatter 34 and 6 times more strongly than C-C pairs, respectively. The majority of the Patterson function is therefore dominated by Br-Br pairs (83%) with a smaller contribution from C-Br pairs (15%) and C-C pairs being almost nonexistent (2%) indicating that, to a first approximation, the PF can be interpreted by considering only Br-Br and molecule-molecule pairs.

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Figure 5.8. High symmetry [1̅11̅̅̅], [1̅11̅] diffraction image manipulated with GIMP to remove the beam stop to (a) Q=21 Å-1 and (b) Q=7 Å-1. 2D Patterson functions of (a) to (c) r=60 Å and (d) r=11 Å. 1 C-Br: 1.91Å. 2 Intramolecular: Br-Br 3.2Å. 3 Intermolecular Br-Br: 3.8Å. 4 1st nearest neighbor: 6.2Å. 5 2nd nearest neighbor: 10.4 Å. 6 3rd nearest neighbor: 12.3 Å.

The 2D Patterson function of α-CBr4 in Figure 5.8 exhibits pair correlations completely consistent with the intramolecular and first nearest neighbor contacts described in numerous structural investigations. The C-Br bond length is 1.91 Å, the intramolecular Br-Br distance

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is 3.2 Å, and the average intermolecular Br-Br distance is 3.8 Å. In addition this PF clearly demonstrates the nearest neighbor intermolecular distance is 6.2 Å along the 1 ⟨110⟩ real-space 2 directions and the second and third intermolecular distances perpendicular to [111] direction are located along the ⟨1 1 0⟩ and ⟨110⟩ real-space directions at 10.4 Å and 12.3 Å, respectively. 2 These six distances are labeled 1-6 in Figure 5.8d. Only one of each symmetrically equivalent distance has been highlighted for clarity, but the remaining five in each set can be visualized from the sixfold rotational symmetry about the image center. The remaining features in the Patterson function can be assigned based on the average molecular construct (Figure 5.3c) which has twenty-four bromine positions with pairs split across the 1 ⟨110⟩ intermolecular directions. In the (111) real space plane, there are twelve 2 bromine positions and the carbon center. A [111] slice through a face-centered cubic lattice containing these molecular constructs is overlaid onto the 2D PF, as shown in Figure 5.9. The carbons, shown in black, and the bromines, shown in orange, are plotted with 0.3 Å radii so that the Patterson function is still visible behind the overlay. The van der Waals radius of 1 bromine in CBr4 is ~ × 3.78Å and so the electron density cloud of bromine is significantly 2 larger than the plotted image suggests. It also must be noted the bromine positions that can be occupied in a real structure are highly influenced by their molecular neighbors. For example, consider the closest intermolecular bromine positions that are shown in Figure 5.9 and highlighted with the black arrow in Figure 5.9d. If these bromine positions were simultaneously occupied, the intermolecular Br-Br distance would be approximately 2.4 Å, or 25% shorter, than the observed intramolecular Br-Br distance. Interpretation via such an average molecular construct must be performed carefully so as to avoid giving weight to these impossibly short distances. The [111] average structure overlay in Figure 5.9a and c is aligned such that a molecular center is coincident with the PF origin. With a carbon in the center of the image any peaks in the PF that are obscured by an atom position in the average structure overlay correspond to C-(C/Br) pair correlations with a real space distance that can be measured from the center of the PF. In Figure 5.9b and d, the overlay is shifted such that a bromine position

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is coincident with the image center. With this overlay it is apparent that the several of the bromine positions along the vertical “arm” are coincident with the observed maxima. These indicate molecular correlations along the nearest-neighbor cubic 1 [202] real space direction 2 for at least 5-6 molecules, or 31 to 37 Å based on an average C-C distance of 6.2 Å. By symmetry, all cubic ⟨110⟩ real-space directions should have this 31-37 Å correlation length which is consistent with previous simulation results.146 To more effectively visualize the pair correlations, the intensity of a cross section of the Patterson function (along the white arrow in Figure 5.9b) is plotted logarithmically against the radial distance in Figure 5.9e. Beginning at 4.6 Å, there are regularly repeating peaks with maxima spaced 1.87 Å apart. The significance of the regularity of this spacing is unclear, though it was initially tempting to try to relate it to the 1.91 Å C-Br bond distance. Instead these peaks can be understood from a structural perspective from a radial pair-pair correlation function constructed by centering a carbon on the origin and counting the number of radial C- C and C-Br contacts, and by centering each of the twelve bromine positions of the overlay on the origin and counting the number of radial Br-C and Br-Br contacts. Additionally, the remaining six split bromine positions of the average molecular construct that are above and below the plane containing the carbon center must be centered and their Br-Br contacts accounted for (not shown, but it was done).

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Figure 5.9. An expanded portion of the 2D Patterson function from Figure 5.8c overlaid with the structure of a ⟨111⟩ slice of the average molecular construct to illustrate the location of various pair-pair correlations. (a), (c) Molecule centered overlay. (b), (d) Bromine centered overlay. (e) (black) Plot of intensity along a cross section of the PF along the white line in (b). (red) Electron-density weighted pair-pair correlations calculated from the average molecular construct.

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To a first approximation, it is assumed that the spatial coherence of each pair correlation can be modeled by the Gaussian function in Equation 4.10, acknowledging that a more complex function is probably necessary given the significant anisotropy in the diffuse scattering. 1 (푟 − 푟 )2 푓(푟) = 푛 ∗ 퐴 ∗ (푍 푍 ) × exp(−푘 ∗ 푟) × × exp (− 𝑖 ) 5.2 𝑖 1 2 2휋휎 2휎2

The distance separating atom pairs is defined as 푟𝑖, the Gaussian width of each pair-pair distance as 휎𝑖 and is probably primarily related to intermolecular orientational correlation functions145, 154 which are not accounted for. The remaining terms are multiplicative factors related to the number of contacts at each distance, 푛𝑖, the electron density of the atom pair,

푍1 ∗ 푍2, and a scaling factor, 퐴𝑖. Additionally, a qualitative exponentially decaying function is incorporated into to account for the experimentally observed decrease in intensity of the pair-pair correlations with increasing radial distance, likely dominated by positional disorder and thermal motion. These terms, combined as shown in Equation 4.10, are computed for all pair correlations for all atom pairs and summed to produce the red line in Figure 5.9. With the scaling factor set to 퐴 = 6 × 105, the pre-exponential decay constant set to 푘 = 0.16 and the Gaussian peak width set to 휎 = 0.4 for all atom-atom pairs, this qualitative approximation fits the observed Patterson function peak intensity remarkably well with the exception of the peaks at 9.2, 15.3, 21.4 and 27.5 Å. However, the apparent peak at 9.2 Å appears to be more of a shoulder between the two maxima highlighted with the two black arrows in Figure 5.9a as opposed to a peak itself. The remaining peaks at 15.3, 21.4 and 27.5 Å are likely due to the same phenomenon, indicating that the Patterson function can be understood quite well from this simple structural perspective. It is important to recognize that the average molecular construct, while helpful in assigning distances, does not allow for direct visualization of intermolecular orientational correlations.

5.3.4. Bragg and Diffuse Scattering in the Patterson Function Diffraction images collected along low symmetry axes result in Patterson functions which possess one rapidly and one slowly varying component with respect to the capillary

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rotation angle. This phenomenon can be reasonably captured by comparison of select sets of images, each separated by a 1° rotation. Such a set of diffraction images (letters) and their corresponding PF (primed letters) is shown in Figure 4.10. In all diffraction images, the beam stop is visible as the translucent black line rotated about 75° clockwise from vertical. The beam stop scar appears in all PFs as a line, rotated about 15° counterclockwise from the vertical and is highlighted by the arrow in Figure 5.10b'. It is visible in all PF images unless obscured by other Fourier components as in Figure 5.10a'. As previously described, the horizontal and vertical scarring highlighted with arrows in Figure 5.10a are also visible in in these PFs. The ordered pair correlations with points of a “star” along the cubic ⟨110⟩ real-space directions are visible in all PFs in Figure 5.10; albeit with the sixfold symmetry being most apparent in images d'-h'. Additionally visible is a background halo that is prominent in Figure 5.10a', c', g' and h' but disappears in images b', d'-f'. It is inconceivable that rotation over such a small angle, 1 about the capillary axis, would give rise to dramatically different distributions of ordered electron density. However, inspection of the 2D diffraction features demonstrates that while the ordered diffuse scattering is consistent across the whole 7  of rotation represented by these images, there is a dramatic variation in the intensity of Bragg diffraction that is observed. The Bragg scattering is highly dependent upon the orientation of the Ewald sphere and thus the diffraction spots are visible in no more than two sequential frames. Therefore, the slowly varying features in the PFs result from the structured diffuse scattering while the rapidly varying features result from the Bragg diffraction peaks.

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Figure 5.10. Experimentally obtained 2D diffraction images (letters) and corresponding Patterson functions (primed letters). Each set is separated by 1° of rotation around the capillary axis.

5.4. Simulation Methods & Results The analysis of the experimental 2D diffraction images with Patterson Function demonstrates that there are strong orientational correlations to the fifth or sixth nearest neighbor along the cubic ⟨220⟩’s. With an average intermolecular distance along these directions of 6.2 Å, the orientational correlation length is between 31 and 37 Å which is 146 consistent with the results of the most recent α-CBr4 simulation. However, the structural features which give rise to the low-Q structured diffuse, the high-Q repeating diffuse hexagons along the [111] projection and the Bragg scattering remains unclear. The development of

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simulations in this section is motivated by the desire to understand the structural origins of each of these three types of diffuse scattering by creating an appropriate simulated plastic crystalline lattice starting from structural principles instead of using pre-existing software to “fit” a structure to the observed diffraction. The appropriateness of the resultant simulated plastic crystal lattices is judged by its relationship to the experimental diffraction images and Patterson Functions. There is very little literature describing the use of Patterson Functions to solve the phase problem in highly disordered systems. The results described in this section are presented as much from the context of developing insight into the highly structured diffuse scattering observed in the twinned plastic crystal of α-CBr4 as they are from the context of exploring the complex relationship between the diffuse scattering in 2D diffraction patterns and the real- space features in the resultant Patterson Functions.

5.4.1. Simulation Methods Simulations were written in Java and developed in the Eclipse IDE.* Diffraction patterns and Patterson functions were calculated with software written in Java, C and CUDA and visualized with the interactive diffraction analysis tool, Ramdog. The diffraction calculation is an inherently parallelizable algorithm because q-vectors depend only on atomic coordinates and so are independent from each other. As such, general purpose computing on graphics processing units (GPGPU) was used to accelerate diffraction pattern calculations by developing algorithms in C and CUDA to interface with Java through the Java Native Interface (JNI). All lattices generated were single crystalline with the global x, y, and z directions aligned with the cubic x, y and z directions. Exploratory lattices were constructed with between five and ten unit cells per crystal axis with four molecules per unit cell and five atoms per molecule, giving between 53 × 4 × 5 = 2 500 and 103 × 4 × 5 = 20 000 atoms.

* All simulation software will be made available once whitepapers are written and the code has been refactored. See http://www.ncsu.edu/chemistry/people/jdm.html or email [email protected] for more information.

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Simulation images presented in this document were calculated from simulated lattices containing between 15 and 30 unit cells per Cartesian direction, or 153 × 4 × 5 = 67 500 to 303 × 5 = 540 000 atoms. Simulations were generally repeated 50-100 times and the diffraction patterns summed to increase the signal to noise ratio.

Table 5.1. Bonds lengths for the bromines surrounding the four unique tetrahedra. Br 1 2 3 4 1 1.827 1.883 1.879 1.857 2 1.903 1.898 1.881 1.935 3 1.928 1.931 1.911 1.936 4 1.952 1.965 1.940 1.969 Avg(s.e.) 1.90(3) 1.919(18) 1.903(14) 1.92(2)

Table 5.2. Br-C-Br bond angles for the four unique tetrahedra. Br-Br 1 2 3 4 1-2 114.36 108.37 109.97 108.30 1-3 109.53 110.92 107.66 106.17 1-4 105.19 108.49 111.00 113.01 2-3 106.61 107.06 113.32 111.41 2-4 109.69 112.63 105.43 108.23 3-4 115.56 109.37 109.50 109.75 Avg(s.e.) 109.5(14) 109.5(8) 109.5(11) 109.5(10)

Table 5.3. Interaction parameters for CBr4. Atom Pair ε (eV) σ (Å) Br-Brinter 0.0116-0.0349 3.82 Br-Brintra ? 3.2 C-Br 0.824 1.91

In half of the simulations the tetraheda were ideal with Br-C-Br bond angles of 109.47° and a C-Br bond length of 1.91Å in accordance with the reported crystal structure of β-CBr4 with perfect tetrahedra.165 In the other half, the four crystallographically unique distorted

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tetrahedra from one of the reported monoclinic crystal structures were used with parameters as reported in Table 5.1 and Table 5.2.161 Molecular distortions can decrease the total energy by about 10%.144 In all simulations, the tetrahedra were treated as rigid molecules. The only interaction potential is therefore intermolecular Br---Br. One previous simulation was able to effectively model the structured diffuse scattering from single crystal synchrotron diffraction after thoroughly relaxing the simulation cell via Monte Carlo reorientations by using the simplest possible intermolecular Br---Br potential, the

Hooke’s law restorative force, 퐹 = −푘(푥 − 푥0). The two-body Lennard-Jones (LJ) potential has previously been unsuccessful in replicating experimental measurements of α-CBr4 due to the polarizability of the bromine electron cloud.144, 177 It has also been shown that no set of parameters in the LJ model was able to produce any orientational disorder below the melting point without shortening the C-Br bond by 20%.152-153 The four-parameter Lennard-Jones152- 153 and Buckingham144, 178-179 models have had success in modeling fundamental molecular vibrations, elastic constants, sound velocities and the structured diffuse scattering observed in neutron diffraction. Nevertheless, the two parameter LJ potential was used initially with the understanding that complexity would be added as needed. 휎 12 휎 6 푈(푟) = 4휀 [( ) − ( ) ] 5.3 푟 푟 12휎12 6휎6 퐹(푟) = 4휀 [− + ] 5.4 푟13 푟7 In the Lennard-Jones model (Equations 5.3 and 5.4), σ corresponds to the distance where the interaction potential is zero and ε corresponds to the maximum depth of the well at

6 푟푚 = √2휎. The intermolecular Br---Br distance (rm) was set to the observed peak in the experimental PDF, 3.82 Å. The depth of the well was approximated by using the enthalpy of -1 180 vaporization, ΔvapH=40.4 kJ mol = 0.418 eV per molecule, as an experimental measure of the intermolecular interaction energy. Electrostatics can contribute an additional 4% to the total lattice energy.144 There are four bromines per molecule which can each interact with between three and nine intermolecular bromines, depending on the amount of closest-packing.

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To a first approximation, the well depth for intermolecular Br---Br interactions is between 0.0349 and 0.0116 eV per bromine, respectively. Though not reported in this chapter, simulations which allowed intramolecular interactions were performed. To model the C-Br interaction, the C-Br bond dissociation energy of 318.0 ± 8.4 kJ mol-1,180 or 0.824 eV per bond was set to ε and σ determined based on the ideal distance of 1.91 Å. The intramolecular Br-Br distance is 3.2Å but its interaction was not determined. Presumably, to obtain a reasonable estimate of the intramolecular Br-Br interaction energy from experimental measurements, one could investigate the vibrational spectrum of gas phase CBr4. These values are collected in Table 5.3. 5.4.1.1. Energy Minimization The plastic crystalline lattices as constructed initially resulted in many impossibly short intermolecular Br-Br contacts. There are many approaches to reducing the energy of a simulated system including Molecular Dynamics (MD) and Monte Carlo (MC). MD simulations allow atoms to move according to the laws of classical mechanics whereby the forces on all atoms are calculated at every time step and atoms and molecules usually retain their velocity from the previous time step. While MD calculations may provide additional insight, the goal of this work is to understand the extent to which the structure of α-CBr4 can be described from a static perspective. Thus, MD simulations do not provide the desired insight into α-CBr4. Traditional MC attempts to move one atom at a time by a random amount in a random direction using the energy before and after as criteria for deciding whether the move is to be kept, though this strategy is not commensurate with maintaining rigid tetrahedra. Another MC approach is to randomly distribute the six orientations shown in Figure 5.3b onto the FCC lattice sites and then randomly change the molecular orientations until no impossibly short distances remain.38 Using MC to change molecular orientations was shown to fit the most intense structured diffuse scattering but was unable to account for the new structured diffuse features shown in Figure 5.6. Reverse MC (RMC) is similar to traditional MC with the exception that the simulation cell is converged to “fit” some experimental measurement, though the RMC results are frequently poorly interpreted as it can be quite challenging to relate RMC results to fundamental structural principles.

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For these reasons, a new energy minimization strategy was desired that focused on enhancing intermolecular orientational correlations while allowing the freedom of movement (rotation + translation) that the MD approach provides which is missing from MC methods. This approach is described as a pseudo-random walk, with its algorithm described in pseudocode in Figure 5.11. The random walk algorithm begins by selecting a molecule (testMolecule) from the simulation cell at random. The pairwise interaction energy with the surrounding twelve nearest neighbors along the ⟨220⟩ directions is determined and the molecule with the most unfavorable interaction energy is selected (worstMolecule). The pairwise intermolecular forces are calculated and the molecules are moved as a result of these forces. The testMolecule was then set to the worstMolecule and the algorithm was repeated some number of times, called numWalkSteps. After each molecule in the simulation cell had been the starting point for a random walk, one “walk” was complete. The lattice which resulted from this approach is dependent upon the number of walk cycles (numWalkCycles), the number of steps per random walk cycle (numWalkSteps) and the interaction energies selected. If too many walk cycles and walk steps were set, the resultant lattices did not produce diffraction patterns that closely resembled the experimental images, as shown in some images presented in §4.3.2. The ideal number of walk cycles was generally between 1 and 5 and the ideal number of walk steps was generally between 256 and 512, depending upon ε, the depth of the LJ well as a deeper well will converge faster than a slower one. Additionally, because this algorithm must resolve physically unreasonable intermolecular contacts, the force calculation often results in extremely large forces. To ensure that the simulation cell did not “explode”, the maximum translational and rotational motions for any given movement attempt were set to 0.25 Å and 1°, respectively.

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Figure 5.11. Pseudocode random walk algorithm.

5.4.1.2. Calculating Diffraction Images Diffraction images were calculated from the simulated structures to judge the appropriateness of the chosen lattice generation and energy minimization algorithms. Three high symmetry diffraction images were calculated for each simulated lattice: [001], [011] and [111]. These three projections are comprised of two orthogonal vectors: [100] × [010], [1̅00] × [011̅] and [11̅0] × [11̅̅̅̅2], respectively. A single calculated diffraction image often has a poor signal to noise ratio (SNR) which can be improved by combining the results of many simulations with the same input parameters because of the randomness inherent in the mechanisms of lattice construction and energy minimization algorithms. The SNR can also be improved by computing and summing the diffraction pattern along similar projections, e.g.

