Surface Freezing in N-Alkanes: Experimental and Molecular Dynamics Studies

Surface Freezing in n-Alkanes: Experimental and Molecular Dynamics Studies

DISSERTATION

Presented in Partial Fulfillment of the Requirements for the Degree Doctor of Philosophy in the Graduate School of The Ohio State University

Viraj Prakash Modak

Graduate Program in Chemical Engineering

The Ohio State University

2015

Dissertation Committee

Barbara E. Wyslouzil, Advisor

Sherwin J. Singer

Isamu Kusaka

David L. Tomasko

Viraj Prakash Modak

2015

Abstract

Crystallization from the melt is a common process encountered in both industrial and natural settings. Nucleation is the first step in the process and hence, understanding where nucleation occurs is critical to controlling the process. For systems with free surfaces, like droplets, nucleation can occur on the surface or throughout the bulk. This aspect of crystallization has been extensively debated for water droplets because of its implications in the atmospheric sciences. For intermediate chain length n-alkanes (14 < n < 50), experiments show that surfaces can freeze above the melting point. Since an organized surface can then template freezing of the bulk, these alkanes are hard to supercool. There are competing theories regarding the physics that dive the phenomenon, but both state that shorter alkanes will not surface freeze.

The goal of this work is to investigate surface freezing in n-alkanes, from both experimental and theoretical perspectives. Experiments will identify if surface freezing occurs for short chain n-alkanes containing 8 to 10 carbon atoms. Molecular dynamics will help identify the driving force.

In experiments, an alkane carrier gas-vapor alkane mixture flows through a supersonic nozzle and cools at a rate of ~106 K/s. Eventually the vapor alkane condenses initially forming liquid nanodroplet aerosols, that can then freeze if the temperatures are cold ii enough. Static pressure measurements characterize the flow; whereas x-ray scattering and

Fourier transform infrared spectroscopy, characterize the aerosol. Experiments yielded evidence for surface freezing in C8H18 to C10H22 droplets as well as estimates for the surface and volume based nucleation rates. In n-decane, decreasing the inlet conditions eventually led to the formation of nanoparticles that had a fractal-like structure and were not fully crystalline.

Molecular Dynamics (MD) simulations at the united atom level provided both visualization of surface freezing as well as estimates of the thermodynamic quantities and understand the driving force behind the phenomenon. Droplet simulations, for example clearly showed an organized monolayer developed within ~4 ns on a supercooled droplet of n-octane followed by freezing of the adjacent liquid in a layer-by-layer manner, confirming our experimental hypothesis.

To understand the driving force quantitatively, MD simulations were done crystals and slabs, to determine surface free energies of the liquid-vapor (LV) and the solid-vapor

(SV) interfaces. The usual pressure tensor approach sufficed for LV interfaces, but a new, less computationally intensive method was developed for the SV surface free energies.

The new method works well for the LJ solid and n-octane, but overestimates SV surface free energy for n-nonadecane. Simulations also provide estimates for the entropic changes associated with surface freezing. Overall these results suggest that the entropic contribution to the driving force is significant for n-octane but not for n-nonadecane.

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To Aai-Baba, who believe in me more than I do myself

Acknowledgements

I would like to thank my advisor, Prof. Barbara Wyslouzil – she is the best advisor, and the aerosol research lab is the best group that I could have asked for. She has been a constant inspiration to me throughout my time in graduate school. She has been instrumental in not just helping me reach my scientific goals, but also in improving my approach and thought process towards solving scientific problems. She has been a great source of knowledge and working with her has been a tremendous learning experience for me. I also greatly appreciate the independence as well as encouragement that she gave me to formulate and express my ideas and opinions freely. It has helped me indentify my own strengths as well as improve on my weaknesses.

I would also like to thank my co-advisor, Prof. Sherwin Singer. I had the opportunity to work with him on Molecular Dynamics simulations – something about which I had very little knowledge, when I started. Dr. Singer was more than welcoming to take a novice like me under his wings. He was extremely patient and I am sure had to endure all sorts of inane questions while I was on the learning curve. I still recall the initial few meetings that went on for hours in Dr. Singer’s office. Had it not been for his calm and encouraging demeanor, it would have been an overwhelming experience for me. It is

v solely due to Dr. Singer that I can at least speak the MD language, if not declare myself as someone who is well versed.

I consider myself lucky to have known and worked with two amazing scientists and advisors in Dr. Wyslouzil and Dr. Singer and I owe all my accomplishments completely to them.

I would like to thank Dr. Harshad Pathak, who was my mentor during the first few months when I joined the research group. He was my go-to person for anything that went wrong and for anything that I could not understand. I continued to seek his advice even after I gained a foothold in my projects, until he graduated in December 2013. I would also like to thank Andrew Amaya – he is an amazing person to work with. I still remember the crazy few months that we had around our trip to Stanford Linear

Accelerator Center (SLAC). Having colleagues like Harshad and Andrew turned it into a great, fulfilling experience.

I greatly appreciate the support and encouragement that I received from Dr. Ashutosh

Bhabhe and Dr. Kelley Mullick, when I joined the lab. I am also thankful my current and former colleagues, Dr. Shinobu Tanimura, Yensil Park, Kehinde Ogunronbi, Matt

Gallovic, Matt Souva, Gauri Nabar and Alyssa Robson for keeping the atmosphere in my workplace lively.

I would like to thank the National Science Foundation and the Ohio Supercomputer

Center for providing the funding and the computational resources respectively, for my research. I would like to thank the scientists at SLAC and Argonne National Lab (ANL) and in particular, our collaborators Dr. Hartawan Laksmono and Dr. Judith Wolk, for vi their help in running the X-ray scattering experiments. I would like to thank Dr. Soenke

Seifert, for the stimulating discussions and the unending supply of coffee during our trip to the Advanced Photon Source at ANL.

I have been lucky to have several friends who have made my life in Columbus enjoyable.

Special thanks to Dr. Nihar Phalak, Dr. Kalpesh Mahajan, Dr. Hrishikesh Munj, Mandar

Kathe, Dr. Niranjani Deshpande Dr. Anshuman Fuller, Prateik Singh, Dr.

Somsumndaram Chettiar, Dr. Shweta Singh, Dr. Shreyas Rao, Dr. Kartik

Ramasubramanian, and Dr. Preshit Gawade. I would also like to thank Sumant, Varun,

Ankita, Sreshtha, Dhruvit, Insiya, Varsha, Nitish, Aamena, Janani, Abhilasha, Deeksha,

Sourabh, Amoolya, Yaswanth, and Atefeh.

Last but not the least; I would like to thank my parents, my brother and my close family for their constant love and support. They are the ones that I turned to during times of stress and frustration. They never once doubted my capability to overcome any obstacle and have always encouraged me to pursue my dreams and for that, I am forever grateful.

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Vita

July 2006 – June 2010………………………. Bachelor of Chemical Engineering,

Institute of Chemical Technology,

University of Mumbai,

Mumbai, India

September 2010 – August 2011…..………… Graduate School Fellow

The Ohio State University

September 2010 – May 2013…..…………… MS in Chemical Engineering

The Ohio State University

August 2014 – December 2015…..…………. Graduate Teaching Associate

Engineering Education Innovation Center,

The Ohio State University

September 2010 – Present………..…………. Graduate Research Associate

The Ohio State University

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Publications

V. P. Modak, H. Pathak, M. Thayer, S. J. Singer, B. E. Wyslouzil, Experimental evidence for surface freezing in supercooled n-alkane nanodroplets, Physical Chemistry Chemical

Physics, 15, 6783-6795, (2013)

Fields of Study

Major Field: Chemical Engineering

Table of Contents

Abstract……………………………………………………………………………… ii Dedication…………………………………………………………………………… iv Acknowledgements………………………………………………………………….. v Vita…………………………………………………………………………………... viii List of Tables………………………………………………………………………… xiii List of Figures……………………………………………………………………….. xvi Chapter 1: Introduction……………………………………………………………… 1 1.1 Liquid-solid phase transitions……………………………………………….... 1 1.2 Background for current work………………………………………………… 2 1.3 Objective……………………………………………………………………… 6 1.4 Thesis outline…………………………………………………………………. 7 Chapter 2: Aerosol generation flow system and experimental techniques………….. 13 2.1 Experimental setup…………………………………………………………… 13 2.2 Pressure Trace Measurements (PTM) ……………………………………….. 16 2.3 Small Angle X-ray Scattering (SAXS) ………………………………………. 17 2.4 Fourier Transform Infrared Spectroscopy……………………………………. 19 2.5 Integrated data analysis………………………………………………………. 24 2.6 Wide Angle X-ray Scattering (WAXS) ……………………………………… 25 2.7 Materials……………………………………………………………………… 25 Chapter 3: Surface freezing in supercooled n-octane and n-nonane nanodroplets….. 30 3.1 Introduction…………………………………….…………………………….. 30 3.2 Experiments…………………………………….…………………………….. 32 3.3 Vapor – liquid condensation of n-nonane……………………………………. 33 3.4 Freezing of n-octane and n-nonane nanodroplets…………………………….. 41

3.5 Conclusion…………………………………….……………………………… 50 Chapter 4: Freezing of supercooled n-decane nanodroplets………………………… 55 4.1 Introduction…………………………………………………………………... 55 4.2 Experiments…………………………………….…………………………….. 57 4.3 Vapor-Liquid phase transitions……………………………………………… 58 4.4 Liquid solid phase transitions………………………………………………… 64 4.5 Conclusion……………………………………………………………………. 79 Chapter 5: Molecular Dynamics (MD) simulation methods and results from droplet simulations…………………………………….…………………………………….. 87 5.1 Introduction …………………………………….……………………………. 87 5.2 Molecular modeling parameters……………………………………………… 88 5.3 Droplet simulations…………………………………………………………... 91 5.4 Conclusion……………………………………………………………………. 94 Chapter 6: Chapter 6: Solid-vapor surface free energy……………………………… 96 6.1 Introduction………………………………………………………………...... 96 6.2 Creation of the free crystalline surface……………………………………….. 97 6.3 Gaussian approximation……………………………………………………… 98 6.4 λ integration…………………………………………………………………... 100 6.5 Umbrella Sampling…………………………………………………………… 101 6.6 Solid-vapor surface free energy of the LJ solid……………………………… 102

6.7 Approximation for P0 V ………………………………………………….... 105 6.8 Solid-vapor surface free energy of n-octane and n-nonadecane……………... 108 6.9 Conclusion……………………………………………………………………. 114 Chapter 7: Mechanism of surface freezing in n-alkanes…………………………….. 120 7.1 Introduction…………………………………………………………………... 120 7.2 Determination of the bulk melting temperature and the surface melting temperature……………………………………………………………………….. 125 7.3 Eliminating the possibility of surface melting for n-octane………………….. 132 7.4 Liquid-vapor surface free energy…………………..………………………… 133 7.5 Solid-vapor surface free energy…………………..………………………….. 138

7.6 Surface properties…………………..…………………..…………………….. 141 7.6.1 Liquid-vapor interface…………………..………………………………. 141 7.6.2 Solid-vapor and surface frozen interfaces………………………………. 143 7.7 Energy density across slabs…………………..………………………………. 145 7.8 Continuum Theory…………………..…………………..…………………..... 148 7.9 Discussion…………………..…………………..…………………………….. 150 Chapter 8: Conclusions and future work…………………..………………………… 160 Bibliography…………………………………………………………………………. 166 Appendix A: Matlab code to calculate phase-wise contribution from FTIR………. 187 Appendix B: Thermophysical properties of materials……………………………… 190 Appendix C: Fortran code to calculate thermodynamic unknowns with p and g as input…………………..…………………..…………………..……………………… 195

Appendix D: PTM and SAXS results for n-nonane pv0 = 489 Pa…………………… 230 Appendix E: Mathematica code to generate an n-alkane crystal…………………..... 231 Appendix F: GROMACS topology file for n-octane…………..……………………. 233 Appendix G: Sample GROMACS simulations parameter file……………………..... 236 Appendix H: Fortran code to calculate the orientational order in a liquid slab from the trajectory file…………………..…………………..…………………………….. 238 Appendix I: Fortran code to calculate the fluctuations along the molecular axis from the trajectory file…………………..…………………..……………………….. 241 Appendix J: Fortran code to calculate the energy densities of either a solid, liquid or a surface frozen slab from the trajectory file as a function of slab length………... 243

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List of Tables

Table 3.1: A summary of the experiments conducted. Here T0 and p0 are the stagnation temperature and pressure, and pv0 is the condensible partial pressure at the start of the a expansion. Ton and pon are the temperature and pressure of the gas mixture at the onset of the vapor to liquid (v→l) phase transition; Texit is the temperature 70 mm downstream of the throat near the nozzle exit. The equilibrium melting temperatures of n-octane and n- nonane are 216 K and 220 K, respectively...…………………………………………… 33

Table 3.2: The nucleation rates for surface and bulk nucleation; t0S and t0V are the times at the onset of surface and bulk freezing, respectively. TS is the temperature at t0S and TV is the temperature at t0V. JS is the nucleation rate for the first nucleation event on the surface. JV can be calculated either by assuming the second nucleation event occurs in the bulk of a surface frozen droplet or heterogeneously at the existing surface monolayer…………………………………………………………………………….… 49

Table 4.1: Summary of the experimental conditions. The inlet conditions are denoted by stagnation temperature T0 , stagnation pressure p0 and condensible partial pressure pv0 .

The onset temperature is the temperature of the vapor-liquid phase transitions. Texit is the temperature at 102 mm downstream of the throat near the nozzle exit. Tm , the equilibrium melting point of n-decane is 243.51 K...…………………………………... 58

Table 4.2: The locations of the relevant peaks observed in the normalized absorptivities for liquid n-octane, n-nonane and n-decane agree with each other well. All the values reported are in the units of cm-1………….……………………………………………... 63

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Table 4.3: The locations of the absorption peaks observed in the solid normalized absorptivities for n-octane and n-decane differ from each other by less than 0.5 cm-1. All the values reported are in units of cm-1………………………………………………..... 67

Table 4.4: Locations of the absorption peaks observed in the n-decane frustrated crystalline spectrum are compared to the corresponding values for solid and liquid decane. All the values reported are in the units of cm-1……………………………..….. 76

Table 5.1: Values of parameters used in Eq. (5.1 – 5.4) governing the interaction potentials…………….………………………………………………………………….. 90

Table 6.1:  SV values calculated for an LJ solid, n-octane and n-nonadecane, using multiple methods. The temperature and for the LJ solid are in reduced units. For n- octane and n-nonadecane they are in K and mN/m, respectively…………………...… 115

Table 7.1: Estimates for solid-vapor, solid-liquid, and liquid-vapor surface tensions as 6 5 estimated by DS and TR for alkanes C16 and C36, where Cn is an abbreviation for     s  CnH2n+2. The latter were estimates based on  sv   , sl  lv cos   sl  lv (zero  l  contact angle for a liquid alkane droplet on the solid).5 In both cases the liquid-vapor surface tension near the melting point, the only interface amenable to experimental measurement, reflects the empirically observed value of 28 mN/m, insensitive to chain length……………………………………………………………...…………………… 122

Table 7.2: Surface free energy values for liquid vapor interface of n-octane as a function of temperature. Pressure tensor values are calculated by averaging data over 150 ns. Tail corrections to incorporate atomic interactions outside the cutoff range are obtained from the density profile of the system, using the approach of Blokhuis et al. Also surface tension values are in units of mN/m..…………………………………………………. 136

xiv obtained from the density profile of the system, using the approach of Blokhuis et al. All surface tensions are given in units of mN/m………………………………………...… 137

Table 7.4: Surface excess energy values for liquid, surface frozen and solid slabs respectively. The values are calculated based on the standard definition of the Gibbs dividing surface………………………………………………………………………... 146

Table 7.5: Surface number densities and surface energies calculated using continuum theory for XZ, YZ and XY surfaces and surface free energies calculated using our novel method for XY, XZ and YZ surfaces. The representative slab has 720 octane molecules, with 6 layers of 10 x 12 = 120 molecules in each layer. The Lx, Ly and Lz dimensions used are for T = 190 K for example………………………………………………….... 150

Table 7.6: Values for the change in entropy for Surface Freezing and Bulk Freezing for octane and nonadecane. SM and SSF are calculated using Eq. (7.18) and (7.19), respectively. SM ' is the change in entropy for bulk freezing calculated at Tm using Eq.

(7.20) and (7.21), respectively. SM ' and SSF are significantly different for octane, suggesting a higher entropy for the surface frozen monolayer. However, the values are close to each other for nonadecane, which implies that the solid bulk is not too different from the surface frozen monolayer………………………………………………….… 154

Table B.1 Thermophysical properties of n-octane……………………………………...190

Table B.2 Thermophysical properties of n-nonane……………………………………..191

Table B.3 Thermophysical properties of n-decane……………………………………..192

Table B.4 Thermophysical properties of Nitrogen……………………………………..193

Table B.5 Thermophysical properties of Argon……………………………………...... 193

List of Figures

Figure 1.1: ΔT as a function of carbon chain length n. The black circles in both (a) and (b) are from experimental studies Ocko et al The solid lines are the predicted behavior based on the theories by (a) Deutsch and coworkers and (b) Tkachenko and Rabin. In both studies, it was found that ΔT approaches zero at n = 14.…………………………... 6

Figure 2.1: Schematic of the continuous flow supersonic nozzle apparatus………….... 15

Figure 2.2: Internals of the supersonic nozzle. The red arrow indicates the direction of the flow. The sidewall shown here has mica windows. Pressure taps are located at the entry, throat and the exit of the nozzle………………………………………………………… 16

Figure 2.3: Scattering intensity I (blue) as a function of scattering vector q, measured

9.51 cm downstream of the throat. The inlet partial pressure pv0 of n-decane was 433 Pa. The data are fit assuming scattering from a Schulz distribution of polydisperse spheres (red)…………………………………………………………………………………...… 18

Figure 2.4: Schematic of the setup for conducting FTIR absorption spectroscopy experiments using Perkin Elmer 100 instrument……………………………………….. 20

Figure 3.1: (a) The temperatures of the condensing flow, Tflow (solid lines), the expected isentropic temperature profile, Tisentrope (dashed lines), and the aerosol droplets Tdrop (open circles) as a function of position. The black short dashed line corresponds to the equilibrium melting temperature of n-nonane. (b) The mean particle radius as a function of position measured by SAXS .………………………………………………………... 34

xvi difference in the curves near 2926 cm-1 and 2855 cm-1 is consistent with the presence of an extra CH2 group in n-nonane relative to n-octane………………………………...… 37

Figure 3.3: (a) The normalized absorptivities of n-nonane liquid from supersonic nozzle for three different partial pressures of condensible. Despite the ~25 K temperature difference, the spectra agree well, validating our assumption that the droplets are not frozen in these experiments. (b) The normalized absortivities of n-nonane liquid from direct transmission measurements and aerosol measurements. The wavenumber ranges for C-H stretch vibrations are as follows; CH3: asymmetric stretch 2962 ± 10, symmetric stretch 2872 ± 10; CH2: asymmetric stretch 2926 ± 10, symmetric stretch 2855 ± 10.... 39

Figure 3.4: (a) The observed normalized aerosol absorptivity 49 mm downstream of the throat for the experiment conducted with pv0 = 625 Pa is well fit by Eq. (2.7). The material balance for this particular position deviates from 1.0 by only ~1.4%. (b) The values of gi/g∞ obtained from the integrated analysis based on PTM and SAXS are compared to those obtained by FTIR. The lower plot illustrates the material balance obtained from fitting FTIR spectra, where M.B.= (gl+gv)/g∞.……………………….... 40

Figure 3.5: Top: The mixture and particle temperatures as a function of position downstream of the throat for (a) n-octane and (b) n-nonane. The long dashed and solid lines represent the temperature profiles for an isentropic expansion and the condensing flow, respectively. The dash dotted lines correspond to the equilibrium melting temperatures. The open circles represent the temperature of droplets. Bottom: The mean particle radius determined from SAXS for (c) n-octane and (d) n-nonane. The vertical dashed lines correspond to the positions downstream of the throat where the phase transitions occur .……………………………………………………………………….. 43

Figure 3.6: Normalized absorptivity of n-octane liquid from direct transmission measurements and aerosol measurements……………………………………………… 44

Figure 3.7: Normalized aerosol absorptivities for solid n-octane and n-nonane. These spectra are determined assuming that the aerosol is entirely frozen near the exit of the nozzle…………………………………………………………………………………… 45

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Figure 3.8: The distribution of the condensible between the vapor, liquid and solid phases as determined by FTIR for n-octane and n-nonane, together with the material balance. The trends observed are as expected, i.e. the condensible is initially all in the vapor phase, the initial condensate is liquid, and as the solid phase grows in the concentration of the liquid phase approaches zero. Here mass balance, M.B.= (gl+gv+gs)/g∞.……..…… 46

Figure 3.9: The fraction of solid alkane present in the condensate as a function of time. The symbols are based on the FTIR measurements. The red dashed lines represent the fraction of solid corresponding to a monolayer of alkane on surface of every droplet. The black dashed lines represent the fits used to determine the freezing rates. The presence of a kink in the experimental data suggests that freezing occurs as a two step process that we interpret as surface ordering followed by bulk freezing………………………………... 47

Figure 4.1: (a) The normalized absorptivity for n-decane vapor is compared to those measured for n-octane and n-nonane in our earlier work. (b) The integrated absorptivity, between 2500 cm-1 and 3400 cm-1, for n-decane follows the linear trend reported by Klingbeil et al…………………………………………………………………...……… 59

Figure 4.2: (a) Temperature profiles as a function of position derived from the integrated analysis based on PTM and SAXS results. The equilibrium melting temperature of decane is indicated by Tm . (b) The mean particle radius (black circles) and σ (red triangles) measured by SAXS . (c) A typical fit of the polydisperse sphere model (red) to the scattering intensity I(q) (blue). (d) The normalized nucleation rate J/Jmax calculated using Classical Nucleation Theory (red solid line) as a function of time. The experimental number densities calculated from SAXS (black circles) are compared to the predicted number densities, calculated by integrating the nucleation rate curve with respect to time and scaling to match the experimental values at the exit.………...……………………...61

xviii compared to that for n-nonane from previous experiments. The peak locations, indicated by the vertical grey dashed lines, match well across the chain length, and the intensity of the CH2 stretches increases with chain length. The temperature of the droplets corresponding to the n-decane spectrum is 235.26 K. The n-nonane spectrum was calculated by averaging three spectra recorded for droplets between 206 K and 234 K………………………………………………………………………………………….62

Figure 4.4: The vapor and liquid mass fractions calculated from integrated data analysis using the results of PTM and SAXS are compared to those obtained from the FTIR data. Agreement between the two approaches is very good. The bottom graph represents the material balance for the mass fractions calculated by FTIR where MB =

(gv+gl)g∞……………………………………………………………………………...….64

Figure 4.5: Temperature profiles from the integrated PTM + SAXS analysis (a) and (c), and mean particle radius from SAXS (b) and (d) for Exps (ii) and (iii), respectively. The black solid line in the temperature profile is T flow , whereas the black long dashed line is

Tisentrope. The horizontal short dashed line indicates the equilibrium melting temperature of n-decane. The vertical dashed lines indicate the onsets of condensation and freezing….65

Figure 4.6: (a) Normalized absorptivities of the near the exit calculated for different experimental conditions match each other well. The red line corresponds to Exp (ii) with

= 13.54 nm and Td = 207.32 K. The blue line corresponds to Exp (iii) with = 10.2 nm and Td = 200.2 K. In both cases the aerosol droplets are assumed to be completely frozen. (b) Normalized absorptivity for solid n-decane (red), agrees well with that measured for n-octane (black) in earlier experiments.………………….………………..66

Figure 4.7: Phase-wise mass distribution calculated from FTIR experiments as a function of position for (a) Exp (ii) with pv0 = 264 Pa and (b) Exp (iii) with pv0 = 219 Pa. The symbols in the upper part of each graph are the mass fractions of vapor (green), liquid (red) and solid (blue). The lower part of each graph shows the unconstrained material balance. The solid lines are meant to guide the eye only, the dashed vertical line indicates the location of the second heat release…………………………………………………...68

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Figure 4.8: Fraction of solid as a function of time for (a) pv0 = 264 Pa and (b) pv0 = 219 Pa. The black dashed lines are fits to the exponential equations used to estimate the surface and/or volume based freezing rates. The red dashed line in (a) indicates the hypothetical fraction for a fully developed monolayer on the surface…………...…….. 69

Figure 4.9: WAXS measurement conducted at 6.9 cm downstream of the throat. The 2-D detector image represents scattering intensity data after a sample to background ratio is taken……………………………………………………...... ………….. 70

Figure 4.10: Temperature profile derived from PTM for pv0 = 124 Pa. The black dashed line is Tisentrope and the black solid line is Tflow. The black circles represent the droplet temperature, Tdrop. At the exit, Tflow is close to 60 K below the equilibrium melting point Tm………………………………………………………………………………………..71

Figure 4.11: Scattering intensity as a function of scattering vector measured at the exit of the nozzle for Exp (v). The SAXS intensities and the fits are shown in blue and red, respectively. (a) The measured data are compared to the expected scattering from polydisperse spheres with = 8.90 nm and polydispersity = 0.3. The parameter choice is discussed in the main test. (b) The fractal polydisperse spheres model provides a better fit to these data………………………………………………………………………….. 73

Figure 4.12: Scattering intensity (blue) as a function of scattering vector measured near the nozzle exit is fit to a unified power law model (red) for Exp (v). Here Rg is 16.9 nm and the power law exponent, P is 2.6 suggesting the particles are mass fractals….….... 74

Figure 4.13: (a) Normalized absorptivity of the “frustrated” crystalline state of the solid calculated 95.1 mm (red), 98.1 mm (black) and 101.1 mm (blue) downstream of the throat. The solid concentration was based on 90% of the vapor having condensed. The grey dashed lines show the locations of the main absorption peaks for the frustrated crystal. (b) FTIR spectra measured near the exit for pv0 = 124 Pa (red), cannot be fit (black line) to a linear combination of the vapor, liquid and solid reference spectra………………………………………………………...………………………… 75

Figure 4.14: Phase-wise mass distribution calculated from FTIR experiments as a function of position for vapor (green), liquid (red) and frustrated crystal (blue). The symbols in the top part are gi / g for the three phases. The lower part shows the material balance…………………………………………………………….……………77

Figure 4.15: Schematic of the fractal aggregate formation for Exp (v). The vapor molecules condense forming particles which freeze immediately. Subsequent growth occurs as molecules condense on the frozen particles forming domains. These domains then freeze forming fractal aggregates...………………………………………………... 79

Figure 5.1: n-octane droplet profiles at different imes in a 250 ns run. These snapshots do not fully convey the degree of chain freezing because the chains do not appear ordered unless they are viewed from the proper angle. For example, at 15ns the droplet surface is uniformly covered by a solid layer, even though only a fraction of the surface appears ordered in the figure. Rotation of the viewing angle brings some regions into alignment and moves other regions out of alignment………………………..…………………….. 92