[111]푐푎푙푐 = [111] + [111̅] + [11̅1] + [1̅11]. Each pixel in the calculated diffraction images was calculated with the generic scattering equation, 푎푙푙 푎푡표푚푠

퐹(푞⃑) = ∑ 푓(푞⃑, 푍푎) × exp(2휋𝑖 × 푞⃑ ∙ 푟⃑⃑⃑푎⃑) . 5.5 푎=1

In Equation 5.5, 푞⃑ is the scattering vector, 푟⃑⃑⃑푎⃑ is the position of atom 푎 in crystal coordinates and 푓 is the scattering factor, dependent upon 푞⃑, the specific element type, 푍푎, and the wavelength of incident photons, 휆. The smallest reciprocal lattice vector represents a sine

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wave that spans the entire box and is equal to the reciprocal of the number of unit cells per crystallographic axis. For a simulation box that has ten unit cells per edge the smallest reciprocal lattice vector is 훥푄 × (1, 0, 0) where ΔQ=10-1, the inverse of the number of unit cells. Assuming the simulation cell is a cube, ΔQ is the same along all three crystal axes. Integral multiples of this fundamental lattice vector can also be used. Two coordinate frames need to be considered: the pixel space of the image and the q space of the diffraction projection. For pixel space, zero is located in the bottom left corner and for q space it is the middle of the image. Pixel coordinates are given in (x, y) and reciprocal space coordinates in 푞⃑. The center of the image will be referred to as c=(xc, yc). The diffraction image to be calculated is comprised of orthogonal vectors whose cross product is the diffraction image zone axis. These orthogonal image vectors will be referred to as 푞⃑⃑⃑⃑푥⃑ and 푞⃑⃑⃑⃑푦⃑ and the zone axis as 푞⃑⃑⃑푧⃑. The formal relationship between the pixel coordinates and reciprocal space coordinates is

푞⃑ = 훥푄(푥 − 푥푐)푞⃑⃑⃑⃑푥⃑ + 훥푄(푦 − 푦푐)푞⃑⃑⃑⃑푦⃑ 5.6 푞⃑ ∙ 푞⃑⃑⃑⃑⃑ 5.7 푥 = 푥 + 푥 훥푄 푐 푞⃑ ∙ 푞⃑⃑⃑⃑⃑ 5.8 푦 = 푦 + 푦 훥푄 푐

푄푚푎푥 5.9 푥푐 = 훥푄 × |푞⃑⃑⃑⃑푥⃑|

푄푚푎푥 5.1 푦푐 = 훥푄 × |푞⃑⃑⃑⃑푦⃑| 0 As an example of determining the values of specific scattering vectors, consider a simulation with ten unit cells per edge (ΔQ=0.1). The desired diffraction image is the [001] -1 projection to Qmax=20 Å which can be composed of 푞⃑⃑⃑⃑푥⃑ = [100] and 푞⃑⃑⃑⃑푦⃑ = [010], though any orthogonal vectors whose cross product is [001] can be used. With these parameters, the center of the image will be located at (200, 200) according to Equations 5.9 and 5.10. The (220) reflection will be located in the image at (20, 20) + (200, 200) = (220, 220). Additional examples are provided in Table 5.4 for selected high symmetry projections.

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The 푞⃑ value of each pixel is determined with Equation 5.6 for every pixel in the diffraction image. The atom positions, 푟⃑⃑⃑푎⃑, are the reduced crystal coordinates, or the Cartesian, real space positions of the atoms divided by the edge length of the unit cell assuming cubic symmetry. The relationship between Cartesian coordinates and crystal coordinates becomes more complicated for lower symmetry systems but can be found in crystallography textbooks or the International Tables for Crystallography.

Table 5.4. Some examples of the coordinate relationships between image (pixel) space and reciprocal space. ΔQ=0.1, Qmax=20.

Projection [ퟎퟎퟏ] [ퟎퟎퟏ] [ퟎퟏퟏ] [ퟏퟏퟏ̅̅̅̅]

풒⃑⃑⃑⃑풙⃑ [100] [110] [100] [110]

풒⃑⃑⃑⃑풚⃑ [010] [1̅10] [011] [11̅2]

풙풄 (pixels) 200 141 200 141

풚풄 (pixels) 200 141 141 82

Pixel coordinates of specific diffraction peaks: value + (xc, yc) (ퟐퟐퟎ) (20, 20) (14,14) (20,14) (14,0) (ퟑퟑퟑ) n/a n/a (30,21) n/a

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dispersion effects for both elements at 90 keV are likely extremely similar to the reported values at 30 keV because neither element has an absorption edge near either energy and anomalous dispersion effects only provide a meaningful contribution to the scattering factor near absorption edges. Therefore, for the diffraction pattern calculations, the contribution from anomalous dispersion effects, 푓′ + 𝑖 ∙ 푓′′, were calculated for λ=0.4 Å as opposed to the experimental value of λ=0.13702 Å. 5.4.1.3. Determining Molecular Alignment

Figure 5.12. Cartoon of the method for projecting C-Br bonds onto the [001] planar surface to visualize the alignment of the C-Br bonds with the [001] direction. The black outline with blue ribs corresponds to the spherical surface where the bromines are located. The green outline corresponds to the [100] projections shown in Figure 5.29 and Figure 5.30.

Determination of the molecular alignment was achieved by considering the alignment of each CBr bond vector with common intermolecular directions in fcc molecular packing: the ⟨100⟩ real-space directions are the cubic unit cell axes, the ⟨110⟩ real-space directions are the intermolecular nearest neighbor vectors and the ⟨111⟩ real-space directions are the molecular closest packing vectors. For each of the three projections, the direction aligned most closely with each CBr bond vector was determined. To visualize these relationships in 2D, each CBr bond vector was projected onto a plane with 0.001 Å2 bins, as cartooned in Figure 5.12. Consider the carbon placed at the origin of the coordinate system. The corresponding bromine

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is then projected onto the surface orthogonal to the [001] direction (black arrow) by calculating the dot product of the C-Br bond vector with the [100] and [010] to obtain the 푥⃗ and 푦⃗ vectors in the orthogonal plane. This projection is computed for each C-Br bond to create the histograms as shown in shown in Figure 5.29 and Figure 5.30. The histograms for the [111] and [110] real-space projections are created analogously. 5.4.1.4. Image Processing: Median Filtering Median filtering is an extremely fast and remarkably simple way to smooth an image and therefore remove the sharp Bragg peaks from the computed diffraction images. The median filter works by inspecting all pixels in the image individually and replacing the intensity of the pixel with the median value of its surroundings if it is greater or less than the median times some multiplicative factor. All computed diffraction images were processed with a 5×5 median filter (2 pixels in each direction from the center pixel) before the 2D Fourier transform (Patterson Function) was computed.

5.4.2. Simulation Results

There are two structural principles in α-CBr4 that are common in the literature: Isotropic molecular orientations over a time/ensemble average and the six orientations that arise from the alignment of molecular 4̅ axes with the cubic ⟨100⟩ real-space directions. Simulations were performed using both of these structural principles by distributing tetrahedra onto the FCC sites with random orientations or randomly selecting one of the six D2d orientations. A new approach presented in this chapter uses the four crystallographically unique molecules from β-CBr4 and their first or second coordination shells as structural motifs to create a disordered α-CBr4 lattice. The simulations are presented in the following order:

Random molecular orientations, D2d orientations, first shell monoclinic and second shell monoclinic 5.4.2.1. Random Molecular Orientations

A few reports have indicated that in the plastic phase of CBr4 molecular orientation “distribution seems to be isotropic.”146 This structural concept can be explored by generating a lattice where the molecules are centered on the fcc lattice but their orientations are completely

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random. Molecular orientations were randomized by rotations around each of the x, y and z axes by three random angles between 0 and π, φx, φy and φz. Intermolecular distance constraints were not imposed when randomly rotating molecules about the Cartesian axes indicating that, as initially constructed, this lattice construction allows bromines of neighboring molecules to be pointing towards each other resulting in intermolecular Br---Br distances up to 1.2 Å shorter than the intramolecular Br---Br distance as previously detailed in §4.2. As constructed with random initial orientations, this lattice does not correspond to α-CBr4 as random, uncorrelated orientations are not a good description of the experimental data135, 153, 169 and many of the resultant Br---Br contacts are impossibly short. Nevertheless, it is a starting point to induce intermolecular correlations through the random walk algorithm to evaluate the proposed isotropic spatial distribution of molecular orientations. The diffraction patterns along the high symmetry [001] and [111] projections corresponding to the initial lattice construction are shown in Figure 5.13a and Figure 5.14a, respectively, for 100 simulations of a 10×10×10 simulation cell. Though no experimental diffraction images were collected along the [001] directions, computing the diffraction patterns along additional high symmetry projections allows for comparison to previous work38, 146 while also serving as a self-consistent check that nothing unexpected exists along different projections. The Bragg scattering in the computed and experimental [111] diffraction images have the same symmetry, though the Bragg diffraction is visible to much higher Q in the simulations than in the experimental images. This is a direct result of the perfect FCC molecular packing in the as-constructed lattice and suggests that the Bragg scattering in the experimental images is reduced in intensity by either significant thermal motion or molecular displacements from the average structure FCC sites. The Bragg scattering in the simulated images results from the molecular translational symmetry since it is the only aspect of this structure that can fulfill the Bragg conditions as the molecules are randomly oriented and have no intermolecular correlations. Nevertheless, this simulation supports the conclusion that the experimental Bragg scattering in α-CBr4 largely results from the average FCC molecular packing.

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Figure 5.13. Diffraction patterns and Patterson functions computed along the ⟨100⟩ directions of an fcc lattice constructed with random molecular orientations. (a-c) Initial lattice construction with no relaxation. (d-f) 1 cycle of random walk relaxation. (g-i) 5 cycles of -1 -1 random walk relaxation. (a,d,g) Qmax=20 Å . (b,e,h) Qmax≈6 Å . Certain Bragg reflections are labeled. (c,f,i) 2D Patterson functions to rmax ≈ 13 Å. Numbers correspond to atomic positions. For each diffraction image, the left half of the image is the raw computed image and right half is the image with a 5×5 median filter mask to remove the Bragg peaks.

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Figure 5.14. Diffraction patterns and Patterson functions computed along the ⟨111⟩ directions of an fcc lattice constructed with random molecular orientations. (a-c) Initial lattice construction with no relaxation. (d-f) 1 cycle of random walk relaxation with a walk length of -1 256. (g-i) 5 cycles of random walk relaxation with a walk length of 256. (a,d,g) Qmax=20 Å . -1 (b,e,h) Qmax≈6 Å . Certain Bragg reflections are labeled. (c,f,i) 2D Patterson functions to rmax ≈ 13 Å. Numbers correspond to atomic positions of intra and intermolecular correlations. For each diffraction image, the left half of the image is the raw computed image and right half is the image with a 5×5 median filter mask to remove the Bragg peaks.

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That no structured diffuse scattering is observed in this initially constructed lattice is consistent with previous work which relates the structured diffuse scattering to intermolecular orientational and displacive correlations.38, 135, 153 The random, uncorrelated positions of the bromines surrounding the carbon centers cause the high-angle diffuse scattering to appear as isotropic rings in the calculated [001] and [111] projections. It is reasonable to suppose that by increasing the intermolecular bromine correlations the diffuse scattering should become more structured by beginning to fulfill the Laue conditions in one, two or three dimensions to create diffuse sheets, rods or Bragg peaks. The Bragg scattering in the initially constructed lattice is significantly more intense than the diffuse scattering and dominates the PF. However, as the Bragg scattering results from the average structure FCC molecular packing, the real-space correlations resulting from the Bragg scattering are of little interest. Therefore, the Bragg scattering was removed with a 5×5 median filter which is shown as the right half of all calculated diffraction images. The 2D PF calculated from the median filtered images is shown in Figure 5.13c and Figure 5.14c and shows little resemblance to the 2D Patterson Function of the experimental data (Figure 5.8). In these computed 2D Patterson functions for randomized molecular orientations, the smallest real feature must be the C-Br bond distance of 1.91 Å, labeled 0 in Figure 5.13c and Figure 5.14c. No maxima or minima are discernible in this inner ring corresponding to the C-Br bond, consistent with the initial lattice possessing no preferred orientational directions. Similarly, the intramolecular Br-Br distance of ≈3.2 Å, labeled 1 in Figure 5.13c and Figure 5.14c shows no discernible maxima or minima, also consistent with there being no initial orientational preference. The nearest molecular neighbors are visible as the ring-like features labeled 2-5 in Figure 5.13c and 2-3 in Figure 5.14c, corresponds to the 1st through 4th nearest intermolecular contacts along the 1 [110], [100], 1 [211] and [110] real-space directions at distances of 6.2, 2 2 8.8, 9.7 and 12.7Å in the [001] projection and the 1st and 2nd nearest intermolecular contacts along the 1 [110] and 1 [211] real-space directions at distances of 6.2 Å and 9.6 Å in the [111] 2 2 projection. Interestingly, along the [111] projection shown in Figure 5.14c, only the 2nd nearest neighbor contact is visible while along the [001] projection, the 4th nearest neighbor

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can clearly be seen. It is unclear why [110] set of nearest-neighbors is clearly defined in the [001] PF but not in the [111] PF. Relaxing the lattice with the random walk algorithm begins to introduce intermolecular displacive and orientational correlations. The displacive disorder causes a significant reduction in the Bragg scattering, even after only one 1 random walk cycle through the lattice with a walk length of 256 steps (1×256 in condensed notation), as shown in the calculated images of Figure 5.13d-e and Figure 5.14d-e. Significant low Q structured diffuse scattering is visible in the [001] projection that connects neighboring {200} reflections along [220] vectors at low Q which is somewhat surprising as the experimental diffuse scattering is along the [111] vectors. That this structured diffuse scattering at low Q is along the [220] direction as opposed to the [111] direction seems to empirically explain why there is no observed diffuse scattering connecting the {220} and {111} reflections in the [111] projection shown in Figure 5.14e. While it would be tempting to argue that there might be diffuse scattering connecting the {220} and {111} that just cannot be seen because of the chosen color scheme, no modification of the color levels could make the inner star appear. At higher Q, structured diffuse scattering is observed to be transversely polarized near the (2̅20) Bragg reflection in Figure 5.13e but is centered at a slightly higher Q value of (-2.15, 2.15, 0), at an identical location to that observed with diffuse neutron scattering.135, 141 The exact location of this diffuse feature in the [111] projection is less clear due to its lower resolution which results from the larger step size required by longer length of the orthogonal vectors that comprise the [111] projection versus the shorter orthogonal vectors that comprise the [001] projection. Additional transversely polarized diffuse scattering is observed near the {440} reflections in both the [001] and [111] projections. The transverse diffuse scattering near the {220} and {440} Bragg peaks has been previously described to result from intermolecular orientational and displacive correlations38 which these results are in agreement with. The corresponding PFs in Figure 5.13f and Figure 5.14f give a significantly different picture of the real space contacts than that for the initially uncorrelated initial lattice. New features that appear result from the intermolecular orientational and displacive correlations at distances up to 14.1Å which are labeled 2-8 and occur at positions similar to those observed

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experimentally (Figure 5.9). The labeled features correspond to the following: 0 The C-Br bond which still appears to be isotropically distributed, though this could be due to the chosen color scale; 1 The Br---Br intramolecular contact at 3.2 Å which appears to have maxima (blue) in the nearest neighbor 1 ⟨110⟩ real-space directions, while the experimentally observed 2 direction seems to be the second nearest neighbor in the 1 ⟨211⟩ real-space direction; 2-8 All 2 have defined maxima along the nearest-neighbor 1 ⟨110⟩ real-space directions similar to those 2 observed experimentally and occur: in Å, parenthesis indicate experimental peak location; at 2 4.5(4.6), 3 6.2(6.1), 4 7.4(7.6), 5 9.2(9.2),* 6 10.5(10.7), 7 12.7 (12.2) and 8 14.1 (13.7), with an exception at 5 which appears to be a shoulder in the experimental PF and may not correspond to a Br---Br distance in the real system. The signal in the simulated PF at these distances is significantly more isotropically distributed than it is in the experimental system, indicating that the intermolecular correlations are significantly stronger in reality than in this initial simulation with randomly distributed starting orientations. The diffraction patterns that result after the 1×256 random walk cycle are further relaxed by four additional 1×256 random walk cycles (5×256 total) is shown in Figure 5.13g- h and Figure 5.14g-h. The primary difference between the 1×256 diffraction and the 5×256 diffraction patterns is further loss of Bragg diffraction at high Q, slightly more pronounced ⟨111⟩ diffraction peaks at low Q as highlighted in Figure 5.14h and the diffuse scattering at low Q becomes more structured. The high Q diffuse rings remain isotropic and unchanged. The PFs computed from the 5×256 random walk diffraction patterns show more anisotropic structure, qualitatively similar to that observed experimentally in Figure 5.8 and is indicative that the random walk algorithm is inducing the desired order into the simulated lattice. The full PF for the 5×256 walk is shown in the larger image in Figure 5.15 where the top left and bottom right quadrants have been overlaid with the average molecular construct based on the D2d orientations with the correlation pattern centered by a carbon atom in the top left and by a bromine atom in the bottom right. Radial distances corresponding to PF features

* The peak at 9.2 Å in the experimental PF corresponds to a shoulder between two peaks and may not correspond to a Br---Br distance in the real system.

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are listed in the top right quadrant. Notably, the correspondence to the experimentally observed PF features is worse with the 5 × 256 random walks than was observed for the 1×256 random walks: in Å, parenthesis indicate experimental peak location; 4.33(4.6), 6.41(6.1), 7.54(7.6), 9.46(9.2), 10.9(10.7), 13.1(12.2/13.7), 14.5(13.7/15.3), 16.4(16.8) and

17.6(16.8/18.3). Given the location of the peaks in the simulated PF relative to the D2d molecular overlay, demonstrates that these D2d orientations arise from the initially random molecular orientations because of intermolecular correlations, consistent with the experimental observations that the plastic phase preferentially adopts specific molecular orientations. However, many differences remain between the experimental and simulated diffraction patterns and PFs based on the initially randomly oriented FCC lattice.

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5.4.2.2. Six D2d Molecular Orientations

Numerous manuscripts have indicated that in the plastic phase of CBr4 the molecules 10-12, 17-18, preferentially adopt one of the six D2d molecular orientations shown in Figure 5.3b.

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24-25, 38, 46 This structural concept can be explored by generating a lattice where the six orientations are randomly distributed across the FCC lattice sites. Similar to the above lattice construction in which the molecules were randomly oriented, the D2d lattice construction allows bromines of neighboring molecules to be pointing directly towards each other resulting in intermolecular Br---Br distances 1.2 Å shorter than the intramolecular Br---Br distance as previously detailed in §4.2. As constructed with randomly distributed D2d orientations, this 135, lattice does not correspond to α-CBr4 as uncorrelated orientations are not a good description 153, 169 and many of these Br---Br contacts are impossibly short. Nevertheless, it is a starting point to study the effects of introducing intermolecular correlations through coupled translational and rotational relaxations. A previous simulation used an identical starting point, but employed a Monte Carlo algorithm to remove the impossibly short Br---Br contacts by reorienting a molecule with the offending contact to one of the other five D2d molecular orientations before allowing relaxation by a displacive motion via a Hooke’s Law potential. This method demonstrated the origin of significant transverse diffuse scattering, though only the {220} and {330} diffuse scattering was observed in the [111] projection.38 The same simulation was repeated during this investigation except the molecules were relaxed via the random walk algorithm and the Lennard-Jones potential after the Monte Carlo orientational changes. No additional structured diffuse scattering was observed, suggesting that the Monte Carlo orientational change algorithm does not induce enough long-range orientational correlations. The diffraction patterns along the high symmetry [001] and [111] projections for Q < 20 Å-1 corresponding to the initial lattice construction of 100 simulations of a 10×10×10 simulation cell are shown in Figure 5.16a-b and Figure 5.17a-b, respectively. Similar to the random orientation simulation, the Bragg scattering in the computed and experimental [111] diffraction images have the same symmetry, for reasons detailed in the text accompanying the random orientation simulation.

That the diffuse scattering observed in this initially constructed D2d lattice with no intermolecular correlations is vastly different from that observed experimentally is consistent with previous work.38, 135, 153 The largest similarity between this diffuse scattering and previous

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experimental observations is that the intense diffuse blob near the (2̅20) Bragg reflection is centered at the slightly higher Q value of ≈(-2.15, 2.15, 0) in Figure 5.16b which is consistent with diffuse neutron diffraction studies.135, 141 As the diffuse scattering of the randomly distributed D2d lattice bears little similarity to the experimental diffuse scattering and the lattice contains impossibly short Br---Br distances, no further effort was made in its interpretation. Similarly, the PFs computed from the initial lattice configuration are uninterpretable as they are swamped by the intense diffuse scattering arising from the uncorrelated D2d orientations, as seen in Figure 5.16c and Figure 5.17c, even after the Bragg peaks were median filtered. Relaxing the lattice with the random walk algorithm introduces intermolecular displacive and orientational correlations as shown in Figure 5.16d-e and Figure 5.17d-e. Similar to the simulation starting with random orientations, the Bragg scattering is significantly reduced after a 1×256 random walk. Also similar to the random initial orientations, a sharp diffuse square appears in the [001] projection which connects the neighboring {200} reflections along ⟨220⟩ reciprocal space vectors. However, in the [111] projection, there are hints of diffuse scattering connecting the {220} and {111} Bragg reflections via the ⟨111⟩ reciprocal space directions, more similar to the experimentally observed diffuse scattering. Moreover, much of the calculated structured diffuse scattering in the 1×256 [111] projection is in qualitative agreement with that observed experimentally. The transversely polarized diffuse scattering near the {220} Bragg reflections in the [111] projection is the most intense scattering observed (ignoring the self-correlation at Q≈0 Å-1), the diffuse scattering connecting the {331} Bragg reflections is present and there is diffuse scattering that corresponds to the “dog teeth” though it is less sharply defined than in the experimental diffraction. Also note that the high Q diffuse hexagons that are present in the experimental [111] diffraction are, though still significantly “blobby”, are starting to form, as seen in the computed [111] diffraction pattern. Because this series of hexagons was not seen in the simulations which started with random initial orientations, that the hexagons appear when the molecules begin from a set of preferential orientations suggests that the may contribute significantly to the shape of this high Q diffuse.

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Figure 5.16. Diffraction patterns and Patterson functions computed along the ⟨100⟩ directions of an fcc lattice constructed by randomly placing one of six molecular orientations on each lattice site. (a-c) Initial lattice construction with no relaxation. (d-f) 1 cycle of random walk relaxation with a walk length of 256. (g-i) 5 cycles of random walk relaxation with a walk -1 -1 length of 256. (a,d,g) Qmax=20 Å . (b,e,h) Qmax=9 Å . (c,f,i) 2D Patterson functions to rmax ≈ 13 Å. For each diffraction image, the left half of the image is the raw computed image and right half is the image with a 5×5 median filter mask to remove the Bragg peaks.

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Figure 5.17. Diffraction patterns and Patterson functions computed along the ⟨111⟩ directions of an fcc lattice constructed by randomly placing one of six molecular orientations on each lattice site. (a-c) Initial lattice construction with no relaxation. (d-f) 1 cycle of random walk relaxation with a walk length of 256. (g-i) 5 cycles of random walk relaxation with a walk -1 -1 length of 256. (a,d,g) Qmax=20 Å . (b,e,h) Qmax=9 Å . (c,f,i) 2D Patterson functions to rmax ≈ 13 Å. For each diffraction image, the left half of the image is the raw computed image and right half is the image with a 5×5 median filter mask to remove the Bragg peaks.

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The PF computed from the 1×256 [111] projection, shown in Figure 5.17f and a larger version shown in Figure 5.18, is remarkably similar to the [111] experimental PF (Inset) at short distances but the similarities fall off rapidly as the distance increases past the peak labeled 8.45 Å. This rapid decay in orientational correlations occurs at the 1st to 2nd coordination shell

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boundary, indicating that the random walk algorithm, while successfully inducing intermolecular correlations to the 1st coordination shell is not inducing the longer-range intermolecular correlations that exist in α-CBr4.

Figure 5.19. Side-by-side comparison between the [001] diffraction images calculated by starting with the lattices constructed with (a) D2d orientations and (b) random orientations and performing 5×256 random walks.

The diffraction patterns that result after the 1×256 D2d lattice is further relaxed by four additional 1×256 random walk cycles (5×256 total) is shown in Figure 5.17g-h and Figure 5.18g-h. These diffraction patterns are remarkably similar to those obtained after 5×256 random walks when starting from random molecular orientations as shown by the side-by-side comparison in Figure 5.19 and indicate that the random walk algorithm and the Lennard-Jones potential is reaching a global minimum after 5×256 random walks. Unfortunately, as detailed in previous simulation manuscripts, the Lennard-Jones model indeed does not converge to a structure commensurate with the known properties of α-CBr4, now including the single crystal diffraction pattern. Nevertheless, if the random walk algorithm with the LJ potential is allowed to run for only one cycle (1×256) and stopped before its global minimum is reached, a simulated plastic crystal that produces a diffraction pattern extremely similar to that observed experimentally can be obtained. 5.4.2.3. First Shell Monoclinic Constructing the plastic crystalline lattice by randomly distributing six starting orientations onto FCC sites and minimizing the energy via a 1×256 random walk algorithm

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does not generate sufficient intermolecular orientational correlations to account for the longer range order experimentally observed in the experimental diffraction patterns and PFs. This is not to imply that the random walk algorithm is ineffective at creating intermolecular correlation. Instead, it is likely that the initial lattice construction is not starting from a sufficiently ordered state such that the system can relax via the random walk algorithm into a configuration that produces a diffraction pattern analogous to the experimental pattern. To increase the intermolecular correlation length of the initial lattice model before random walk relaxation is allowed, the intermolecular relationships in the lower temperature monoclinic polymorph of CBr4 (β-CBr4) are used, including the reported tetrahedral 161 distortions. In β-CBr4, there are four crystallographically unique molecules, each possessing unique first shell coordination environments in which the molecules are arranged in an approximately FCC pattern with nearest-neighbors along the real space 1 ⟨110⟩ directions. 2

Because β-CBr4 is centrosymmetric, there are an additional four unique, non-superimposable coordination shells giving eight total molecular coordination environments (first shells). These

β-CBr4 first shell constructs each contain thirteen molecules which have a preferential orientation given by the unique axis of the monoclinic crystal structure. Half of the molecules in β-CBr4 are preferentially aligned such that one of their 4̅ axes is coincident with bm. As all experimental and simulation evidence suggests that α-CBr4 is cubic, the preferential alignment of β-CBr4 must be removed in order to construct an isotropic α-CBr4 simulation cell from these four β-CBr4 first shell constructs.

The monoclinic bm is coincident with one of the -cubic ⟨110⟩ directions, thus to remove the preferential orientation from the β-CBr4 starting model, a set of first shell constructs, each oriented such that its monoclinic b-axis is aligned along one of the twelve [110] directions, was generated. These rotations give 8×12=96 unique first shell constructs. The specific rotational symmetry operations that will rotate the [110] into the other eleven cubic ⟨110⟩ directions is summarized in Table 5.5.

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Table 5.5. Symmetry operations to generate the twelve cubic ⟨110⟩ directions from a single direction. To obtain the new direction, perform the operation listed in the row labels then the operation in the column labels.

푬 푪ퟐ,풛 푪ퟐ,풚 푪ퟐ,풙 푬 110 1̅10 11̅0 11̅̅̅̅0 ퟏ ̅ ̅ ̅̅̅̅ 푪ퟑ,풙풚풛 011 011 011 011 ퟐ ̅ ̅ ̅ ̅ 푪ퟑ,풙풚풛 101 101 101 101

Rotating the eight unique first shells such that the monoclinic b-axis is pointed along all twelve cubic ⟨110⟩ directions results in 96 possible first shells relative to the global coordinate system. The initial approach to constructing a plastic crystalline lattice with these ninety-six first shells was to tessellate them in a space filling arrangement. Random walk simulations with first-shell tessellated arrangements did not relax into realistic structures, as evidenced by additional supercell Bragg peaks that appeared in the computed diffraction patterns which could only be indexed based on the tessellation motifs. That little relaxation occurred is likely because the first shells were too stable with respect to their surroundings so there is little driving force to disorder the system. However, the diffuse scattering in the calculated diffraction patterns was in better agreement with the experimentally observed diffuse than in either the initially random orientations or the random D2d orientations, likely because of the specific orientational correlations “built in” to the first shells. To maintain the intra-first shell packing while also building into the model a driving force to disorder the system a “first shell overwrite” method was designed to generate a starting structure. In this first shell overwrite method, an FCC lattice point and its surrounding twelve lattice points were randomly selected. If any molecules had previously been placed on these FCC lattice points, they were removed. A first shell was randomly selected from the ninety- six possible constructs and then placed in these now vacant thirteen FCC lattice positions. This was repeated until a first shell had been placed on every lattice point within the simulation cell

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once. Additional overwrite cycles through the lattice points showed no change in the calculated diffraction patterns. Calculated [001] and [111] diffraction patterns of the initial first shell overwrite construction are shown Figure 5.20a-b and Figure 5.21a-b for 100 simulations of a 103 unit cell periodic simulation cell. The diffuse scattering seen in these images is significantly more structured than for either of the previously shown initial lattices, but still lacking qualitative agreement with the experimentally observed diffraction. The PFs corresponding to the initial lattice construction, shown in Figure 5.20c and Figure 5.21c are significantly structured, though distinctly different from the experimentally observed [111] PF. Allowing this lattice to relax via a 1×256 random walk steps results in the diffraction patterns shown in in Figure 5.20d-e and Figure 5.21d-e. These projections and those calculated with the 1×256 random walk with randomly distributed D2d initial orientations both significantly resemble the experimental pattern. The ⟨220⟩ diffuse streaks are intense though a little too spherically symmetric and the ⟨331⟩ diffuse streaks reasonably replicate the experimentally observed binary distribution of diffuse intensity. The “dog teeth,” though slightly sharper than the D2d simulation, are not as sharp as what is observed experimentally. As a guide to the eye, the cartoon of the structured diffuse scattering shown in Figure 5.6e has been scaled and overlaid on the bottom half of the as-constructed lattice along the [111] projection to distinguish the features that are in reasonable agreement from those where significant modification is required. The overlay highlights what is most lacking: The sharp inner star of diffuse connecting the {220} to the {111}, good separation between the {220} and the {331} diffuse, sharper “dog teeth” and the structured diffuse hexagon corresponding to the blue hexagon.

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Figure 5.20. Diffraction patterns and Patterson functions computed along the ⟨100⟩ directions of the first shell monoclinic fcc lattice (see text for details). (a-c) Initial lattice construction with no relaxation. (d-f) 1 cycle of random walk relaxation. (g-i) 5 cycles of random walk -1 -1 relaxation. (a,d,g) Qmax=20 Å . (b,e,h) Qmax=9 Å . (c,f,i) 2D Patterson functions to rmax ≈ 13 Å. For each diffraction image, the left half of the image is the raw computed image and right half is the image with a 5×5 median filter mask to remove the Bragg peaks.

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Figure 5.21. Diffraction patterns and Patterson functions computed along the ⟨111⟩ directions of the first shell monoclinic fcc lattice (see text for details). (a-c) Initial lattice construction with no relaxation. (d-f) 1 cycle of random walk relaxation with a walk length of 256. (g-i) 5 -1 cycles of random walk relaxation with a walk length of 256. (a,d,g) Qmax=20 Å . (b,e,h) -1 Qmax=9 Å . (c,f,i) 2D Patterson functions to rmax ≈ 13 Å. For each diffraction image, the left half of the image is the raw computed image and right half is the image with a 5×5 median filter mask to remove the Bragg peaks.

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The PFs computed from these diffraction patterns are given in Figure 5.20f and Figure 5.21f. There is excellent agreement at short length scales (<8 Å) but the correlation length rapidly decays past the first coordination shell, suggesting that the system still lacks the experimentally observed medium-range correlations (10-30Å). An expanded view of the [111] PF is shown in Figure 5.22 and is sufficiently similar to the PF computed from the 1×256 random walk starting from a random distribution of D2d orientations, that little new insight is gained. The reader is directed to the text accompanying Figure 5.18.

If the random walk energy minimization is allowed to continue, the β-CBr4 orientational correlations begin to be lost. Shown in Figure 5.20g-h and Figure 5.21g-h are the computed diffraction patterns after allowing four more cycles of the random walk to be performed such that a random walk algorithm has been started from every lattice point 5 times and allowed to run for 256 steps. As additional random walk cycles are performed, the inner structured diffuse scattering becomes less structured and the diffraction pattern, as a whole, becomes extremely similar to that obtained after 5×256 walk from the lattices constructed with randomly oriented and randomly distributed D2d molecules. The Patterson functions calculated from these 5×256 random walk lattices, shown in Figure 5.20i and Figure 5.21i, reflect this and are again extremely similar to those obtained from the random orientations and randomly distributed D2d simulations.

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Figure 5.22. Large version of Figure 5.17f with labeled distances. Compare to Figure 5.8e and Figure 5.15.

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Figure 5.23. Comparing the (a) diffraction patterns and Patterson Functions (b) for the randomly distributed D2d orientations and monoclinic first shell constructs after 1×256 random walks.

It is notable how similar the [111] diffraction patterns are for the random walk relaxation of the randomly distributed D2d orientations and the randomly distributed monoclinic first shell constructs, given their significantly different starting configurations. The two 1 × 256 random walk diffraction images are directly compared in Figure 5.23a, with the random-D2d construct on the left and the monoclinic-First Shell simulations on the right. The similarity of these diffraction images suggests the random-D2d and monoclinic-First Shell constructs equally treat first shell molecular correlations; in noted contrast to the completely Random construct which considered only single molecule correlations. The most notable difference between the D2d and monoclinic-First Shell constructs is observed in the high Q broad diffuse. Whereas the overall character of this diffuse pattern is similar in both models, the high Q diffuse pattern from the monoclinic First Shell construct exhibits broad sheets of diffuse (seen as the hexagon in the [111] projection], as opposed to the more “blobby” pattern observed from the random-D2d construct. This difference, however, is unlikely due to different starting orientations. Rather it appears to be a signature of the distorted tetrahedra employed

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in the monoclinic-First Shell construct as opposed to the ideal tetrahedra employed in the random-D2d construct. The other discernible difference between these two diffraction patterns is a subtle difference in the {220} transverse diffuse scattering which appears to be slightly more rounded in the monoclinic-First Shell simulations than in the random-D2d simulation. The round vs. transverse diffuse pattern must imply differences in the respective translational and rotational molecular displacement about the ⟨200⟩ lattice vectors.38 The corresponding PFs shown in Figure 5.23b are also extremely similar, though many of the features of the monoclinic-First Shell simulation appear more rounded than their counterparts from the random-D2d simulation; a result of the more rounded {220} diffuse scattering in the first shell simulation. Because the {220} diffuse scattering is the most intense diffraction feature, it is the dominant component in the resultant PFs even though the difference between the higher Q diffraction appears to be more significant at first glance. 5.4.2.4. Second Shell Monoclinic As an attempt to induce longer-range correlations in the simulations of plastic crystalline carbon tetrabromide, the β-CBr4 crystal structure was once again exploited by creating the set of second shell constructs, i.e. constructs that include the above described monoclinic-First Shell and all molecules that are in direct contact with the first coordination shell. While the center of all molecules in the first coordination shell are the same distance from the molecular center (≈6.2Å), the molecules in the second coordination shell are located along the 1 ⟨211⟩ or ⟨110⟩ real-space directions, at to ≈10.7 and ≈12.4 Å, respectively. The 2 second coordination shell contains 42 molecules, which together with the 1 + 12 molecules of the first shell construct results in a construct containing 55 molecules. Based on the symmetry arguments outlined in the previous section, 8 unique second shell constructs are extracted from the monoclinic crystal structure. These eight unique second shell constructs were rotated with the symmetry operations outlined in Table 5.5 such that the monoclinic b-axis of each is aligned to one of the twelve cubic ⟨110⟩ directions to remove the anisotropy present in β-CBr4. Again this results in a total of 96 second shells constructs that can be used to construct a starting model lattice for simulation. Simulation cells were constructed by selecting a lattice site, removing any pre-existing molecules from the 55 FCC lattice sites corresponding to the second

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shell environment and placing a randomly selected second shell in the cleared lattice sites. This was repeated until a second shell construct had been centered on all lattice sites once. Additional cycles through the lattice points showed no change in the calculated diffraction patterns. Calculated [001] and [111] diffraction patterns of the initial second shell overwrite construction are shown Figure 5.24a-b and Figure 5.25a-b for 50 simulations of a 30×30×30 simulation cell. Similar to the diffraction patterns and corresponding PFs of the initial lattice from the monoclinic first shell simulations, the diffuse scattering and PFs for the initial construction of the monoclinic second shells is of limited utility with regard to understanding the diffuse scattering in the experimental α-CBr4 diffraction images. Allowing this lattice to relax with a 1×256 random walk produces the diffraction patterns shown in in Figure 5.24d-e and Figure 5.25d-e. The calculated [111] projection is qualitatively very similar to that observed experimentally with high Q diffuse hexagons that are almost perfectly regular, sharp “dog teeth”, and well-defined diffuse scattering connecting the {331} Bragg peaks. Most importantly, this simulation finally begins to reproduce the inner diffuse pattern, with sheets of diffuse scattering connecting the {111} and {220}, and {100} and {113} Bragg reflections, seen as a square or star in the [100] and [111] projection, respectively. However, the {220} diffuse features in Figure 5.25d-e are even more rounded than they were in the monoclinic-First Shell simulation. As seen with the monoclinic-First Shell simulation, the {220} transverse diffuse scattering is the most dominant feature with regards to the resultant Patterson Functions because it is the most intense feature, an effect that becomes quite clear in the PF computed from the median filtered Figure 5.25d. The PF computed from the median filtered [100] and [111] diffraction images are shown in Figure 5.24f and Figure 5.25f for which the real-space features are observed to be very rounded compared with the experimental [111] PF (Figure 5.8). Nevertheless, it appears that the goal of increasing the intermolecular orientational correlation length was successful using the monoclinic-Second Shell construct, as the intensity of the PF features suggests strong correlations to approximately 12 and 18 Å in the [111] and [100] projections.