Figure 5.2: Lennard-Jones, dihedral potential, angle-bending, and bond-stretching contributions to the potential energy of the droplet as a function of time. The initial decrease is due to short range relaxation as the configuration initially equilibrated at 230K is quenched to 190K, and surface freezing. Two steps in the potential energy at roughly 25 and 85ns track the layer-by-layer freezing of the droplet from the exterior surface…………………………………………………………………………………... 93

Figure 6.1: Creation of the crystal-vapor surface from initial state “0” (left) to the final state “1” (right)………………………………………………………………………..... 97

Figure 6.2: Crystal-vapor free energy of the (111) face of the Lennard-Jones crystal in

* reduced units,  SV , as calculated (curve) by Broughton and Gilmer using thermodynamic integration from 0K1 ,2 are estimated here by digitizing Fig. 9 of ref.2. as calculated (black diamonds) by thermodynamic integration using either the Bennett acceptance ratio method, Eq. (6.10-6.11) or the Zwanzig formula with P0 V  obtained from umbrella

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* sampling and the WHAM method.  SV using the first cumulant (black squares), second cumulant (black triangles) approximation, or Gaussian approximation (open circles) in Eq. (6.5)…………………………………………...... 103

1  V  V  Figure 6.3: (a)  ln P V  and (b) e 0 P V  calculated for an LJ solid using  0 0

V WHAM. varies linearly with V away from 0 ………….……. 104

1 Figure 6.4: Qualitative behavior of (a) f V    ln P V  and (b) f V  V ……...105 0  0 0

Figure 6.5: (Left) Schematic of the six-layered n-octane crystal used to calculate the lattice parameters and the solid-vapor surface free energy. (Right) Box dimensions of the n-octane crystal as a function of temperature. The box increases in size only in the x and the y directions, whereas the box height along z remains relatively constant………… 109

Figure 6.6: Solid-vapor surface free energies of n-octane calculated using a Gaussian approximation. As described in the text the value at 210 K (32.75 mN/m) is in good agreement with a value obtained using the BAR method (34.74 mN/m)………………111

Figure 6.7: (a) Box dimensions of the n-nonadecane crystal as a function of temperature. The box increases in size only in the x and the y directions, whereas the box height along z remains relatively constant. (b) Solid-vapor surface free energies of n-octane calculated using a Gaussian approximation………………………………………………………. 112

Figure 6.8: Layer spacing in a six-layered (a) n-nonadecane slab and (b) n-octane, compared to that for a crystal. The x-axis in the plot represents the location of the spacing within the system. Location 1 on the x-axis represents the difference between the Z coordinate of the centers of masses of layers 1 and 2. Location 2 represents the difference for layers 2 and 3, and so on. The surfaces in the slab are in the XY plane adjacent to layers 1 and 6. The layer observed in drift in the slab for n-nonadecane is more pronounced than that for n-octane…………………………………………………….. 113

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Figure 6.9: Difference in the density for each layer as a function of position, calculated when the layer is in a slab and when the layer is in a crystal. Each color represents a layer. The peaks and the troughs observed for each layer suggest that in a slab, suggests layer “broadening”…………………………………………………………………….. 114

Figure 7.1: (a) Crystal-Melt starting configuration for determining the bulk melting temperature. (b) Potential Energy (PE) profiles for the system in (a) is simulated for 3 ns. The PE of the system increases with respect to time for 216 K (red) and decreases for 214 K (blue), whereas, it remains relatively flat for 215 K (black)……………………...… 126

Figure 7.2: (a) Starting configuration of a slab that is partially melted and partially surface frozen. This system is simulated to determine the surface freezing temperature. The slab at 211 K (b) does not show a tendency to transition into being completely surface frozen or completely melted. The final configuration after 7 ns shows that the slab is completely surface frozen at 210 K (c) and progressing towards complete surface disorder at 212 K (d). This shows that the surface freezing temperature for octane is 211 K……………………………………………………………………………………….. 128

Figure 7.3: Potential Energy profiles to estimate the surface freezing temperature. PE of the system increases at 212 K, decreases at 210 K and remains flat at 211 K, which is the surface freezing temperature…………………………………………………………... 129

Figure 7.4 (a) PE plot for n-nonadecane for PYS to estimate the bulk melting temperature where we use a partially frozen system of 960 molecules. The PE of the system increases with time for 326 K (red) and decreases with time for 321 K (blue). It remains relatively flat for 323 K (black) and 324 K (grey). (b) PE of the partially surface-frozen system of 960 n-nonadecane molecules tends to increase at 329 K (red) and decrease at 327 K (blue) and relatively stable at 328 K. Similar results are shown for the TraPPE model in (c) and (d). The PE of the partially frozen slab is stable at 324 K, whereas that for the partially surface frozen the PE is stable at 323 K……………………….…………….. 131

Figure 7.5: (a) n-octane slab with melted surface layers and crystalline bulk. The surface is on both sides of the slab in the Z directions. (b) The final trajectory after configuration

xxiii in (a) is simulated for 15 ns with the entire system at 215 K. The melted surface layers go back to being crystalline, implying that surface melting cannot be observed for short chain alkanes…………………………………………………………………………... 133

Figure 7.6: A representative reduced density profile of an octane slab simulated at 205 K as a function of reduced slab length. The dots represent the data from MD simulations and the line is the tanh fit according to Eq. (7.7). The fit parameters for this data set are:  =2.025, a = 18.35, b = 6.25,  =0.26………………………..……………………. 135

Figure 7.7: Temperature dependence of the liquid-vapor surface free energy ( ) for octane (a) and nonadecane (b,c) as listed in Tables 7.2 and 7.3. C19 results for the PYS and TraPPE interaction potentials are given in (b) and (c), respectively. From the linear fits indicated in the plots, we obtain the surface excess entropy of the liquid-vapor interface which is reported in the text……………..…………………………………………….. 138

Figure 7.8: Orientational order represented as number density of vectors as a function of Z and cosθ, where θ is the angle of the vector with the Z axis, (a) for octane at 217 K, . The peaks at ~4 and ~11 nm imply that the molecules near the surface have a tendency to orient themselves perpendicular to the surface. Orientational ordering is also observed for nonadecane using (b) PYS model at 333 K, but is significantly reduced for (c) TraPPE model at 325 K. All the temperatures here are slightly higher than Tm as calculated using the respective models.…………………...…………………………………………….. 143

Figure 7.9: Probability density function of difference between the Z coordinates of the center of masses of nearest neighbor molecules. The red symbols represent an ordered monolayer of a surface frozen slab , and the black symbols molecules in the interior of a completely frozen solid slab calculated at TSF for the molecule and model. The blue symbols for nonadecane represent the interior of a solid slab at a supercooled temperature 2 of 300 K. The lines in the plots represent the Gaussian fits for the histogram data. σB and 2 σSF for (a) octane at 211 K is 0.0198 and 0.005, respectively. Those for (b) nonadecane, using PYS at 329 K are 0.0433 and 0.0331, respectively and for (c) nonadecane, using

xxiv

2 TraPPE at 323 K are 0.0428 and 0.0361, respectively. σB for nonadecane at 300 K is 0.0166 for PYS and 0.0182 for TraPPE………………………...... 145

Figure 7.10: Energy density (kJ/nm3) of solid (blue), liquid (red) and surface frozen (black) slabs as a function of Z, shown for octane (a) and nonadecane (c) shows that there is a significant energy gain when the slab surface freezes. Number density (nm-3) of united atoms as a function of Z for octane (b) and nonadecane (d) shows that the density of the surface frozen layer is lower than that of the solid. The Gibbs dividing surface for each of the slabs is located at the x-axis origin and the data is shifted accordingly.….. 147

Figure 7.11: (a) Schematic used to develop continuum theory for estimating the surface excess energy, Eex of a slab consisting of molecules stacked in a layered format. Eex is calculated via the missing LJ interactions upon creation of an imaginary surface. (b) The distance between two layers along this plane, ay is used to calculate Eex, for a surface in the XZ plane. A similar approach is used for the YZ plane. (c) For an XY surface the parameters are given by the layer thickness lz and the distance, az, between the bisecting planes of the layers.…………………………………………………………………..... 149

Figure 7.12: Enthalpies of bulk solid and liquid calculated from simulations as a function of temperature. (a) and (b) are for the octane liquid and solid respectively. Data for nonadecane liquid and solid is shown for the PYS model in (c) and (d) and for the TraPPE model in (e) and (f). The slopes of the linear fit to the enthalpies represent the specific heat capacities CP…………………………………………………………….. 155

Figure D.1: (a) The temperatures of the condensing flow, Tflow (solid lines), the expected isentropic temperature profile, Tisentrope (long dashed lines), and the aerosol droplets Tdrop (open circles) as a function of position for experiments conducted with pv0 = 489 Pa. The short dashed line corresponds to the equilibrium melting temperature Tm of n-nonane. The droplet temperature is above Tm and hence they are not frozen. (b) The mean particle radius as a function of position measured by SAXS………………..230

xxv

Chapter 1: Introduction

1.1 Liquid-solid phase transitions

Liquid-solid phase transitions are important and fundamental processes1 that we routinely encounter in the natural environment and in industry. In the atmosphere, the most common example of this process is freezing of water droplets. Another example is the formation of stalactites/stalagmites in caves, on bridges and in mines. In the industry, controlling liquid-solid phase transitions have found a wide range of applications. For example, is critical in the manufacture of semiconductors and for the production of high purity chemicals in the pharmaceutical industry.2 An example of an undersired liquid- solid phase transition is the formation of wax deposits in crude oil transportation pipelines. Wax deposits increases pressure losses, which then translates directly into increased operating costs.

In order to control crystallization, it is important to understand the key parameters that can affect this process, i.e. the degree of supersaturation, the sample size and purity, the cooling rates etc. Achieving this goal is difficult because liquid-solid phase transitions are not as well understood as vapor-liquid transitions. Some of the challenges include the presence of intermediate metastable states between the liquid and the most stable crystal, and the possibility for the system to get trapped in a glassy, amorphous or a frustrated

1 crystalline state.3 Given its complexity and importance, crystallization is an interesting phenomenon to study, from a fundamental as well as practical perspective.

Nucleation is the first step in crystallization. Nucleation is interpreted as the formation of a critical nucleus or cluster that is large enough to grow. Nucleation can be either heterogeneous or homogeneous. In heterogeneous nucleation the critical nucleus is formed on a surface present in the system. In homogeneous nucleation, the molecules in the liquid form a critical nucleus, in the absence of any foreign surfaces or particles.

Since nucleation is the first step in the process, it is often the rate-determining step and can influence how freezing progresses. In systems with free surfaces such as droplets, the critical nuclei can form at or near the surface, or in the bulk. Hence, understanding which nucleation pathway dominates is critical to controlling these phase transitions.

Furthermore, identifying the thermodynamic driving forces that determine the preferred location for nucleation can deepen our knowledge about this process.

1.1 Background for current work

As noted above, in systems with free surfaces, the critical nucleus can be formed near the surface. For water droplets, this has been a subject of some controversy4-9 in the atmospheric science community. For micron-sized droplets of water the nucleation rates scale well with the droplet volume.4 For smaller droplets, the the current experimental techniques are not capable of distinguishing between nucleation on the surface and in the bulk.7 Furthermore, the most definitive evidence for surface induced freezing comes from experiments with chain-like molecules, in particular the n-alkanes and n-alcohols.10-16

Extensive surface tension and X-ray measurements on bulk samples of intermediate chain

2 length alkanes (15 ≤ n ≤ 50) show that they form a dense frozen monolayer on the surface at temperatures up to 3 K above the equilibrium melting point.11, 13 Supercooled microdroplets of odd n-alkanes in the range of 14 ≤ n ≤ 17, also appear to form an ordered monolayer on the surface that may serve as the site for templated freezing of the entire droplet.17 None of these studies, however, reported surface freezing rates.

Although surface induced freezing has been established for these molecules, the driving force behind this process is still a matter of debate, with two distinct arguments prevailing.18-20

The first point of view championed by Deutsch and co-workers is based on a “wetting” argument. In this theory surface freezing is thought to occur due to the solid wets the liquid surface. The wetting free energy can be expressed as:

(0)    sv  sl  lv , (1.1)

where sv ,  sl and  lv are the surface free energies of the solid-liquid, liquid-solid, and liquid-vapor interfaces, respectively. According to this argument, surface freezing will occurs when,

(0)   0 (1.2)

Deutsch and coworkers estimated , and and concluded that  sv   sl was below . in the range of chain lengths where surface freezing was observed.

The second point of view proposed by Tkachenko and Rabin, maintains that surface freezing arises from the inherent differences between the properties of the surface monolayer and the bulk solid. In particular they suggest, that fluctuations along the molecular axis lead to an entropic stabilization of the surface monolayer. As a result surface freezing can occur even when the wetting condition in Eq. (1.1) and (1.2) is not satisfied. The condition for surface freezing to occur via such a mechanism is:

(0) S sl   N(T  Tm )   s  0 , (1.3) A0 where, the first two terms are the energy costs to create the surface frozen layer – the wetting free energy cost and the bulk free energy cost for creating surface frozen layers as a linear expansion about the melting point where it is zero. In addition, A0 is

the area per molecule, Ssl is the gain in entropy per molecule for a bulk solid to liquid transition and Δs is the entropic stabilization per unit area for N surface layers relative to the bulk, where fluctuations are suppressed. They further derived that for surface freezing, N = 1, and for surface freezing to occur at T > Tm, Eq. (1.3) is effectively reduced to:

(0)   s  0 (1.4)

Tkachenko and Rabin also estimated the surface free energy values and found that  sv

was less than  lv in the range of chain lengths, , where surface freezing was observed.

It is clear from the literature, that the main point of the disagreement between the two

groups is the value for the solid-vapor surface free energy,  sv . Although there are methods21, 22 which calculate , from molecular simulation data, these have been tested only for the Lennard-Jones (LJ) solid. Furthermore, being computationally intensive, they are not convenient for more complex solids. Thus, developing alternative methods that can estimate , for more complex molecules, including straight chain n-alkanes is a worthwhile task. This is a critical step toward understanding and quantifying the energetic contributions to the driving force to surface freezing.

Finally, in both experimental and theoretical studies, the existing literature reports that surface freezing phenomenon is restricted to intermediate chain alkanes, and to temperatures above or near the equilibrium melt temperature.14, 15, 18, 23-25 Current papers also suggest that short chain alkanes will exhibit the more common surface melting behavior, as for n < 14. Since the difference between the equilibrium melting point and the surface freezing point, ΔT approached zero for n = 14. Fig. 1.1 illustrates the behavior of ΔT is plotted as a function of n, as measured in experiments, by Ocko et al.14 The solid lines in (a) and (b) are the predicted behaviors based on the theories proposed by Deutsch and coworkers and Tkachenko and Rabin,18 respectively. Both theories agree with the experimental data well. As a result, it is hard to identify which approach is more plausible.

3.5 3.5

3.0 3.0

2.5 2.5

2.0 2.0

-T

s 1.5 s 1.5

1.0 1.0

0.5 0.5

0.0 0.0

-0.5 10 20 30 40 10 20 30 40 n n

Figure 1.1: ΔT as a function of carbon chain length n. The black circles in both (a) and (b) are from experimental studies Ocko et a.l14 The solid lines are the predicted behavior based on the theories by (a) Deutsch and coworkers and (b) Tkachenko and Rabin.18 In both studies, it was found that ΔT approaches zero at n = 14.

1.2 Objective

We have tried to extend these studies using both experimental and theoretical approaches, but with a focus on short chain n-alkanes. This work has two key objectives One of the

The first objective is to determine experimentally, whether we can observe surface freezing for short chain alkanes (C8 – C10). Experiments will use a continuous flow supersonic nozzle apparatus, because nucleation is suppressed in small liquid nanodroplets and hence we can study droplets that are highly supercooled. Furthermore, since the surface to volume ratio is high, the signal from the surface can be a large fraction of the total signal. Thus we can investigate if experimental conditions affect the nucleation site (surface/bulk) as well as the crystal structure of the frozen droplets. The

6 second objective of this work is to understand the behavior of n-alkanes at a molecular level and identify the driving force and the mechanism behind surface freezing. In particular we would like to estimate the energetic and the entropic aspects of surface freezing to determine, which term dominates. Here, we use use Molecular Dynamics

(MD) simulations with the interaction potentials described by a united atom model, i.e. the CH3 and CH2 groups are considered as single atoms.

1.3 Thesis outline

This thesis is organized as follows. Chapter 2 summarizes the experimental methods used in this work. It begins with a description of the supersonic nozzle setup and flow system used to generate and freeze nanodroplet aerosols of n-alkanes. Then, the complementary experimental techniques used to characterize the flow and the aerosol, are presented.

These include Pressure Trace Measurements (PTM), Fourier Transform Infrared

Spectroscopy (FTIR), Small Angle X-ray Scattering (SAXS) and Wide Angle X-ray

Scattering (WAXS). Finally the integrated data analysis method that we use to interpret data obtained from multiple techniques and transform them into a single set of consistent results, is outlined.

Chapter 3 focuses on the experiments conducted using n-octane and n-nonane as condensibles. Using PTM, FTIR and SAXS we followed the nanodroplets of n-octane and n-nonane, as they froze in a two step manner. Since, the droplets are subjected to a high degree of supercooling, the surface freezes first, followed by the freezing of the rest of the drop. We also estimated the surface and the volume based nucleation rates for freezing.

Chapter 4 presents the results from experimental studies with n-decane as condensible, run at different inlet conditions. The goal was to see how the droplet size and temperature affect the freezing process. n-decane follows a similar two-step freezing process observed for n-octane and n-nonane. At a lower inlet mole fraction of n-decane to the supersonic nozzle, surface induced freezing was not observed. Pioneering WAXS experiments confirmed the crystalline nature of these frozen n-decane particles. At the lowest inlet mole fraction of n-decane, the particles are no longer spherical; rather they are better described by a complex fractal aggregate-like structure. Furthermore, these particles were not fully crystalline. FTIR showed presence of peaks characteristic of both the crystalline and the liquid states but the relative peak intensities differed.

Chapter 5 briefly summarizes the MD simulation methods used in the computational studies of surface freezing. It presents the models used to describe the intra and intermolecular interactions for n-alkanes as well as other control parameters in the simulations. Our initial results from simulations on nanodroplets are shown and the results confirm the hypothesis presented in Chapters 3 and 4, i.e. freezing in supercooled nanodroplets of short chain n-alkanes is initiated on the surface followed by freezing inside the drop. Simulations suggest that freezing inside the drop occurs in a layer-by- layer basis

Chapter 6 highlights our new method to estimate sv , for fully frozen systems from MD simulations. We initially, test the method for a simple system of LJ hard spheres and compare it to values available in the literature obtained from other standard methods published in the literature. We then apply it to two n-alkanes: n-octane and n-nonadecane.

Our new method works well for the LJ system as well as for n-octane but is less accurate for n-nonadecane. We also investigate the origin of the discrepancy for n-nonadecane by getting a better appreciation of the fluctuations across the crystal vapor interface.

Chapter 7 presents the results of further MD studies on n-octane and n-nonandecane bulk crystals, as well as liquid, surface frozen and crystalline slabs. From the simulations we were able to show that there are both energetic and entropic contributions to the driving force to surface freezing in n-alkanes. In this chapter we also tie in these results to those described in Chapter 6. We then describe how our results compare with the existing arguments in the literature, as to what drives surface freezing.

The concluding remarks for this work are made in Chapter 8 along with potential opportunities to extend this work in the future.

References:

1 Kelton KF. Solid State Physics. Vol. 45. 1991; 75-177.

2 Arkenbout G. Melt Crystallization Technology: CRC Press 1995.

3 Shintani H, Tanaka H. Frustration on the way to crystallization in glass. Nature

Physics. 2006; 2: 200-06.

4 Duft D, Leisner T. Laboratory evidence for volume-dominated nucleation of ice

in supercooled water microdroplets. Atmospheric Chemistry and Physics. 2004; 4:

1997-2000.

5 Tabazadeh A, Djikaev YS, Reiss H. Surface crystallization of supercooled water

in clouds. Proceedings of the National Academy of Sciences of the United States

of America. 2002; 99: 15873-78.

6 Vrbka L, Jungwirth P. Homogeneous freezing of water starts in the subsurface.

Journal of Physical Chemistry B. 2006; 110: 18126-29.

7 Sigurbjornsson OF, Signorell R. Volume versus surface nucleation in freezing

aerosols. Physical Review E. 2008; 77.

8 Kuhn T, Earle ME, Khalizov AF, Sloan JJ. Size dependence of volume and

surface nucleation rates for homogeneous freezing of supercooled water droplets.

Atmospheric Chemistry and Physics. 2011; 11: 2853-61.

9 Kay JE, Tsemekhman V, Larson B, Baker M, Swanson B. Comment on evidence

for surface-initiated homogeneous nucleation. Atmospheric Chemistry and

Physics. 2003; 3: 1439-43.

10 Ofer E, Sloutskin E, Tamam L, Ocko BM, Deutsch M. Surface freezing in binary

alkane-alcohol mixtures. Physical Review E. 2006; 74. 10

11 Sloutskin E, Wu XZ, Peterson TB, et al. Surface freezing in binary mixtures of

chain molecules. I. Alkane mixtures. Physical Review E. 2003; 68.

12 Sloutskin E, Gang O, Kraack H, et al. Surface freezing in binary mixtures of

chain molecules. II. Dry and hydrated alcohol mixtures. Physical Review E. 2003;

68.

13 Earnshaw JC, Hughes CJ. Surface-induced phase-transition in normal alkane

fluids. Physical Review A. 1992; 46: R4494-R96.

14 Ocko BM, Wu XZ, Sirota EB, Sinha SK, Gang O, Deutsch M. Surface freezing in

chain molecules: Normal alkanes. Physical Review E. 1997; 55: 3164-82.

15 Wu XZ, Sirota EB, Sinha SK, Ocko BM, Deutsch M. Surface crystallization of

liquid normal-alkanes. Physical Review Letters. 1993; 70: 958-61.

16 Wu XZ, Ocko BM, Sirota EB, et al. Surface-tension measurements of surface

freezing in liquid normal-alkanes. Science. 1993; 261: 1018-21.

17 Weidinger I, Klein J, Stockel P, Baumgartel H, Leisner T. Nucleation behavior of

n-alkane microdroplets in an electrodynamic balance. Journal of Physical

Chemistry B. 2003; 107: 3636-43.

18 Tkachenko AV, Rabin Y. Fluctuation-stabilized surface freezing of chain

molecules. Physical Review Letters. 1996; 76: 2527-30.

19 Tkachenko A, Rabin Y. What drives the surface freezing in alkanes? Reply.

Physical Review Letters. 1997; 79: 532-32.

20 Sirota EB, Wu XZ, Ocko BM, Deutsch M. What drives the surface freezing in

alkanes? Physical Review Letters. 1997; 79: 531-31.

21 Broughton JQ, Gilmer GH. MOLECULAR-DYNAMICS INVESTIGATION OF

THE CRYSTAL FLUID INTERFACE .4. FREE-ENERGIES OF CRYSTAL

VAPOR SYSTEMS. Journal of Chemical Physics. 1986; 84: 5741-48.

22 Broughton JQ, Gilmer GH. SURFACE FREE-ENERGY AND STRESS OF A

LENNARD-JONES CRYSTAL. Acta Metallurgica. 1983; 31: 845-51.

23 Deutsch M, Wu XZ, Sirota EB, Sinha SK, Ocko BM, Magnussen OM. Crystalline

bilayers on the surface of molten alcohol. Europhysics Letters. 1995; 30: 283-88.

24 Gang O, Ocko BM, Wu XZ, Sirota EB, Deutsch M. Surface freezing in hydrated

alcohol melts. Physical Review Letters. 1998; 80: 1264-67.

25 Tkachenko AV, Rabin Y. Theory of surface freezing of alkanes. Physical Review

E. 1997; 55: 778-84.

Chapter 2: Aerosol generation flow system and experimental techniques

This chapter describes the continuous flow supersonic nozzle setup and the complementary analytical techniques used in my experimental studies. Sections of the chapter were adapted from the publication titled “Experimental evidence for surface freezing in supercooled n-alkane nanodroplets” authored by Viraj P. Modak, Harshad

Pathak, Mitchell Thayer, Sherwin J. Singer and Barbara E. Wyslouzil. Section 2.6, on

Wide Angle X-ray Scattering has been adapted from a manuscript titled “Freezing of supercooled n-decane nanodroplets: from liquid to frustrated crystallization” authored by

Viraj P. Modak and Barbara E. Wyslouzil.

2.1 Experimental setup

The schematic of the supersonic nozzle apparatus is shown in Fig. 2.1. The incoming stream to the supersonic nozzle is a gas mixture consisting of a carrier gas and n-alkane condensible vapor. Argon was used as carrier gas for experiments with n-octane and n- nonane was Argon (Ar), whereas N2 was used for experiments with n-decane. The carrier gas is supplied by two liquid Dewars. A portion of the carrier gas is used to transport the liquid condensible through the vaporizer where it evaporates. The condensible flow is supplied using a peristaltic pump. The condensible-carrier gas mixture then flows through a water bath where the flow achieves the desired inlet temperature. Following the water bath, the mixture then enters the plenum which is essentially a chamber with a large

13 cylindrical cross section, where the final temperature and pressure are monitored The mixture then enters the supersonic nozzle. In the nozzle, downstream of the throat, the flow achieves supersonic speeds and pressure and temperature decrease continuously.

The effective cooling rates achieved in the nozzle are of the order of ~105 – 106 K/s. The

 pc  degree of supersaturation, log  , increases continuously as pressure and temperature  pe  decrease. Here pc is the condensible pressure in the flow and pe is the equilibrium vapor pressure at that temperature. Beyond a particular supersaturation, the n-alkane vapor condenses forming nanodroplet aerosol releasing latent heat to the flow. This, combined with the vapor depletion effectively quenches nucleation after ~10 μs. Subsequently, the liquid droplets continue to grow and once most of the vapor has condensed, the temperature of the droplets starts to decrease again. If the droplets reach cold enough temperatures, they can also freeze. The pressure drop necessary to drive entire flow in a continuous manner is provided by two rotary vane vacuum pumps. The flow and the aerosol are characterized using up to four experimental techniques including Pressure

Trace Measurements (PTM),1-3 Fourier Transform Infrared Spectroscopy (FTIR),4-7

Small Angle X-ray Scattering (SAXS)8-12 and Wide Angle X-ray Scattering (WAXS).

The functioning of and objective behind each of these experimental techniques is described in detail below.