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Similar to all other simulation construction methods, continued structural relaxation with the random walk algorithm results in the C-Br bonds becoming isotropically oriented as evidenced by the diffuse rings at high Q and the loss of discrete maxima in the PFs shown in Figure 5.24i and Figure 5.25i at the distances corresponding to the C-Br bond and intramolecular Br---Br contacts. Shown in Figure 5.26 is an expanded view of the PF corresponding to the [111] diffraction projection for 50 1×256 random walk simulations of a 30×30×30 supercell, where atomic correlations are visible to approximately 27.8 Å. Recall the description of the experimental [111] PF where a crude model was developed by counting the number of pair- pair correlations based on the molecular construct overlay and summing Gaussian’s applied to each pair as defined in Equation 4.10. That model appeared to have absences at 9.2, 15.3, 21.4 and 27.5 Å while the experimental PF had peaks at those locations. It was proposed in the text accompanying Figure 5.9 that the peaks in the experimental PF resulted from the overlap of two peaks on either side of the [110] real space direction indicated by the white arrow in Figure 5.9. This proposal is partially confirmed by the PF analysis shown in Figure 5.26 where the translucent white arrow passes through a valley between the two peaks at 8.82 Å and 20.7 Å, though there does appear to be some intensity near the models’ absences as indicated by the labels at 16.0 Å and 27.6 Å.

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Figure 5.24. Diffraction patterns and Patterson functions computed along the ⟨100⟩ directions of the first shell monoclinic fcc lattice (see text for details). (a-c) Initial lattice construction with no relaxation. (d-f) 1 cycle of random walk relaxation with a walk length of 256. (g-i) 5 -1 cycles of random walk relaxation with a walk length of 256. (a,d,g) Qmax=20 Å . (b,e,h) -1 Qmax=9 Å . (c,f,i) 2D Patterson functions to rmax ≈ 13 Å. For each diffraction image, the left half of the image is the raw computed image and right half is the image with a 5×5 median filter mask to remove the Bragg peaks.

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Figure 5.25. Diffraction patterns and Patterson functions computed along the ⟨111⟩ directions of the first shell monoclinic fcc lattice (see text for details). (a-c) Initial lattice construction with no relaxation. (d-f) 1 cycle of random walk relaxation with a walk length of 256. (g-i) 5 -1 cycles of random walk relaxation with a walk length of 256. (a,d,g) Qmax=20 Å . (b,e,h) -1 Qmax=9 Å . (c,f,i) 2D Patterson functions to rmax ≈ 13 Å. For each diffraction image, the left half of the image is the raw computed image and right half is the image with a 5×5 median filter mask to remove the Bragg peaks.

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Figure 5.26. Expanded view of Figure 5.25f with labeled distances.

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5.4.2.5. Analyzing the Simulated Structures Simulations are a powerful tool for developing an understanding of into disordered structures. However, they can be equally misleading. Many approaches are reported in the literature to analyze the results of simulated crystal structures for which reasonable agreement with some measurable quantity provides a point for evaluation of the validity of the model. One popular technique is to construct a “condensed view” of the simulation supercell, for which the atomic coordinates for all atoms in the simulation box are collapsed and plotted in a single crystalline Bravais cell. Shown in Figure 5.27a is a reprinted figure of such a condensed view of a 12×12×12 simulation supercell from a RMC fit to powder neutron diffraction data 146 of α-CBr4. The authors used this image to conclude an isotropic distribution of molecular orientations is observed for α-CBr4. By way of comparison, similar condensed views from the initial build and single and five random walk monoclinic-Second Shell simulations, respectively, are presented Figure 5.27b-d. From such a presentation of the results, it is not possible to elucidate the structural variation that gives rise to their distinctly different 2D diffraction images and Patterson functions. Thus any interpretation of the molecular orientation from such an evaluation of simulation data is not appropriate. It is somewhat more useful to present the final simulation cell in its entirety, as shown in Figure 5.28 for a single 1×256 monoclinic-Second Shell simulation. Though still difficult to obtain concise descriptions of molecular orientation and position, one can observe the orientational disorder, or presume to see a majority of C-Br contacts oriented along [110] directions, in the view shown in Figure 4.28b. Alternatively, the amount of displacive disorder in the system is observed by presentation of the position of the carbon molecular centers as shown in Figure 4.28c. While a more accurate representation of the simulation it is still challenging to elucidate the structural principles from such presentation, thus precluding understanding of which features give rise to the observed structured diffuse scattering.

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Figure 5.28. One 103 simulation supercell for a single random walk monoclinic-Second Shell, shown as a stick drawing (a) full simulation cell and (b) expanded view to emphasize the orientational disorder and (c) full simulation cell showing only the carbon centers.

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Another popular technique for analysis of simulations describes a mutual orientation probability177, 185 where the resultant structure is described in terms of the number of ligands between parallel planes defined by the molecular centers. This classification scheme results in description of tetrahedral structures in terms of the number of 1-1, 1-2, 1-3, 2-2, 2-3 and 3- 3 intermolecular pairs based on the number of ligand-ligand contacts between neighboring molecules. While such structural analysis has been shown to be an effective tool for quantifying the difference in intermolecular orientational correlations between the liquid and plastic crystalline phases of tetrahedral molecules, it does not capture finer detail of the simulations, such as the extent to which bond vectors of different molecules are aligned with common intermolecular directions. The experimentally observed, highly structured transverse diffuse scattering observed for -CBr4, gives strong indication that the C-Br bond vectors are preferentially aligned with respect to specific lattice directions, notably the ⟨220⟩ directions. This is confirmed by the above simulations which clearly indicate the ordered diffuse is lost with random molecular organization. Thus it seemed important to probe the extent to which C-Br bond vectors are aligned with specific lattice directions in the simulation cells.

Based on the well-established FCC molecular packing described for -CBr4, evaluation along three intermolecular directions is particularly instructive. The ⟨100⟩ real-space directions are the cubic unit cell axes. The intermolecular nearest neighbor contacts are along the 1 ⟨110⟩ real-space directions. The ⟨111⟩ real-space directions correspond to the FCC 2 molecular closest packing. The relative alignment of all C-Br bond vectors with respect to each of these sets of lattice directions was defined by finding the smallest angle between a member of the set and a given C-Br bond vector. This defines the most aligned C-Br bond vector. Subsequently the angle between each of the other three C-Br bond vectors and the closest member of the given set of lattice directions was computed to identify and sort the bond vectors such that the 2nd, 3rd and least aligned bond vectors per molecule were identified. To visualize this, consider, for example, the alignment of a C-Br bond vector with respect to the 푣⃑⃑⃑⃑1⃗ = [01̅0] real-space direction, consider projecting the C-Br bond onto a plane orthogonal to the [01̅0] vector by computing the dot product of the C-Br bond vector with two

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vectors 푣⃑⃑⃑⃑2⃗ and 푣⃑⃑⃑⃑3⃗ that are orthogonal to each other and to the [01̅0] vector, such that 푣⃑⃑⃑⃑2⃗ ×

푣⃑⃑⃑⃑3⃗ = 푣⃑⃑⃑⃑1⃗. One possible choice to satisfy this criterion would be 푣⃑⃑⃑⃑2⃗ = [001] and 푣⃑⃑⃑⃑3⃗ = [1̅00]. Furthermore, by choosing three vectors that produce a right-handed coordinate system* and making sure that 푣⃑⃑⃑⃑2⃗ and 푣⃑⃑⃑⃑3⃗ are the unit vectors, one obtains the relative real-space projection of the evaluated C-Br bond vector with respect to which ever member of the set of directions (in this case, the ⟨100⟩ set) is actually best aligned. Collected in Table 5.6 are one set of orthogonal vectors for the ⟨100⟩ directions that can be used to compute the (x,y) location of the C-Br bond projected onto the orthogonal plane according to Equations 5.12 and 5.13. Similar sets of vectors can be determined for the ⟨111⟩ and ⟨110⟩ real-space directions.

푥 = 푟⃗퐶퐵푟 ∙ 푣⃗2 5.12

푦 = 푟⃗퐶퐵푟 ∙ 푣⃗3 5.13 Visualization of the complete set of projections of the C-Br bonds relative to the ⟨100⟩, ⟨111⟩ and ⟨110⟩ real-space directions was performed by computing a 2D histogram of the individual C-Br bond projections based on the (x,y) coordinates as defined by Equations 5.12 and 5.13. Such 2D histograms are given in Figure 5.29. These histograms can be further evaluated with respect to the 1st to 4th most aligned C-Br bond vector.

Table 5.6. Relative axes to compute the C-Br bond vector projection.

풗⃑⃑⃑⃑ퟏ⃗ = 풗⃑⃑⃑⃑ퟐ⃗ × 풗⃑⃑⃑⃑ퟑ⃗ [100] = [010] × [001] [010] = [001] × [100] [001] = [100] × [010] [1̅00] = [01̅0] × [001̅] [01̅0] = [001̅] × [1̅00] [001̅] = [1̅00] × [01̅0]

* Left handed is ok too, just be consistent.

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The different shapes of the histograms can be understood by considering the surface of a sphere as a Voronoi diagram where the points are equal to the intersection of all vectors of the projection family with the sphere surface. The intersection of the horizontal grid line with the edge of each figure panel represents an alignment of +/- 45° with respect to the projection (z)-axis. The intersection of the vertical line with the edge of each figure panel represents an alignment of +/- 54.7° with respect to the projection (z)-axis. The size of each projection family is directly proportional to the surface area of the sphere divided by the number of vectors in each family, which is why the [001] projection is larger than the [111] projection which is larger than the [110] projection. The shape of the [111] projection should have three-fold

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symmetry but the apparent six-fold symmetry results from an overlay of two projections with three-fold symmetry due to an imperfect projection determination algorithm. From each of the projections shown in Figure 4.29 it is evident that a majority of C-Br bond vectors are aligned in the vicinity of a ⟨110⟩ direction. As shown most clearly in panel f, the regions of maximum intensity are all located within approximately 20° of the ⟨110⟩ real- space directions. The observed ±20° split of C-Br bond vector orientations is due to the difference between the molecular 109.47° tetrahedral bond angles and the 90° angle between ⟨110⟩ directions. If one C-Br bond is directly aligned along a ⟨110⟩ direction, the closest that any other C-Br bond can be aligned with any other ⟨110⟩ is 109.47°-90°=19.47°. If all bonds are equally aligned with their respective closest ⟨110⟩ directions, then all C-Br bond vectors will be 19.47/2≈9.7° away from the nearest ⟨110⟩. Shown in Figure 5.29g is the projection of the C-Br bond of each molecule most aligned with a ⟨110⟩ from each molecule where the heaviest concentration forms a dark gray ring that is ≈9° in diameter. The next most aligned C-Br bond from each molecule is shown in Figure 5.29h. The black spots in Figure 5.29g and Figure 5.29h are in similar places resulting from molecules that are equivalently aligned such that all C-Br vectors are closely aligned with four different ⟨110⟩ directions. There is also a hole in the middle of the histogram as a result of the 1st C-Br bond vector being perfectly aligned with a ⟨110⟩, since two C-Br bonds cannot be simultaneously within less than ≈10° of two different [110] real-space directions. This results in a corresponding dark gray football- shaped shaped region of 2nd most aligned C-Br vectors about 20 off the [110] projection axis. The hole that appears in the third most aligned C-Br bond vectors from each molecule in Figure 5.29i corresponds to the football-shaped feature of Figure 5.29h. The dark black spots in Figure 5.29i are 15-19° from the center of the image, suggesting that they are on molecules where one of the C-Br bonds is aligned more closely to one of the [110] directions. The set of least aligned C-Br bonds with respect to any [110] direction is shown in Figure 5.29j. There is a substantial hole in the middle of the histogram. However, the persistence of some of intense peaks (black spots) indicate that there is a significant population of tetrahedra that exhibit the equivalent molecular alignment where the C-Br bonds are all within ≈20° of different [110] directions.

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As the second shell lattice is allowed to relax by an additional 4 random walk cycles for the 5×256 monoclinic-Second Shell simulation, the C-Br bond alignments become significantly more isotropic, as seen in the corresponding histograms in Figure 5.30. This significantly more isotropic molecular orientation is reflected in the nearly radially symmetric higher order diffuse in the diffraction images presented for the 5×256 random walk relaxations of the random-D2d, the monoclinic-First shell and the monoclinic-Second Shell simulations.

5.5. Discussion

Three primary kinds of scattering are observed in α-CBr4: Bragg, high-Q semi- structured diffuse and low-Q highly-structured diffuse. The origins of the Bragg scattering is

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understood to arise from the regularly repeating molecular FCC packing. From the simulation results, the two types of diffuse scattering can also be understood from the first set of simulations presented. When the molecular orientations are completely random and uncorrelated, all diffuse scattering appears as isotropic rings. As intermolecular correlation is introduced through the random walk algorithm, the low-Q, structured diffuse begins to appear.

At higher Q the diffuse scattering remains as isotropic rings. Similarly, for the random-D2d construct, the starting random distribution of the six molecular orientations exhibits no transversely polarized diffuse scattering because there is no intermolecular correlation. As the random walk algorithm is activated and nearest neighbor intermolecular correlations are created, transverse diffuse scattering is also observed; at both low- and high-Q. As noted in the results above, the random walk algorithm does not converge to the plastic crystal structure. Instead with subsequent random walk cycles the structure becomes more isotropic. Nevertheless, this more gradual randomization process demonstrates that the high-Q structured diffuse scattering is lost prior to the low-Q structured diffuse scattering.

5.5.1. High-Q diffuse scattering. The specific shape of the high-Q broad diffuse scattering observed in the diffraction images (experimental in Figure 4.6a and 4.8a, and simulation Figures 4.16/17d, 4.20/21d and 4.24/25d) is a result of broad diffuse sheets transverse to the [111] direction with slightly less intense diffuse observed transverse to the [100] directions. These sheets of diffuse essentially comprise polyhedra of diffuse about the reciprocal space origin, with vertices at (2h h 0) lattice points. These diffuse bands exhibit intensity between h=odd to h=even, significantly starting with h = 3 to 4; intersecting the [111] vectors between the (333) and (444) and the [100] vectors between the (700) and (800). The width of these broad diffuse sheets is essentially the same width as the features of low-Q diffuse described above as dogteeth. These result in diffraction images with apparent octagonal and hexagonal shapes for the [001] projection and [111] projections, respectively. In real space the [111] planes correspond to the direction of molecular FCC packing as seen in Figure 5.31. Interestingly too, the Br atoms from the six

D2d molecular orientations are oriented in planes orthogonal to the [111] directions shown in

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Figure 5.31a. Disorder resulting from the random occupancy of the six possible molecular orientations should give rise to significant diffuse scattering transverse to the reciprocal space ⟨111⟩ direction. The bromine positions are not quite as ordered in planes orthogonal to the [100], but as shown in Figure 5.31c, they still exhibit sufficient order along the [100] to exhibit scattering, but sufficient disorder within the planes for the scattering to be realized as diffuse sheets. In the simulations the high-Q diffuse become significantly more isotropic as the random walk algorithm is allowed to proceed which, based on the analysis of the C-Br bond alignment with respect to intermolecular directions (Figure 5.29 and Figure 5.30), occurs as the molecular orientations become significantly more isotropic across the ensemble. It is therefore reasonable to conclude that the high-Q diffuse scattering results primarily from the global average of bromine scattering, structurally restricted by the six possible D2d molecular orientations.

Figure 5.31. Average D2d molecular constructs oriented to emphasize the expected diffuse scattering along the (left) [211], (mid) [111] and (right) [100].

5.5.2. Tetrahedral distortion in α-CBr4 That the organization of the bromine atoms substantially determines the nature of the high Q-diffuse is further supported by consideration of the effect of distortion of the CBr4 tetrahedra. In the random-D2d simulations, the tetrahedra were perfectly regular with bond lengths of 1.91 Å and bond angles of 109.47°. By contrast, monoclinic-First and Second-Shell

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constructs were generated with tetrahedra distorted equivalently to those observed in one of 161 the single crystal structure of the monoclinic β-CBr4. Comparison of the high-Q diffuse scattering in the calculated diffraction patterns from these three models after relaxation by a

1×256 random walk demonstrates significantly greater “blobby” character for the random-D2d simulation (Figures 4.16d and 4.17d) with regular CBr4 tetrahedra than is observed for the monoclinic-First and Second-Shell simulations (Figures 4.20d and 4.21d, and Figures 4.25d and 4.25d, respectively) with distorted tetrahedra. A direct comparison of these is shown in

Figure 5.23a for the random-D2d and monoclinic-First Shell simulations after relaxation with 1×256 random walks, which demonstrate remarkably similarity except for the homogeneity of the high-Q diffuse patterns. While the resolution of the average structure diffraction and PF techniques is not sufficient to precisely define the extent of distortion of the CBr4 tetrahedra in the plastic -phase, based on the above simulations, the experimental observation of the reasonably homogeneous high-Q diffraction (Figure 6a and 8a) suggests that the CBr4 molecules are similarly distorted in both the - and β-CBr4.

5.5.3. Low-Q Diffuse Scattering It is interesting to consider the contribution of each specific diffraction feature to the Patterson function. This can be visualized with the aid of FTL-SE, free software that allows for rudimentary editing and visualization of 2D Fourier transforms. Shown in Figure 5.32 is a series of low resolution diffraction images crudely manipulated with the mouse to remove specific diffraction features and the resultant Patterson Functions calculated from the manipulated images. The images shown in Figure 5.32 have been cropped to show the structured diffuse scattering at low Q, but for computing the Patterson functions, the full images to Q=20Å-1 were used. The starting image shown in Figure 5.32a is the experimentally observed twinned [1̅11̅̅̅], [1̅11̅] projection manipulated with GIMP (Figure 5.8) and the Patterson function shown in Figure 5.32aʹ corresponds to the Patterson function shown in Figure 5.8 and Figure 5.9. Removal of the transversely polarized ⟨220⟩ streaks in Figure 5.32b has an extremely significant impact on the real space correlations, showing that most of the real-space

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correlation has vanished. Careful inspection of Figure 5.32bʹ relative to the dashed overlay suggests that the first, second and third nearest neighbor correlations have all but vanished with only a few intermolecular bromine correlations remaining at higher distances. At shorter distances, the CBr correlation, the intramolecular Br-Br and intermolecular Br-Br correlations remain. The correlation length seems to have been reduced from 30 Å to ~12 Å. That the ⟨220⟩ streaks have such a strong impact on the resultant Patterson function explains the poor match between the experimental [111] Patterson function and that calculated from the simulated second shell monoclinic lattice because those ⟨220⟩ streaks in the simulated lattice poorly reproduce the experimentally observed ⟨220⟩ streaks, being significantly rounder. Removal of the transversely polarized streaks connecting the {331} Bragg reflections in Figure 5.32c has a substantially smaller impact on the real space correlations than removal of the ⟨220⟩. Inspection of Figure 5.32cʹ suggests that the second nearest neighbor correlation, along the cubic ⟨221⟩ directions and, , the C-Br pair correlation have been removed. The correlation length has been reduced from 30 Å to ~15-18 Å. Moving to higher Q, the “dog teeth” have been removed in Figure 5.32d which appears to be mostly related to the short inter/intramolecular Br-Br and C-Br bond distances in the corresponding Patterson Function in Figure 5.32dʹ. The C-Br contacts are significantly more visible after the removal of the dog teeth than in the original image. There appears to be little impact on the intermolecular correlations but the correlation length has been reduced to ~15Å. Removing the structured hexagon that forms the outer border of the “dog teeth” as shown in Figure 5.32e and Figure 5.32eʹ seems to strongly impact the C-Br pair correlation and imposes a degree of radial symmetry to the Patterson function.

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Other than the C-Br pair-correlation, it is rather unclear which pair correlations are affected by removal of the high-Q broad diffuse scattering, as shown in Figure 5.32f and Figure 5.32fʹ. Perhaps the first intermolecular Br-Br distance is slightly impacted, but that also could be a result of the crude image manipulation. The correlation length also appears to be unaffected. However, removal of the structured diffuse scattering has an extremely significant impact, as shown in Figure 5.32g and Figure 5.32gʹ. Careful inspection of the resultant Patterson function does suggest that the intramolecular Br-Br pair correlation is still present, but all other pair correlations have vanished. This further supports that the high-Q diffuse features are related to the molecular orientations. Finally, removal of all scattering with the exception of the transversely polarized ⟨220⟩ and ⟨331⟩ streaks is shown in Figure 5.32h. Inspection of the Patterson function in Figure 5.32hʹ demonstrates that many of the features are still present, with the exception of the bonded C-Br pair correlation. The longer distance pair correlations appear to be weaker and the correlation length is smaller than the original image but the similarity between the two images is remarkable given the significant amount of diffraction information that has been removed before the Patterson function is calculated.

5.5.4. Twinning in α-CBr4 It is interesting that the macroscopic twin operation is the [110] since the molecular closest packing directions are in the ⟨111⟩ real-space directions. Prior work has indicated that carbon bromine bonds are preferentially aligned along the intermolecular 1 ⟨110⟩ real-space 2 directions, a condition also observed in this work. In β-CBr4, when such orientational preference is combined with the unique [110] aligned along the monoclinic b-axis the bromines are arranged in pseudo-close packed layers with varying degrees of “close-packed- ness” along the distinct ⟨110⟩ directions. In β-CBr4 the orientational preference is based on the complex relationship between molecular closest-packing and bromine pseudo-closest-packing.

However, in α-CBr4 there is no orientational preference for a single [110] direction, as evidenced by every experimental measurement supporting cubic symmetry.

However, the above simulations suggest that α-CBr4 is effectively described by extracting the second shell coordination environments from the β-CBr4 crystal structure and

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rotating them such that the preferred [110] orientation along the monoclinic b-axis of each second shell construct points in all cubic ⟨110⟩ directions. It therefore seems reasonable to describe α-CBr4 as microscopically twinned groups of β-CBr4 with correlation lengths of >30Å by C4 operations around the cubic ⟨100⟩ directions. The macroscopic twinned crystal, as evidenced by the twinned Bragg diffraction and structured diffuse scattering in the experimental images is twinned about a cubic [110] by a

C4 operation. Under cubic symmetry, this is not a symmetry-preserving twin operation and would result in a significant high-energy stacking fault for the packing of spheres. However, the cubic ⟨110⟩ directions in the crystalline phases of CBr4 are directions of bromine pseudo- closest packing and it is likely that the macroscopic twin started as little more than an orientational correlation defect from a pseudo-close packing mismatch.

Figure 5.33. Remarkable similarity between the “streaky” Bragg scattering in β-CBr4 and the diffuse maxima in α-CBr4.

The similarity between the diffuse scattering seen in the 2D diffraction of polycrystalline β-CBr4 and α-CBr4 further supports the strong relationship between β-CBr4 and

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α-CBr4. Shown in Figure 5.33a is a 2D powder diffraction image of β-CBr4 collected at 321

K during a temperature ramp and of α-CBr4 collected at 327 K approximately one minute later -1 (5 K min ramp rate). The intense, streaky Bragg diffraction in β-CBr4 (black arrows) turns into the intense, streaky Bragg diffraction in α-CBr4. The higher Q streaky Bragg diffraction in β-CBr4 (white arrows) turns into the broad diffuse features with maxima in α-CBr4 at the same location as in β-CBr4.

5.6. Conclusion The best agreement between simulated and experimental diffraction patterns results from using the molecular arrangements from the low-temperature non-plastic polymorph, β-

CBr4, to construct a plastic crystalline lattice and allow it to slightly relax to alleviate the physically unreasonable contacts generated by the construction algorithm as seen by the comparison between Figure 5.6a and Figure 5.25e. The corresponding experimental and simulated Patterson Functions shown in Figure 5.8 and Figure 5.25f suggests that the orientational correlation length is longer in the real plastic crystalline system than in the simulated plastic crystalline system. It is possible that a longer correlation length could be obtained by using third, fourth or fifth shell monoclinic constructs, though based on the results of using first and second shell constructs, the {220} transverse diffuse scattering becomes increasingly rounded as the number of shells increases. Perhaps the poor replication of the {220} transverse diffuse streaks while all other diffraction features are improved is indicative that our static modeling is incomplete and requires dynamic motion to induce streaking along the [111] directions towards the {311} Bragg peaks. Or perhaps this is indicative that the

Lennard-Jones model is truly not sufficient for modeling the disorder in α-CBr4 as has been suggested previously,144, 177 though the results of the simulations presented in this chapter do not agree with the conclusion of a previous MD study that claimed the LJ model was unable to produce orientational disorder.152-153 The unique properties of the plastic crystalline phase are often thought to arise from dynamic rotational disorder. While dynamical rotational disorder certainly exists in some molecular systems, the plastic crystalline phase of α-CBr4 is well described as a static

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disordered cousin of its low temperature polymorph, β-CBr4. With the evidence from the experimental diffraction images and simulations, the molecules in α-CBr4 have their C2 axes aligned with one of the cubic [100] directions such that the C-Br bond vectors are nearly aligned with the intermolecular nearest neighbor ⟨110⟩ directions. Dynamic rotational motion, when it occurs, must either preserve the symmetry, i.e. C3 or C2 rotations, or rotate molecules from one of the six possible global orientations to another via C4 operations around the C2 axes. However, molecular rotations resulting in a new orientation would be extremely energetically unfavorable because a substantial number of molecules within the 30 Å correlation length would need to be moved to allow rotational freedom.

Plastic crystalline α-CBr4 is simply a disordered cousin to its lower temperature, non- plastic β-CBr4 polymorph. The orientational correlations present in β-CBr4 are also present in

α-CBr4, just over a much shorter correlation length of at least 30 Å, or 5-6 coordination shells, as shown by the 2D Patterson functions. The relationship between α- and β-CBr4 can be described as a series of microscopic C4 twins about the cubic ⟨100⟩ real-space directions where the molecular FCC packing is preserved. The macroscopic twins are related by a C4 about the cubic [110] direction where the molecular FCC packing is broken, but it is extremely likely that the macroscopic twin preserves the bromine pseudo-closest packing. As suggested by the simulations and diffuse scattering knockouts (Figure 5.32) the molecular orientations, on average, are aligned such that a molecular C2 is, or nearly so, coincident with a cubic ⟨110⟩ direction and are therefore not isotropic over a time or position average as suggested by previous manuscripts.

Finally, the C4 twin law indicates that the unique [110] in β-CBr4 becomes 12-fold degenerate in α-CBr4 due to the six unique ⟨110⟩ directions plus the C2 symmetry operations about each. If this ordering is described as going from W=1 in β-CBr4 to W=12 in α-CBr4 that would correspond to a change in entropy of R ln(12/1)=20.7 J mol-1 K-1. The reported entropy -1 -1 137-139 between α- and β-CBr4 is between 20.3 and 20.66 J mol K , indicating that the entropic difference between these two solid polymorphs can be described entirely from the unique [110] direction in β-CBr4 becoming 12-fold degenerate.

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154. Descamps, M., Dynamics of Colliding Molecules - A Theory of Coherent Neutron- Scattering for the Plastic Phase of CBr4. J Phys-Paris 1984, 45 (3), 587-596.

155. Hohlwein, D., NUMERICAL STRUCTURE FACTOR CALCULATION OF ORIENTATIONALLY DISORDERED MOLECULES - ANISOTROPIC LIBRATIONS AND REORIENTATION IN THE PLASTIC PHASE OF CBR4. Z Kristallogr 1984, 169 (1-4), 237-247.

156. Hohlwein, D., ANISOTROPIC ROTARY OSCILLATIONS OF CBR4 IN THE PLASTIC PHASE. Z Kristallogr 1982, 159 (1-4), 61-61.

157. Alderdice, D. S.; Iredale, T., NUCLEAR QUADRUPOLE SPECTRA OF IODINE IODIC ACID AND CARBON TETRABROMIDE. T Faraday Soc 1966, 62 (522P), 1370-&.

158. Pettitt, B. A.; Wasylishen, R. E., C-13 NMR-STUDY OF CARBON TETRABROMIDE. Chem Phys Lett 1979, 63 (3), 539-542.

159. Derollez, P.; Lefebvre, J.; Descamps, M.; Press, W.; Fontaine, H., Structure of Succinonitrile in Its Plastic Phase. J Phys-Condens Mat 1990, 2 (33), 6893-6903.

160. Post, B., The Cubic Form of Carbon Tetrachloride. Acta Crystallogr 1959, 12 (4), 349- 349.

161. More, M.; Baert, F.; Lefebvre, J., Solid-State Phase Transition in Carbon Tetrabromide CBr4. I. The Crystal Stucture of Phase II at Room Temperature. Acta Crystallogr B 1977, B33, 3862.

162. Tse, W. S.; Liang, N. T.; Lin, W. W.; Chiang, P. Y., A RAMAN-STUDY OF THE PHASE-TRANSITIONS OF SOLID CARBON TETRABROMIDE. Chem Phys Lett 1985, 119 (1), 67-70.

163. Koga, Y., Growth-Kinetics of Polymorphic Transitions in some Molecular Solids. J Cryst Growth 1984, 66 (1), 35-44.

164. Xiao, R. F.; Rosenberger, F., CBr4 Vapor Growth Morphologies near the Polymorphic Transition Point. 2. Crystals with Large-Angle Grain-Boundaries. J Cryst Growth 1991, 114 (4), 549-560.

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165. Powers, R.; Rudman, R., Polymorphism of the Crystalline Methylchloromethane Compounds. 7. Structures of the Ordered Phases of the Carbon Tetrahalides. J Chem Phys 1980, 72 (3), 1629-1634.

166. Debeau, M.; Pick, R. M., DIFFUSED RAMAN-SPECTRA OF MONOCLINIC CBR4. J Raman Spectrosc 1980, 9 (3), 157-161.

167. Creighton, J. A.; Sinclar, T. J., Raman spectra of crystalline group IV tetrabromides. Spectrochimica Acta Part A: Molecular Spectroscopy 1979, 35 (2), 137-140.

168. Hird, K. M.; Binbrek, O. S.; Anderson, A.; Torrie, B. H., Raman and Far-Infrared Spectra of Solid Carbon Tetrabromide. J Raman Spectrosc 1988, 19 (2), 79-83.

169. Powell, B. M.; Dolling, G., ORIENTATIONAL DISORDER IN THE PLASTIC PHASES OF SF6 AND CBR4. Mol Cryst Liq Cryst 1979, 52 (1-4), 27-34.

170. Hammersley, A. P., FIT2D: An Introduction and Overview". ESRF Internal Report 1997, ESRF97HA02T.

171. Coulon, G.; descamps, M., J. Physique Coll 1979, 3, 132.

172. Descamps, M.; Coulon, G., Intermolecular Correlations in Plastic Crystals - One More Step in the Calculation of the Static Susceptibility of Chi(Q). J Phys C 1981, 14 (17), 2297-2304.

173. Welberry, T. R.; Goossens, D. J., The interpretation and analysis of diffuse scattering using Monte Carlo simulation methods. Acta Crystallogr A 2008, 64, 23-32.

174. Patterson, A. L., A Fourier Series Method for the Determination of the Components of Interatomic Distances in Crystals. Phys Rev 1934, 46 (5), 372-376.

175. Patterson, A. L., A direct method for the determination of the components of interatomic distances in crystals. Z Kristallogr 1935, 90 (6), 517-542.

176. Hendrickson, W., Evolution of diffraction methods for solving crystal structures. Acta Crystallogr A 2013, 69 (1), 51-59.

177. Rey, R., Orientational Order and Rotational Relaxation in the Plastic Crystal Phase of Tetrahedral Molecules†. The Journal of Physical Chemistry B 2007, 112 (2), 344-357.

178. Gavezzotti, A.; Simonetta, M., Molecular-Rearrangements in Organic Crystals .1. Potential-Energy Calculations for Some Cases of Reorientational Disorder. Acta Crystallogr A 1975, 31 (Sep1), 645-654.

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179. Williams, D. E., Nonbonded Potential Parameters Derived from Crystalline Aromatic Hydrocarbons. J Chem Phys 1966, 45 (10), 3770-&.

180. CRC handbook of chemistry and physics. Chapman and Hall/CRCnetBASE: Boca Raton, FL, 1999.

181. Cromer, D. T.; Mann, J. B., X-Ray Scattering Factors Computed from Numerical Hartree-Fock Wave Functions. Acta Crystall a-Crys 1968, A 24, 321-&.

182. Henke, B. L.; Gullikson, E. M.; Davis, J. C., X-Ray Interactions - Photoabsorption, Scattering, Transmission, and Reflection at E=50-30,000 Ev, Z=1-92. Atom. Data Nucl. Data Tables 1993, 54 (2), 181-342.

183. Gullikson, E. The Atomic Scattering Factor Files. http://henke.lbl.gov/optical_constants/sf/sf.tar.gz (accessed 4/20/2011).

184. Humphrey, W.; Dalke, A.; Schulten, K., VMD: Visual molecular dynamics. J. Mol. Graph. 1996, 14 (1), 33-38.

185. Rey, R., Quantitative characterization of orientational order in liquid carbon tetrachloride. J Chem Phys 2007, 126 (16), 7.

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APPENDICES

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Appendix A. Differential Scanning Calorimetry Measurements

2.0x10-3 145°C 150°C 155°C 160°C 162°C

-3

1.5x10 ) 1.0x10-4

-3 1.0x10 -5

Heat Flow (W/g) Heat Flow

5.0x10

5.0x10-4 0.0 10 20 30 40 50 Heat Flow (W/g) Flow Heat Time (min)

0.0

exo up

0 5 10 15 Time (min) Figure A.1. Heat flow curves resulting from a quench of a 12.2 mg CZX-1 sample to various isotherms. The initial peak is an artifact of the instrumental quench response and was subtracted out for analysis. Exotherm is the positive y-direction.

0.6 150°C 155°C 157°C 159°C 160°C 0.6 0.6

0.4 0.4

0.4 0.2 0.2

Heat Flow (W/g) Heat Flow

Heat Flow (W/g) Heat Flow 0.0 0.0

-0.2 -0.2

Heat Flow (W/g) Flow Heat 0.2

0 3 6 9 0 5 10 15

Time (min) Time (min)

exo up 0.0 0 25 50 75 100 125 Time (min) Figure A.2. Heat flow curves resulting from a quench of a 26.5 mg CZX-1 sample to various isotherms. The initial peak is an artifact of the instrumental quench response and was subtracted out for analysis. Exotherm is the positive y-direction.

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Appendix B. 2D diffraction images of crystalline CZX-1.

Figure B.1. Final frames from the X-1 XRD experiments (BNL 0.5 mm capillaries)

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Figure B.2. Final frames from the X-2 XRD experiments (BNL 0.7 mm capillaries)

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Figure B.3. Final frames for the X-3 XRD experiments (ANL 0.5 mm capillaries)

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Figure B.4. Final frames for the X-4 XRD experiments (ANL 0.7 mm capillaries)

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Appendix C. Numerical crystal growth simulation: CrystalSim C.1. Introduction CrystalSim is a stand-alone piece of software written in Java to simulate the geometric aspects of crystal growth in the condensed phase according to the fundamental assumptions of the Kolmogorov-Johnson-Mehl-Avrami (KJMA) condensed phase reaction model,12-16 given in Equation C.1.

푛 훼(푡) = 1 − exp{−[푘 ∗ (푡 − 푡0)] } C.1 The KJMA model is commonly used to fit condensed phase reactions even though 18-22 many have recognized that the KJMA parameters (k, t0 and n) are empirical, significantly affected by experimental factors,23-26 and do not necessarily reflect the true reaction mechanism.27-30 Additionally, during our analysis of isothermal crystallization kinetics we observed a strong correlation between the sample size and the KJMA rate constant which supported that the KJMA parameters might be empirical. For these reasons, we needed to understand the relationship between the KJMA parameters and physically meaningful, material-specific parameters before we used this model to understand the isothermal crystallization kinetics in our model system, CZX-1. To specifically evaluate the relationship between the sample volume and the KJMA rate constant, the simulations were designed to strictly obey the fundamental assumptions of the KJMA model and then compared the resultant fits of the simulated transformations to Equation C.1 with the known simulation parameters. The KJMA assumptions are that (1) Nucleation occurs randomly and homogeneously throughout the sample, (2) the rate of crystal growth is independent from the extent of the transformation and (3) the rate of growth is independent of spatial location. Additionally, simplifying assumptions were made that crystal growth is phase- boundary controlled and thus requires no appreciable mass transport, in agreement with our experimental system, CZX-1.36, 85 The adjustable simulation parameters are:  the size and shape of the sample volume,  the shape and axial growth rates of the individual crystallites,  the number of initial “seed” crystallites and the nucleation rate.