Figure 2.1: Schematic of the continuous flow supersonic nozzle apparatus

The supersonic nozzle has a rectangular cross section and is machined from aluminum. It consists of a straight section, followed by a converging-diverging profile. There are pressure taps in the straight section, including at the inlet, the throat and near the exit.

The nozzle block is held between two aluminum side walls and the entire assembly is held together by bolts. The sidewalls have windows for spectroscopy and X-ray scattering experiments. For PTM and FTIR we use sidewalls with CaF2 windows. For

SAXS and WAXS, we use windows made of mica and Kapton respectively. All the nozzles have an effective expansion rates, d(A(z))/dz = ~0.074 - 0.078 cm-1. The internals of the nozzle block are shown in Fig. 2.2.

Figure 2.2: Internals of the supersonic nozzle. The red arrows indicate the direction of the flow. The sidewall shown here has mica windows. Pressure taps are located near the entrance and the exit and at the throat of the nozzle.

2.2 Pressure Trace Measurements (PTM)

The flow is characterized by six key thermodynamic variables that vary with position or flow time. They include pressure (p), temperature (T), density (ρ), velocity (u), area ratio

(A/A*), and mass fraction condensed (g). A* is the flow area at the throat. The system is governed by four equations i.e. conservation of mass, momentum and energy and the ideal gas law. We conduct position resolved static pressure measurements using a long pressure probe with a pressure tap, connected to a pressure transducer. The probe can be moved relative to the nozzle using a linear stage. Pressure traces are measured with (dry trace) and without (wet trace) the condensible. For the dry trace there are only five unknowns since g is zero. Thus by measuring p, we can solve the diabatic flow equations

16 to obtain the remaining thermodynamic variables for the dry trace. We then assume that

A/A* for the wet trace is equal to that calculated from the dry trace, and measure p for the wet trace. Using p and A/A* as input, we then calculate the remaining thermodynamic variables for the wet trace. One weakness of this approach is that as the vapor condenses and latent heat is released to the system, the increasing density of the flow compresses th boundary layers.13 Thus the assumption that A/A* is equal for the dry and the wet trace is invalid. By incorporating data from FTIR and SAXS we can refine these estimates via an integrated data analysis method and relax the assumption regarding the area ratio.14 This method is described in more detail in Section 2.5.

2.3 Small Angle X-ray Scattering (SAXS)

We characterize the aerosol size distribution using position resolved SAXS measurements. The experiments are conducted at the Advanced Photon Source, Argonne

National Labs, Argonne IL, at the 12-ID_C beam line. The X-ray has a cross section of

0.2 mm x 0.2 mm and a wavelength λ, of 0.1 nm and a spread Δλ/λ of 10-4. The beam passes perpendicular to the flow and the scattering pattern is recorded on a 2D detector.

The nozzle and the plenum are mounted on a stage, which can be moved relative to the beam. Manka and coworkers7S described the procedures used to calibrate the range in the scattering vector q and place the spectra on an absolute intensity scale; q = (4π/

λ)sin(θ/2), where θ is the scattering angle. The 2D scattering pattern is then converted to a

1D spectrum which represents scattering intensity I as a function of q. This spectrum is then fit to scattering from a Schulz distribution of polydisperse spheres to obtain the

17 mean particle radius r , standard deviation σ, and the intensity as q tends to zero. A sample spectrum along with the Schulz fit is shown in the Fig. 2.3.

-1 10 n-decane -2 P0 = 152 torr 10 o

) -3 T0 = 55 C

-1 10 pv0 = 433 Pa -4 10 -5

10 I(q) (cm I(q) -6 10 data -7 10 fit -8 10 5 6 2 3 4 5 6 2 3 4 5 6 0.1 -1 1 q (nm )

Figure 2.3: Scattering intensity I (blue) as a function of scattering vector q, measured

9.51 cm downstream of the throat. The inlet partial pressure pv0 of n-decane was 433 Pa. The data are fit assuming scattering from a Schulz distribution of polydisperse spheres (red).

If the condensate consists of spherical droplets (Schulz distribution) of single phase, the number density of the aerosol can be calculated as follows:

2  3  (Z 1)5 I N   0 (2.1)   2 6  4  (Z  6)(Z  5)(Z  4)(Z  3)(Z  2)  SLD  r 

2  r    where, Z    1and ΔρSLD is the difference between the scattering length density of    the condensate and the gas mixture. The scattering length density of the condensate depends on the density of the condensate ρc and is, therefore, a function of the droplet temperature, phase, and droplet size through the Young-Laplace equation. In particular,

 2     1  (2.2) c 0    r 

where ρ0 is the density of the bulk condensate, γ is the surface tension, and β is the compressibility, and all properties are evaluated at the temperature of the droplet. Using

N, r and Z, we can express the volume fraction of condensate ϕ as follows:

4 3 (Z  3)(Z  2)   N r (2.3) 3 (Z 1)2

The molar concentration of condensate ci is given by

~ ci  ii , (2.4)

~ where i is the molar density of the condensed alkane in phase i (solid or liquid), and the

mass fraction of condensate gi can be expressed as

 g  i  (2.5) i 

Here i is the mass density of the alkane in phase i (solid or liquid) and ρ is the density of the gas mixture including the condensate. If the aerosol consists of partially frozen particles, we can only bound scattering length density of these particles. Hence we cannot determine the exact number density N of the particles using only this approach.

2.4 Fourier Transform Infrared Spectroscopy (FTIR)

When the aerosol freezes, a new variable – the fraction of the solid, gs must also be included in the analysis. To determine the distribution of condensible between the 19 coexisting vapor, liquid and solid phases, we use FTIR absorption spectroscopy. Since we cannot place our sample in the standard instrument beam path, we guide the beam emitted from the source using 6 plane and 2 focusing mirrors, so that the beam passes through the nozzle perpendicular to the direction of the flow. A schematic of the FTIR setup is shown in Fig. 2.4.

Plenum

Nozzle

Source Detector

Perkin Elmer 100

Figure 2.4: Schematic of the setup for conducting FTIR absorption spectroscopy experiments using Perkin Elmer 100 instrument (adapted from ref. 4)

The mirrors have a focal length of 20 cm. The focal point of the beam is at the center of the nozzle and is ~4 mm wide. The nozzle and the plenum assembly are on a moving plate to enable position resolved measurements. In these experiments, we use a Perkin

Elmer 100 FTIR instrument. The beam source is a quartz halogen light bulb. The residual

20 intensity is measured by a liquid nitrogen cooled Mercury Cadmuium Telluride (MCT) detector.

In an FTIR measurement, we determine the spectrum of the carrier gas – condensible

-1 mixture Is, as well as that of carrier gas alone Ie, for wavenumbers between 900 cm and

4000 cm-1 with 1 cm-1 resolution. The absorbance A is expressed as

I s A  log10 , (2.6) Ie

A is obtained using the Perkin Elmer software version 6.3.4 while suppressing the absorption from atmospheric CO2 and H2O. The decreased transmission through the gas- condensible mixture relative to the carrier gas alone is primarily the result of absorption.

The scattering from the sample is negligible because even the biggest particles (diameter

= 50 nm) are about two orders of magnitude smaller than the wavelength of the IR radiation (~4000 nm).

To analyze the FTIR data, we assume that the measured absorbance can be described by a linear combination of the normalized absorptivity of each phase present i.e.

A  (ai i ) (2.7) i

th Here A is the measured absorbance, and εi is the normalized absorptivity of the i phase

(m2/mol). We are interested in wavenumber range of 2800 – 3000 cm-1, because this spectral region includes the characteristic C-H stretches (2850 – 3000 cm-1). Beer-

Lambert’s law is valid for our experiments because the maximum measured absorbances

2 are below 0.06. We used a least-squares fit to determine the coefficients ai (mol/m ), 21 without imposing any restrictions based on satisfying mass balance, and then calculate the concentrations (mol/m3) of each species as follows:

a c  i (2.8) i l

Here l is the path length through the nozzle. Finally, we convert the concentrations to mass fraction of solid, liquid, or vapor in the flow as follows: g c RT i  i (2.9) g py0

Here y0 is the initial mole fraction of the condensible in the gas mixture, g∞ is the total mass fraction of condensate (vapor + liquid + solid) and R is the universal gas constant.

To determine the normalized absorptivity of the vapor εv, we measure the absorbance downstream of the throat prior to condensation and normalize it by the concentration of the vapor using

ART  v  (2.10) py0l

To determine the normalized absorptivity of the liquid aerosol εl for we either conduct experiments at conditions under conditions where the presence of solid is unlikely and for which there is no other evidence for freezing. We obtain mass fractions of liquid and vapor for these spectra from PTM and SAXS and the integrated analysis method (Section

2.5 of this chapter). We then subtract the vapor contribution from the measured

22 absorption spectra and normalize the difference by the liquid molar concentration as follows:

A  cvlv l  (2.11) cll

To confirm that our approach is reasonable we can compare εl, obtained this way to a direct transmission measurement using a solution of n-alkane in carbon tetrachloride

(CCl4). Although the direct transmission spectrum is close to the aerosol spectrum, the two spectra are not identical. This is because, when the particles are much smaller than the wavelength of light, The absorbance in an aerosol is expressed as15:

6l  6nk  A ( )    (2.12) aerosol  2 2 2 2  ln(10)  (n  k  2)  (2nk) 

Here ν is the wavenumber and, n and k are the real and imaginary parts of the refractive index. In contrast, transmission spectra of a bulk liquid sample is expressed as,

4kl A ( )  (2.13) bulk ln(10)

To obtain the normalized solid absorptivity εs we conduct experiments where we can assume that at the exit, the condensible exists only in the solid and vapor states. The solid-vapor distribution can therefore be determined from the SAXS data as described previously and we can obtain εs using

A  cvlv  s  (2.14) csl

The code for performing the fits to the intermediate spectra using the values of εv, εl and

εs was written in MATLAB and is provided in Appendix A.

2.5 Integrated data analysis

The first step in the integrated data analysis is to determine the offset between the physical throat, the minimum geometrical cross sectional area, and the effective throat, the minimum in the flow area, where the difference arises due to boundary layer formation along the nozzle walls. To determine the offset, we measure the pressure at the physical throat using a 0.34 mm diameter hole in the nozzle block, connected to the pressure tap at the throat shown in Fig 2.2. We then compare it to the pressure measured by the static pressure probe. The actual throat is ~0.9 – 1.1 mm downstream of the physical throat and all measurements made with respect to the physical throat (SAXS,

FTIR) are corrected to align with the actual throat.

If there is only a single phase transition, i.e. vapor to liquid, the next step is to use the initial estimates of T and ρ derived from the pressure trace measurements to analyze the

SAXS data and determine an improved estimate of g. We then solve the equations that describe supersonic flow with a phase transition using p and g as the known variables, thereby relaxing the assumption regarding the stability of the boundary layers to condensation. Using the improved values of T and ρ we reanalyze the SAXS data and iterate until the solution converges.

If there is more than one phase transition, i.e. if freezing occurs, we rely on position resolved FTIR measurements to determine the mass fraction of condensible in the liquid gl, and solid gs phases. The values of gl, gs and p, are then used to rerun the analysis and 24 generate new estimates for the other flow variables. The procedure is repeated until the values converge. The inversion code for calculating the thermodynamic variables from the pressure trace measurements is provided in Appendix C.

2.6 Wide Angle X-ray Scattering (WAXS)

We can also confirm if the particles formed in the nozzle are crystalline by using WAXS.

These experiments are conducted at the Coherent X-ray Imaging (CXI) Beamline of the

Linac Coherent Light Source (LCLS) at the Stanford Linear Accelerator Center (SLAC).

The beam has a focus of 0.1 μm2 at the focus and has an intensity of 1012 photons/pulse.

The pulse duration is < 10 fs. We measure WAXS spectra at positions where the particles are most likely to be completely frozen. Hence, we can see a diffraction pattern of high intensity diffraction rings, which can confirm the crystalline nature of the particles. These experiments are conducted for the sole purpose of confirming the crystalline nature of the frozen n-alkane particles. Hence we do not conduct full, position resolved experiments as we do for PTM, SAXS and FTIR. WAXS spectra are measured only at positions near or slightly upstream of the nozzle exit.

2.7 Materials

We use liquid Nitrogen (N2) or Argon (Ar) from Dewars to generate the carrier gas for our experiments. The liquids are purchased from Praxair and have a purity of greater than

99.99%. The higher heat capacity ratio of Ar (γ = 1.67) relative to N2 (γ = 1.4) means that expansions in the same nozzle reach lower temperatures when Ar is used as carrier gas than when N2 is used. For n-nonane and n-octane, the temperatures necessary to observe

25 the droplets freeze are more easily reached with Ar is the carrier gas. For n-decane we can observe freezing even when N2 is the carrier gas.

The n-octane and n-decane, purchased from Sigma-Aldrich, have a purity of at least

99.9% and are not purified further. The n-nonane, purchased from ChemSampCo has a purity of at least 99% and is not purified further. The relevant physical properties of these materials, with the exception of liquid compressibilities of n-nonane and n-decane are listed in Appendix B. The liquid compressibilities are derived from the pressure-volume data published by National Institute of Standards and Technology (NIST).16 They are 7.0 x 10-10 Pa-1 and 7.78 x 10-10 Pa-1 for n-nonane and n-decane respectively.

References:

1 Gharibeh M, Kim Y, Dieregsweiler U, Wyslouzil BE, Ghosh D, Strey R.

Homogeneous nucleation of n-propanol, n-butanol, and n-pentanol in a supersonic

nozzle. Journal of Chemical Physics. 2005; 122.

2 Heath CH, Streletzky K, Wyslouzil BE, Wolk J, Strey R. H(2)O-D(2)O

condensation in a supersonic nozzle. Journal of Chemical Physics. 2002; 117:

6176-85.

3 Wyslouzil BE, Heath CH, Cheung JL, Wilemski G. Binary condensation in a

supersonic nozzle. Journal of Chemical Physics. 2000; 113: 7317-29.

4 Laksmono H, Tanimura S, Allen HC, et al. Monomer, clusters, liquid: an

integrated spectroscopic study of methanol condensation. Physical Chemistry

Chemical Physics. 2011; 13: 5855-71.

5 Modak V, Pathak H, Thayer M, Singer S, Wyslouzil B. Surface Freezing of n-

octane Nanodroplets. 19th International Conference on Nucleation and

Atmospheric Aerosols (ICNAA). Vol. 1527. Colorado State Univ, Ctr Arts, Fort

Collins, CO 2013; 89-92.

6 Pathak H, Woelk J, Strey R, Wyslouzil B. Co-condensation of Nonane and D2O

in a Supersonic Nozzle. 19th International Conference on Nucleation and

Atmospheric Aerosols (ICNAA). Vol. 1527. Colorado State Univ, Ctr Arts, Fort

Collins, CO 2013; 51-54.

7 Manka A, Pathak H, Tanimura S, Woelk J, Strey R, Wyslouzil BE. Freezing

water in no-man's land. Physical Chemistry Chemical Physics. 2012; 14: 4505-16.

8 Ghosh D, Bergmann D, Scwering R, et al. Homogeneous nucleation of a

homologous series of n-alkanes (CiH2i+2, i = 7 - 10) in a supersonic nozzle.

Journal of Chemical Physics. 2010; 132: 024307.

9 Wyslouzil BE, Wilemski G, Strey R, Seifert S, Winans RE. Small angle X-ray

scattering measurements probe water nanodroplet evolution under highly non-

equilibrium conditions. Physical Chemistry Chemical Physics. 2007; 9: 5353-58.

10 Wyslouzil BE, Wilemski G, Strey R, Seifert S, Winans RE. Small angle X-ray

scattering measurements probe water nanodroplet evolution under highly non-

equilibrium conditions (vol 9, pg 5353, 2007). Physical Chemistry Chemical

Physics. 2008; 10: 7327-28.

11 Tanimura S, Dieregsweiler UM, Wyslouzil BE. Binary nucleation rates for

ethanol/water mixtures in supersonic Laval nozzles. Journal of Chemical Physics.

2010; 133: 174305-1 - 05-14.

12 Tanimura S, Wyslouzil BE, Wilemski G. CH3CH2OD/D2O binary condensation

in a supersonic Laval nozzle: Presence of small clusters inferred from a

macroscopic energy balance. Journal of Chemical Physics. 2010; 132: 144301-1 -

01-22.

13 Tanimura S, Zvinevich Y, Wyslouzil BE, et al. Temperature and gas-phase

composition measurements in supersonic flows using tunable diode laser

absorption spectroscopy: The effect of condensation on the boundary-layer

thickness. Journal of Chemical Physics. 2005; 122: 194304-1 - 04-11.

14 Modak VP, Pathak H, Thayer M, Singer SJ, Wyslouzil BE. Experimental

evidence for surface freezing in supercooled n-alkane nanodroplets. Physical

Chemistry Chemical Physics. 2013; 15: 6783-95.

15 Signorell R, Reid J. Fundamentals of Aersosol Spectroscopy. CRC Press 2010.

16 NIST Reference Fluid Thermodynamic and Transport Properties - REFPROP.

2002.

Chapter 3: Surface freezing in supercooled n-octane and n-nonane nanodroplets

This chapter was adapted from the publication, “Experimental evidence for surface freezing in supercooled n-alkane nanodroplets”, authored by Viraj P. Modak, Harshad

Pathak, Mitchell Thayer, Sherwin J. Singer and Barabara E. Wyslouzil. Harshad Pathak and Barbara E. Wyslouzil performed the vapor-liquid phase transition experiments, highlighted in Section 3.3 as well as, the X-ray scattering experiments in Section 3.4.

Harshad Pathak developed the methods for analyzing the experimental data in Section

3.3. Viraj P. Modak and Barbara E. Wsylouzil performed the PTM and FTIR experiments to study vapor-liquid-solid phase transition in Section 3.4. Viraj P. Modak enhanced the analytical methods for Section 3.4 and integrated the results from multiple studies into a single report.

3.1 Introduction

As noted in Chapter 1, the question whether crystallization is preferentially initiated near a surface or throughout the volume of the sample has been an intensely debated phenomenon in the scientific community. For the case of water droplets, this is of particular importance in atmospheric science.1-6 For micron-sized droplets, the volume- based nucleation rates scale with the droplet volume.1 For smaller droplets (<1 μm), experiments usually involve polydisperse aerosol samples, and current techniques have not been accurate enough to distinguish surface induced freezing from volumetric

30 freezing in water droplets.4 For straight chain molecules including n-alkanes and n- alcohols, surface freezing has been observed experimentally in macroscopic samples using X-ray Scattering and surface tension measurements at temperatures up to 3oC

7, 8 above the bulk melt temperature, Tm. Since the surface freezing temperature was only higher than Tm for alkanes with carbon numbers in the range of 15 ≤ n ≤ 50, the general perception was that n-alkanes with chain length less than 14 transition should exhibit the more common surface melting phenomenon. 7-15

The objective of this study is to identify if surface freezing can occur for superccoled droplets of short chain n-alkanes, in particular for n-octane and n-nonane Using a supersonic nozzle apparatus, we conduct position resolved PTM, SAXS and FTIR experiments to characterize the flow and the aerosol, and we analyze the data using an iterative data analysis method. This chapter highlights the key results from these experiments. The details of the supersonic nozzle apparatus, complementary experimental techniques and the data analysis method are provided in Chapter 2. Details of the experiments conducted and the conditions are reported in Section 3.2. In Section

3.3 we report PTM, SAXS and FTIR results for studies with n-nonane, at conditions where freezing was highly unlikely. These experiments provide the data to confirm the validity of the quantitative FTIR analysis method for a a single phase transition, before extending the approach to a case where freezing occurs. Section 3.4, highlights the results of n-octane and n-nonane experiments where the aerosol samples froze. Concluding remarks are presented in Section 3.5.

3.2 Experiments

We conducted two sets of experiments in this study and the experimental conditions are summarized in Table 3.1. The first set of experiments was conducted as a part of a separate study by Pathak and coworkers.16 Here, we re-analyze those results from a different perspective and thus are included as part of this work as well. In the first set of experiments, nitrogen was the carrier gas and n-nonane the condensible. Freezing did not occur, under these conditions because the droplet temperatures were always higher than the melt temperature of n-nonane, or the degree of supercooling was too low to initiate freezing on the timescale of the experiment. The purpose of these experiments was to develop and prove the accuracy of the quantitative FTIR analysis method for the simple case of vapor – liquid condensation before extending the approach to the more complicated case where vapor, liquid, and solid are all present. The second set of experiments was conducted with Argon as the carrier gas and n-octane or n-nonane as the condensible. In both cases, the droplet temperature at the nozzle exit was well below Tm, and all of the experimental techniques detect freezing.

a a Carrier Gas Condensible p0 T0 (K) pv0 Ton (K) pon (Pa) Texit (K) (Pa) (Pa) (v→l) (v→l) Nitrogen n-nonane 30130 308.16 322 196.2 60 202.8 489 204.6 103 219.7 625 209.3 137 229.8 Argon n-octane 30130 308.16 209 180.2 51 178.8 n-nonane 328.16 196 195.6 49 171.2 a The onset of condensation is defined as that point in the flow where the temperature of the condensing flow deviates from the expected isentropic expansion by 0.5 K.

3.3 Vapor – liquid condensation of n-nonane

Fig. 3.1(a) illustrates the temperature profiles of the expanding gas mixture determined using the integrated analysis, for two of the n-nonane in N2 experiments. The experimental droplet temperatures Tdrop are also shown, where these are determined by solving the energy balance between the growing droplets and the surrounding gas as described in Appendix B of reference. 42. Fig. 3.1(b) summarizes the mean particle radii measured by SAXS.

For either set of conditions in Fig. 3.1(a), the temperature of the gas mixture initially follows the isentropic temperature profile dropping well below the equilibrium melting point of the bulk solid. Eventually, however, the vapor condenses, forming particles that

33 are significantly warmer than the surrounding gas mixture. The dramatic increase in the temperature of the gas mixture reflects the latent heat release due to the vapor – liquid phase transition.

(a) n-nonane in N2 260 p0 = 30.2 KPa T0 = 35oC

240

Tm T (K) T 220

pv0 625 Pa 322 Pa 200 Tflow

Tisentrope

Tdrop 180 25 (b) 625 Pa 20

15 322 Pa

Radius (nm) Radius 10

0 10 20 30 40 50 60 70 Position downstream of the throat (mm)

As illustrated in Fig. 3.1(b), the location where the temperature profile deviates from that expected for the corresponding isentropic expansion, is the location where we first observe particles using SAXS. The particles continue to grow rapidly as they move through the nozzle, and even near the nozzle exit the droplet temperature is still slightly higher than that of the gas mixture. The results for pv0 = 489 Pa are similar and are not shown here for ease of readability but can be found in Appendix D.

For the experiment at the highest value of pv0, the droplet temperatures are all distinctly higher than Tm and the droplets cannot be frozen. For the experiment at pv0 = 489 Pa, in

Appendix D, droplet temperatures slightly above the melting point near the nozzle exit and hence are assumed to be liquid. For the lowest pv0, both Tflow and Tdrop are well below

Tm throughout the expansion. A number of experimental and theoretical studies, however, report that direct nucleation of the solid from the vapor is not favored, even when the solid is the most stable phase under the prevailing experimental conditions. 18-22 Hence, even when Tdrop is below Tm near the onset of condensation, the initial fragments of the new phase are assumed to be liquid. We will use spectroscopy to confirm our hypothesis that the particles formed are indeed liquid under all conditions in the three n-nonane – N2 experiments. To do so we turn to FTIR.

We conduct position resolved FTIR measurements and assume the measured absorbance can be described by a linear combination of the normalized absorptivity of each phase as per Eq. (2.7). The normalized absorptivity of vapor εv, is determined from a measurement downstream of the throat prior to condensation. We normalize this using the concentration of vapor using Eq. (2.10).

For both n-nonane and n-octane, there was little difference between the normalized absorptivity derived from different experiments at temperatures that ranged from 199.9 K to 247.5K for n-nonane and 186.9 K to 231.8 K for n-octane. Thus, we assumed that over the temperature range of our experiments εv is independent of temperature, and we averaged three spectra from different experiments to determine the final normalized absorptivity for each alkane. The averaged εv spectra for n-octane and n-nonane are illustrated in Fig. 3.2(b). We cannot directly compare our data to those published by the

NIST23 since the NIST spectra are saturated in the region of interest, i.e. for wavenumbers ν between 2800 cm-1 and 3000 cm-1. We can, however, compare the integrated normalized absorptivities for these species with those reported in the literature.

In particular, Klingbeil et al calculated the integrated normalized absorptivities of several gaseous n-alkanes including n-pentane, n-heptane and n-dodecane. Although their data are at much higher temperatures, 298 K to 773 K, and over such a wide range spectral shape does depend on T, the integrated normalized absorptivities are independent of temperature. As illustrated in Fig. 3.2(a), Klingbeil et al24 found that the integrated absorptivities for the n-alkane vapors vary linearly with n, and our points line up well with his correlation. In particular, the correlation predicts values of 6.43×105 m/mol and

5.80×105 m/mol for n-nonane and n-octane, respectively, and our measurements are

6.46×105 m/mol for n-nonane and 5.64×105 m/mol for n-octane.

(a)

(b)

Figure 3.2: (a) The integrated absorptivities measured in the current work agree well with the correlation established by Klingbeil et al. (b) The normalized absorptivities of n- octane vapor and n-nonane vapor from supersonic nozzle FTIR measurements. The difference in the curves near 2926 cm-1 and 2855 cm-1 is consistent with the presence of an extra CH2 group in n-nonane relative to n-octane.

To determine the normalized absorptivity of the liquid aerosol εl for n-nonane, we first determined the mass fractions of condensate and vapor as a function of position by combining the results from PTM and SAXS as described in Section 2.5 of Chapter 2. We then took one FTIR spectrum near the nozzle exit from each n-nonane in N2 condensation

37 experiment, and subtracted the vapor contribution from the measured absorption spectra based on the results of integrated analysis approach described in Section 2.5 of Chapter 2.

We then normalized the resultant spectra by the liquid molar concentration obtained from the integrated analysis using Eq. (2.11).

The three spectra, shown in Fig. 3.3(a), are in good agreement. Since the spectrum corresponding to the highest pv0 corresponds to a droplet temperature well above Tm, we concluded that the droplets in all three experiments were liquid and, therefore, averaged the three spectra to determine εl. To further confirm that our approach is reasonable we made a direct transmission measurement using a 6 µm thick film of a 15 mol% solution of n-nonane in carbon tetrachloride (CCl4). Fig. 3.3 (b) compares the aerosol εl to the direct transmission measurement. Although there are some differences between the two spectra, the overall shape, location, and intensity of the peaks is very close. As explained in Chapter 2, we do not expect the spectra to be identical due to inherent differences in the absorption characteristics between an aerosol and a bulk sample.