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All parameters, including those that are more “administrative” and less related to the question of the relationship between sample volume and KJMA rate constant are described in §C.5. C.2. Software and Hardware Environment Execution Requirements.  CrystalSim requires the Java Runtime Environment (JRE) version 1.7 which takes 30-40 MB of disk space and is freely available from Oracle.*  The CrystalSim.jar runnable file is a 2.5 MB file.  All other files in the github repository are less than 10 MB.  Multi-threaded simulations are available in the ‘src\archivedUI’ folder. These multi- threaded simulations require that the source code be edited and recompiled for changes to the simulation parameters, but the total simulation time is reduced by a factor of the number of total processors used versus the single-threaded GUI implementation (i.e. total_time = single_threaded_time / num_processors). These files in the “archived_UI” folder are not currently supported and have some compile errors throughout. C.3. Program specifications CrystalSim is expected to run similarly on all platforms which support the Java Runtime Environment (JRE), though the look and feel of the GUI may vary between platforms based on the operating system default behaviors. This software was written in Java 1.7 (JDK 1.7.0_25) with the aid of the Eclipse IDE (Version: Kepler Service Release 1. Build id: 20130919-0819). All relevant simulation parameters are accessible from the GUI. The output data is automatically fit to the KJMA model (Equation C.1). C.4. Program Documentation and Availability The program source code is available free of charge at https://github.com/ericdill/CrystalSim. Documentation, is available in the “doc” folder at the aforementioned location, including a readme demonstrating complete program usage and a class diagram, demonstrating object relationships to aid you in further program development.

* http://www.oracle.com/technetwork/java/javase/downloads/java-se-jre-7-download-432155.html

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Additionally, a generic programming library is required for source code manipulation: https://github.com/ericdill/GlobalPackages. Both of these github repositories can be cloned into the Eclipse IDE as described in “doc\Readme.docx” in the CrystalSim github repository. C.5. Program Screenshots

Figure C.2. Screenshot of the CrystalSim GUI on Windows 7.

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Figure C.3. Screenshot of the final arrangement of cubic crystallites grown in a cylindrical sample geometry, as visualized with VMD.184 (a) Condensed View. (b) Exploded View. Colors correspond to different crystallites. Distinct groups that are the same color are not the same crystallite as VMD has a limited default atom-color palette with which to work.

C.6. Explanation of Simulation Parameters The CrystalSim GUI consists of three main panels: modify-able simulation parameters, a list of the current simulation parameters, and a message box at the bottom of the screen where relevant output is visible, as shown in Figure C.1. The list of current simulation parameters will auto-update as the toggle-able parameters in the modify-able simulation parameters window are changed, and the changed parameter will be noted in the message box. The simulation parameters are further delineated by color in Figure C.2 and described subsequently.

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Figure C.4. Three main sections of the UI highlighted in color.

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Figure C.5. Simulation parameters grouped by color.

Figure C.6. Nucleation and growth parameters grouped by color.

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C.6.1 Nucleation and Growth Parameters -- Figure C.3 Crystal Shape. There are five options for the shape of the crystallites to be grown in the simulation: Spherical, Cylindrical, Cubic, Tetragonal and Orthorhombic. Axial Growth Velocity. Based on the radio button that is selected for the crystal shape, the appropriate axial growth velocities, in green, will be editable. The axial growth velocity is the number of distance units that each axis increases during one unit time. One distance unit is the distance between two of the simple cubic lattice points. Nucleation. The sample will be “seeded” with the “Initial number of seed crystals.” These crystallites are placed in random locations and will begin growing at t=100. An initial time of 100 is used because the non-linear fitting algorithm does not behave when fitting parameters drop below zero and tinit=100 is usually sufficient to ensure that the t0 fitting parameters do not drop below zero. The parameter “Maximum number of crystallites in simulation” provides an upper bound for the number of crystallites that can grow during a simulation. After this limit has been reached, no more nucleation will occur. The “Probability of subsequent nucleation” (Pnuc) parameter provides a method to allow nucleation to occur after the initial nucleation events that occur at t=100. The value provided in this box is per unit time and is scaled appropriately based on the value provided in the “Time step” text box, such that the effective nucleation probability is Peff = Pnuc × Δt. A random number, R, is generated every time step and compared with Peff. If R < Peff, then a nucleation event is allowed to occur in the untransformed volume. The nucleated crystallite has an orientation according to the Crystal Orientation settings. Crystal Orientation. The Kolmogorov-Johnson-Mehl-Avrami condensed phase reaction framework assumes that nucleated crystallites have no preferred orientation, so the default mechanism for determining crystallite orientation is to give them random orientations. It might be useful in certain cases for the user to be able to provide a set of pre-determined crystal orientations from which the system should choose. This functionality has not yet been implemented. If the “SetOfOrientations” radio button is selected this currently results in the

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crystallites being oriented only along the Cartesian axes in the positive and negative directions, for six total orientations.

Figure C.7. Parameters controlling the size and shape of the sample

C.6.2 Sample Parameters -- Figure 14 The simulation volume is a 3D volume gridded as a simple cubic lattice of size a × b × c. The shape of this lattice is determined by the radio buttons, currently allowing the sample shape to be Spherical, Cylindrical, Cubic, Tetragonal or Orthorhombic. Based on the radio button selection, the appropriate axes will be editable. The total number of lattice points which are inside the sample shape with the specified units is given after “Vol:”

Figure C.8. Parameters controlling how the simulation ends.

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C.6.3 Termination Condition Parameters -- Figure C.5 The user is given the option to specify how the simulation is terminated. By default, the simulation ends after the entire simulation volume is transformed. However, there additional selectable options for “Time elapsed” which ends the simulation after the specified amount of time has elapsed and also to end the simulation after some number of crystallites have nucleated. Note that the number of nucleated crystallites also includes the number of crystallites that the sample was seeded with. C.6.4 Fitting Parameters -- Figure C.6 to Figure C.8 It is strongly encouraged that multiple simulations are run with each set of parameters. For the initial paper where this software was used, I did ~600 simulations for each set of parameters that I used. Because of the randomness built in to the simulation, specifically the way in which new crystals are nucleated, that they are nucleated in random locations and with random (or a small set of) orientations, no two simulations will produce the same output, unless the same random number seed is used for the random number generator. Dozens to hundreds of simulations may be required to establish the reliability of simulation results. An automated fitting routine is available to automatically fit the individual crystallite transformations and output all results into a tab-delimited text file that can be directly copied and pasted into Microsoft Excel (or your favorite spreadsheet software) for further analysis and visualization. This text file is described in a subsection of Input/Output Parameters -- Figure 20 on page 16. The KJMA model is used to fit the crystallite transformations, as given in Equation C.1. This model has three fitting parameters, k, t0 and n. The parameters are hideously correlated (>0.9) and so it may be desirable to fix some or all of them to a specific value and fit the others. This option is controlled by the set of toggle buttons as given in Figure C.6.

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Figure C.9. Toggle buttons to control which parameters are fit.

Figure C.10. Toggle buttons to control how n is determined.

Figure C.11. Specific values to use when fixing various parameters in the KJMA model as highlighted in green (the lower part of the figure).

When the dimensionality (n) parameter is fixed, there are a few options as to how its value is determined. This is controlled by the toggle buttons given in Figure C.7. When the “FIXED_TO_VALUE” button is selected, the fitting algorithm will use the value given in the “Fixed n =” text box, shown in Figure C.8. When the value “APPROXIMATE_BY_SAMPLE_SHAPE” is selected, the dimensionality is approximated based on the dimensions of each of the sample shape axes (a, b, c), as given in Equation C.2. In the isotropic limit when all sample axes are equivalent nest degenerates to 3. 푎 + 푏 + 푐 푛 = C.2 푒푠푡 max(푎, 푏, 푐) When the value “APPROXIMATE_BY_ACTUAL_SHAPE” is selected, the dimensionality is estimated based on the shape of the actual crystallite when it was 50% crystallized. The 50% mark was selected because the algorithm is automatically fitting from

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0% crystallized to 50% crystallized and thus the sample shape after 50% is irrelevant for the fit. This part of the code could be updated to give the option for the user to specify the region of alpha where the crystallite is being fit. If you need this feature implemented, please let us know at [email protected] or [email protected]. The shape of the individual crystallites is approximated by computing the center of mass of the crystallite at its 50% transformed shape and then finding the longest distance from the center of mass of the crystallite to the surface of the crystallite. The smallest distance from the center of mass to the surface of the crystallite that is orthogonal to the long axis defines the 2nd axis. The third axis is defined as the cross product of the long and short axes. The estimated dimensionality is determined with an expression similar to Equation C.2 except a, b, and c are the lengths of the three crystallite axes. Finally, the “initial k” variable allows the user to tweak the starting guess for my non- linear fitting algorithm. As non-linear fitting can be very sensitive to the starting conditions, and the fit value of the rate constant is modified by very many parameters, this is essentially an empirical value that allows the user to help me automatically fit the resultant transformations more effectively. If you find that when you use this program your crystallites are not being automatically fit, adjust this term until the crystallites are being automatically fit.

Figure C.12. Specific values to use when fixing various parameters in the KJMA model as highlighted in yellow (upper 2/3 of the Figure).

C.6.5 Misc Parameters -- Figure C.9 The first option is self-explanatory: It determines how many simulations are run with the set of selected parameters.

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The second option controls the time step of the simulation. This option is useful in conjunction with the Axial Growth Velocity on page 11. As the axial growth velocity increases, the time step should be reduced to ensure that enough time points are samples during the simulated transformation. Similarly, as the axial growth velocity decreases, the time step can be increased to reduce the simulation time. A reasonable guideline is that the inverse of the time step should equal the maximum of the growth velocity in any of the three dimensions.

Figure C.13. Control the simulation output location and choose the specific things to output.

C.6.6 Input/Output Parameters -- Figure C.10 Clicking the “Set output folder” button will open a file chooser that allows the user to select a specific location to store simulation results. The “Project file root” text box allows the user to change the name of the simulation. Changes to this name will be reflected in the text below. There are five output options: Structures. This option will output an xyz file which is simply a list of the coordinates of each of the lattice points in the simulated volume. An xyz file contains three coordinates and the Z value of each atom, all are tab delimited. The Z value of each atom, in this case, is the crystallite index. Because of the large number of data points in these simulations, I would strongly advise opening these with VMD, which is extremely effective at handling a large number of coordinates to plot. See “Loading structures and movies with VMD” in “doc/PlasticSim Readme.docx” in the PlasticSim github repository.

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Movies. This option will output a file which is formatted similarly to the xyz files but contains a list of coordinates of each of the lattice points in each of the time steps of the simulation. For this reason, these files can be extremely large (multiple gigabytes) and so this option is disabled by default. Similar to the structure files, these files work well with VMD. See “Loading structures and movies with VMD” in “doc/PlasticSim Readme.docx” in the PlasticSim github repository. Fits. This software will automatically fit the transformations of all simulated crystallites and the total transformation of the simulated volume when this option is selected. The first row of these output files are the fitting parameters in order k, t0, n. The remainder of this file are three tab-delimited columns which contain the simulation time, the simulated transformation and the fit transformation, respectively. If your fits are not very good, consider adjusting the initially estimated rate constant to a more appropriate value. If you still have trouble, email me. Java Objects. This software is written in Java and, if this option is selected, will output the native Java objects which can be used at a later date to re-analyze the simulations. This is an advanced feature which currently requires coding knowledge to utilize. Transformation Files. The transformation files contain all relevant information for the simulation. It is strongly recommended to output this file. The header of this file is the majority of the simulated parameters that were output. The next component of this file is a series of columns which correspond to the simulation time, the number of untransformed lattice points, the number of lattice points transformed during that time step, the total number of transformed lattice points. The remaining columns are the number of lattice points that each individual crystallite transformed during that time step. The next component of this file is the crystal information, which contains the nucleation location, crystallite orientation, axial growth rates, initial dimensions and the nucleation time for each crystallite that contributed to the transformation.The next component of this file is a fitting summary which gives the fitting parameters and the standard deviation of these fitting parameters.Finally, a summary of each crystallite is given in a row-tab-delimited format which can easily be copied into excel for further analysis. The header of each column should be sufficiently descriptive. If it is not,

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please contact me and I will clarify that. This is the same information that is provided in the “[simulation name] -- automated fitting.txt” file.

Figure C.14. Section of the UI dedicated to updating the user as to the runtime status of the simulation

C.6.7 Simulation Runtime -- Figure C.11 To run the simulation, click the “Run Simulation” button. Once this button is clicked, the two progress bars will begin to update. The first progress bar shows the number of simulations that are complete and the number of simulation remaining. The second progress bar shows the progress of the current simulation. What is currently showing as “Sample shape” will change to show the currently chosen shape of the sample container. The “Sample axes” text will change to show the unique axes and the number of units in those directions. Finally, the “??” after “Approximate Sample Volume” will change to show the approximate sample volume of the simulation.

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C.7. Class Diagram

Figure C.15. Class Diagram for CrystalSim.

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Appendix D. Interactive 2D Diffraction Analysis: Ramdog D.1. Introduction Ramdog is a stand-alone piece of software written in Java to provide an interactive image analysis program for X-ray diffraction images. Ramdog can read some raw X-ray diffraction images that were collected with flat-plate detectors, assuming the user knows certain properties of the file (header size in bytes, image dimensions and the size of each data entry in bytes). Otherwise, users can load pre-rendered diffraction images (png, gif, etc.). Once these images are loaded into Ramdog, the images can be analyzed for the presence of Bragg diffraction peaks. Our primary use of this software was to follow the intensity of Bragg diffraction peaks as a function of time during an isothermal crystallization experiment and then output the time-dependent intensity for each of the diffraction spots that were found. Ramdog can compute the 2D Fourier Transforms (Patterson Functions) of diffraction images if you have a CUDA-capable graphics processing unit. Ramdog can also display diffraction images that were calculated with the software described in Appendix E. Five of the use cases of Ramdog are described in §D.6 D.2. Software and Hardware Environment Ramdog requires the Java Development Kit (JDK) version 1.7 which takes approximately 200 MB of disk space and is freely available from Oracle. Ramdog is available from the github repository, http://www.github.com/ericdill/Ramdog. Ramdog was designed in the Eclipse integrated development environment (IDE) and, as such, it is recommended that you install Eclipse and run Ramdog from within that. See Section 2 in the “Ramdog Readme.docx” in the “doc” folder in the github repository. To use the 2D Fourier Transform capability of Ramdog, a CUDA-capable GPU is required. D.3. Program specifications Ramdog is expected to run similarly on all platforms which support the Java Runtime Environment (JRE), though the look and feel of the GUI may vary between platforms. This software was written in Java 1.7 (JDK 1.7.0_25) with the aid of the Eclipse IDE (Version: Kepler Service Release 1. Build id: 20130919-0819).

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D.4. Program Documentation and Availability The program source code is available free of charge at https://github.com/ericdill/Ramdog Documentation, is available in the “doc” folder at the aforementioned location, including a readme demonstrating complete program usage. Additionally, a generic programming library that I also authored is required for source code manipulation: https://github.com/ericdill/GlobalPackages. Both of these github repositories can be cloned into the Eclipse IDE as described in “doc\Ramdog Readme.docx” in the Ramdog github repository. D.5. Program Screenshots

Figure D.2. Screenshot of the Ramdog GUI on Windows 7.

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D.6. Use Cases. D.6.1 Use Case 1: Spotpicking a series of diffraction images

Figure D.3. Flowchart of the Spotpicking use case. Orange indicates a step that requires user input. Gray indicates a step that is optional. Blue indicates a step that is automated by the software.

D.6.1.1. Loading an Image. Ramdog supports loading xray images that are either in their native format (usually a binary file with a header and your xray data) or images that have been saved as common file formats (.png, .bmp, etc.). Loading the raw image file is preferred for a number of reasons but loading a pre-rendered image is significantly simpler. The raw image file is preferred because each pixel in the detector corresponds to one data entry while the pixels in the pre-rendered image do not necessarily correspond to one pixel in the detector. This can cause a headache when trying to relate the detector calibration info to the displayed image in Ramdog. Automatically determine the image color scale. On the Menu bar under Image is a menu item called “Auto-scale Image?” which tells Ramdog to use the min and max intensity values of the loaded image to generate a greyscale image which goes from white at the min

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value to black at the max value. The default image view is a logarithmic scale. This can be changed with the buttons on the right-hand side of the GUI. Loading a pre-rendered image. Click File->“Load image (png, jpg, bmp, etc.)”, select your image in the popup file selection window and click Open. Loading a raw x-ray image. To load the raw image you need to know the image dimensions (256×256, 512×512, 2048×2048, etc.), the size of the header (if present) in bytes and the size (in bytes) of each pixel in the diffraction image. Additionally, the diffraction data, when stored in a binary format, can be little endian or big endian. If you’re unsure, choose one and see what the image looks like, then choose the other and see what it looks like. Pick the one that looks more appropriate. These options (image dimensions, header size, data element size and endianness) are options that appear when you select File->Load Xray Image and are required parameters for Ramdog to open the file. These fields will be automatically populated by a guess, but these are not necessarily correct. D.6.1.2. Manipulate Image Color Model. To manipulate the mapping from xray intensity to color, click on Image->Color Model and the color model window will appear. The color model is set up as a set of intensity-color pairs which serve as the color levels. Colors are then interpolated from these color levels and displayed accordingly. There are six possible operations that the user can perform on the color model: 1. Change the intensity of the level (Value), 2. Change the color of the associated intensity level, 3. Move the intensity-color pair up, 4. Move the intensity-color pair down, 5. Remove the currently selected intensity-color pair and 6. Insert a new intensity-color pair below the currently selected level. Operations 3-6 are performed with the buttons in Figure D.4b. Operations 1-2 are performed with the snippet from the image color model window shown in Figure D.4a. To change the intensity level, double click on the desired cell under the “Value” column and edit it to be the new desired value. When finished, press enter. To change the color of a level,

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click on the desired color level and select a new color by simply clicking on the desired color at the bottom of the window. To see the effect that your changes have on the displayed image, click the “Apply Filter” button on the main window (It’s about 1/3 of the way down the buttons on the right side of the window).

Figure D.4. Relevant portions of the Image color scale window.

D.6.1.1. Detector Calibration. To extract any meaningful information from a diffraction image, five parameters are needed: Wavelength of the incident radiation, beam center (x,y) in pixels, the physical size of each pixel (μm) and the distance between the sample and the detector (mm). With these five parameters, the pixels in the diffraction image can be converted to Q (Å-1) according to Equations D.1-D.3 and further analysis becomes possible.

( )2 ( )2 2휃 = tan−1 [푝𝑖푥푒푙푆𝑖푧푒 ∗ √ 푥−푥0 + 푦−푦0 ] D.1 1000 푠푎푚푝푙푒퐷𝑖푠푡푎푛푐푒

2휃 4휋 sin( ) 푄 = 2 D.2 휆 푦−푦 180 휑 = tan−1 ( 0) ∗ D.3 푥−푥0 휋 The calibration section of the GUI shown in Figure D.5 can be used as a temporary store for the calibration information which is not saved when the program is closed, or the calibration information, once saved, can be saved as a new calibration file which will be automatically loaded each time Ramdog starts.

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Figure D.5. Detector calibration portion of the UI.

To use calibration information in a temporary state simply enter the five parameters (wavelength, beam center, pixel size and sample-detector distance) and click “Submit.” After this is done, when your cursor is over the diffraction image, the coordinate info in the top right corner of the GUI will now have values for Q and phi. To use the calibration information as a permanent state, enter the five parameters plus information about the calibration parameters (Synchrotron, Date, Calibrant and Notes). A file will then be generated called “[Synchrotron]--[Date]--[Calibrant].calib” in a folder called “Calibration” in your Eclipse directory. This file will then be automatically loaded the next time Ramdog is loaded and the parameters will be accessible from the load calibration window. The calibration files that have been saved are accessed by clicking on the “Load Calibration File” button which opens a new window, shown in Figure D.6. Each calibration file that was loaded is given a row in the table and an associated button at the top of the window. To load the new calibration file, simply click the button at the top of the window that is associated with the calibration file that you want to load. The number in the first column is associated with the buttons at the top of the window. The calibration files can be edited in this window and saved to file. The “Add new calibration file” currently does nothing and the “Delete calibration file” is a temporary delete and does not remove the file from disk.

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Figure D.6. Window to load previously saved calibration files.

D.6.1.1. Determine regions of interest. As the goal of this use-case is to extract information about your diffraction images as a function of time, there are a few tools available to choose these interesting regions of your diffraction image. These options are: 1. Single pixels from mouse clicks, 2. Multi-line paths based on mouse clicks, 3. Elliptical or square regions based on two mouse clicks, 4. Find a diffraction spot in a small region around the users mouse click, 5. Find diffraction spots in the entire image. Manual selection of regions of interest (Options 1-4). These options are straightforward and require the user to click on the desired option, given as a button near the bottom of Figure D.7. Based on the button clicked, the set of buttons labeled “Shape Selection” in Figure D.7 will change or vanish. The currently shown buttons correspond to the “Add new region” button and allow the user to select a rectangular or elliptical region of the diffraction image with a two-click scheme (i.e., clicking on opposite corners of the shape). Once the shape has been selected (it will show up as the border of a rectangle/ellipse) clicking on the “Save

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Region” button will save all pixels within that shape to a new Region object within the “Regions” tree structure at the top of Figure D.7. Double clicking on the word “Regions” will expand it to show a new entry with some incomprehensible name along the lines of “Lanalysis.Pixel;@6e584751” which can also be double clicked on to show all the pixels that are contained within that new Region object. Clicking on any of those pixels will highlight it on the diffraction image. The mechanism to add a path is similar to that for the addition of new regions. Clicking on “Add new path” will change the buttons under “Shape Selection” to “Close Path”, “Save Path”, “Clear Path”, and “Center Click”. To start a new path, either click “Clear Path” to remove any memory of previous clicks or just click on the diffraction image. When Ramdog registers the click, a circle will appear around where the click occurred. To add a segment to the path, click elsewhere on the image. Now a line will appear between the original click and the new click. Line segments can be added with further clicks on the diffraction image. To save this path so that Ramdog will extract these specific pixels out of each image in the time series, click “Save Path.” To close the path (join the most recent click with the first click) click “Close Path.” Finally, clicking on the button “Center Click” will move the path to the center of the image, as denoted by the currently loaded calibration. This is especially useful when radial cuts are of interest. Once the path is saved, a new entry will appear under the “Paths” heading at the top of Figure D.7.

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Figure D.7. Pixel/Region selection portion of the GUI.

The mechanism to add a pixel was defined such that additional pixels could be added to spots, regions or paths after they had initially been chosen. To add pixels to an existing grouping of pixels, select the specific region/path/spot that you wish to add pixels to, click on “Add a pixel” and then select the pixels that you wish to add to that existing grouping of pixels. To plot the green overlay that corresponds to the pixels within that object, click on something else in that window and then click back to the region/path/spot that you wish to see the overlay for. The new pixels that you added should now be seen in the overlay. Ramdog can automatically find spots near a mouse click by calculating the curvature in x and y (2nd derivatives in the x- and y-directions). This is more thoroughly described in §4.3.3 on page 70.

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Automatic selection of Bragg diffraction spots in the entire image (Option 5). To automatically find the Bragg diffraction spots in the selected diffraction image, click on Image- >“Spot pick this image” and the window shown in Figure D.8 will appear. First, uncheck “Spot pick all images?”. There are a number of options that allow for a rather customizable way to set the criteria for finding the “active pixels” in the image that will then be coalesced into diffraction spots if at least some number of active pixels are touching (the default is 5). These selection criteria are (where ## represents the user input). 1. Threshold: ## 2. Pixel > ## 3. Pixel < ## 4. ## < Pixel < ## 5. Pixel == ## 6. Pixel != ##

Figure D.8. Spotpicking control window.

To enable/disable these selection options click on the check box next to that option. Additionally, there are two buttons next to each of these options labeled “!=” and “==”. If the “!=” button is selected, then when active pixels are found with these criteria, those pixels are rejected. When the “==” button is selected, an active pixel found with the specific criteria is kept. Once the desired criteria have been selected, clicking the “Run Spotpicking” button at

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the bottom of the window shown in Figure D.8 and the algorithm will process the image and save a number of spots to the Pixel Grouping called “Target Spots” shown in Figure D.7. By clicking on “Target Spots” the overlay will be shown. It must be noted that this is not a perfect algorithm and the picked spots may not be exactly the same as what can be seen visually, but it is usually a pretty good approximation. If better spots are desired then perhaps playing around with the selection criterion will deliver the results you desire. Once you’re satisfied with those diffraction spots, proceed to the next step. Select Corresponding File Series. The objective of this use case is to follow the time-evolution of pixels, regions, spots, paths, etc. At this point, you now need to tell Ramdog what the file series is. This is done by clicking Output->”Get Histories For All Regions” and then selecting all files that you wish to extract these pixel groupings from. Once you have selected these files and clicked “Save,” a window similar to that shown in Figure D.9 will appear. This is the order that Ramdog thinks your files belong in. If this is not the correct order, you need to rename the files such that Ramdog understands your file naming scheme. There is an included file renaming tool that works if your files are numbered sequentially and, other than the numbering index, all parts of the file name are the same. As you can see in Figure D.9, my files are named such that the only difference is the file index which comes almost at the end of the file name. The problem here is that Ramdog sorts by the first character of the file index and does not treat the file index as a number. To trick Ramdog into sorting the correct way, your files will be renamed to include leading zeroes into the index such that 7 becomes 007 and Ramdog sorts correctly. To do this, click on the “Rename” button and a new window will pop up asking for every part of the file name before the index in one box and every part of the file name after the index in the second box. In the example given here, I would put “D:\Data\aps 09\eric\czxfun\raw 2D\czx1- 10_230-135-cooled_90kev_1s_350\czx1-10_230-135-cooled_90kev_1s_350f.cor.” into the first box and “.cor” into the second box. Make sure to click “Set Prefix” and “Set Suffix” after placing the correct text into the box. Also note that ctrl+c and ctrl+v work here so that you do not have to type all of that text in by hand. Once you’ve entered the correct file names, click

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“Rename” and your files will be renamed (ON DISK, so make sure you have a backup of these files as I will not be held responsible for harming your raw data) and sorted in the correct order in Figure D.9.

Figure D.9. File series display.

After your files have been renamed, click on “Get Histories” and a save window will appear allowing you to select an output folder and a file root. Clicking on “Save” will then load each of the images, extract the relevant pixels and move on to the next image. After all images have been processed, the results will be written to disk in the specified location. Output File Format.

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Multiple files will be output (where root is the file root that you input and type is region/path/spot): root_type.txt, root_normalized_type.txt, root_QandPhiandI.txt. root_type.txt. This file is structured so that each column corresponds to one of the diffraction paths/spots/regions of interest and the rows in each column correspond to the total intensity of the path/spot/region in each frame. root_normalized_type.txt. This file is structured such that every column after the first corresponds to one of the paths/spots/regions of interest. The first row is the path/spot/region index. The second row is the starting intensity and the third row is the final intensity. The fourth row is Q and the fifth row is phi. Rows 6 and 7 are the average x and y coordinates of the spot/path/region. The remaining rows are the integrated and normalized (between 0 and 1) regions/spots/paths as a function of time and can be fit with the KJMA model to extract the rate constant and nucleation time for each diffraction spot. Root_QandPhiandI.txt. This file is simply a summary of the properties of the spot/path/region in the final frame. Each row corresponds to one of the spots/paths/regions of interest. The first column is Q of the spot, the second column is phi, third is the intensity in the last frame, the fourth column is the average x coordinate and the fifth column is the average y coordinate. D.6.2 Use Case 2: Computing the Patterson Function The Patterson function is the Fourier transform of the raw intensities of the diffraction images and gives the real-space electron density weighted pair correlations in two dimensions. The pixels in the diffraction image, assuming a flat plate detector, do not have a constant ΔQ which is a problem for the Fourier transform, as it expects an even coordinate grid. Thus, the raw images need to be first interpolated into a set of data with constant ΔQ. This is performed automatically but requires the detector calibration to determine the Q- coordinates of each of the pixels and currently requires a CUDA-capable GPU to perform the Discrete Fourier Transform (DFT). An OpenCL version is on the to-do list but has not yet been finished. Note: The Fast Fourier Transform, FFT, is orders of magnitude faster but requires the data dimensions to be a multiple of two and therefore requires the data to be

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truncated or padded, while the DFT works on data sets of arbitrary dimensions without any extra manipulation. Technically your measured diffraction image includes a host of factors other than the diffraction of your sample including the sample container, air scattering, the beam stop and anything else in the beam which need to be subtracted from the measured intensities before the Patterson Function is calculated. However, as many of these factors are isotropic, if your measured image is highly anisotropic (i.e., not powder rings) then the background subtraction may not be that critical. Conversely, the beam stop is highly anisotropic in your diffraction image and will present a problem unless your data is highly isotropic. Essentially, if you know you data is anisotropic/isotropic then you can ignore experimental artifacts that are the opposite (isotropic/anisotropic) since while the experimental artifacts will appear in the Patterson Function, you will be able to quickly discard them and explore the experimentally relevant pieces of the Patterson Function. Though, it must be noted that the Patterson Function must be the measured intensities less the background. All of this is to say that you may be able to get almost as much information without a background file. Assuming that you have a detector calibration, a measured background image and a CUDA-capable GPU, then you can calculate the Patterson Function of your 2D images, as detailed below.

Figure D.10. Calibration file information.

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D.6.2.1. Loading a background file. A background file can be loaded by clicking on the “Yes” button in the “Subtract Background?” portion of the Calibration panel, as shown in Figure D.10 which will cause an open file dialog to appear. To load your raw background file you will need to know the image dimensions, data element size and the file header size. Enter those values into the corresponding text boxes and clicking “Open” will result in your background image appearing in the image display window. The background subtraction is scaled based on the text box labeled “Background Scaling Factor.” To switch between the background and the diffraction image, click on the buttons labeled “BACKGROUND” or “INPUT_IMAGE”, respectively, as shown in Figure D.11 and then click “Apply Filter” on the right side of the UI.

Figure D.11. Image display options.

D.6.2.1. Loading the X-ray Image. See “Loading an Image” in D.6.1. D.6.2.2. Loading the Calibration Data. See “Input Detector Calibration” in D.6.1. D.6.2.3. Calculating the Patterson Function. Before the Patterson Function can be calculated, you need to tell Ramdog what your desired ΔQ is. This is done by entering the desired value into the “q per pix” text box in the top right corner of the GUI in the “Coordinate Info” section (Figure D.12). By clicking the “Fourier Transform” button (located in the buttons along the diffraction image), Ramdog compute the Discrete Fourier Transform of your 2D diffraction image which has been linearly interpolated onto a grid whose step size is defined by the entered value of ΔQ. After the Fourier

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Transform is complete, the new image will be visible, though the color model will probably need to be adjusted (See “Manipulate Image Color Model in D.6.1). To switch between the Patterson Function, the original image and the background image simply click on the corresponding button, as shown in Figure D.11, and then click “Apply Filter” on the right side of the UI. You will also need to update the coordinate info by clicking on the appropriate buttons (Figure D.12) so that Ramdog can calculate the coordinates of your mouse cursor according to the proper coordinate space. Finally, the Patterson Function can be viewed in four ways, shown in Figure D.13. The real and imaginary buttons simply show the real and imaginary parts of the Patterson Function, the modulus is √푟푒2 + 𝑖푚2 and the power spectrum is 푟푒2 + 𝑖푚2. After clicking on the desired button, click “Apply Filter”. You may need to adjust the color model after changing the Fourier view.

Figure D.12. Coordinate Info Panel.

Figure D.13. Fourier view options.

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D.6.2.1. Saving the Patterson Function as an image. See §D.6.3. D.6.2.2. Fourier Transform a series of images. Ramdog has the capability to automatically compute the Patterson Function for a series of diffraction images and save the result in a png formatted image to disk. This is done with the menu bar by clicking “Output”->“Fourier Transform Series”. An open dialog will appear asking you to select the files you wish to automatically Fourier Transform. Once you select your images and click “Open” the files will be automatically loaded, saved to disk as an image with the extension “-auto.png”, Fourier Transformed and saved to disk as an image with the extension “--FT-raw.png”. It would probably be beneficial to load your raw image, set the color levels that you want and then Fourier transform the image and set the color levels for that as well. If you do this then your output images will be correctly formatted. D.6.3 Use Case 3: Image Output Click on “Output” on the menu bar and then click “Save Image”. The window shown in Figure D.14 will appear which provides flexible options for saving your image. The three buttons along the top, “Select Entire Image”, “Select Specific Region” and “Select Region From Image Center” control the mechanism by which images are saved.

Figure D.14. Image saving options window.

Select Entire Image. This option simply saves the entire image in png format based on the Java ImageIO output standard. Select Specific Region. This option allows the user to input a set of pixel coordinates to save only a specific region of the image to disk. Image is saved in png format.

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Select Region From Image Center. This option allows the user to select a region of the image based on the center of the image. Image is saved in png format. Save Image Series. The “Save Image Series” button pops up a file open dialog that asks for the series of raw diffraction images that are to be opened and saved to disk as images in png format. You will need to set the color model (“Image”-> “Color Model”) for your images before you try to automatically save them to disk unless you have selected the auto-scaling option (“Image”->“Auto-scale Image?”). The automatically generated images will be saved according to the options selected in Figure D.14. D.6.4 Use Case 4: Loading calculated diffraction images Diffraction images that were calculated with PlasticSim (http://www.github.com/ericdill/PlasticSim) can be natively loaded with RamDog with “File”- >“Load calculated xray images” which will show a file open dialog asking you to select a calculated diffraction image. Ramdog will automatically parse the other files that are in the same folder as the file that you just loaded to find files that contain the character sequences [100], [110] and [111]. Ramdog will automatically place those in the appropriate tree structure shown in Figure D.15. All other files will be placed under the “other” node in the tree structure in Figure D.15. By clicking on any of the calculated xray images that have now been loaded into Ramdog, the calculated image will appear in the image display portion of the UI. This image can now be manipulated identically to the experimental diffraction images and can be Fourier Transformed to show the associated Patterson Function. Additionally, the Coordinate Info panel (Figure D.12) will show the appropriate x/y/I/Q/phi values if the “CALCULATED_SPACE” button is selected. The header of the images calculated with the PlasticSim software contains the diffraction axes and ΔQ which Ramdog uses to determine the Q value of each of the pixels in the diffraction image. These calculated images can be saved as png images by following the instructions in §D.6.3.

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Figure D.15. Calculated xray images

D.6.5 Use Case 5: Comparing the diffraction image to calculated Bragg points/rings To access the Bragg point/ring window, click on “Calculation”->“Setup Bragg Reflection Calculation” and the window shown in Figure D.16 will appear. This window allows you to calculate reciprocal lattice points based on a set of atoms in crystal coordinates. These reciprocal lattice points can then be plotted on top of your input diffraction image. Alternatively, you can give Ramdog a file that contains a set of information in five tab- delimited columns (Q, h, k, l and intensity). These rings can then be overlaid onto a diffraction image.

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Figure D.16. Bragg reflection calculation setup window.

D.6.5.1. Plotting Bragg Rings. To overlay the Bragg rings onto a diffraction image, you need to provide Ramdog with calibration information so that it can translate pixels to Q and a tab-delimited file with five columns of data in the order Q, h, k, l and intensity. The intensity column can be filled with zeroes, but it still needs to be there. The (h, k, l) columns are so that when you mouse over a ring, Ramdog can translate that ring into hkl values as opposed to just Q. To load this information into Ramdog, click the “Read Q values from file” button and provide Ramdog with the file that contains this information, which, if the file is set up correctly will appear in the text area in Figure D.16 once Ramdog has parsed it. After these values have been successfully parsed by Ramdog, clicking on the “Plot Bragg rings” button will overlay rings at the specified Q values onto your diffraction image. Additionally, if you click on the “Permanent” toggle button on the right side of the UI then the Bragg rings will become part of the image and will not disappear when you change the zoom level of the image. D.6.5.2. Plotting Bragg Points. NOTE: This feature is still in development and is only barely functional.

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We frequently use the program “Atoms” to view our crystal structures. Atoms can output the atoms in the crystal in crystal coordinates (i.e., normalized to between 0 and 1) to an .atd file. This file is the target for reading in the crystal coordinates into Ramdog. To load one of these .atd files, click the button “Load Atoms .atd file” and an open dialog will appear. Select your .atd file and Ramdog will parse the atom coordinates and present its parsed information in the text box as shown in Figure D.16. Once you tell Ramdog what the unit cell parameters are (a, b, c, α, β, γ) then you can click on “Calculate Bragg Reflections” and Ramdog will use those atoms and unit cell parameters to calculate all Bragg points out to the value given in “Maximum Q to calculate”. Once this calculation is complete, a number of “Calculated Bragg Reflections” will appear in the “Pixel Groupings” portion of the UI (Figure D.16). Clicking on the “Calculated Bragg Reflections” node will overlay the Bragg points oriented along one of the [001] directions. The eventual goal is to be able to rotate these Bragg points such that they can be aligned with the Bragg peaks in the diffraction image.

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Appendix E. PlasticSim: Simulations of plastic crystalline carbon tetrabromide and GPU-accelerated diffraction calculations.

Figure E.1. General program flow of PlasticSim.

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E.1. Introduction PlasticSim is designed to provide a single entry point (RunSimulationThread.java) for a set of simulation codes that allow the user to simulate the plastic crystalline lattice of CBr4 and calculate the corresponding 2D diffraction patterns along generic reciprocal lattice directions. The general program flow is illustrated in Figure E.1 with the three main components of PlasticSim being the construction of the lattice, the minimization of the lattice and the calculation of the corresponding diffraction patterns. Visualization of the calculated diffraction patterns is handled by Ramdog (http://www.github.com/JamesDMartin/Ramdog). E.2. Software and Hardware Environment PlasticSim requires the Java Development Kit (JDK) version 1.7 which takes approximately 200 MB of disk space and is freely available from Oracle. PlasticSim is available from the github repository, http://www.github.com/ericdill/PlasticSim. PlasticSim was designed in the Eclipse integrated development environment (IDE) and, as such, it is recommended that you install Eclipse and run PlasticSim from within that. See Section 2 in the “PlasticSim Readme.docx” in the “doc” folder in the github repository. To compute 2D diffraction patterns with PlasticSim, a CUDA-capable GPU is required. E.3. Program specifications PlasticSim is expected to run similarly on all platforms which support the Java Runtime Environment (JRE), though the look and feel of the GUI may vary between platforms. This software was written in Java 1.7 (JDK 1.7.0_25) with the aid of the Eclipse IDE (Version: Kepler Service Release 1. Build id: 20130919-0819). E.4. Program Documentation and Availability The program source code is available free of charge at https://github.com/ericdill/PlasticSim Documentation, is available in the “doc” folder at the aforementioned location, including a readme demonstrating complete program usage. Additionally, a generic programming library that I also authored is required for source code manipulation: https://github.com/ericdill/GlobalPackages. Both of these github repositories can be cloned into the Eclipse IDE as described in “doc\PlasticSim Readme.docx” in the Ramdog github repository.