50 n-nonane liquid at (a) /mol

2 625 Pa 40 489 Pa 322 Pa

10 normalized absorptivity m 0 2800 2850 2900 2950 3000 -1 wavenumber (cm ) (b)

Thus, to determine the vapor and liquid mass fractions from the spectra measured in the nozzle we use the normalized absorptivities measured in the nozzle. We then determined the fraction of vapor condensed as a function of position using Eq. (2.7) and the

39 experimental values of εv and εl. A typical fit is shown in Fig. 3.4(a) and the agreement between the observed and fitted spectrum is very good, i.e. R2 = 0.99.

(a)

(b)

Finally Fig. 3.4(b) summarizes the position resolved mass fractions of vapor and liquid derived from FTIR and compares these to the values of gi/g∞ obtained from the integrated data analysis described in Section 2.5 of Chapter 2. The overall agreement between the two approaches is excellent, and, furthermore, the FTIR measurements are always within

5% of mass balance. Thus, given an appropriate normalized absorptivity for the solid aerosol εs, it should be possible to extend this approach to three phases.

3.4 Freezing of n-octane and n-nonane nanodroplets

Fig. 3.5 shows the temperature profiles obtained from the integrated PTM/FTIR analysis for (a) n-octane and (b) n-nonane condensing in Ar as well as the average particle radii

(c,d) measured using SAXS under identical experimental conditions. The dash dotted lines indicate the equilibrium melting temperatures, Tm, and the dashed lines represent the expected isentropic temperature profile for the expanding gas mixtures. In each condensing flow curve (solid line) there are clearly two “bumps”, corresponding to two separate phase transitions, and for both phase transitions Tdrop is well below Tm. We assume that the first phase transition corresponds to vapor – liquid condensation and, as in the n-nonane-N2 experiments, SAXS first detects particles in this region. The second bump in the condensing flow curve, that we interpret as freezing, is more subtle since the heat of fusion is only ~ 30% of the heat of condensation.17 Finally, the position of the second bump coincides with the position (50 mm for n-octane and 42.5 mm for n- nonane) where the radius of the particles begins to decrease. Since the density of solid alkane is higher than that of the liquid, the decrease in particle size serves as independent evidence for droplet freezing.

For n-nonane, the ~2.5% decrease in particle size is less than the ~5% decrease expected based on a simple volume balance and a ~16% increase in the density of the solid over that of the liquid. This difference is consistent with droplets that are simultaneously freezing and growing. In contrast, the 12% decrease in particle size for n-octane is far larger than the 4% decrease expected based on the ~14% increase in density. Although it was not the focus of this study, it is noteworthy, that the decrease in particle size is accompanied by an increase in the particle number density so that the mass fraction of condensate does not decrease, i.e. net droplet evaporation is not detected. Two possibilities for increasing the particle number density include a second nucleation burst or particle break-up. Determining which process dominates is the subject of further investigation.

250 250 n-octane n-nonane 240 (a) 240 (b) P = 30.2 KPa p = 30.2 KPa 230 0 230 0 o T T0 = 35 C m T0 = 55oC 220 Tm 220

210 210 Tdrop T 200 drop 200 Tflow

T (K)

T (K) 190 190

180 Tflow 180 170 170 Tisentrope 160 160 Tisentrope 150 150 26.9 mm 50 mm 25 mm 42.5 mm 140 140 9 9 (c) (d) 8 8

7 7

6 6

5 5

Radius (nm)

Radius (nm) 4 4

3 3 26.9 mm 50 mm 25 mm 42.5 mm 2 2 0 10 20 30 40 50 60 70 0 10 20 30 40 50 60 70 Position downstream of throat (mm) Position downstream of throat (mm)

To further elucidate the sequence of transitions from vapor to liquid to solid, we need to determine the fraction of the three phases as a function of position using FTIR spectroscopy. To do so we must first establish the normalized absorptivity for liquid n- octane εl, and the normalized absorptivities for the n-octane and n-nonane solids, εs.

For n-octane we did not have an experiment where the aerosol was unfrozen at the nozzle exit. To obtain a normalized absorptivity spectrum for the liquid, we therefore analyzed the spectrum measured 34 mm downstream of the throat. At this position the droplets are sub-cooled by only ~16 K and are, therefore, unlikely to be frozen. Here, the SAXS data give gl/g∞ = 0.4. To confirm that the spectrum derived this way is reasonable, we compared it to a direct transmission measurement at 298 K made with a 6 µm thick film of a 16.7 mol % solution of n-octane in carbon tetrachloride. As illustrated in Fig. 3.6, the two n-octane spectra are very similar and the overall shape, intensity, and peak locations agree quite well.

Figure 3.6: Normalized absorptivity of n-octane liquid from direct transmission measurements and aerosol measurements.

The normalized aerosol liquid absorptivity εl for n-octane is noisier than that for n- nonane in Fig. 3.3(b) both because the concentration of condensate at this point is much lower than in the n-nonane experiments and because we did not have multiple spectra to average over.

To obtain the normalized solid absorptivity εs we assume that at the exit, the condensible exists only in the solid and vapor states. This assumption is reasonable because, as illustrated in Fig. 3.5, the average particle radius does not change near the exit of the nozzle and the temperatures at the nozzle exit are ~40 K to ~50 K below Tm. The solid- vapor distribution can therefore be determined from the SAXS data and we can obtain εs using Eq. (2.14). The normalized absorptivities for the n-octane and n-nonane solids derived this way are shown in Fig. 3.7.

Figure 3.7: Normalized aerosol absorptivities for solid n-octane and n-nonane. These spectra are determined assuming that the aerosol is entirely frozen near the exit of the nozzle.

Using the normalized absorptivities εv, εl, and εs, we determined g/g∞ for each phase by fitting the intermediate spectra using a least squares approach without imposing material balance constraints during the fit. Fig. 3.8 illustrates the derived distribution of the condensible between the vapor, liquid and solid states, as well as the material balances.

Initially gv/g∞ = 1, and as condensation begins, gv/g∞ decreases monotonically until it

45 attains the constant value that is determined from the integrated PTM/SAXS analysis. In contrast, gl/g∞ starts at zero, increases to reach a maximum, and then decreases to zero at the nozzle exit. Finally, gs/g∞ starts at zero, increasing slowly shortly after the appearance of the liquid, and then more rapidly, before reaching a constant value near the nozzle exit.

Although mass balance was not imposed, it is always satisfied to within 5%.

1.2 1.2 n-octane n-nonane 1.0 1.0 vapor vapor solid 0.8 liquid solid 0.8 liquid

inf 0.6 0.6

inf

i 0.4 g 0.4

0.2 0.2

0.0 0.0

-0.21.1 -0.21.1 1.0 1.0

M.B.

M.B. 0.9 0.9 0 10 20 30 40 50 60 70 0 10 20 30 40 50 60 70 Position downstream of the throat (mm) Position downstream of the throat (mm)

(gl+gv+gs)/g∞.

One interesting aspect of the FTIR results is that spectroscopy first detects the solid phase well upstream of the position inferred from the PTM and SAXS experiments. Based on the temperature profiles and the change in particle size, n-octane and n-nonane freezing appeared to start ~50 mm and ~42.5 mm downstream of the throat, respectively. In

46 contrast, FTIR experiments suggest the solid phase first appears ~37 mm and ~32 mm downstream of the throat, respectively.

To investigate the kinetics of the liquid to solid phase transition, we used the velocity of the flow to convert from position to time. Fig. 3.9 illustrates the fraction of the aerosol that is solid Fs as a function of time, where Fs = gs/(gs+gl) and t=0 corresponds to the throat.

1.2 1.2 n-octane n-nonane 1.0 1.0

0.8 0.8 1-exp(-JvV(t-t0)) 1-exp(-JvV(t-t0)) 0.6 0.6

F 0.4 0.4

0.2 Monolayer 0.2 Monolayer 0.32(1-exp(-J S(t-t ))) 0.35(1-exp(-JsS(t-t0))) s 0 Coverage Coverage 0.0 0.0

-0.2 -0.2 0.00 0.05 0.10 0.15 0.20 0.00 0.05 0.10 0.15 0.20 time (ms) time (ms)

Here we initially see a slow increase in the solid mass fraction before the expected rapid increase begins. Furthermore, the rapid increase starts at the same time (location) as the second heat release is observed to start in the PTM and the droplet shrinkage measured by SAXS begins. We interpret this two-stage behavior as the development of a frozen monolayer on the surface of the droplets prior to crystallization in the remainder of the 47 droplet. PTM does not easily detect the heat release due to surface ordering since the latent heat of fusion is significantly smaller than the latent heat of condensation, and the droplets are still growing slowly. Likewise, droplet growth makes it difficult for SAXS to detect any potential changes in particle size associated with surface freezing.

As illustrated in Fig. 3.9, n-octane and n-nonane both exhibit distinctly different nucleation behavior than we observed in our recent work on ice nucleation from supercooled water nanodroplets, where we found that the fraction of solid Fs, i.e. ice, in the condensed water was well described by the expected functional form

Fs 1 exp(B(t t0 )) , (19)

where B is a fit parameter and t0 is the time corresponding to the onset of liquid-solid nucleation. The fit parameter B can be used to derive the volume or surface based liquid

- solid nucleation rates, JV and JS respectively, by setting BV = JVV or BS = JSS. Here V and S are the characteristic volume or surface area, often taken as the average droplet volume or surface area, although other choices are possible.25

Table 3.2 summarizes the fit parameters BS, t0S, BV and t0V derived by fitting the Fs data to a two-step crystallization process. Table 3.2 also reports the freezing rates derived from the fits, assuming surface nucleation is followed by bulk nucleation. We provide nucleation rates for crystallization of the droplet interior based on two possible mechanisms for the bulk crystallization process. First, we give the rate suppose that crystal nuclei are formed anywhere in the droplet below the (already frozen) surface layer. The bulk nucleation rates so calculated for the alkanes are comparable to those we

48 observed for water. Second, we report a “heterogeneous” nucleation rate, i.e. the rate associated with nucleation in the interior of the droplet catalyzed by the presence of the surface layer.

Surface Nucleation: S = 4 r 2

-1 -2 -1 BS (ms ) t0S (ms) JS (cm .s ) TS (K) n-octane 19.9 0.097 3.2E+15 181.6 n-nonane 20.9 0.075 6.5E+15 194.3

4 3 Volume Nucleation Mechanism: V =  r , where ri = r - lm; 3 i 19 lm is the length of an alkane molecule -1 -3 -1 BV (ms ) t0V (ms) Jv (cm .s ) Tv (K) n-octane 61.2 0.13 2.6E+22 182.1 n-nonane 59.0 0.11 3.3E+22 199.0

4 4 Heterogeneous Nucleation Mechanism: V =  r 3   (r l )3 3 i 3 i m -1 -3 -1 BV (ms ) t0V (ms) Jv (cm .s ) Tv (K) n-octane 61.2 0.13 9.0E+22 182.1 n-nonane 59.0 0.11 9.0E+22 199.0

Following Weidinger et al, the “heterogeneous” rates are calculated supposing that nuclei for bulk nucleation form in the layer immediately below the surface frozen layer. The heterogeneous rates reported here are 12 orders of magnitude higher than the heterogeneous nucleation rates reported by Weidinger et al. This difference is consistent with the fact our droplets are much smaller and are subjected to a much higher degree of supercooling than the micron sized droplets used by Weidinger et al. Although the data by themselves offer quite a compelling case in favor of surface nucleation, to strengthen our interpretation of the experimental results we turned to molecular dynamics simulations. The details of the MD simulations and selected droplet simulations are described in Chapter 5. In Chapter 5 we will also describe how both models for crystallization of the interior described here, are likely to be oversimplifications.

3.5 Conclusion

In this study, we followed the freezing of highly supercooled n-octane and n-nonane nanodroplets generated in a supersonic nozzle apparatus. We characterized the flow by combining the results of static pressure measurements and SAXS. SAXS also characterized the particle size distribution, whereas FTIR absorption measurements determined the distribution of the condensable between the vapor, liquid and solid states.

The presence of two bumps in the temperature profile of the condensing flow was the first indication of multiple phase transitions. SAXS measurements showed that particles grew rapidly from 4 nm to ~9 nm before distinctly decreasing in size. This size decrease mirrored the 2nd heat addition and served as independent evidence of freezing. Both PTM

50 and SAXS detected onset of freezing at about the same position downstream of the throat.

In contrast to PTM and SAXS, FTIR first detected the presence of solid at a position distinctly further upstream. Furthermore, a distinct kink in the fraction of solid versus time curve suggested that freezing occurred as a two-step process that we interpret as the development of an ordered surface monolayer followed by freezing of the rest of the droplet. Analyzing the time dependence of the fraction of solid let us determine nucleation kinetics of freezing. The surface nucleation rates Js for the outer layer of n- octane and n-nonane droplets were 8.8 x1014 /cm2.s and 3.1 x1015 /cm2.s, respectively.

Interpreting our data for crystallization of the interior as a volume nucleation rate, we

22 3 22 3 find Jv = 2.4 x10 /cm .s for n-octane and Jv = 2.3 x10 /cm .s for n-nonane, comparable to the bulk nucleation rate determined for water droplets of comparable size.26 Alterrnatively interpreting the data for the interior with a heterogeneous nucleation model, the nucleation rates are 9.0 x1022 for both n-octane and n-nonane. Our values are about 12 orders of magnitude higher than those reported by Weidinger et al, consistent with the current experiments using much smaller droplets that are far more highly supercooled.

References:

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in supercooled water microdroplets. Atmospheric Chemistry and Physics. 2004; 4:

1997-2000.

2 Tabazadeh A, Djikaev YS, Reiss H. Surface crystallization of supercooled water

in clouds. Proceedings of the National Academy of Sciences of the United States

of America. 2002; 99: 15873-78.

3 Vrbka L, Jungwirth P. Homogeneous freezing of water starts in the subsurface.

Journal of Physical Chemistry B. 2006; 110: 18126-29.

4 Sigurbjornsson OF, Signorell R. Volume versus surface nucleation in freezing

aerosols. Physical Review E. 2008; 77: 051601.

5 Kuhn T, Earle ME, Khalizov AF, Sloan JJ. Size dependence of volume and

surface nucleation rates for homogeneous freezing of supercooled water droplets.

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6 Kay JE, Tsemekhman V, Larson B, Baker M, Swanson B. Comment on evidence

for surface-initiated homogeneous nucleation. Atmospheric Chemistry and

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7 Earnshaw JC, Hughes CJ. Surface-Induced Phase-Transition in Normal Alkane

Fluids. Physical Review a. 1992; 46: R4494-R96.

8 Sloutskin E, Wu XZ, Peterson TB, et al. Surface freezing in binary mixtures of

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Liquid Normal-Alkanes. Physical Review Letters. 1993; 70: 958-61.

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Bilayers on the Surface of Molten Alcohol. Europhysics Letters. 1995; 30: 283-

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chain molecules: Normal alkanes. Physical Review E. 1997; 55: 3164-82.

13 Gang O, Wu XZ, Ocko BM, Sirota EB, Deutsch M. Surface freezing in chain

molecules. II. Neat and hydrated alcohols. Physical Review E. 1998; 58: 6086-

100.

14 Kraack H, Sirota EB, Deutsch M. Measurements of homogeneous nucleation in

normal-alkanes. Journal of Chemical Physics. 2000; 112: 6873-85.

15 Sloutskin E, Gang O, Kraack H, et al. Surface freezing in binary mixtures of

chain molecules. II. Dry and hydrated alcohol mixtures. Physical Review E. 2003;

68: 031606.

16 Pathak H, Woelk J, Strey R, Wyslouzil BE. Co-condensation of nonane and D2O

in a supersonic nozzle. Journal of Chemical Physics. 2014; 140: 034304-1 - 04-

14.

17 Tables of Physical and Thermodynamic Properties of Pure Compounds.

University Park, PA: Pennsylvania State University 1983.

18 Chen B, Kim H, Keasler SJ, Nellas RB. An aggregation-volume-bias Monte Carlo

investigation on the condensation of a Lennard-Jones vapor below the triple point

and crystal nucleation in cluster systems: An in-depth evaluation of the classical

nucleation theory. Journal of Physical Chemistry B. 2008; 112: 4067-78.

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crystallization. Physical Review Letters. 2006; 96: 046102.

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Chapter 4: Freezing of supercooled n-decane nanodroplets

This chapter is based on a manuscript titled, “Freezing of n-decane nanodroplets: from complete crystallization to frustrated crystallization”. The authors of the manuscript are

Viraj P. Modak and Barbara E. Wyslouzil. Both authors performed the experiments and worked on the interpretation of the results.

4.1 Introduction

As discussed in Chapter 1, if crystallization occurs via homogeneous nucleation, i.e. in the absence of solid seed particles, the location of the first fragments of the new phase, at an interface, near an interface (within ~ 1 nm), or throughout the bulk is both a topic of fundamental interest and of some controversy.1-5 In particular, the importance of ice formation in the atmosphere has led to intense debate regarding the location of the first fragments of the emerging solid phase in supercooled atmospheric water droplets.6-8 For micron-sized droplets, experimental nucleation rates scale well with the volume of the droplets, suggesting nucleation occurs throughout the droplet.5 Experiments with smaller droplets generally involve polydisperse aerosol samples, and here Sigurbjornsson and

Signorell6 concluded that current experimental techniques cannot easily distinguish between freezing that occurs preferentially near the surface and freezing that occurs

6 9 throughout the droplet volume. In our own recent studies on freezing of D2O nanodroplets (radius ~ 3-9 nm) in a continuous flow supersonic nozzle, we found that

55 interpreting the nucleation results is further complicated by the fact that for droplets in this size range the fraction of the droplet volume that lies within 1 nm of the surface can approach 70%.

Despite these challenges, in the experiments described in Chapter 3 with n-octane and n- nonane nanodroplets (radius ~ 6-9 nm) we found spectroscopic evidence10 that suggests rapidly cooled short chain n-alkane droplets start to freeze on the surface well before bulk freezing takes over. Prior to our work, surface freezing had only been observed in longer chain alkanes,11-18 and short chain alkanes were thought to revert to the more usual surface premelting behavior. Our data also let us estimate the surface, surface-templated, and volumetric freezing rates for these droplets.10 Given the limited number of experiments, however, we were not able to explore the observed phenomena as a function of droplet temperature or size.

The goals of the current work are, therefore, twofold. The first is to extend our work to a third alkane, n-decane, and confirm the surface-freezing phenomenon for a molecule closer in size to the n-alkanes used in “bulk” surface freezing experiments (C14 – C55).

The second is to investigate whether the nature of the liquid – solid phase transition changes as we tune the droplet size and temperature. We should expect that droplets cooled rapidly enough may not crystallize completely and may instead form a partially disordered or amorphous solid state.19 Bartell and co-workers found that large molecular clusters, formed in supersonic expansions and then evaporatively cooled, did not all freeze into uniform crystalline solids. Rather, some remained liquid-like, or crystallized into different structures that depended on the inlet conditions.20-23 Our experiments

56 should further our understanding of the conditions required to enhance surface freezing over other solidification pathways.

This chapter is organized as follows. In Section 4.2, we present the experimental conditions investigated. In Sections 4.3 and 4.4 we establish the FTIR absorption spectra required to quantify the amount of decane in the vapor, liquid and solid phases. We first test the quantitative FTIR analysis for the simpler case of a vapor-liquid phase transition

(Sec. 4.3). In Section. 4.4 we extend the analysis to a system with coexisting vapor, liquid and solid phases analyze the kinetics of freezing, and explore the effect of inlet conditions on whether surface nucleation precedes bulk nucleation. For one set of conditions, the crystalline nature of the particles is confirmed using WAXS. Finally, this section explores the lowest temperature experiments, where the aerosol particles freeze into a partially ordered state. Concluding remarks are mentioned in Section 4.5.

4.2 Experiments

We performed experiments under the five conditions summarized in Table 4.1. In Exp (i), there was no evidence for freezing. Although the droplets were supercooled, the degree of supercooling was not enough to induce freezing on the time scale of the experiment.

The data from Exp (i) yield the liquid normalized absorptivity that is used in the quantitative FTIR analysis.

Table 4.1: Summary of the experimental conditions. The inlet conditions are denoted by stagnation temperature T0 , stagnation pressure p0 and condensible partial pressure pv0.

The onset temperature Ton is the temperature of the vapor-liquid phase transitions. Texit is the temperature 102 mm downstream of the throat near the nozzle exit. Tm , the equilibrium melting point of n-decane is 243.51 K.24

p0 (Pa) (K) pv0 (Pa) Ton (K) Texit (K) (i) 20265 328.16 433 218.3 231.7 (ii) 30200 318.16 264 208.9 206.3 (iii) 30200 318.16 219 204.6 200.0 (iv) 30200 308.16 305 210.9 195.8* (v) 30200 318.16 124 194.8 184.0

* Texit for this experiment is not corrected for boundary layer compression, since we did not have FTIR measurements for this experiment. Based on the corrections for other conditions, we do not expect the temperature to be more than 10 K higher than the reported value, which is still more than 35 K below Tm.

4.3 Vapor-liquid phase transitions

The normalized absorptivity for n-decane vapor was determined by averaging 2 measurements made at positions upstream of condensation during Exp (i), where the pressure and temperature used in Eq. (2.10) come from the PTM. The averaged normalized absorptivities obtained for different inlet conditions agree well with each other, and in Fig. 4.1(a) we compare the averaged n-decane spectrum to the vapor spectra obtained previously for n-octane and n-nonane. All of the peaks in this spectral region correspond to C-H stretches, and our measurements agree well with those reported25 for the alkane CH3 asymmetric (2962 ± 10), CH3 symmetric (2872 ± 10), CH2 asymmetric

(2926 ± 10) and CH2 symmetric (2855 ± 10) stretches. As expected the CH2 peaks 58 increase in intensity with the chain length of the molecule, whereas the CH3 peak intensities are essentially26 constant. Furthermore, as shown in Fig. 4.1(b), the integrated absorptivity of n-decane vapor, 6.91 x 105 (m/mol), is consistent with our previous measurements and matches the linear trend reported by Klingbeil et al.27

60 Vapor

/mol 50 decane 2 nonane octane 40 30 (a)

10 normalized normalized absorptivity m 0 2800 2850 2900 2950 3000 -1 Wavenumber (cm ) 9e+5

8e+5

7e+5 (b) 6e+5

m/mol

5e+5

Integrated Absorptivities Absorptivities Integrated Klingbeil et al. decane (2015) 4e+5 nonane (2013) octane (2013)

3e+5 4 6 8 10 12 14 carbon number Figure 4.1: (a) The normalized absorptivity for n-decane vapor is compared to those measured for n-octane and n-nonane in our earlier work.10 (b) The integrated absorptivity, between 2500 cm-1 and 3400 cm-1, for n-decane follows the linear trend reported by Klingbeil et. al.27

The main purpose of Exp (i) was to establish the normalized aerosol absorptivity of liquid n-decane. To prevent droplet freezing, T0 was increased by 10 K, the partial pressure of the condensable was increased, and the stagnation pressure was reduced by 30% relative to the values used in the freezing experiments. Fig. 4.2(a) illustrates the temperature profile for this experiment derived from an integrated data analysis that combined the

PTM and SAXS results, and Fig. 4.2(b) summarizes the relevant SAXS data. A typical fit to the scattering intensity, that assumes the aerosol is a collection of polydisperse spheres, is shown in Fig. 4.2(c). Fig. 4.2(d) depicts the normalized nucleation rate calculated from

Classical Nucleation Theory.28 Initially, the temperature profile of the condensing flow closely follows that of the isentrope with a cooling rate of ~ 106K/S. About 5.8 cm downstream of the throat, the decane begins to condense, forming an aerosol, and the latent heat released by the growing droplets increases the temperature of the gas mixture above that expected for isentropic expansion of the same gas mixture. This is also very close to the position where we first detect particles using SAXS. During the period of

29 rapid particle growth, an energy balance shows that Td is considerably higher thanT flow .

Although the liquid droplets are ~10 K colder than Tm at the nozzle exit, this degree of supercooling is unlikely to be enough to initiate freezing on the timescale of the experiment. In our previous experiments10 with n-octane and n-nonane we only observed the presence of solid when the droplets were supercooled by at least 15 K.

280 -1 (a) 10 (c)

T0 = 328 K )

p = 20.265 kPa -1 -3 0 10 260 p = 433 Pa v0 -5 10 Tm I(Q) I(Q) (cm 240 -7 10 Scattering Intensity

T (K) Polydisperse Spheres Model -9 10 220 5 6 2 3 4 5 6 2 3 4 5 6 0.1 1 -1 Q (nm ) T isentrope 1.2 2.00e+18 Tflow 200 (d) Tdrop 1.0 1.50e+18 24 0.8 (b)

20 0.6 1.00e+18 )

-1

max

J/J

(nm) 16 0.4 N (kg 5.00e+17 12 0.2

and and 8 0.0 0.00 4 0.00 0.05 0.10 0.15 0.20 0 2 4 6 8 10 time (ms) Position downstream of the throat (cm) Figure 4.2: (a) Temperature profiles as a function of position derived from the integrated analysis based on PTM and SAXS results. The equilibrium melting temperature of decane is indicated by Tm . (b) The mean particle radius (black circles) and σ (red triangles) measured by SAXS . (c) A typical fit of the polydisperse sphere model (red) to the scattering intensity I(q) (blue). (d) The normalized nucleation rate J/Jmax calculated using Classical Nucleation Theory28 (red solid line) as a function of time. The experimental number densities calculated from SAXS (black circles) are compared to the predicted number densities, calculated by integrating the nucleation rate curve with respect to time and scaling to match the experimental values at the exit.

We can therefore use the data from Exp (i) to calculate the normalized absorptivity for the liquid εl. In particular, we used the vapor and liquid concentrations derived from the

PTM+SAXS analysis, the measured vapor and the total absorption spectra, and Eq. (2.11) to determine the normalized liquid absorptivity. Fig. 4.3(a) illustrates three normalized 61 liquid absorptivities measured near the nozzle exit. The three spectra agree quite well with each other with respect to peak locations and intensities.

60 P0 = 20.265 kPa (a)

pv0 = 433 Pa /mol 2 50 T0 = 328 K 101.1 mm 92.1 mm 40 85.1 mm

normalized normalized absorptivity m 0 2800 2850 2900 2950 3000 -1 Wavenumber (cm )

60 decane nonane (b)

/mol 2 50

normalized absorptivity m 0 2800 2850 2900 2950 3000 -1 Wavenumber (cm )

Figure 4.3: (a) Liquid normalized absorptivity obtained from absorbance measured at 101.1 mm (red), 92.1 mm (black) and 85.1 mm (blue) downstream of the throat. The spectra agree well with each other with respect to peak positions and intensities. (b) The liquid normalized absorptivity for n-decane at 92.1 mm downstream of the throat compared to that for n-nonane from previous experiments. The peak locations, indicated by the vertical grey dashed lines, match well across the chain length, and the intensity of the CH2 stretches increases with chain length. The temperature of the droplets corresponding to the n-decane spectrum is 235.26 K. The n-nonane spectrum was calculated by averaging three spectra recorded for droplets between 206 K and 234 K.

When we compare the normalized liquid absorptivity for n-decane to our previous result for n-nonane in Fig. 4.3(b) the peaks line up nicely, (see also Table 4.2 for peak locations), and the intensities the CH2 peaks increase with chain length. The best agreement in the absolute intensity of the CH3 peaks for these two alkanes corresponds to the n-decane spectrum measured 92.1 mm downstream of the throat. Hence, this εl is used in all subsequent data analysis.

Table 4.2: The locations of the relevant peaks observed in the normalized absorptivities for liquid n-octane, n-nonane and n-decane agree with each other well. All the values reported are in the units of cm-1 n-octane n-nonane n-decane 2856.8 2855.2 2854.8 2872.5 2873.0 2872.2 2923.2 2925.0 2925.0 2959.0 2958.2 2957.8

To test the consistency between the FTIR measurements and the integrated PTM+SAXS analysis, we calculated the vapor and liquid mass fractions by fitting the measured absorbance spectra to Eq. (2.7), using the established values of εv and εl, without enforcing mass balance. Fig. 4.4 illustrates the result and shows that mass balance is generally satisfied within 5%. Thus, for the vapor to liquid phase transition, the agreement between independent experimental techniques is good and validates our analytical approach.

1.0

0.8

0.6

inf PTM+SAXS - Liquid

i FTIR - Liquid g 0.4 PTM+SAXS -Vapor FTIR - Vapor

0.2

0.0 1.1

MB 1.0 0.9 0 2 4 6 8 10 Position downstream of the throat (cm) Figure 4.4: The vapor and liquid mass fractions calculated from integrated data analysis using the results of PTM and SAXS are compared to those obtained from the FTIR data. Agreement between the two approaches is very good. The bottom graph represents the material balance for the mass fractions calculated by FTIR where MB = (gv  gl ) / g

4.4 Vapor-liquid-solid phase transitions

To extend this approach to the more complex system of coexisting vapor, liquid and solid, requires εs. Freezing is expected in Exps (ii) and (iii) since Texit is ~40 K below the equilibrium melting point, and this degree of supercooling is comparable to that observed when octane was fully frozen in our earlier experiments. Figs. 4.5(a) and 4.5(b) illustrate the results for Exp (ii). In Fig. 4.5(a) the signature of nanodroplet freezing includes a subtle second bump in the condensing flow temperature, ~ 7 cm downstream of the throat. The data in Fig. 4.5(b) show that the particles also start to shrink at about the same position. For Exp (iii) we cannot identify a second bump in the condensing flow temperature in Fig. 4.5(c), nor can we see a distinct decrease in the mean particle size after the particles stop growing in Fig. 4.5(d). Nevertheless, temperatures near the exit are

~6K colder than in experiment (ii), and, thus, the aerosol droplets should also be fully frozen. Assuming that they are, we use the aerosol mass fraction derived from SAXS experiments to determine the partitioning of decane between the vapor and condensed phases, and use Eq. (2.14) to calculate εs.

T = 318 K 260 (a) T0 = 318 K 260 (c) 0 p0 = 30.200 kPa p0 = 30.200 kPa pv0 = 264 Pa pv0 = 219 Pa Tm Tm 240 240

Tdrop 220 220 Tdrop Tflow

Temperature (K) Temperature Temperature (K) Temperature Tflow 200 200

Tisentrope 180 180 Tisentrope 16 (b) 14 (d) 14 12 12 10 10

Radius (nm) 8 Radius (nm)

6 6

4 4 0 2 4 6 8 10 0 2 4 6 8 10 Position downstream of throat (cm) Position downstream of throat (cm)

Tisentrope. The horizontal short dashed line indicates the equilibrium melting temperature of n-decane. The vertical dashed lines indicate the onsets of condensation and freezing.

100

P0 = 30.2 kPa /mol T = 318 K (a) 2 80 0 pv0 = 264 Pa pv0 = 219 Pa 60

20 Normalized absorptivity m 0 2800 2850 2900 2950 3000 -1 Wavenumber (cm ) 100 decane /mol (b) 2 octane 80

20 Normalized absorptivity m 0 2800 2850 2900 2950 3000 -1 Wavenumber (cm )

Figure 4.6: (a) Normalized absorptivities of the near the exit calculated for different experimental conditions match each other well. The red line corresponds to Exp (ii) with

Fig. 4.6(a) illustrates that the normalized aerosol absorptivities for solid n-decane, εs, derived from Exps (ii) and (iii) near the nozzle exit, agree quantitatively with each other.

10 When we compare εs for n-decane to that of n-octane measured in earlier experiments, in Fig. 4.6(b), the overall shape of the spectra for these two even numbered alkanes agrees well. The dashed grey lines show that the peak locations line up well, (see Table

4.3 for exact peak locations) and the n-octane peak at 2853.5 cm-1 corresponds to the n- decane shoulder at 2848.5 cm-1. Since the n-decane spectra are measured at temperatures

~30 K higher than the n-octane spectrum, the absence of a sharp peak in the n-decane spectrum may be reflect thermal broadening.

Table 4.3: The locations of the absorption peaks observed in the solid normalized absorptivities for n-octane and n-decane differ from each other by less than 0.5 cm-1. All the values reported are in units of cm-1. n-octane n-decane 2848.5 2849.0 2870.8 2871.0 2919.0 2918.8 2953.2 2953.5 2962.5 2962.5

Using the normalized absorptivities of the vapor ( v ), liquid ( l ) and solid ( s ), we

determined the mass fractions ( gi / g ), of the individual phases as a function of position using Eq. (2.7) and (2.9). The results are summarized in Fig. 4.7(a) for Exp (ii) and 4.7(b)

for Exp (iii). As expected, in both cases the vapor mass fraction gv / g initially equals one, decreases as the liquid and solid phases appear, and reaches a small finite value at

the exit. The liquid mass fraction gl / g starts from zero, goes through a maximum and

reaches zero at the exit. The solid fraction gs / g starts from zero and increases monotonically to reach a finite value at the exit. In all cases the unconstrained mass balance is within 10%.

The progression of the phase transitions in Exp (ii), Fig. 4.7(a), closely follows the behavior reported in our earlier work for n-octane and n-nonane. In particular, the FTIR experiments detect the presence of the solid phase at least 1 cm upstream of PTM+SAXS analysis. As in our earlier work, we interpret this difference as an indication that surface freezing occurs prior to freezing throughout the rest of the droplet.

1.0 (a) 1.0 (b) vapor vapor solid solid 0.8 0.8

0.6 0.6

inf

g 0.4 g 0.4

0.2 0.2 liquid liquid 0.0 0.0 1.1 1.1 1.0 1.0

MB 0.9 0.9 0 2 4 6 8 10 0 2 4 6 8 10 Position downstream of the throat (cm) Postion downstream of the throat (cm)

The progression of the phase transitions in Exp (iii), Fig. 4.7(b), is more difficult to separate. In particular, in Exp (iii) the liquid mass fraction peaks much sooner after the

onset of condensation and at a much lower level ( gi / g ~20%) compared to Exp (ii).

Furthermore, in Exp (iii) the solid fraction increases at essentially a constant rate after the solid first appears, rather than exhibiting the two distinct slopes observed in Exp (ii). One

68 reason we may not observe the heat release due to fusion in Exp (iii) is simply that this smaller heat release is swamped by the much larger signal from the latent heat of condensation.

To examine the freezing kinetics, we plot the fraction of solid (Fs) in the condensed phase as a function of flow time in Fig. 4.8(a) and 4.8(b), where t = 0 corresponds to the nozzle throat. In Exp (ii), the fraction of the solid increases in the same two-step manner we previously observed for n-octane and n-nonane, consistent with our hypothesis that the droplets first freeze on the surface followed by freezing of the bulk. The surface and the

14 -2 -1 21 -3 -1 bulk freezing rates for these droplets, JS = ~10 cm s and JV = 10 cm s , respectively, are however, an order of magnitude lower than those measured for n-octane and n-nonane. Turning to Exp (iii), Fig. 4.8(b) shows that under these conditions Fs simply increases monotonically before reaching a plateau near the exit.

Here the behavior is consistent with our observations of H2O and D2O droplets freezing, and there is no evidence supporting surface based nucleation. The volume based

21 -3 -1 nucleation rate determined for Exp (iii), is JV= 10 cm s , comparable to the value derived in Exp (ii).

Prior to this point, to detect the presence of solid, we have relied on changes occurring in the system including, latent heat addition, density increase, or the change in shape of the infrared absorption spectra. Exp (iv) in Table 4.1 pertains to pioneering WAXS measurements made at positions where the particles are likely to be crystalline. Fig. 4.9 shows the 2-D detector image of a WAXS measurement made 6.9 cm downstream of the throat.

Figure 4.9: WAXS measurement conducted at 6.9 cm downstream of the throat. The 2-D detector image represents scattering intensity data after a sample to background ratio is taken.

The image represents the intensity after the scattering from the sample has been divided by that from the background. SAXS measurements to estimate the size of the droplets

70 formed under these conditions were reported previously.30 The particles have a radius of

~12 nm. The concentric rings seen on the detector suggest that the particles are crystalline in nature.

It was challenging to obtain similar WAXS results for lower partial pressures of n- decane, or for n-nonane and n-octane, since the particles formed were not big enough to get a good signal to background ratio. We can see that even for Exp (iv) where we were able to confirm the presence of crystalline particles, the scattering from the crystalline particles is only 3-4% of the scattering from the carrier gas and ambient air. These data are reported in this chapter to provide qualitative evidence that under certain conditions, the particles formed in our system are crystalline. Further analysis is required to obtain more information about the crystal structure of n-decane.

n-decane 260 P0 = 30.2 KPa o T0 = 45 C pv = 124 Pa Tm 0 240

220

Temperature (K) Temperature

200

T 180 flow Tisentrope

0 2 4 6 8 10 12 Position downstream of the throat (cm)

Tm. 71

The final experiment, Exp (v), used a very low partial pressure of condensable, pv0 = 124

Pa. Here, as illustrated in Fig. 4.10, condensation does not start until Tflow ~ 194.84 K, i.e, almost 50 K below Tm. Under these extreme conditions we observe two significant differences in aerosol evolution compared to any of our previous alkane experiments. The first difference is that we cannot fit the SAXS spectra to scattering from a polydisperse collection of spheres. Fig. 4.11(a) illustrates a typical SAXS spectrum (blue line) measured near the nozzle exit for pv0 = 124 Pa, together with the spectrum we would have expected (red line) if the aerosol were comprised of compact spherical particles/droplets and aerosol evolution had followed the usual trends. In particular, the anticipated scattering spectrum was calculated assuming that N for Exp (v) is the same as that measured in Exp(iii), that polydispersity is ~ 0.3, and that the mean particle size

1/3 31 scales with (pvo) . Ghosh et al demonstrated that this scaling matches the basic trend for liquid droplets, but that is overestimated because N generally increases as pvo decreases. In Fig. 4.11(a) it is clear that the basic shape of the measured scattering spectrum deviates significantly from the expected curve. All attempts to force a fit to scattering from polydisperse spheres failed to give reasonable results.

-2 -2 10 (a) 10 (b) -3 -3

10 10 ) ) -4

-1 -4 -1 10 10 -5 -5 10 10 -6 -6

10 I(Q) (cm I(Q) (cm 10 -7 10 -7 Scattering Intensity 10 Scattering Intensity -8 Polydisperse Spheres Fractal Spheres 10 -8 10 5 6 2 3 4 5 6 2 3 4 5 6 5 6 2 3 4 5 6 2 3 4 5 6 0.1 -1 1 0.1 -1 1 Q (nm ) Q (nm )

As shown in Fig. 4.11(b), however, assuming the particles are fractals composed of a polydisperse distribution of spherical primary particles yields a better, although not perfect, fit. The fit parameters found for the spectrum in Fig 4.11(b) include the average monomer particle radius, ~0.69 nm, primary particle polydispersity, σ of ~0.13, correlation length – a measure of the overall size of the particles – ξ of ~7.3 nm and fractal dimension Df of ~2.62. Furthermore, the polydisperse fractal spheres model fit the data measured near the onset of nucleation much better and becaming progressively worse as the exit is approached.

Another way to analyze the scattering data is to use the unified power law approach

32 developed by Beaucage. This fit yields the radius of gyration, Rg and the power law

73 exponent, P. We use a two level fit, where the first level corresponds to scattering from the primary particles and the second level corresponds to scattering from the aggregates.

The unified power law fits the data much better than the fractal spheres model as shown in Fig. 4.12.

-2 10 -3

10 )

-1 -4 10 -5 10 -6 I(Q) I(Q) (cm 10 -7 Scattering Intensity 10 Unified power law -8 10 0.01 0.1 1 -1 Q (Å )

The scattering intensity data in Fig. 4.12 have been recorded near the exit. Rg as calculated from the unified power law is 16.9 nm and P is 2.6. The data seem to indicate that the particles are branched mass fractals, with a fractal dimension of 2.65.32

The difficulty in fitting the scattering intensities make it more challenging to calculate the mass fraction of condensate purely from the SAXS data – values required to perform the quantitative FTIR analysis. Nevertheless, we can still constrain the fraction condensed near the exit as follows.

For Exp (ii) and (iii), the fraction of incoming vapor condensed near the exit based on

PTMs alone was 90 %, whereas the corrected value from the integrated PTM + SAXS 74 data analysis was close to 95%. For Exp (v) the fraction of incoming vapor condensed based on PTM alone is 85%. Although we would expect this value to increase after correcting for boundary layer compression, it is unlikely that it will exceed 95%. For the purpose of the quantitative FTIR analysis, we therefore assumed the fraction of vapor condensed at the exit was 90% and calculated the normalized absorptivities for the solid using this value.

The second difference between Exp (v) and those at higher temperature is seen in the

FTIR spectra. As illustrated in Fig. 4.13(a), near the nozzle exit the normalized absorbance spectra agree well with each other.

140 95.1 120 98.1 (a) (b)

/mol) 101.1 2 100

Normalized Absorptivity (m 0 2800 2850 2900 2950 3000 -1 Wavenumber (cm )

Although most of the absorption peaks lie very close to those for the crystalline solid, as noted in Table 4.4, a peak close to the liquid peak at 2958 cm-1 is also present. This 75 suggests that the crystallization is not complete, and, thus, we will refer to this state as

“frustrated” crystallization. Another key difference between the frustrated crystal and the crystal spectrum in Fig. 4.6, is that the intensity ratios between peaks are different. For example, the intensity ratio for the peaks around ~2848 cm-1 and ~2918 cm-1 is 0.52 for the crystalline solid and 0.76 for the frustrated crystal. One consequence of this difference is that, as illustrated in Fig. 4.13(b), these spectra cannot be fit to a linear combination of the εv, εl, and εs values determined previously.

Frustrated crystal Crystalline Solid Liquid 2848.5 2849.0 2854.8 2918.5 2918.8 2925.0 2953.5 2953.5 -- 2957.2 -- 2957.8 2962.5 2962.5 --

Finally from the normalized absorptivities, we find that the particles in the frustrated crystalline state absorb more strongly, than either the solid or liquid. The integrated normalized absorptivity for the crystal is ~8.2 x 105 m/mol whereas for the frustrated crystal, it is ~1.02 x 106 m/mol. This difference cannot be explained by uncertainty in the mass fraction of condensate used to normalize the raw spectra. If we take the spectrum of the frustrated crystal as representative of the final stable state, we can fit the spectra to a linear combination of normalized absorptivities of vapor, liquid and the frustrated crystal

76 and obtain the fraction of these three phases as a function of position in the nozzle. This is shown in Fig. 4.14. The good mass balance suggests our approach is not unreasonable.

We observe that at no point of time is the liquid fraction greater than the solid fraction.

Generally the contribution of the “liquid” state is less than 23% of the condensate.

1.2 frustrated 1.0 crystal vapor 0.8

0.6

inf

g/g 0.4

0.2

0.0 liquid -0.21.11.1 1.0 MB 0.9 0 2 4 6 8 10 Position downstream of the throat (cm) Figure 4.14: Phase-wise mass distribution calculated from FTIR experiments as a function of position for vapor (green), liquid (red) and frustrated crystal (blue). The symbols in the top part are gi / g for the three phases. The lower part shows the material balance.

Fractal structures can form by multiple mechanisms including, cluster-cluster aggregation,33-35 monomer-cluster aggregation34, 35 and turbulent aggregation.36, 37

Cluster-cluster aggregation yields fractal dimensions of 1.7-1.8, and is, therefore, inconsistent with the fractal dimension of ~2.6 observed here. Although the other two mechanisms can lead to the high fractal dimensions observed here, both mechanisms need a large number of monomers – or a continued source of monomers – to ensure a high probability of a monomer particle collision. This is unlikely given the finite length of the nucleation pulse.

An alternate explanation for the structure of the particles formed in our system is as follows. Although the most stable phase at these temperatures is solid, the first particles formed via vapor-liquid homogeneous nucleation are usually liquid.38-42 Owing to the high degree of supercooling, it is likely that once the critical nuclei are formed they freeze immediately. In our setup, nucleation is quenched in ~* μs due to the combined effect of vapor depletion and latent heat release. The residual vapor condenses as the particles grow and coagulation plays almost no role in particle growth due to the relatively low number densities. If the nuclei freeze as soon as they are formed, the particles may not be able to sinter effectively as the residual vapor condenses onto the solid nuclei. As a result, condensing molecules may form domains that are only partially ordered. Subsequent particle growth will occur in a similar fashion. This can potentially cause the particles to assume a fractal cluster-like shape. A pictorial representation of this mechanism is presented in Fig. 4.15.

The fractal structure of the particles comprising the aerosol may also explain the enhanced normalized absorptivities of the particles. Dipole coupling in the particle largely depends on particle shape and structure and hence, these factors can significantly affect the infrared absorption characteristics.43 In particular, the magnetic and electric dipole absorption in clustered particles is stronger as compared to in isolated particles.44,45 Prior studies46, 47 of composite aggregates of metal particles (~10 nm) have shown enhanced infrared absorption intensities as compared to the isolated particles.

Although a direct comparison between our straight chain alkanes and a metal aggregates, is not completely valid, it may still be reasonable to interpret the data in a similar manner;

78 i.e. the enhanced absorptivity stems from the fractal structure of the particles generated in the experiments with the lowest flow rate of n-decane.

4.5 Conclusion

In this study we investigated the effect of n-decane inlet mole fraction on the freezing of nanodroplets formed in a supersonic nozzle. We characterized the thermodynamics of the flow using position resolved static pressure measurements, monitored the particle size using SAXS, and calculated the distribution of the condensible in vapor, liquid and solid phases using FTIR absorption spectroscopy. We found that the manner by which the nanodroplets freeze is governed by the inlet mole fraction of the n-decane. For an inlet pv0 = 264 Pa, FTIR spectroscopy detects freezing prior to PTM or SAXS and we attribute this difference to ordering of the surface prior to nucleation throughout the droplets.

These results are similar to those of our recent studies with n-octane and n-nonane.10 The surface and the bulk nucleation rates for the n-decane droplets are ~1014 cm-2.s-1 and 1021 79 cm-3.s-1, and are an order of magnitude lower than those measured for n-octane or n- nonane. This may reflect the fact that the n-decane droplets are larger than the octane and nonane droplets. For an inlet pv0 = 219 Pa, freezing occurred so soon after nucleation that it was not possible to separate surface induced freezing from freezing throughout the drop. Thus, freezing appear to proceed in a single step. In this case the volume-based freezing rate was 1021 cm-3.s-1.

If the particles are large enough, WAXS can confirm that they are crystalline. For particles with radius of ~12 nm, we observed powder diffraction patterns characteristic of crystalline particles. These experiments are strongly constrained by background scattering from the carrier gas, ambient air, and other unidentified sources. Consequently, obtaining similar results was challenging for smaller particles. Changes to the experimental setup as well as a more thorough optimization of the inlet conditions would be necessary to enhance the signal to background ratio, to gain further insight into the crystal structure of n-decane.

When pv0 = 124 Pa, the nanodroplets cannot be characterized by a Schulz distribution of polydisperse spheres. Rather, SAXS suggests particles exhibit a fractal structure with a fractal dimension ~2.6. Further analysis is ongoing. The FTIR data suggests the particles are not in a fully crystalline state, but are in a frustrated crystalline state. The enhanced absorption measured for these particles is consistent with enhanced absorption observed in other particle systems that exhibit fractal structure.

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Chapter 5: Molecular Dynamics (MD) simulation methods and results from droplet

simulations

This chapter has been adapted from the publications titled “Experimental evidence for surface freezing in supercooled n-alkane nanodroplets”, authored by Viraj P. Modak,

Harshad Pathak, Mitchell Thayer, Sherwin J. Singer and Barabara E. Wyslouzil and a manuscript in preparation titled “Identifying the mechanism of surface freezing in n- alkanes using molecular dynamics simulations”, authored by Viraj P. Modak, Barbara E.

Wyslouzil and Sherwin J. Singer. Sherwin J. Singer and Barbara E. Wyslouzil identified the interaction parameters to use. Mitchell Thayer conducted the simulations on the droplets. Viraj P. Modak analyzed the results from the simulations.

5.1 Introduction

We conduct Molecular Dynamics (MD) simulations on systems including droplets, crystals and slabs to (a) complement our experimental results and (b) identify the mechanism and driving force behind surface freezing. To run our simulations we use the

GROMACS code. The molecular interactions are described by a united atom model where the -CH3 and -CH2 groups are treated as single entities with identical interaction parameters. Section 5.2 summarizes the equations governing the intra and intermolecular interactions, the relevant simulation parameters and the procedures for conducting MD simulations on supercooled droplets. Section 5.3 presents the droplet modeling results.

Further studies conducted to obtain insight into the driving force behind surface freezing and the mechanism by which it occurs will be described in Chapters 6 and 7.

5.2 Molecular modeling parameters

Bonded intramolecular potentials include:

(1) the bond stretching potential between two united atoms is given by

1 V  k (l  l )2 (5.1) b 2 b b b0

where, lb is the bond length.

(2) the bond angle vibration potential given by

1 V  k (  )2 (5.2) a 2  0 where, θ is the complement of the bond angle.

(3) and the dihedral potential: Ryckaert-Bellemans function

3 n Vrb  Cn (cos( )) (5.3) n0 where ψ is the torsion angle.

For the inter and the intramolecular non-bonded interactions we use the 6-12 Lennard

Jones potential

 12 6   LJ   LJ  VLJ  4 LJ       (5.4)  r   r  

where εLJ is the depth of the potential well and and σLJ is the distance at which the LJ potential is zero.

We investigated two sets of interaction parameters and those are summarized in Table

5.1. The first set was that of Paul, Yoon and Smith1 (PYS) and these were used by Yi and

Rutledge2 to simulate n-octane freezing in bulk samples. For nonadecane, the PYS model overestimated the liquid surface tention by more than 25% as compared to the experiments. Hence, for our analysis of C19, we also used the TraPPE model proposed by

Martin and Siepmann3 because this potentially reproduces the liquid surface tension quite well.

The cutoff distance chosen to truncate a potential can to significantly affect the physical property values derived from the simulations including bulk density and the surface tension. Furthermore, in droplet simulations, we found that truncating the non-bonded interactions near the usual value of 1.0nm produced severe artifacts in the droplet shape.

Although standard energy and pressure corrections for cutoff r–6 interactions are very effective for uniform systems, they do not apply to non–homogeneous systems. Since many of the studies described here, and in subsequent chapters, are done with systems with interfaces, choosing the correct cutoff scheme is critical. While tail corrections for inhomogeneous systems can be incorporated to derive more accurate results, they usually employ approximations.

Table 5.1: Values of parameters used in Eq. (5.1 – 5.4) governing the interaction potentials.

PYS TraPPE 5 5 kb (kJ/mol) 2.92 x10 2.92 x10 lb0 (nm) 0.153 0.153 2 kθ (kJ/mol∙deg ) 502.08 519.6543

θ0 (deg) 109.526 114.000

C0 (kJ/mol) 6.505 8.39736

C1 (kJ/mol) 16.995 16.78632

C2 (kJ/mol) 3.620 1.13393

C3 (kJ/mol) -27.12 -26.316

CH3 – CH3: 0.815

εLJ (kJ/mol) 0.469 CH2 – CH2: 0.382

CH3 – CH2: 0.558

CH3 – CH3: 0.375

σLJ (nm) 0.401 CH2 – CH2: 0.395

CH3 – CH2: 0.385

Using a large long range cutoff yields accurate total energy values and tail corrections for surface free energy that are no more than a few percent. To implement the long range cutoff without extensive computational cost, we used a twin range cutoff. Here, forces between particles separated by 1.0 nm or less were updated every step, whereas forces up separations of 2.4 nm were updated every 5 steps. These values were chosen after running simulations with different values for the outer cutoff for our droplet simulations.

In particular, we found that the energy dependence of the droplet shape converged for an outer cutoff set at 2.4 nm. For simulations on crystals, when the simulation box length

90 was less than twice this value, we decreased the long range cutoff to 2.3nm with negligible effect. Temperature control was achieved using the v-rescale thermostat. The pressure for calculations in the NPT ensemble was always set to 1bar and was maintained using the Parrinello-Rahman barostat.

5.3 Droplet Simulations

All the droplet simulations used the PYS potential. To start the droplet simulations, we formed a crystal containing 3840 n-octane molecules and placed it in box with dimensions 20x20x20 nm3. The code for generating the starting crystal is included in

Appendix E. In GROMACS the interaction potential and the simulation parameters are entered in the “.top” and “.mdp” files respectively. Samples of these files are provided in

Appendix F and Appendix G, respectively.

Periodic boundary conditions were employed for numerical efficiency in order to use the efficient link-cell algorithm, with the periodic replicas of droplets separated by a distance greater than the range of the interaction potential. We melted the crystal by setting the system temperature to 300K and running for 1.7 ns to ensure complete melting. We then re-equilibrated the droplet 230K, still well above the melting temperature, Tm for this potential, 2161K. The procedure for obtaining the melting temperature from simulations is described in Chapter 7. Finally, we quenched the system to 190K and followed the droplet for 130 ns as it froze. During the droplet simulation, the angular momentum4, 5 drifted away from its initial value of zero and was, therefore, quenched every nanosecond.

~12 nm

Fig. 5.1 illustrates snapshots of the n-octane droplets at intermediate times in a 130ns run at 190K. Within the first few nanoseconds, the intermolecular distances and intramolecular coordinates relax from their initial state characteristic of 230K to values characteristic of 190K. Also, within a few nanoseconds the surface freezes with a single layer of molecules aligned perpendicular to the bulk liquid.