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E.5. PlasticSim Program Design PlasticSim is designed to provide a single entry point (RunSimulationThread.java) for a set of simulation codes that allow the user to simulate the plastic crystalline lattice of CBr4 and calculate the corresponding 2D diffraction patterns along generic reciprocal lattice directions. The general program flow is illustrated in Figure E.1 with the three main components of PlasticSim being the construction of the lattice, the minimization of the lattice and the calculation of the corresponding diffraction patterns. Visualization of the calculated diffraction patterns is handled by Ramdog (http://www.github.com/JamesDMartin/Ramdog). E.5.1 Lattice Construction Simulating the plastic crystalline lattice begins by constructing a (10×10×10 to 30×30×30) supercell where each tetrahedral molecule is placed on an FCC lattice site with an orientation based on one of a few ordering motifs, more fully described in Chapter 5. Five ordering motifs are currently implemented as part of the InitialLatticeTypes enum:  Random orientation

 Six D2d orientations  1st shell Monoclinic  2nd shell Monoclinic  3rd shell Monoclinic “star” Random Orientation. The random orientation lattice motif distributes tetrahedra onto all FCC lattice sites in the supercell and then independently rotating each tetrahedron by three randomly generated angles around each of the Cartesian axes.

Six D2d Orientations. There are six orientations in which the 4̅ axis of a tetrahedron can be aligned with the Cartesian axes. These six orientations are randomly distributed onto all FCC lattice sites in the supercell. 1st shell Monoclinic. The low temperature solid phase of carbon tetrabromide adopts a non-plastic monoclinic (퐶2/푐) superstructure (Z=32), with cell parameters a=21.441(10) Å, b=12.116(6) Å, c=21.012(8) Å and β=110.91(3)°, and four molecules in the asymmetric unit.135, 162-164 Molecules in the first coordination shell of each of the four crystallographically

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unique molecules (13 total molecules) were extracted from this low temperature crystal structure and rotated such that the [110] direction was brought into coincidence with the other 11 members of the ⟨110⟩ family, resulting in 4 × 12 = 48 total first shell molecular arrangements. These first shells were randomly distributed onto all FCC lattice sites in the supercell, with new molecules overwriting existing molecules when a collision occurred. See src/basisSets/FirstShell.java for the source code which extracts the first shell monoclinic constructs from the monoclinic crystal structure and prints to a file called “sorted_eight.lattice”. 2nd shell Monoclinic. The exact same principle as the 1st shell monoclinic ordering motif except the 2nd molecular coordination shell was used for a total of 1 + 12 + 42 = 55 total molecules. 3rd shell Monoclinic “Star”. The monoclinic “star” ordering motifs were constructed by starting from the four crystallographically unique molecules in the monoclinic unit cell and finding the neighboring molecules whose centers of mass were well-aligned with the intermolecular ⟨110⟩ directions. The black molecules in the illustration given in Figure E.2a show a slice through the center of a 3rd shell star construct. There are six additional ⟨110⟩ directions that are above and below the plane shown in Figure E.2a that are not shown. The intermolecular ⟨110⟩ directions are defined by the vectors connecting the central molecule with its first shell molecules. The criteria for determining if higher-shell molecules were “well- aligned” with these intermolecular directions is illustrated in Figure E.2b. In this illustration, consider molecule 1 to be the central molecule, molecule 2 to be in the first shell and molecule 3 to be in the second shell. If the angle between the vectors 12⃑⃑⃑⃑⃗ and 23⃑⃑⃑⃑⃗ was less than 15°, then the second shell molecule was considered to be well-aligned with the first shell molecule and thus part of the shell structure. Interestingly, this algorithm produced shell structures that were not equivalent in all of the intermolecular ⟨110⟩ directions but instead were more planar, as illustrated in Figure E.3. Similar to the 1st and 2nd shell monoclinic structural motifs, the four 3rd shell star aggregates were rotated such that the [110] direction was aligned with the remaining 11, producing 4 × 12 = 48 unique molecular aggregates which were then randomly placed on the

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FCC lattice sites with new molecules overwriting existing molecules when a collision occurred. See src/basisSets/Star_110_Lattices.java for the source code which computes this star pattern to an arbitrary shell distance and prints to a file called “110_stars- [numShells]shells.lattice” where numShells is the coordination shell to compute the star constructs. See output/3 shell star/ for the atoms files and the corresponding rendered .png images.

Figure E.2. a. Illustration of the [111] plane of one of the 3rd shell monoclinic star ordering motif. The remaining six members of the ⟨110⟩ family are out of the plane and not shown. b. Illustration of the criteria to decide if a higher-shell molecule is “sufficiently aligned” with the intermolecular ⟨110⟩ directions as defined by the central molecule and its first shell.

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Figure E.3. 3rd shell monoclinic star constructs based on the four crystallographically unique molecules.

E.5.2 Lattice Relaxation Once the simulation lattice is constructed according to one of the ordering motifs, it is in a high-energy state due to the numerous physically impossible contacts that result from the randomness inherent to the construction mechanism. The potential used was the 2-body Lennard-Jones potential (src/defaultPackage/LennardJonesPotential.java), given in Equations E.1 and E.2, manipulated to be slightly less computationally intensive than the original analytical form. 24휀휎6 2휎6 퐹(푟) = [ − 1] E.1 푟7 푟6 4휀휎6 휎6 푈(푟) = [ − 1] E.2 푟6 푟6 CrystalSim also has the Hooke potential available for use (src/defaultPackage/HookePotential.java) and an interface for implementing other isotropic potentials (src/defaultPackage/Potential.java). These potentials are pre-calculated to significantly reduce computational time and stored in a HashMap with a distance precision of 0.001 Å (src/defaultPackage/PotentialLookup.java). The plastic crystalline lattices as constructed initially resulted in many impossibly short intermolecular Br-Br contacts. There are many approaches to reducing the energy of a simulated system including Molecular Dynamics (MD) and Monte Carlo (MC). MD simulations allow atoms to move according to the laws of classical mechanics whereby the forces on all atoms are calculated at every time step and atoms and molecules usually retain their velocity from the previous time step. While MD calculations may provide additional insight, the goal of this work is to understand the extent to which the structure of α-CBr4 can be described from a static perspective. Thus, MD simulations do not provide the desired

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insight into α-CBr4. Traditional MC attempts to move one atom at a time by a random amount in a random direction using the energy before and after as criteria for deciding whether the move is to be kept, though this strategy is not commensurate with maintaining rigid tetrahedra.

Another MC approach is to randomly distribute the six D2d orientations onto the FCC lattice sites and then randomly change the molecular orientations until no impossibly short distances remain.38 Using MC to change molecular orientations was shown to fit the most intense structured diffuse scattering but was unable to account for the new structured diffuse features. Reverse MC (RMC) is similar to traditional MC with the exception that the simulation cell is converged to “fit” some experimental measurement, though the RMC results are frequently poorly interpreted as it can be quite challenging to relate RMC results to fundamental structural principles. For these reasons, a new energy minimization strategy was desired that focused on enhancing intermolecular orientational correlations while allowing the freedom of movement (rotation + translation) that the MD approach provides which is missing from MC methods. This approach is described as a pseudo-random walk, with its algorithm described in pseudocode in Figure E.4. The random walk algorithm begins by selecting a molecule (testMolecule) from the simulation cell at random. The pairwise interaction energy with the surrounding twelve nearest neighbors along the ⟨220⟩ directions is determined and the molecule with the most unfavorable interaction energy is selected (worstMolecule). The pairwise intermolecular forces are calculated and the molecules are moved as a result of these forces. The testMolecule was then set to the worstMolecule and the algorithm was repeated some number of times, called numWalkSteps. After each molecule in the simulation cell had been the starting point for a random walk, one “walk” was complete. The lattice which resulted from this approach is dependent upon the number of walk cycles (numWalkCycles), the number of steps per random walk cycle (numWalkSteps) and the interaction energies selected. If too many walk cycles and walk steps were set, the resultant lattices did not produce diffraction patterns that closely resembled the experimental images, as shown in some images presented in §4.3.2. The ideal number of walk cycles was generally between 1 and 5 and the ideal number of walk steps was generally between 256 and 512,

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depending upon ε, the depth of the LJ well as a deeper well will converge faster than a slower one. Additionally, because this algorithm must resolve physically unreasonable intermolecular contacts, the force calculation often results in extremely large forces. To ensure that the simulation cell did not “explode”, the maximum translational and rotational motions for any given movement attempt were set to 0.25 Å and 1°, respectively. See src/defaultPackage/Simulate.java for all methods related to the energy minimization strategies.

Figure E.4. Pseudocode random walk algorithm.

Once the lattice has been sufficiently relaxed, as determined by the criteria that the user sets in the primary control file, , the lattice is output as a .xyz file. The .xyz file is structured such that the first line contains a single integer representing the number of atoms contained in the file followed by a line for comments which is ignored by most .xyz file readers. The remainder of the file consists of rows representing a single atom as four numbers: Z, x, y, z, where Z is the atomic number and (x, y, z) are the coordinates of that atom. Note that Z, x, y, z must be tab-delimited to adhere to the .xyz file standard. Also note that Z can be the string (C, Br, etc) or the atomic number (6, 35, etc.). The .xyz files output by PlasticSim gives atoms in terms of the crystal coordinates and not Cartesian coordinates. The crystal coordinates, in

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the case of a cubic crystal are the Cartesian coordinates divided by the edge length of the unit cell edge. E.5.3 Calculating the diffraction patterns A 2D diffraction pattern calculated from a single simulated lattice has a very high signal-to-noise ratio. However, because no two simulated lattices are identical due to the randomness inherent to the lattice construction and relaxation, 2D diffraction patterns can be calculated for multiple simulated lattices with the same input parameters and then summed together to increase the signal-to-noise ratio, as shown in Figure E.5 for a series of D2d simulations with 10×10×10 supercells after a 1×256 random walk.

Figure E.5. Visually demonstrating the increase in signal-to-noise ratio as the diffraction patterns from an increasing number of simulations are summed together.

CrystalSim can compute diffraction patterns along any reciprocal lattice direction with the constraint that three orthogonal vectors must be given (see runDiffractionCalc(…,…) in src/simulationTypes/RunSimulationThread.java). The [001], [011], [111], [211], [210], [010] and [100] directions are built in. CrystalSim computes diffraction patterns by determining the number of pixels to be calculated based on the size of the .xyz input lattices, the desired ΔQ and the maximum Q to calculate, computing the Q vectors in 3-space for those pixels and pre-calculating the scattering

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factors for each element in the input lattices for each pixel at the given wavelength. The scattering factors are pre-computed for each of the pixels and each element in the input lattices and then CUDA-capable GPUs are leveraged to compute the scattering equation, as given in E.3. (See c&cuda/cuDiffraction.cu and associated files for the CUDA implementation) 푎푙푙 푎푡표푚푠

퐹(푞⃑) = ∑ 푓(푞⃑, 푍푎) × exp(2휋𝑖 × 푞⃑ ∙ 푟⃑⃑⃑푎⃑) . E.3 푎=1

The elemental scattering factors, 푓(푞⃑, 푍푎), are a combination of the scattering power, 0 ′ ′′ 푓 (푞⃑, 푍푎), and anomalous dispersion effects, 푓 (푞⃑, 푍푎) + 𝑖 ∙ 푓 (푞⃑, 푍푎), which are large near elemental absorptions. Numerous models have been proposed to describe 푓0 but the most popular was computed with Hartree-Fock wavefunctions and fit to the nine parameter expression in Equation E.4 where the nine parameters are called the Cromer-Mann coefficients.181 4 0 2 푓 (sin 휃/휆 ) = ∑ 푎𝑖 exp[−푏𝑖(sin 휃⁄휆) ] + 푐. E.4 𝑖=1 The wavelength dependence of the anomalous dispersion effects are available for incident X-ray energies between 50 eV (λ=24.8 Å) and 30 keV (λ=0.4 Å) for most elements.182- 183 The experimental images presented in this chapter were collected at λ=0.13702 Å (90keV), significantly higher than the measured energies. The wavelength dependence of the anomalous dispersion effects for both elements at 90 keV are likely extremely similar to the reported values at 30 keV because neither element has an absorption edge near either energy and anomalous dispersion effects only provide a meaningful contribution to the scattering factor near absorption edges. Therefore, for the diffraction pattern calculations, the contribution from anomalous dispersion effects, 푓′ + 𝑖 ∙ 푓′′, were calculated for λ=0.4 Å as opposed to the experimental value of λ=0.13702 Å. E.5.4 Visualizing the diffraction patterns As PlasticSim is set up to automate much of the simulation and diffraction calculation process, tens to hundreds of diffraction patterns may be calculated for a single set of simulation parameters. As shown in Figure E.5, the sum of those diffraction patterns provides

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significantly more information than a single diffraction pattern. To sum those diffraction patterns, have a look at the main method of src/input_output/SumXrayFilesInFolder.java. This java file allows you to simply provide a path to the calculated diffraction patterns and it will sum them into a single file. This file can then be loaded in to Ramdog (http://www.github.com/ericdill/Ramdog). Ramdog will automatically parse the header information which contains the diffraction axes and the ΔQ so that the coordinate info panel (in the top right of the Ramdog UI) will provide accurate Q-coordinate information as the mouse is moved across the image.

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Appendix F. 2D Savitzky-Golay mask coefficients

Table F.1. Coefficients for smoothing

x-3 x-2 x-1 x x+1 x+2 x+3 y-3 0.0425 -0.0359 -0.0049 0.0183 -0.0049 -0.0359 0.0425 y-2 -0.0359 -0.0575 0.0074 0.0421 0.0074 -0.0575 -0.0359 y-1 -0.0049 0.0074 0.0928 0.1342 0.0928 0.0074 -0.0049 Y 0.0183 0.0421 0.1342 0.1779 0.1342 0.0421 0.0183 y+1 -0.0049 0.0074 0.0928 0.1342 0.0928 0.0074 -0.0049 y+2 -0.0359 -0.0575 0.0074 0.0421 0.0074 -0.0575 -0.0359 y+3 0.0425 -0.0359 -0.0049 0.0183 -0.0049 -0.0359 0.0425

Table F.2. Coefficients for 푑 푑푥

x-3 x-2 x-1 x x+1 x+2 x+3 y-3 0.0380 -0.0210 -0.0244 0.0000 0.0244 0.0210 -0.0380 y-2 0.0125 -0.0380 -0.0329 0.0000 0.0329 0.0380 -0.0125 y-1 -0.0028 -0.0482 -0.0380 0.0000 0.0380 0.0482 0.0028 y -0.0079 -0.0516 -0.0397 0.0000 0.0397 0.0516 0.0079 y+1 -0.0028 -0.0482 -0.0380 0.0000 0.0380 0.0482 0.0028 y+2 0.0125 -0.0380 -0.0329 0.0000 0.0329 0.0380 -0.0125 y+3 0.0380 -0.0210 -0.0244 0.0000 0.0244 0.0210 -0.0380

Table F.3. Coefficients for 푑 푑푦

x-3 x-2 x-1 x x+1 x+2 x+3 y-3 0.0380 0.0125 -0.0028 -0.0079 -0.0028 0.0125 0.0380 y-2 -0.0210 -0.0380 -0.0482 -0.0516 -0.0482 -0.0380 -0.0210 y-1 -0.0244 -0.0329 -0.0380 -0.0397 -0.0380 -0.0329 -0.0244 y 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 y+1 0.0244 0.0329 0.0380 0.0397 0.0380 0.0329 0.0244 y+2 0.0210 0.0380 0.0482 0.0516 0.0482 0.0380 0.0210 y+3 -0.0380 -0.0125 0.0028 0.0079 0.0028 -0.0125 -0.0380

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2 Table F.4. Coefficients for 푑 푑푥2

x-3 x-2 x-1 x x+1 x+2 x+3 y-3 -0.0212 0.0363 -0.0018 -0.0265 -0.0018 0.0363 -0.0212 y-2 -0.0070 0.0363 -0.0103 -0.0379 -0.0103 0.0363 -0.0070 y-1 0.0015 0.0363 -0.0154 -0.0447 -0.0154 0.0363 0.0015 y 0.0043 0.0363 -0.0171 -0.0469 -0.0171 0.0363 0.0043 y+1 0.0015 0.0363 -0.0154 -0.0447 -0.0154 0.0363 0.0015 y+2 -0.0070 0.0363 -0.0103 -0.0379 -0.0103 0.0363 -0.0070 y+3 -0.0212 0.0363 -0.0018 -0.0265 -0.0018 0.0363 -0.0212

2 Table F.5. Coefficients for 푑 푑푥푑푦

x-3 x-2 x-1 x x+1 x+2 x+3 y-3 -0.0302 0.0146 0.0177 0.0000 -0.0177 -0.0146 0.0302 y-2 0.0146 0.0329 0.0234 0.0000 -0.0234 -0.0329 -0.0146 y-1 0.0177 0.0234 0.0152 0.0000 -0.0152 -0.0234 -0.0177 y 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 y+1 -0.0177 -0.0234 -0.0152 0.0000 0.0152 0.0234 0.0177 y+2 -0.0146 -0.0329 -0.0234 0.0000 0.0234 0.0329 0.0146 y+3 0.0302 -0.0146 -0.0177 0.0000 0.0177 0.0146 -0.0302

2 Table F.6. Coefficients for 푑 푑푦2

x-3 x-2 x-1 x x+1 x+2 x+3 y-3 -0.0212 -0.0070 0.0015 0.0043 0.0015 -0.0070 -0.0212 y-2 0.0363 0.0363 0.0363 0.0363 0.0363 0.0363 0.0363 y-1 -0.0018 -0.0103 -0.0154 -0.0171 -0.0154 -0.0103 -0.0018 y -0.0265 -0.0379 -0.0447 -0.0469 -0.0447 -0.0379 -0.0265 y+1 -0.0018 -0.0103 -0.0154 -0.0171 -0.0154 -0.0103 -0.0018 y+2 0.0363 0.0363 0.0363 0.0363 0.0363 0.0363 0.0363 y+3 -0.0212 -0.0070 0.0015 0.0043 0.0015 -0.0070 -0.0212

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Appendix G. Number of spots versus 1D transformation kinetics

Figure G.1. BNL 0.5 mm capillary

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Figure G.2. ANL 0.7 mm capillary

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Figure G.3. ANL 0.7 mm capillary, continued.

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