In addition, as shown in Fig. 5.2, these processes are reflected by a steep drop in the

Lennard-Jones cohesive energy of the droplet, as well as by a drop in the dihedral energy as alkane chain links are able to assume trans configurations. The average angle-bending potential hardly changes, and the bond stretching potential in fact rises, indicating that the

92 bonds are stretched in the ordered state. This is followed by a time period where we observe just a single ordered layer on the surface. Subsequently the droplet starts to freeze inward from the surface frozen layer to the center in a layer-by-layer manner.

Two major events of this type are visible in the configurations shown in Fig.5.1, and the energy curves of Fig. 5.2: formation of a second frozen layer after 25ns, and a third frozen layer after 85ns. The implication of these results is that a surface frozen layer appears virtually instantaneously as soon as the droplet temperature dips below the surface freezing temperature, while “bulk” freezing appears to proceed by heterogeneous nucleation at a previously frozen surface.

5.4 Conclusion

The droplet simulations complement the experimental work related to surface freezing discussed in Chapters 3 and 4 and support our claim of surface-based nucleation. We simulated an n-octane droplet with 3840 molecules at 190 K and described the molecular interactions using a united atom model. We found that upon cooling a liquid droplet to below the surface melting temperature, a monolayer developed on the surface of the droplet within a few nanoseconds. This structure persisted for tens of nanoseconds before layer-by-layer freezing of the entire droplet began. This behavior was also reflected in the different contributions to the potential energy of the droplet with respect to time. The freezing mechanism suggested by our droplet simulation does not follow either of the two simple models used to interpret the freezing data in Table 3.2. Nucleation by layer-by- layer freezing is distinctly heterogeneous, in that the successive freezing steps always begin in the fluid immediately under the last frozen layer. However, the heterogeneous mechanism developed by Weidinger et al, and utilized in Table 3.2 assumes that the rate- determining step is nucleation of the layer beneath the surface. However, in the simulation, crystallization of the second layer beneath the surface took longer to nucleate than the first layer below the surface. Therefore, extensive simulation and further theoretical development, as well as further experiments, are required to fully understand the mechanism of alkane crystallization in the presence of a free surface. These studies are described in Chapters 6 and 7.

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Advances in Polymer Science. 2005; 173: 105-49.

Chapter 6: Solid-vapor surface free energy

Sections of this chapter have been adapted from a manuscript titled “Determination of the crystal-vapor surface free energy”, authored by Viraj P. Modak, Barbara E. Wyslouzil and Sherwin J. Singer. Sherwin J. Singer developed the methods for the calculations presented in the chapter. Viraj P. Modak implemented the methods and performed the data analysis and interpretation.

6.1 Introduction

As was mentioned in Chapter 1, there exists a need for a simple method to calculate the solid-vapor surface free energies using molecular dynamics (MD) simulations. Taking cue from previous methods,1, 2 thermodynamic integration from full to no interactions is an obvious choice and will be explored here. In addition, we will consider multiple levels of approximations. For example, we consider the interaction –ΔV, the interaction between two slabs and assume that the probability distribution against which exp(–βΔV) is averaged is Gaussian. We found that the results from this approximation, yielded accurate results for a system of particles interacting via a Lennard-Jones (LJ) potential. Upon further examination we found that, this approximation turns out to be accurate only because the contribution of the Gaussian average to the overall surface free energy is small. We dissected this case to understand the true, non-Gaussian probability distribution. Besides a better appreciation of fluctuations at the crystal-vapor interface, this points us toward ne approximations for the surface free energy. 96

This chapter is organized as follows. In Section 6.2, we will introduce the virtual cleaving process for creating surfaces from crystals. In Section 6.3, we will describe the Gaussian approximation for obtaining surface free energies. In Section 6.4 and 6.5, we will talk about other methods which we use to perform more rigorous calculations for the surface free energies. Results for the LJ system are presented in Section 6.6. In Section 6.7, we describe another approximation for calculating the surface free energies. The theory in

Section 6.7 is a result of the analysis performed for the LJ system. In Section 6.8, we present our results from testing these multiple methods on n-octane and n-nonadecane.

Concluding remarks are presented in Section 6.9.

6.2 Creation of the free crystalline surface

Figure 6.1: Creation of the crystal-vapor surface from initial state “0” (left) to the final state “1” (right).

Consider the cleavage process in a single-component system, shown schematically in Fig.

6.1, in which a solid-vapor interface is created. The initial state, state “0”, consists of Nc molecules in the crystalline phase, and Nv ≈ 0 molecules in the vapor phase , both in a periodically replicated volumes V. In the final state, state “1”, the two volumes contain

NA and NB molecules, respectively, such that N = Nc + Nv = NA + NB. The final state

97 systems, which contain a solid- vapor interface, are only stable when the pressure and temperature are determined by coexistence conditions. However, the surface free energy of metastable systems may also be studied.

We anticipate situations of greatest interest where there are essentially zero vapor molecules. In these cases, the molecules of the initial state are broken into A and B subsets, N = NA + NB, according to their destination in the final state, and are also chosen so that the boundary between A and B molecules forms a crystal plane for which the surface free energy is required. The cleavage process is equivalent to turning off the interactions between the A and B molecules in a volume V, creating four interfaces of area Λ. This leads to several well-known routes to the surface free energy.

6.3 Gaussian approximation

The Zwanzig3 formula expresses the free energy difference between two states as the average of the difference of the Hamiltonian in the final state minus the initial state.

N  A0,1 V (r ) V e  e  dV e P0 V  (6.1) 0 

N NA NB N Vr  VAr VB r Vr  (6.2)

N P V    V r  V  (6.3) 0 0

1   A (6.4) 4 0,1

A0,1 is the Helmholtz free energy difference between the initial and final states of Fig.

... 1. The notation 0 means average in the initial, “0” state which is governed by the full

N N A NB potential V r . The potentials VA r  and VB r  include interactions among A and B molecules, respectively. Straightforward evaluation of the Zwanzig formula is problematic for the reasons published before.4 The probability distribution for V in the

V  initial state P0 V  peaks far from the function e P0 V . Therefore, a simulation conducted in the “0” state does not sample configurations that most significantly contribute to the integral in Eq. (6.1). Below, we demonstrate that taking to be

Gaussian even in the tails of the distribution that are effectively unsampled in a simulation of the “0” state, gives a surprisingly accurate estimate of the surface free energy in some situations.

1   2    V  V 2  V (6.5) C2  0  0  4  2 0 

The Gaussian approximation represents a truncation of the Cumulant series,5

2 3 4   2  3   4 2 2   V  V  V   V 3 V  1 0 2! 0 3! 0 4!  0 0   Cn    (6.6) 4   5    V 5 10 V 2 V 3 ....  5! 0 0 0 

V  V  V where, 0 . Calculations beyond cumulant of the second order are difficult because of the large statistical error associated with averaging higher powers of V .

6.4 λ integration

Another route to the surface free energy is available if we stipulate a series of potentials

N N N V r  parameterized by  such that   0 is the full potential V0 r Vr  of the

N NA NB initial state, and V1r VA r VB r  of the final state. An obvious choice is:

N N NA NB V r  1 Vr  1 VA r VB r  (6.7)

However, this choice is not optimal for pair interactions because the potential disappears discontinuously at both ends of . Hence we use a soft-core potential scheme which has been developed previously.6-8 In addition to the overall scaling by as in Eq. (6.7), the potential that grows in with has a modified argument,

1 sc m p m m v r  v   r  (6.8) with a corresponding expression with replaced by 1  for the potentials that disappear with λ. We employed soft-core potentials for LJ interactions with m = 6; p = 1;

α = 0.7. For the non-bonded LJ interactions in n-nonadecane, we changed α to 0.6. The symbol means the usual scaled potential without the use of soft-core interactions.

sc N The symbol V r  denotes the scaled potential with soft-score interactions. With some

-dependent evolution between initial and final state defined, the surface free energy can be accurately calculated by defining k intermediate values.

4   A  A  A ... A (6.9) sv 0,1 0,1 1,2 k ,1

100

Passing to the limit of many steps gives the standard  integration, but a more powerful approach is to use the Bennett algorithm9 to estimate the stepwise free energy differences with soft core potentials.

sc sc f  V V  C j, j1  A j 1 j e  j , j 1  j sc sc (6.10) f  V V  C j, j1  j 1 j j1 n A j1 C j , j 1  j , j 1 e  e (6.11) n j

The soft-core potentials even out the free energy differences Aj,j+1 among all the intervals and provide a very efficient route for calculation of free energy differences. Henceforth, in this chapter, this method is referenced as the Bennett acceptance ratio (BAR) method.

6.5 Umbrella Sampling

While the Gaussian approximation to P0 V in Zwanzig’s formula in Eq. (6.1) is poor, the formula is viable as an route to the surface free energy with an accurate estimate of the probability distribution . In the full crystal, only a narrow range of the reaction

V  coordinate V is sampled, and near the maximum of e P V  cannot be P0 V  0 determined. Introducing V r N , with V r N defined in Eq. (6.2), as a biasing potential within an umbrella sampling10 procedure the required values of ∆V can be sampled. The

biased probability distributions are then combined to form the overall P0 V  according to the WHAM method.11-13

101

6.6 Solid-vapor surface free energy of the LJ solid

The solid-vapor interface of a system of LJ particles was studied by Broughton and

Gilmer.1, 2, 14-18 To compare with their results for the crystal-vapor surface free energy of the (111) surface (Fig. 9 of Ref.2), we used, with very slight modification of the Cn coefficients described below, the same smoothed cut-off from the work of Broughton and

Gilmer.

12 6        4        C1 , r  2.3    r   r     12 6 2          vr  C2    C3    C4   , 2.3  r  2.5    r   r   r   0, r  2.5       

C1 = 0.016316565128809635; C2 = 3136.832254427915; C3 = –68.0740402750283;

C4 = –0.08331678930715568; C5 = 0.7469338898479347.

The negative sign on C4 is consistently missing in the articles by Broughton and Gilmer.

Also, because the matching conditions were noticeably off, even in single precision arithmetic, we extended the precision of the coefficients to exactly match the value and first derivative of the potential at 2.3σ and 2.5σ.

To run the simulations, we used a crystal with 9504 LJ atoms. The crystal itself was made of two groups, to facilitate virtual cleaving as described in Section 6.2. We ran the simulations at constant volume and temperature (NVT) at four different temperatures.

The simulations were 5 ns long each. The reduced crystal lattice parameters used in the

102 simulations, were calculated as a function of reduced temperature using the relation published by Broughton and Gilmer.19

* The reduced solid-vapor surface free energy  SV , using the Gaussian approximation for an LJ solid is illustrated in Fig. 6.2, as a function of reduced temperature. The statistical error in the plots is negligible. We can observe that the Gaussian approximation works very well up to the onset of crystal surface pre-melting.

Figure 6.2: Crystal-vapor free energy of the (111) face of the Lennard-Jones crystal in reduced units, , as calculated (curve) by Broughton and Gilmer using thermodynamic integration from 0K1 ,2 are estimated here by digitizing Fig. 9 of ref.2.

as calculated (black diamonds) by thermodynamic integration using either the

Bennett acceptance ratio method, Eq. (6.10-6.11) or the Zwanzig formula with P0 V  obtained from umbrella sampling and the WHAM method. using the first cumulant (black squares), second cumulant (black triangles) approximation, or Gaussian approximation (open circles) in Eq. (6.5).

As mentioned previously, we also calculated the value of the solid-vapor surface free energy for an LJ solid, using the BAR method as well as using umbrella sampling 103 according to the WHAM method. We performed these calculations at a single reduced

* temperature of 0.2. For the BAR method, we estimated the value of  SV to be 1.99, which agrees very well with the Gaussian approximation value of 2.02.

We also calculated the exact P0 V  curve, for an LJ solid using umbrella sampling according to the WHAM method. We observed a linear “tail”20 in the behavior of

1  ln P V  as is shown in Fig. 6.3(a). Furthermore, the slope of the linear region is  0 almost exactly equal to –1. We explain the presence of this linear tail as follows.

 V  V  0  Figure 6.3: (a) and (b) e P0 V  calculated for an LJ solid

V using WHAM. varies linearly with V away from 0 .

From Eq. (6.3), we can express P0 V  as

104

 N N V r N  P0 V  dr  V r  V e (6.12) 

 N N N N N V (r )  VA r VB r  P0 V  dr  V r  V e e (6.13) 

 N N V N N  VA r VB r  P0 V  e dr  V r  V e (6.14) 

In regions where the integral in Eq. (6.14) is independent of V , we will observe that

1  ln P V  will vary as a linear function of V , with a slope of –1. From  0

 V  V  0  * e P0 V , Eq. (6.1) and Eq. (6.4) we calculate  SV to be 1.99, which in exact agreement with the value from the BAR method.

6.7 Approximation for P0 V 

1 f0 V    ln P0 V   f0 V  V

(a) (b)

V V V V 1 m 0

1 Figure 6.4: Qualitative behavior of (a) f V    ln P V  and (b) f V  V 0  0 0

105

In Fig. 6.1, the free energy surface governing fluctuations in V , when full interactions

1 govern the system can be described as f V    ln P V . As was discussed in 0  0

Section 6.6, the qualitative behavior for the LJ solid is shown in Fig. 6.4. The fluctuations

V are almost perfectly Gaussian around 0 , which is the average for the system with full interactions. Hence we can write,

1 2 1 f V  V  V  ln 2 2 0   2  0   0  (6.15) 2 0 2

 2  V  V 2 where 0  0  , the variance for the fluctuations in , for a system with full interactions.

Now, the integrand in Eq. (6.1) is within a constant, the probability distribution for , when there are full interactions between the two sub-systems. Using this and the relation derived by Bennett, we can write,

Z0 V P1V   e P0 V  (6.16) Z1

 N N V r  where, Z  dr e , the configuration integral where the interaction 

between the subsets is multiplied by 1  as in Eq. (6.7). P1V  is peaked around

V  dV P V V . The data presented in Section 6.6 suggested that is 1  1

less perfectly Gaussian as compared to P0 V .Hence, we can write,

106

V   V  e P (V ) V 1 0 dV e P0 V   e P0  V  dV  (6.17)  1   V e 1 P V 0  1 

  V  V 1 P1 V  dV e P0 V   e P0  V  d(V ) (6.18)  1  P V 1  1 

  f ( V ) V V  0 1 1 2 dV e P0 V   e 2 1 (6.19) 

2 Eq. (6.15) follows from Eq. (6.14) and Eq. (6.13). In Eq. (6.15), the symbol 2 1

 1 stands for the integral d(V )P1 (V )P1 ( V ) . Only if PV  is close to Gaussian,  1 1 would  2  (V  V )2 . However, we have retained the symbol reserved for the 1 1 variance as a reminder of its relation to the Gaussian limit.

A simple approximation for the free energy surface is generated by supposing there is a

value of V , where we can match the quadratic behavior of f0 V  described in Eq.

V (6.15) with a line of slope –1 expected for values of further away from 0 in the direction of V . Denoting this value of as , the matching value shown in Fig. 1 Vm

6.2, we choose it to be the point where the slope of the quadratic function in Eq. (6.15) becomes –1.

V  V   2 m 0 0 (6.20)

1 2 1 2 f V     ln2  (6.21) 0 m 2 0 2 0

107

f V The value of 0  1 , required in Eq. (6.15), is estimated by extending a line with a

slope of –1 back from f0 Vm .

1 1 f  V  f V  V  V   2  V  V  ln2 2  (6.23) 0 1 0 m 1 m 2 0 1 0 2 0

f V Inserting the estimate for 0  1  from Eq. (6.23) into Eq. (6.19), we obtain an

approximation for the surface free energy that incorporates the linear behavior of f0 V 

V V over most of the range from 0 to 1 .

1 2 2    0  V A0,1 V 2 0 1 e  dV e P0 V   e  (6.24)  0

1 1   1      A  V   2  ln 1  0,1  0 0   (6.25) 4 4  2   0 

This expression is identical to the second cumulant expression of Eq. (6.5) with the

 exception of addition term dependent on 1 . The results from this theory are mentioned  0 in Section 6.9 for an LJ solid. However, preliminary results indicate that the assumption

that f0 V  is quadratic up to the matching point is not accurate.

6.8 Solid-vapor surface free energy of n-octane and n-nonadecane

We obtained the solid-vapor surface free energy,  SV of n-octane and n-nonadecane using the Gaussian approximation as a function of temperature. The temperature range

108 was chosen to be in the vicinity of the respective equilibrium melting points, Tm. The procedures for calculating Tm are highlighted in Chapter 7.

For n-octane we use a six-layered crystal with 720 molecules, made of two groups to create a virtual surface as described in section 6.2. The virtual surface is created in the

XY plane. We run this crystal at constant pressure and temperature (NPT), to estimate the lattice parameters (Lx, Ly and Lz) as a function of temperature. We use a pressure of 1 bar for the simulations. For the NPT simulations, we averaged data over 10 ns after 5 ns worth of equilibriation. A schematic of the crystal is illustrated in Fig. 6.5(left) along with the lattice parameters as a function of temperature in Fig. 6.5(right).

4.95

7.3 4.90 Ly

4.85 Lz 7.2

(nm) y 4.80

(nm)

or L or

L 4.75 7.1

4.70 Lx

z 4.65 7.0 190 200 210 220 x Temperature (K) y

109

We can observe that the crystal size increases only in the X and the Y direction, whereas across Z it is fairly constant. We then use these lattice parameter values and run

simulations at constant volume and temperature (NVT) to calculate  SV . For the

Gaussian approximation, we averaged data over 5 ns after an equilibriation run of 2.5 ns.

The  SV values, calculated using Eq. (6.5) for n-octane are illustrated in Fig. 6.6. We also calculate using the BAR method at a single temperature close to Tm, 210 K, as

N described in Section 6.4. For the BAR simulation, we specify the potential V r  for

0   1. We divide the range of  into 10 intervals and calculate A for each of 1 ,2 the intervals. Each simulation was 1 ns long after allowing the system to equilibriate for

300 ps. We obtain the surface free energy from the sum of the free energy difference of the individual intervals as illustrated in Eq. (6.9). We found that the BAR value (34.74 mN/m) was in good agreement with the Gaussian approximation value (32.75 mN/m).

110

We adopt a similar procedure for n-nonadecane. Once again, we used a six-layered crystal for our simulations. In this case the crystal consisted of 1620 n-nonadecane molecules. In the NPT simulations to obtain the lattice parameters, we averaged data over

5 ns after allowing the system to equilibriate for 2.5 ns. The lattice parameters are illustrated in Fig. 6.7(a) as a function of temperature. We observe that for n-nonadecane,

Lz does not remain constant as was observed for n-octane, but increases with temperature.

Similar to the procedure adopted for n-octane, we use the lattice parameters thus obtained

to set up the NVT simulations from which we calculate  SV . Once again, we average data over 5 ns after allowing the system to equilibriate for 2.5 ns. as a function of temperature is illustrated in Fig. 6.7(b).

111

(a) (b)

Similar to the procedures adopted for n-octane we use the BAR method to calculate  SV at Tm which is 323 K for n-nonadecane. The BAR value for n-nonadecane is 23.55 mN/m whereas the Gaussian approximation value is 28.78 mN/m. The agreement between the two methods is worse for n-nonadecane as compared to what we observed for n-octane.

We try to investigate this further by understanding how a crystal system differs from a system with surfaces present i.e. slabs. From independent simulations, we found that the spacing between adjacent layers in the interior of the slab is higher as compared to the spacing between adjacent layers in a bulk crystal. This is illustrated in Fig. 6.8(a). Such a layer “drift” was much less pronounced for n-octane, as is illustrated in Fig. 6.8(b).

112

2.68 1.23

(a) Slab (b) Slab Crystal Crsytal 2.67 1.22

2.66 1.21

2.65 1.20

Layer spacing (nm) Layer

2.64 1.19

2.63 1.18 0 1 2 3 4 5 6 0 1 2 3 4 5 6 Spacing location Spacing location

We also observed that the layer drift increased further for the BAR simulations with no interactions within the two subsystems. It can be noted that such a case essentially represents simulating two non-interacting three-layered slabs.

Furthermore, we also investigated the effect on layer “broadening”, on going from a crystal to a slab. We quantified layer roughening as a difference in the density of each layer when in a slab and the density of the same layer in the crystal. The quantities in the calculations are as functions of slab length. This is illustrated in Fig. 6.9. Each color in

Fig. 6.9 represents the density difference for a particular layer. As we can observe from

113 the trends, the layers tend to broaden on going from a crystal to a slab. Both layer “drift”

and layer “broadening” can be reasons for why Gaussian approximation fails.

crystal

on on going from slab to

Layer

ρ Δ

We tested the effect of layer “drift” on the surface free energies by repeating the calculations for the BAR method. In this case we rigorously constrained the motion of the

layers. However, this had negligible effect on the final  SV value calculated. As a result we can conclude that it is unlikely that the layer drift as the reason for the discrepancy between the Gaussian approximation and the BAR method.

6.9 Conclusion

In this chapter we have highlighted results from multiple approaches to calculate by virtually cleaving a solid crystal into two subsystems. The salient results from these calculations are listed in Table 6.1.

114

Gaussian linear approx. System Temperature  integration WHAM Approx. theory

LJ 0.2 2.02 1.99 1.99 2.02 n-octane 210 K 32.75 34.74 -- -- n-nonadecane 323 K 28.78 23.55 -- --

We started with a simple LJ solid and calculated using full  integration (BAR

method), umbrella sampling (WHAM) as well as by assuming that P0 V  demonstrates

Gaussian behavior. For a LJ solid, the Gaussian approximation works really well for an

LJ system. Using WHAM, we calculated the curve for the LJ solid and observed

1 a linear tail in  ln P V , with a slope almost exactly equal to -1. This led to another  0 theory which took into account this linear behavior and allowed us to come up with an approximation for . After examining the LJ solid closely, we turned to n-octane and n-nonadecane and calculated using a Gaussian approximation and the BAR method. For n-octane we found that the Gaussian approximation worked well, when compared to a full  integration. The Gaussian approximation did not perform as well for n-nonadecane. To account for the disagreement between the Gaussian approximation and the integration, which is predomnant for n-nonadecane, we explored the characteristic differences between a crystal and a slab with surfaces. For example, we 115 noticed considerable layer drift as well as layer broadening in an n-nonadecane slab as compared to a crystal. BAR simulations on a system where the motion of the center of

mass of the layers was severely constrained led to a negligible change in  SV . This effectively ruled out layer drift as a factor but layer broadening might play a role.

Quantifying the effect of the same, on , is challenging using the current methods, and is yet be explored.

116

References:

1 Broughton JQ, Gilmer GH. MOLECULAR-DYNAMICS INVESTIGATION OF

THE CRYSTAL FLUID INTERFACE .4. FREE-ENERGIES OF CRYSTAL

VAPOR SYSTEMS. Journal of Chemical Physics. 1986; 84: 5741-48.

2 Broughton JQ, Gilmer GH. SURFACE FREE-ENERGY AND STRESS OF A

LENNARD-JONES CRYSTAL. Acta Metallurgica. 1983; 31: 845-51.

3 Zwanzig RW. HIGH-TEMPERATURE EQUATION OF STATE BY A

PERTURBATION METHOD .1. NONPOLAR GASES. Journal of Chemical

Physics. 1954; 22: 1420-26.

4 Beck T, Paulaitis M, Pratt L. The Potential Distribution Theorem and Models of

Molecular Solutions. New york: Cambridge University Press 2006.

5 Kubo R. GENERALIZED CUMULANT EXPANSION METHOD. Journal of

the Physical Society of Japan. 1962; 17: 1100-20.

6 Pham TT, Shirts MR. Optimal pairwise and non-pairwise alchemical pathways for

free energy calculations of molecular transformation in solution phase. Journal of

Chemical Physics. 2012; 136.

7 Pham TT, Shirts MR. Identifying low variance pathways for free energy

calculations of molecular transformations in solution phase. Journal of Chemical

Physics. 2011; 135.

8 Beutler TC, Mark AE, Vanschaik RC, Gerber PR, Vangunsteren WF.

AVOIDING SINGULARITIES AND NUMERICAL INSTABILITIES IN FREE-

ENERGY CALCULATIONS BASED ON MOLECULAR SIMULATIONS.

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9 Bennett CH. EFFICIENT ESTIMATION OF FREE-ENERGY DIFFERENCES

FROM MONTE-CARLO DATA. Journal of Computational Physics. 1976; 22:

245-68.

10 Torrie GM, Valleau JP. NON-PHYSICAL SAMPLING DISTRIBUTIONS IN

MONTE-CARLO FREE-ENERGY ESTIMATION - UMBRELLA SAMPLING.

Journal of Computational Physics. 1977; 23: 187-99.

11 Kumar S, Bouzida D, Swendsen RH, Kollman PA, Rosenberg JM. THE

WEIGHTED HISTOGRAM ANALYSIS METHOD FOR FREE-ENERGY

CALCULATIONS ON BIOMOLECULES .1. THE METHOD. Journal of

Computational Chemistry. 1992; 13: 1011-21.

12 Souaille M, Roux B. Extension to the weighted histogram analysis method:

combining umbrella sampling with free energy calculations. Computer Physics

Communications. 2001; 135: 40-57.

13 Roux B. THE CALCULATION OF THE POTENTIAL OF MEAN FORCE

USING COMPUTER-SIMULATIONS. Computer Physics Communications.

1995; 91: 275-82.

14 Broughton JQ, Gilmer GH. MOLECULAR-DYNAMICS INVESTIGATION OF

THE CRYSTAL FLUID INTERFACE .6. EXCESS SURFACE FREE-

ENERGIES OF CRYSTAL LIQUID-SYSTEMS. Journal of Chemical Physics.

1986; 84: 5759-68.

15 Broughton JQ, Gilmer GH. MOLECULAR-DYNAMICS OF THE CRYSTAL

FLUID INTERFACE .5. STRUCTURE AND DYNAMICS OF CRYSTAL

MELT SYSTEMS. Journal of Chemical Physics. 1986; 84: 5749-58.

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16 Broughton JQ, Gilmer GH. MOLECULAR-DYNAMICS INVESTIGATION OF

THE CRYSTAL FLUID INTERFACE .1. BULK PROPERTIES. Journal of

Chemical Physics. 1983; 79: 5095-104.

17 Broughton JQ, Gilmer GH. MOLECULAR-DYNAMICS INVESTIGATION OF

THE CRYSTAL FLUID INTERFACE .2. STRUCTURES OF THE FCC (111),

(100), AND (110) CRYSTAL VAPOR SYSTEMS. Journal of Chemical Physics.

1983; 79: 5105-18.

18 Broughton JQ, Gilmer GH. MOLECULAR-DYNAMICS INVESTIGATION OF

THE CRYSTAL FLUID INTERFACE .3. DYNAMICAL PROPERTIES OF

FCC CRYSTAL VAPOR SYSTEMS. Journal of Chemical Physics. 1983; 79:

5119-27.

19 Broughton JQ, Gilmer GH, Weeks JD. CONSTANT PRESSURE

MOLECULAR-DYNAMICS SIMULATIONS OF THE 2DR-12 SYSTEM -

COMPARISON WITH ISOCHORES AND ISOTHERMS. Journal of Chemical

Physics. 1981; 75: 5128-32.

20 Beck TL. Hydration Free Energies by Energetic Partitioning of the Potential

Distribution Theorem. Journal of Statistical Physics. 2011; 145: 335-54.

119

Chapter 7: Mechanism of surface freezing in n-alkanes

This chapter has been adapted from a manuscript titled “Mechanism of surface freezing in n-alkanes using molecular dynamics simulations” authored by Viraj P. Modak,

Barbara E. Wyslouzil and Sherwin J. Singer. Viraj P. Modak and Sherwin J. Singer developed the methods discussed in the chapter. Viraj P. Modak ran the simulations and analyzed the data from the simulations.

7.1 Introduction

Surface freezing is known to occur for chain like molecules including intermediate chain length n-alkanes (15 ≤ n ≤ 50). Up until now, the general perception was, below this limit, i.e. for n ≤ 14 a transition to pre-melting would be observed. However, in Chapters

3-5 we also provided evidence for surface freezing in short chain alkanes C8 – C10, as well. As was mentioned in Chapter 1, the idea behind the mechanism of surface freezing has been vigorously debated. The two main points of view, which were mentioned briefly in Chapter 1, are described here in more detail.

The starting point is generally taken to be the wetting free energy,  (0) to introduce a bulk solid phase between the vapor and liquid.

(0)    sv   sl   lv (7.1)

120

Here  sl ,  sv and  lv are the surface free energies of the solid-liquid, solid-vapor, and liquid-vapor interfaces, respectively. In a series of publications, Deutsch, Sirota and co- workers1-3 (DS) advanced the idea that surface freezing occurs as a result of the solid

“wetting” the liquid surface upon the condition,

 (0)  0 (7.2)

The assumption is that the structure of the surface frozen layer is same as the bulk rotator

phase. These workers estimated , and , concluding that  sv   sl below coincided exactly with the range of chain lengths where surface freezing was observed.1

Tkachenko and Rabin (TR) suggest that the structure of the surface frozen layer is very different from the bulk solid.4, 5 According to their model, large fluctuations in the direction of the molecular axis provide an entropic stabilization to the surface frozen layer leading to surface freezing even when the wetting condition, Eq. (7.2), is not satisfied. TR represent the condition for surface freezing for N ordered layers in the following form:

(0) S sl   N(T  Tm )   s  0 (7.3) A0

In TR’s expression, the first two terms are the energy costs to create the surface frozen layer, the wetting free energy cost  (0) and the bulk free energy cost for creating surface frozen layers as a linear expansion about the melting point where it is zero.

121

Here A0 is the area per molecule, andSsl change in the entropy per molecule for a bulk liquid to solid transition. ∆s is the entropic stabilization per unit area for the stack of layers when they are on the surface relative to the bulk, where fluctuations are suppressed. The most favorable transition occurs for just one surface frozen layer,

. For surface freezing to occur anywhere above Tm, the surface freezing condition reduces to:

(0)    s  0 (7.4)

The sign of  (0) is a crucial quantity in deciding between the DS and TR mechanisms of surface freezing, and arguments in both directions were advanced.6, 7 Values proposed by both groups have been listed in Table 7.1.

γsv (mN/m) γsl (mN/m) lv (mN/m) (mN/m) 6 DS : C16 23.07 3.87 28 1.06 6 DS : C36 21.20 2.00 28 4.80 5 TR : 37.03 9.03 28 +18.06

122

The difficulty in proposing a convincing argument for either of the theories stems from the lack of data. Only the liquid-vapor surface tension is amenable to direct experimental

observation, so naturally the debate is centered on the values of  sv and  sl . Of these,

 sv is much larger, and is established for two alkane models in our calculations, described in Chapter 6. Both DS and TR calculate a range of chain length n where surface freezing is expected to occur, and the surface freezing temperature TSF. In both cases, the calculations rely on some experimental data as input. DS take the independence of

at TSF from n as an experimental fact. Arguing that arises from “missing” long range

-6 r dispersion force, they fit parameters in their  sv to experimental data. This provides sufficient parameterization to predict the range of n values where it is possible to satisfy the wetting condition, Eq. (7.2), and the difference between TSF and Tm. TR develop an effective Hamiltonian for height variations and obtain the free energy for a system governed by it by a variational approximation. The bulk freezing temperatures Tm and densities are taken from experiment, while parameters in the effective Hamiltonian are obtained by best fit to experimental surface freezing results. In their theory, surface freezing at low n is suppressed because the short alkanes have low Tm where entropy makes smaller contribution to the free energy. At large n surface freezing is suppressed by the larger energy cost for chains to slide relative to each other. In contrast, DS attribute the disappearance of surface freezing for low n to a transition to pre-melting.

Pre-melting is also understood in terms of a wetting criterion,8, 9 this time driven by

 lv  sl  sv  0 . However, reversing the inequality of Eq. (7.2) does not yield the

123 wetting condition for pre-melting, and in this chapter we demonstrate that pre-melting does not occur when TSF falls below Tm. At the upper n limit for surface freezing, DS point to the likelihood of defects near the higher Tm of longer alkanes, frustrating the formation of a solid surface layer.

We were interested in this problem because our experiments on the freezing of alkane droplets provided evidence for surface freezing for C8 – C10 – well outside the range of alkanes where surface freezing occurs above the melting point. Although the surface frozen state is only metastable and not an equilibrium state in the C8 – C10 experiments, the fundamental physics that drives the SF process should still be the same. We used molecular dynamics (MD) simulations to gain more insight into the mechanism of surface freezing. We chose to conduct extensive simulations on C8, which is of experimental interest to us and C19, since, for this case, the surface frozen state is known to be an equilibrium state and not just metastable.

For our simulations, we use simple united atom interaction models in which methylene and methyl groups are replaced by a single effective atom. Whether from a stable or a metastable liquid phase, surface freezing occurs quite robustly in the united atom models, and therefore they are a suitable starting point for studying the mechanism and quantifying the driving forces associated with surface freezing even though they may be inadequate for describing rotator transitions. Simulations methods similar to the ones described in Chapter 5 are followed. The methods in which we obtain the starting trajectories for the simulations as well as the simulation parameters depend on the kind of simulations which we run. These are described in the relevant sections. Phase boundaries for the bulk liquid-solid and surface liquid-surface frozen transitions at 1 bar are reported 124

10, 11 in Section 7.2. We treated C8 with the PYS interaction model, and C19 with both

PYS and TraPPE12 models. In Section 7.3 we mention the results from our preliminary simulations to eliminate the possibility of surface premelting. In Section 7.4, we provide results for the liquid-vapor interfacial free energies,  lv . Results from independent

simulations to obtain the solid-vapor interfacial free energy,  sv have been described in

Chapter 6. Pertinent results from those studies which are critical in understanding the mechanism of surface freezing are mentioned in Section 7.5. In Section 7.6 we report orientational and translational fluctuation properties of liquid-vapor, solid-vapor and surface frozen interfaces near TSF. We conclude with an analysis in Section 7.7, finding that the mechanism of surface freezing for C8 and C19 are distinct. Neither the DS nor TR picture applies universally to both these cases. Also, factors not included to this point may play a role. Furthermore, while surface freezing is quite facile in both united atom models considered in this work, the ordering of the melting and surface freezing temperatures, and the interfacial properties are quite sensitive to the model.

7.2 Determination of the bulk melting temperature and the surface melting temperature

The bulk melting temperature Tm and the temperature TSF at which surface freezing occurs are central quantities in our study. We used the two-phase interface method13-15 to determine both bulk and surface phase transition temperatures. Tm for n-octane was determined using a system of 960 molecules with the NPT ensemble. An initial configuration with a rotator phase crystal adjacent to a liquid (Fig 7.1(a)) was prepared by thermostating half the system at 200 K and the other at 300 K. Subsequently the

125 system was thermostated at various trial temperatures. The potential energy was used as an order parameter to track whether the solid phase grew at the expense of the liquid, or vice versa. Results at a few trial temperatures are shown in Fig. 7.1(b). The initial slope of the potential energy as a function of time was obtained by a linear fit to data like that shown in Fig. 7.1(b).

(a) (b)

Then, this initial slope as a function of trial temperature was fitted to a line which was interpolated to estimate the transition temperature, the point where the initial slope would be zero. For our octane crystal melt system governing by the PYS potential cut off at

2.4nm, the transition temperature was estimated to be 2151K. When the potential is cut off at 1 nm, from a similar calculation, we found Tm to be 2121K which was in near perfect agreement with 2122K as reported for octane with the same interaction potential and cutoff by Yi and Rutledge.16 The value with 2.4nm cutoff is not significantly

126 different from Yi and Rutledge and is also in excellent agreement with the experimental value of 216 K.17

To determine the surface freezing temperature of n-octane, we simulate a slab of 1920 C8 molecules, with surfaces on either side of the slab in the XY plane. We employ an NVT ensemble and periodic boundary conditions in all three directions. In this case, the slab is prepared with a surface that is partially frozen and partially melted as shown in Fig.

7.2(a). For a temperature lower than the surface freezing temperature TSF we expect the partially surface frozen slab to be completely surface frozen and for a temperature greater than TSF, we expect the slab to be completely melted. At the surface freezing temperature, we expect the two sections to remain in equilibrium with each other. The initial state was prepared by a sequence of steps beginning with an initial crystalline configuration. First, half the surface layer was thermostatted at 170K while the rest of the system was maintained at 320K for 36ps. Then the temperature of the remainder of the system outside the frozen surface patch were maintained at 240K for 62ps, and finally at 213K for 100ps. The frozen surface patch began to disorder past these relatively short times when the remainder was held at 320K and 240K. We verified that these were sufficient times for order parameters like the potential energy to stabilize. Fig. 7.2(b), 7.2(c), and

7.2(d), show the system configuration after 7 ns run at 211, 210 and 212 K, respectively.

127

(a) (b)

Figure 7.2: (a) Starting configuration of a slab that is partially melted and partially surface frozen. This system is simulated to determine the surface freezing temperature.

The slab at 211 K (b) does not show a tendency to transition into being completely surface frozen or completely melted. The final configuration after 7 ns shows that the slab is completely surface frozen at 210 K (c) and progressing towards complete surface disorder at 212 K (d). This shows that the surface freezing temperature for octane is 211

128

At 210 K, the slab is completely surface frozen, whereas at 212 K, it is almost fully melted. Although, the final configuration at 211 K is slightly different than what we start with, we can see that the slab does not show a particular tendency to completely surface freeze or melt. This suggests that the surface freezing temperature for n-octane is 211 K.

We can confirm this by following PE profiles at different temperatures which are shown in Fig. 7.3. As expected, the PE of the system tends to decrease at 210 K, increase at 212

K, and remains relatively flat for 211 K. From this we can conclude that the surface freezing temperature for n-octane for our model is 211 K.

-44000 212 K

-46000 211 K

-48000 210 K

Potential Energy (kJ/mol) Energy Potential -50000

-52000 0 2000 4000 6000 8000 Time (ps)

129

We repeated this procedure for n-nonadecane by running a crystal-liquid system of 960 molecules for calculating the bulk melting temperature. For the surface freezing temperature, we used a system of 960 molecules as opposed to 1920 molecules for n- octane. For nonadecane, we found that for the PYS model, TSF and Tm, are 328 K and

323-324 K respectively. For the TraPPE model the values were, 323 K and 324 K respectively. The experimental values are 3101 K and 3071 K. PYS captures this particular behavior of nonadecane i.e. TSF > Tm, better than TraPPE.

130

(a) (b)

Figure 7.4 (a) PE plot for n-nonadecane for PYS to estimate the bulk melting temperature where we use a partially frozen system of 960 molecules. The PE of the system increases with time for 326 K (red) and decreases with time for 321 K (blue). It remains relatively flat for 323 K (black) and 324 K (grey). (b) PE of the partially surface- frozen system of 960 n-nonadecane molecules tends to increase at 329 K (red) and decrease at 327 K (blue) and relatively stable at 328 K. Similar results are shown for the TraPPE model in (c) and (d). The PE of the partially frozen slab is stable at 324 K, whereas that for the partially surface frozen the PE is stable at 323 K.

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7.3 Eliminating the possibility of surface melting for n-octane

Ocko et al. in their study of surface freezing in normal alkanes1, 18 using x-ray reflectivity and surface tension measurements, establish that the lower limit on the chain length for surface freezing to occur is n ≈ 14. They also state that below this limit, the more common surface premelting phenomenon would be encountered. Using simulations, we were able to confirm that surface melting is not observed, even in the case of n-octane.

We simulate a slab of 720 octane molecules with the NVT ensemble. The slab has two surfaces on either side along the direction perpendicular to the length of the n-octane molecules. Periodic boundary conditions are employed in all three directions.

Starting with a well equilibrated slab, we then set the surface group to 600 K and the bulk at 180 K. The objective here is to get a configuration close to the premelted situation, where the surface layers are melted, but the bulk is still crystalline. We found that a very high temperature was required in induce surface melting, presumably because of a templating effect of the lower layers and effective energy transport into the bulk stolid.

After 105 ps the system reached the desired surface-molten state, which was then further equilibrated holding the surface at 300 K and the bulk at 180 K for 1 ns. We then simulated this entire system (Fig. 7.5(a)) at 215 K, the bulk melting point. Within 15 ns the slab went back to being completely crystalline (Fig. 7.5(b)), which implies that surface melting will not be observed even for short chain alkanes.

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Figure 7.5: (a) n-octane slab with melted surface layers and crystalline bulk. The surface is on both sides of the slab in the Z directions. (b) The final trajectory after configuration in (a) is simulated for 15 ns with the entire system at 215 K. The melted surface layers go back to being crystalline, implying that surface melting cannot be observed for short chain alkanes.

7.4 Liquid-vapor surface free energy

The liquid-vapor surface free energies,  LV is obtained from a slab of 480 octane molecules with two free surfaces, such that the surface normal is parallel to the z-axis.

We use the NVT ensemble and periodic boundary conditions in all directions. We collect the pressure tensor during the simulation and calculate values using the relation,

Lz  1   LV   Pzz   Pxx  Pyy    t (7.5) 2  2 

where Lz is the slab length in the z-direction, Pxx, Pyy and Pzz are the diagonal pressure

tensor values in x-, y- and z-directions, and  t is the tail correction term that has to be 133 added to incorporate the interactions which lie outside the simulation cutoff distance.

Blokhuis et al have derived the formula for tail corrections:

1   sr    12 2 ds r 3 3s 3  scoth dr (7.6) t    2  0 rc  

where  is the difference between the bulk liquid and vapor densities l and  v , respectively and rc is the cutoff distance used in the simulation.  is the interface thickness parameter slab obtained from a fit to a tanh profile of the following form.

   a  z   z  b    z  tanh   tanh  (7.7) 2   2   2 

The parameters a and b are the midpoints of the upper and lower surface, respectively. A typical example of the density profile along with the tanh fit is shown in Fig 7.6.

134

2.0

1.5

1.0

Density (reduced) Density 0.5

0.0

0 5 10 15 20 25 Z (reduced)

We ran 150 ns simulations at 205, 210 and 215 K to obtain the statistical averages of the pressure tensors. They are summarized in Table 7.2 along with the tail corrections

obtained from the density profiles.  LV for octane deviate from experimental values by

~7%, close to the melting point. We also ran simulations at 325 K and found that even at higher temperature the deviation was around ~7%.

135

Temperature Lz  1   Pzz   Pxx  Pyy   t  t  LV  Experiment (K) 2  2 

205 29.30 2.6 31.90 30.00 210 29.13 2.5 31.63 29.52 215 28.69 2.5 31.19 29.05

For octane, the variation of  LV with temperature is shown in Fig. 7.7(a). We fit the data

ex to a linear equation and obtain the surface excess entropy S LV for the liquid as the negative slope of the linear fit.1

d S ex   (7.8) LV dT

For n-octane, = 0.072 mN/(m K). This agrees reasonably well with the experimental value of 0.095 mN/m.K.19

We adopt a similar approach to estimate for nonadecane using the same PYS interaction model as we used for octane. At 325 K the surface free energy predicted by

PYS overestimates the experimental by more than 25%. Hence we decided to also

investigate the TraPPE model as it was known to predict values accurately for C6,

20 # C8, C10 and C17. For the TraPPE model, the disagreement between experiment and 136 simulations is ~10%. The  LV values for nonadecane according to the PYS and TraPPE models are listed in Table 7.3.

Table 7.3: Surface free energy values for liquid vapor interface as a function of temperature for nonadecane. Pressure tensor values are calculated by averaging data over 120 ns. Tail corrections to incorporate atomic interactions outside the cutoff range are obtained from the density profile of the system, using the approach of Blokhuis et al. All surface tensions are given in units of mN/m.

Lz  1  Temperature (K)  LV   Pzz   Pxx  Pyy    t    Experiment 2  2  t LV

PYS 313 31.38 3.08 34.46 26.92 318 31.16 3.06 34.22 26.51 323 30.84 3.04 33.88 26.09 328 30.34 3.01 33.35 25.67 TraPPE 313 26.84 2.30 29.14 26.92 318 26.55 2.28 28.83 26.51 323 26.22 2.26 28.48 26.09 328 25.75 2.24 27.99 25.67

137

From a linear fit to the data shown in Fig. 7.7(b) and 7.7(c), the surface excess entropy

ex S LV value for nonadecane is 0.074 mN/(m K) for the PYS model and 0.083 mN/(m K) for the TraPPE model. The TraPPE surface excess entropy is in near perfect agreement with the experimental value of 0.084mN/m.K.19

(a) (b) (c)

Figure 7.7: Temperature dependence of the liquid-vapor surface free energy ( LV ) for octane (a) and nonadecane (b,c) as listed in Tables 7.2 and 7.3. C19 results for the PYS and TraPPE interaction potentials are given in (b) and (c), respectively. From the linear fits indicated in the plots, we obtain the surface excess entropy of the liquid-vapor interface which is reported in the text.

7.5 Solid-vapor surface free energy

We calculate the solid-vapor surface free energy,  SV , using the Bennett acceptance ratio

(BAR) method. For this purpose, we simulate a six-layered alkane crystal at constant volume and temperature. The lattice parameters used for the constant volume simulations are calculated from independent simulations run at constant pressure.

The six-layered crystal is divided into two groups, each consisting of three adjacent layers. The interaction between the two groups is modulated by a parameter , such that

138

 = 0 indicates full interaction between the groups and  = 1 indicates no interaction between the groups. The intra-group interactions are at full strength throughout the process. At = 1, there are two non-interacting slabs and four solid-vapor surfaces. We

obtain SV , from the Helmholtz free energy difference calculated as the crystal goes from system  = 0 to  = 1.

A  A( 1)  A(  0)  exp 4 SVLxLy  (7.9)

1 Here,   (kBT) and Lx Ly is the area of each of the four surfaces created in the process.

Lennard-Jones interactions between the atoms are turned off as  increases according to the soft-core potential scheme,21

12 6        1 v (r,)  4(1 )     , where r ()  p 6  r 6  6 , (7.10) sc  r ()   r ()  sc  sc   sc   which avoids the sudden disappearance of the singularity at r = 0 at the end of the  interval. Here α = 0.6 and p = 1. We divide the  range into 10 intervals, and calculate

22 A(i )  A(i 0.1), i 1,2,10 by the Bennett’s acceptance ratio (BAR) method. For octane, = 34.74 mN/m, at 210 K for the PYS model, and for nonadecane =

23.15 mN/m, at 323 K for the TraPPE model.

Furthermore, we have also explored an approximate method to estimate . The free energy difference Ain Eq. (7.9), by standard statistical mechanical manipulations (e.g.

Ref. 22) can be cast as an expectation value in a perfect solid.

139

A 4 SVLxLy  V V  e  e  e 1 0 (7.11) 0

In this equation, the subscript “0” refers to a solid with full interactions, while “1” denotes a system in which there are no interactions between two groups and four solid- vapor interfaces are created. The subscripts 0 and 1 coincide with the  = 0 and  = 1 systems, respectively, discussed in relation to Eq. (7.9) and (7.10), although here we are

not considering intermediate  values. The symbols V0 and V1 indicate total potential energy function with and without interactions between the two groups. The angle brackets in Eq. (7.11) denote an average of the exponentiated interaction between two solid slabs that comprise the full solid, taken in the full solid reference system.

In Chapter 6 we investigated the properties of expression (7.11) for the solid-vapor free energy. If we assume that the probability distribution governing the fluctuations of the group-group interactions in Eq. (7.11) is Gaussian, then we obtain the following approximation for the solid-vapor free energy.

1   2    V V  (7.12) SV  1 0 0  4Lx Ly  2 

V V In Eq. (7.12), 1 0 0 is the average value of (minus) the Lennard-Jones interaction energy between the two groups and σ2 is the variance of the same. Elsewhere, we show

that Eq. (7.12) provides an excellent approximation for  SV a Lennard-Jones solid even up to the region where surface premelting occurs, and also for the octane solid-vapor interfaces. However, the Gaussian approximation is less accurate for the nonadecane solid-vapor interface, largely because C19 melts at much higher temperature than C8, 140 where surface fluctuations are much stronger. Even where Eq. (7.12) gives a good result,

 2 it succeeds because V V is much larger in magnitude than and all higher-order 1 0 0 2 corrections. Indeed Eq. (7.12) can be recognized as the first two terms of a cumulant expansion. In principle we could add further corrections in Eq. (7.12). However, such

terms, involving higher moments of V1 V0 , are notoriously difficult to calculate because of large statistical error.

For octane,  SV calculated using the Gaussian approximation is 32.75 mN/m at 210 K, which agrees reasonably well with the BAR value of 34.74 mN/m. For nonadecane at 323

K, Eq. (7.12) estimates the value of to be 28.78 mN/m, whereas the BAR value is

23.55 mN/m.

7.6 Surface properties

7.6.1 Liquid-vapor interface

Special properties of the surface frozen layer above the liquid in the temperature range near TSF and Tm have been invoked to explain the thermodynamic preference for a surface frozen layer under certain conditions.1,4 Even when the bulk solid is the lowest free energy phase, there can exist a kinetic preference for surface freezing, as we have noted in our droplet studies of aerosol droplets.23 Besides properties of the surface frozen layer for temperatures near TSF and Tm, an overlooked feature is the behavior of the liquid surface in this region. Closing the free energy gap between the liquid surface and one with a frozen layer may be achieved either through disorder of the surface-frozen layer,4

141 or incipient order of the liquid surface. This is the motivation for examining the orientation of molecules near the liquid.

We will explain this procedure for one octane molecule as a representative. 1 We track orientational order using six vectors that join alternate united atoms, 2 3 as shown in the adjacent schematic. We then calculate the z coordinates of the 4 center of mass of these pairs, and the projection the vectors on the XY plane, 5 6 that is, the cosine of the angle that the vector makes with the surface normal. 7 This data is then normalized and represented as a 3-D histograms as shown in 8

Fig. 7.8. The Fortran code for making these calculations is provided in Appendix H.

For octane we observed peaks near the location of the surface, which suggests that the molecules close to the surface have a natural tendency to lie perpendicular to the surface.

We observed this behavior even for temperatures slightly higher than Tm as is shown for

217 K in Fig. 7.8(a). This tendency weakens as the intensity of the peaks decreases with respect to temperature. We observe this ordering for the surface of a nonadecane slab, simulated using the PYS model, as shown in Fig. 7.8(b). However, the extent of ordering is significantly reduced when we switch to the TraPPE model as can be seen in Fig.

7.8(c). The PYS and TraPPE slabs were simulated at 333 K and 325 K respectively.

Similar results with a lower degree of ordering for TraPPE model, have been observed by

Hernandez et al.

142

(a) (b) ) (c)

) )

cos

(Z,

ρ ρ

TraPPE model at 325 K. All the temperatures here are slightly higher than Tm as calculated using the respective models.

7.6.2 Solid-Vapor and surface frozen interfaces

Tkachenko and Rabin24 from their studies found that the relative vertical displacement between nearest neighbor molecules at the surface in the surface frozen layer is of the order of the bond length l, i.e.

2 2 h  h ' ~ l  r r  (7.13)

They also found that for the molecules in the interior layers of a solid this value is much smaller than l. Furthermore, they also neglect the difference between the fluctuations between the surface and the interior layers of a solid.

For our system, we calculate difference between the Z coordinates of the centers of mass of all the nearest neighbor molecules. This data is then represented as a histogram as shown in Fig. 7.9, for octane (a) and nonadecane using both PYS (b) and TraPPE (c) 143 models. The Fortran code for these calculations from the simulation trajectories can be found in Appendix I. Here the red symbols correspond to the molecules in the surface frozen layer and the black symbols represent molecules in the solid bulk. The data is estimated at TSF for the particular molecule and the model, i.e. 211 K for octane 329 K for nonadecane PYS and 323 K for nonadecane TraPPE. The blue symbols represent corresponding data for nonadecane calculated at a supercooled temperature of 300 K. The lines in all the plots are the Gaussian fits for the symbols.

The variance σ2 is given by:

2 2 2   h  h  h  h  h  h '  r r' r r'   r r  (7.14)

2 2 For octane, σSF for the surface frozen slab is 0.0198 nm which compares reasonably

2 2 25 2 well with l =0.0161 nm . Compared to this, σB for the interior layers in the solid is

2 0.005 which is four times lower than that for the σSF . Although the trends and values are not in perfect agreement, they match qualitatively with those predicted by Tkachenko and

2 2 Rabin. However, for nonadecane, the difference between σSF and σB is not as prominent.

The values for the PYS model are 0.0433 and 0.0331, respectively, whereas, the values for the TraPPE model are 0.0428 and 0.0361 respectively. Hence, for nonadecane, there is no significant entropic stabilization of the surface monolayer over the solid bulk due to

2 fluctuations of the molecules along their axes.Furthermore, σB for nonadecane solid at

300 K is 0.0166 and 0.0182 for PYS and TraPPE respectively. This suggests that the solid interior becomes more mobile and resembles the surface frozen layer at TSF. This behavior is in stark contrast as shown by octane.

144

Figure 7.9: Probability density function of difference between the Z coordinates of the center of masses of nearest neighbor molecules. The red symbols represent an ordered monolayer of a surface frozen slab , and the black symbols molecules in the interior of a completely frozen solid slab calculated at TSF for the molecule and model. The blue symbols for nonadecane represent the interior of a solid slab at a supercooled temperature of 300 K. The lines in the plots represent the Gaussian fits for the histogram 2 2 data. σB and σSF for (a) octane at 211 K is 0.0198 and 0.005, respectively. Those for (b) nonadecane, using PYS at 329 K are 0.0433 and 0.0331, respectively and for (c) 2 nonadecane, using TraPPE at 323 K are 0.0428 and 0.0361, respectively. σB for nonadecane at 300 K is 0.0166 for PYS and 0.0182 for TraPPE.

7.7 Energy density across slabs

We can calculate the energy and number density of the slab along the surface normal direction, which is shown in Fig. 7.10 for octane in (a) and (b) and for nonadecane, using the TraPPE model in (c) and (d) The Fortran code for calculating the energy density from the simulation trajectories file, can be found in Appendix J. The trends for the solid

(blue), liquid (red) and surface frozen (black) slabs are shown in blue, red and black lines respectively. The simulations are run at Ts i.e. 211 K for octane and 323 K for nonadecane. The data are adjusted so that the origin on x-axis represents the location of the Gibbs dividing surface for each of the slabs. If we compare the liquid and surface

145 frozen slabs we observe that the energy density near the surface for the surface frozen slab is much lower than that for the liquid slab. Also, the interiors of the surface frozen and liquid slabs match each other closely. This suggests there is significant energy gain associated with surface freezing. We also estimate the surface excess energies for the slabs with respect to the Gibbs dividing surface, which are shown in Table 7.4

Table 7.4: Surface excess energy values for liquid, surface frozen and solid slabs respectively. The values are calculated based on the standard definition of the Gibbs dividing surface.

Surface Excess Energy (mN/m) Liquid Surface Frozen Solid octane 51.42 -13.46 51.24 nonadecane 55.26 -277.72 64.85

If we compare the energy density and the number density of the surface frozen layer to that of the solid surface, we find that the difference is more prominent for octane than for nonadecane. This is in agreement with the fact that the molecules in the surface frozen layer are prone to larger fluctuations vertically, than those in the bulk solid, and which are more pronounced for octane. These results suggest that unlike Deutsch et al’s hypothesis, the ordered monolayer does not behave identically as a layer in the bulk solid, with octane being more distinct than nonadecane.

146

(a) (b)

147

7.8 Continuum Theory

We also developed a theory to calculate the surface excess energies by estimating the missing Lennard-Jones (LJ) interactions on creation of a surface. Consider the schematic shown in Fig. 7.11(a), for a system where the molecules are stacked in a layered format.

The energy cost associated with a surface is created between the layer j = 0 and j = 1, would be due to the missing LJ interactions between all pairs of layers for j ≤ 0 and j’ >

0. This is the surface excess energy of such a slab which is given by:

0  Eex   f ( j  j') (7.15) j j'1

Where, the function f(j) calculates the missing Lennard Jones interaction between any single pair of layers located on either side of the surface. For our system, if the surface created is perpendicular along the XZ plane, then f(j) is given by:

   2C f ( j)  dx dz xz 6   2 2 2 6/ 2 (7.16)   (x  ( ja y )  z )

Where, ay = distance between two layers, as shown in Fig. 7.11(b), σxz is the surface number density of united atoms on an XZ surface and C6 is the constant corresponding to the attractive long-range LJ interactions. A similar analysis is also valid for a surface perpendicular to the X axis. However, for a surface in the XY plane, we have to modify the equation to incorporate the thickness of the layers arising due to the length of the n- octane molecules, as shown in Fig. 7.11(c). In such a case, f(j) is given by:

148

  l / 2 l / 2 ( / l )2 C f ( j)  dx dy dz dz' xy z 6     2 2 2 6 / 2 (7.17)   l / 2 l / 2 (x  ( ja z  z'z)  y )

Here, lz is the thickness of a layer, az is the distance between two layers if they are hypothetically collapsed along their respective bisecting planes, with surface number density, σxz.

j = ∞ ay az j = 3

j = 2 lz j = 1 j = 0 j = -1 j = -2

j = -∞ (a) (b) (c)

The surface number densities and the surface excess energies for the three surfaces are listed in Table 7.5. Furthermore, we can also use our Modak-Singer method discussed in

149

Chapter 6 to estimate the surface free energies for the XZ and YZ surfaces. These values calculated for an n-octane slab are also listed in Table 7.5.

Surface Energy using Surface free energy using Surface Surface number density theory (mN/m) Modak-Singer (mN/m)

10 8 6 2 XZ σxz =  14.24 / nm 180.2 54.48 4.68 7.2

12 8 6 2 YZ σyz =  16.42 / nm 134.1 59.8 4.87  7.2

12108 2 * XY σxy =  42.12/ nm 57.1 37.10 4.684.87

* Calculated via numerical integration

We can observe that the energies associated with XZ and YZ surfaces are significantly higher than those for an XY surface. In other words, a surface where the molecules lie with their axes parallel to the surface is not feasible.

7.9 Discussion

Using molecular dynamics simulations, we have tried to investigate the phenomenon of surface freezing in n-alkanes. Our initial simulations on n-octane suggest that unlike previous reports in the literature this is not restricted to n > 14 and we do not observe a

150 crossover to pre-melting below this point. As regards to the driving force, we study both the surface wetting and entropic aspects.

TR make a simple density scaling argument to calculate . Furthermore,

they mention that the liquid wets the solid almost completely, resulting in δγ(0) ≤ 0 and

 SV ≥  LV . Our results for the surface free energies for octane agree well with this explanation. We estimate for octane to be 34.74 mN/m and 32.75 mN/m using BAR method and our new method respectively. These values match reasonable well with

= 37.03 mN/m as proposed by TR. More importantly, our results qualitatively agree well with TR’s proposition, ≥ . For nonadecane, however, our values find much better agreement with DS’s values. DS have suggested that surface freezing is primarily an energy driven process. However the condition for surface freezing proposed by them that is, δγ(0) > 0, is opposite to the one proposed by TR. For nonadecane we found that = 23.55 mN/m as calculated using the BAR method, which agrees well with = 23.07 mN/m, as proposed by DS. Our new method does not work for nonadecane as it does not take into account for the layer broadening which can be a significant factor. TR’s calculations to estimate also do not take into consideration

this effect, which can be the reason why even in their explanation, >  lV . The argument that surface freezing is driven purely by surface wetting does seem to hold for octane but it not for nonadecane. It is interesting to note that for octane TSF < Tm.

Furthermore, we do find from our simulations that there is a significant energy gain associated with surface freezing. However, for octane at least, since DS’s wetting

151 argument cannot explain surface freezing we turn to the entropic contribution to surface freezing. We can estimate the entropy changes for surface freezing and bulk freezing on a per mol basis independently from simulations as follows. For surface freezing, we run a liquid slab and a surface frozen slab at the surface melting point and constant volume, where,

E  T S SF SF SF (7.18)

Where, E  ESF  Eliq , is the internal energy change for surface freezing which we can directly obtain from simulations and S is the change in entropy.

Furthermore, to obtain the entropy for bulk freezing we can write, at the bulk melting point:

H  T S M M M (7.19)

Where, H  H  H , is the enthalpy change for bulk freezing which we can M S liq directly obtain from simulations on liquid and crystal at constant pressure and

temperature and SM is the change in entropy for bulk freezing. However, to compare it

to the entropy of surface freezing, we need to find SM ' at TSF. From a Taylor series approximation of the first order, we can write:

 S  SM '  SM    TSF TM  (7.20)  T P,T

Furthermore, we know that 152

 S  C p    (7.21)  T P,T T

Where, ΔCp is the difference between the specific heat capacities for soild and liquid. We can estimate the specific heat capacities from the enthalpies as follows:

  H  C    (7.22) p  T    P

We ran simulations with bulk solid and liquid in a range of temperatures in the neighborhood of TSF and Tm to obtain the enthalpies as shown in Fig. 7.12. The ΔCp values estimated are -0.0069 for octane and 0.2293 and 0.2107 for nonadecane for PYS

and TraPPE models respectively. SM ' is then calculated using Eq. (7.20) and (7.21).

The values for SM ' and SSF are listed in Table 7.6.

153

Molecule Model Tm TSF S S S M M ' SF octane PYS 215 211 -0.0600 -0.0599 -0.0385 nonadecane PYS 323 329 -0.1315 -0.1272 -0.1271 nonadecane TraPPE 324 323 -0.1384 -0.1390 -0.1247

154

(a) (b)

(e) (f)

For n-octane SM and SSF were found to be -0.0600 and -0.0385 kJ/mol.K respectively. This difference can be attributed to either partial ordering of the liquid surface, or the enhanced fluctuations along the molecular axis in the surface frozen monolayer, or both. However, quantifying the relative importance of partial ordering of the liquid surface versus partial disordering of the surface frozen layer, merits further

careful analysis. In conclusion, we can say that although a difference in SM and SSF is essential for surface freezing to occur, the significant energy gain on surface freezing must also be considered an important driving force.

In addition to the driving force, we also investigated the why molecules always lie perpendicular to the surface. Approximate calculations using interatomic LJ interactions show that the surface excess energies for molecular axes parallel to the surface are much higher than if they are perpendicular.

156

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liquid normal-alkanes. Phys Rev Lett. 1993; 70: 958-61.

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Liquid Normal-Alkanes. Physical Review Letters. 1993; 70: 958-61.

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159

Chapter 8: Conclusions and future work

This chapter summarizes the results from my research, which are described in detail in previous chapters and discusses the potential impact that it can have on the field. It also presents the opportunities that can be explored to expand this research in the future.

In this research, we have studied the surface freezing phenomenon for n-alkanes using both experimental and computational approaches. As was discussed in Chapter 1, several factors provided the motivation for this research. One of them was the debate behind surface/volume based nucleation for droplets. Furthermore, prior studies on surface freezing on n-alkanes and in particular, the perception that this phenomenon cannot be observed for short chain n-alkanes posed an interesting scientific problem. In addition the intense discussion in the scientific literature about what drives surface freezing motivated us to explore this problem from a theoretical perspective as well.

In Chapter 2, we described our experimental flow system to generate an aerosol of n- alkane by rapidly cooling a gas mixture in a converging-diverging supersonic nozzle.

Chapter 2 also illustrated the functioning of the complementary experimental techniques used to characterize the flow and the aerosol. These include position resolved static pressure measurements (PTM), small angle x-ray scattering (SAXS), wide angle x-ray scattering (WAXS) and Fourier transform infrared spectroscopy (FTIR). In addition,

160

Chapter 2 also highlighted the self-consistent data analysis method to compile data from these multiple techniques into a single consistent set of results.

We presented the first sets of results from our experimental studies in Chapter 3. We initially followed the vapor-liquid phase transition of n-nonane in N2 using PTM, SAXS and FTIR and tested the applicability of our data analysis method to a simple system. We then extended this approach to studies where we also observe freezing i.e. when there is coexisting vapor, liquid and solid in the system. In these studies, we focused our attention on n-octane and n-nonane with Ar as the carrier gas. We observed two “bumps” in the flow temperature profile as a function of position. These correspond to the latent heat release, which occurs on condensation and subsequently, freezing. From the SAXS experiments, we obtained the particle sizes. This allowed us to follow the formation and growth of the aerosol particles. We also observed the decrease in particle size as freezing occurred. Both PTM and SAXS offered independent evidence for phase transitions in our system. Furthermore, these techniques were also consistent with each other in terms of the positions at which they detected the phase transitions. We calculated the individual mass fractions of vapor, liquid and solid phases in our system from FTIR. We found that

FTIR detected the presence of solid more than 1 cm upstream as compared to PTM or

SAXS. Further analysis revealed the presence of a two-step freezing process that we interpreted as surface freezing followed by bulk freezing. The surface and volume freezing rates are ~1015 cm-2s-1 and ~1022 cm-3s-1 respectively.

We then extended this study to n-decane in N2, and observed a similar two-step process for a particular inlet mole fraction of n-decane. These results are presented in Chapter 4.

161

Furthermore, using n-decane we also investigated the effect of inlet mole fraction of n- decane on the freezing process as well as the particle shape and structure. We found that for a reduced inlet mole fraction, there is no evidence to suggest that freezing is initiated at the surface of the droplets. We were also able to confirm the presence of crystalline particles in our system, using WAXS. We found that for a very low inlet fraction of n- decane, the droplets are no longer spherical but have a complex fractal aggregate-like structure. This structure is a result of formation and freezing of n-decane domains on already frozen n-decane nuclei. The primary particle or domain radius is ~0.69 nm and the fractal dimension is ~2.45. The fractal structure also results in enhanced IR absorptivities potentially because of stronger dipole-dipole interactions.

Following Chapters 3 and 4 that primarily explored surface freezing in short chain n- alkanes, the next three chapters focused on studies involving Molecular Dynamics (MD) simulations. In Chapter 5, we introduced the MD simulation methods as well as list the interaction parameters to estimate the inter-atomic and intra-atomic potentials for n- alkanes. In Chapter 5, we also highlighted the results from our n-octane droplet simulations. We found that, within the first few nanoseconds, a well-ordered monolayer develops on the surface of the supercooled n-octane droplet and subsequent freezing occurs in a layer-by-layer manner. This confirmed the hypothesis presented in Chapters 3 and 4, i.e. freezing in supercooled n-alkane droplets is initiated at the surface. Chapter 5 also served as a segue to Chapters 6 and 7, where we described our results from our in- depth analysis of surface freezing in n-alkanes, using MD simulations.

162

Chapter 6 illustrated our calculations for the surface free energy of the solid-vapor

interface,  SV . We calculated  SV using multiple methods, including λ integration and

Weighted Histogram Analysis Method (WHAM), and by assuming a Gaussian

approximation for P0 (V ) . Here P0 (V ) is the probability distribution for the energy interactions of molecules, V , on either side of a virtual interface, created in a crystal.

Initially, we used these methods for a Lennard-Jones (LJ) solid and found that the

Gaussian approximation provides a very good estimate for . Out of the systems of our interest, i.e. n-alkanes, the Gaussian approximation worked well for n-octane, but not so well for n-nonadecane. We attributed this difference in n-nonadecane to the layer

“broadening”, which is observed on going from a crystal to a slab with surfaces.

In Chapter 7, we initially presented simulations that are used to calculate the salient transition temperatures for our models. These include the equilibrium melting point and surface freezing point for both n-octane and n-nonadecane. This is followed by results from simulations that effectively rule out the possibility of surface melting for n-octane.

We then listed our liquid-vapor and solid-vapor interfacial free energy values for both n- octane and n-nonadecane. For the liquid-vapor surface free energies, we used the pressure tensor data from MD simulations. The solid-vapor surface free energy values reported are from Chapter 6. In addition, we also estimated the properties of the multiple types of surfaces existing in our systems. For the n-octane liquid-vapor surface, we found that the molecules on the surface have higher orientational order as compared to the bulk.

However, for n-nonadecane the orientational order of the surface molecules was not too different from that of the bulk molecules. We also found that for n-octane the fluctuations

163 along the molecular axis were stronger as compared to a bulk solid, whereas for n- nonadecane, the difference is not as prominent. We also calculated the energy density across the length of a liquid, solid and a surface frozen slab and observed a significant energy gain on surface freezing. After carefully considering the interfacial free energy values and the surface properties, we concluded that there is a strong entropic contribution to the driving force to surface freezing for n-octane, whereas for n- nonadecane it is not overly significant. This was confirmed when we calculated the entropy changes associated with bulk and surface freezing i.e. ΔSM and ΔSSF respectively, using independent simulations. We found that for n-octane ΔSM was greater than ΔSSF, by a factor of 1.5. We attributed this difference to the partial ordering of the liquid surface and the enhanced fluctuations along the molecular axis in the surface frozen monolayer. For n-nonadecane, the values are not significantly different from each other. Furthermore, in Chapter 7 we also estimated the surface free energies for a hypothetical surface with the molecules parallel to the surface. These approximate calculations showed that it is energetically unfavorable for such a surface to exist and the molecules will always be perpendicular to the surface.

In this research, we have attempted to investigate surface freezing in n-alkanes from a fundamental perspective. We have pushed the boundaries for observing surface freezing experimentally. In particular, we were able to observe surface freezing for short chain n- alkanes, for which it was considered unlikely. We have also tried to obtain further insight into the hitherto unresolved issue of the mechanism of surface freezing. Understanding how these small basic molecules behave will no doubt influence how we perceive or

164 predict the behavior of more complex molecules. Furthermore, this research has also opened up several opportunities to probe this phenomenon further.

From an experimental standpoint, it will be interesting to investigate whether even shorter n-alkanes surface freeze, under the proper conditions. Achieving extremely cold temperatures to probe these phase transitions pose a challenge as regards to experimental design and optimization. In addition, using MD to observe the various nucleation pathways and crystal structures, as a function of temperature and droplet radius, is an interesting prospect. These studies, done at a molecular level, will further strengthen our understanding about the results from the studies on n-decane highlighted in Chapter 4.

Furthermore, current methods make it extremely cumbersome to estimate the solid-liquid interfacial free energies. Identifying ways to estimate these quantities is also critical to develop a better understanding of what drives surface freezing. Even so, from a purely theoretical perspective, these studies have been able to identify that the mechanism for surface freezing is not unique across the entire domain of n-alkanes. Quantifying which of the two contributions, i.e. energetic and entropic, dominate as a function of chain length is still to be explored.

165

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15 Wu XZ, Sirota EB, Sinha SK, Ocko BM, Deutsch M. Surface crystallization of

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8 Kay JE, Tsemekhman V, Larson B, Baker M, Swanson B. Comment on evidence

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11 Wu XZ, Ocko BM, Sirota EB, et al. Surface-tension measurements of surface

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13 Earnshaw JC, Hughes CJ. Surface-induced phase-transition in normal alkane

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ENERGY CALCULATIONS BASED ON MOLECULAR SIMULATIONS.

Chemical Physics Letters. 1994; 222: 529-39.

22 Bennett CH. EFFICIENT ESTIMATION OF FREE-ENERGY DIFFERENCES

FROM MONTE-CARLO DATA. Journal of Computational Physics. 1976; 22:

245-68.

23 Modak VP, Pathak H, Thayer M, Singer SJ, Wyslouzil BE. Experimental

evidence for surface freezing in supercooled n-alkane nanodroplets. Physical

Chemistry Chemical Physics. 2013; 15: 6783-95.

24 Tkachenko AV, Rabin Y. Theory of surface freezing of alkanes. Physical Review

E. 1997; 55: 778-84.

25 Weidinger I, Klein J, Stockel P, Baumgartel H, Leisner T. Nucleation behavior of

n-alkane microdroplets in an electrodynamic balance. Journal of Physical

Chemistry B. 2003; 107: 3636-43.

185

References for Appendix B:

1. D. Ghosh, D. Bergmann, R. Schwering, J. Woelk, R. Strey, S. Tanimura and B. E.

Wyslouzil, Journal of Chemical Physics, 2010, 132, 024307-024301 - 024307-

024317.

2. T. E. Daubert and R. P. Danner, Hemisphere Pub. Corp New York1989.

186

Appendix A: Matlab code to calculate phase-wise contribution from FTIR clc clear all istart=3000; iend=2800; % n=19; FullData = xlsread('Data_5.2.xlsx'); wn=FullData(:,1); a=FullData(:,2); %normalized gas absorptivity b=FullData(:,3); %normalized liquid absroptivity s=FullData(:,4); %normalized solid absroptivity c=FullData(:,5:end); n=size(c,2); for k=1:n sumaa(k)=0; sumbb(k)=0; sumss(k)=0; sumac(k)=0; sumbc(k)=0; sumsc(k)=0; sumab(k)=0; sumas(k)=0; sumbs(k)=0; for i=1:12401; d(i,k)=0; end for i= (4000-istart)*4+1 : (4000-iend)*4+1; aa(i)=a(i).^2; bb(i)=b(i).^2; ss(i)=s(i).^2; bc(i)=b(i).*c(i,k); ac(i)=a(i).*c(i,k); sc(i)=s(i).*c(i,k); ab(i)=a(i).*b(i); as(i)=a(i).*s(i); bs(i)=s(i).*b(i); sumaa(k)=sumaa(k)+aa(i); sumbb(k)=sumbb(k)+bb(i); sumss(k)=sumss(k)+ss(i); 187

sumac(k)=sumac(k)+ac(i); sumbc(k)=sumbc(k)+bc(i); sumsc(k)=sumsc(k)+sc(i); sumab(k)=sumab(k)+ab(i); sumas(k)=sumas(k)+as(i); sumbs(k)=sumbs(k)+bs(i); end

A = [sumaa(k) sumab(k) sumas(k); sumab(k) sumbb(k) sumbs(k); sumas(k) sumbs(k) sumss(k)]; Z = [sumac(k); sumbc(k); sumsc(k)]; t = A\Z; avalue(k)=t(1); bvalue(k)=t(2); svalue(k)=t(3);

%bvalue(k/2)=(sumbc(k/2)-sumac(k/2)*sumab(k/2)/sumaa(k/2))/(sumbb(k/2)- sumab(k/2)^2/sumaa(k/2)); %avalue(k/2)=(sumac(k/2)-bvalue(k/2)*sumab(k/2))/sumaa(k/2);

% goodness of fit-Pearson's chi-squared test http://en.wikipedia.org/wiki/Pearson%27s_chi-square_test cd(k)=0; for i=(4000-istart)*4+1:(4000-iend)*4+1; d(i,k)=a(i)*t(1)+b(i)*t(2)+s(i)*t(3); cd(k)= ((c(i,k)-d(i,k))^2)/d(i,k)+(cd(k)); end

figure(k) plot(wn,d(:,k),'k') hold on plot(wn,c(:,k),'g') hold off legend('fitted','measured') xlabel('wavenumber(cm-1)') ylabel('absorbance') axis([2800 3000 0 0.035]) j=j+1;

188

% p=reduction in degrees of freedom http://en.wikipedia.org/wiki/Pearson%27s_chi- square_test p=3; % degrees of freedom=df df=4*(istart-iend)-p; chisquarereduced(k)=cd(k)/df; end avalue=transpose(avalue); bvalue=transpose(bvalue); svalue=transpose(svalue); chisquarereduced=transpose(chisquarereduced); vapcon=79.365*avalue; %1/0.0126 kgperm3=11.292*bvalue; % bvalue*mol wt(kg)/0.0126 xlswrite('absvals.xlsx', avalue)% bvalue)% bvalue, svalue, chisquarereduced) xlswrite('absvals.xlsx', vapcon, '', 'B1') xlswrite('absvals.xlsx', bvalue, '', 'C1') xlswrite('absvals.xlsx', kgperm3, '', 'D1') xlswrite('absvals.xlsx', svalue, '', 'E1') xlswrite('absvals.xlsx', chisquarereduced, '', 'F1')

189

Appendix B Thermophysical properties of materials

Table B.1 Thermophysical properties of n-octane

Thermophysical Property Ref Molecular Weight 114.23 1 1 Critical Temperature Tc (K) 568.91 1 Triple Point Ttr (K) 216.37 1 Critical Pressure pc (MPa) 2.490 Specific isobaric liquid 2.22 1 -1 -1 heat capacity cpl (J g K )

Cp  132.490098  (3048.8270/T)2 exp(3048.8270/T) Specific isobaric vapor heat 166.06550  (1 exp(3048.8270/T))2 1 capacity C (J mol-1K-1) p (1378.6073/T)2 exp(1378.6073/T) 254.85474 (1 exp(1378.6073/T))2

Specific isobaric solid heat C  2.4 10 4 1.9472 103 T  ps 2 -1 -1 2 2 3 5 4 capacity Cps (J kmol K ) 8.536T 1.34 10 T  2.0940 10 T p 101.325  Equilibrium vapor pressure e exp[(1 398.793/T) exp(2.9015  1 p (kPa) e 2.046204 10 3 T  2.010759 10 6 T 2 )]

1/ 3 l  0.237[11.969770(1T /Tc)  -3 2 / 3 1 Liquid density ρl (g cm ) 1.100623(1T /Tc )  6.364172(1T /Tc )  4 / 3 5/ 3 8.693475(1T /Tc )  4.420047(1T /Tc ) ]

-3 3 2 Solid density ρs (kmol m ) 8.3409  3.15110 T

-1 5 1 Surface tension γl (N m ) 0.02375  9.826 10 (T  273.15)

3 H vap  R  exp(2.9015  2.046204 10 T  Heat of vaporization ΔHvap 2.010759 10 6 T) [398.793  T(T  398.793) (J mol-1) (2.046204 10 3  2.010759 10 6 T)] Heat of fusion ΔH fus 2 2.074 107 (J kmol-1) 190

Table B.2 Thermophysical properties of n-nonane

Thermophysical Property Ref Molecular Weight 128.26 1 1 Critical Temperature Tc (K) 594.90 1 Triple Point Ttr (K) 219.65 1 Critical Pressure pc (MPa) 2.290 Specific isobaric liquid 2.217 1 -1 -1 heat capacity cpl (J g K )

C p  148.150 36  (1380.8003/T) 2 exp(1380.8003/T) Specific isobaric vapor heat 288.249 04  (1 exp(1380.8003/T))2 1 capacity C (J mol-1K-1) p (3051.1566/T) 2 exp(3051.1566 /T) 178.574 91 (1 exp(3051.1566 /T))2

Specific isobaric solid heat C  3.19104  2.3720103T  ps 2 -1 -1 2 2 3 capacity Cps (J kmol K ) 1.2440T  3.018010 T p 101.325 Equilibrium vapor pressure e exp[(1 423.932/T)exp(2.946 90  1 p (kPa) e 2.051 933103T 1.903 6839106 T 2 )]

1/ 3 l  0.237[11.927 780(1T /Tc)  -3 2 / 3 1 Liquid density ρl (g cm ) 0.930 218 9(1T /Tc ) 1.334 128(1T /Tc )  4 / 3 1.392 823(1T /Tc ) ]

-3 3 2 Solid density ρs (kmol m ) 7.4283 2.369910 T

-1 5 1 Surface tension γl (N m ) 0.024 84 -9.41710 (T  273.15)

3 H vap  Rexp(2.9469  2.05193310 T  Heat of vaporization ΔHvap 1.903683106 T)[423.932 T(T  423.932) (J mol-1) (2.051933103 1.903683106 T)]

Heat of fusion ΔHfus 2 1.547107 (J kmol-1)

191

Table B.3 Thermophysical properties of n-decane

Thermophysical Property Ref Molecular Weight 142.286 1 1 Critical Temperature Tc (K) 617.61 1 Triple Point Ttr (K) 243.50 1 Critical Pressure pc (MPa) 2.105 Specific isobaric liquid 2.217 1 -1 -1 heat capacity cpl (J g K )

Cp 163.7837  (1379.9706/T)2 exp(1379.9706/T) Specific isobaric vapor heat 320.24325  (1 exp(1379.9706/T))2 1 capacity C (J mol-1K-1) p (3024.7636/T)2 exp(3024.7636/T) 191.23849 (1 exp(3024.7636/T))2

Specific isobaric solid heat C  3.49104  2.528103T  ps 2 -1 -1 2 2 3 capacity Cps (J kmol K ) 1.265T  2.8410 T p 101.325 Equilibrium vapor pressure e exp[(1 447.269 /T)exp(2.9669  1 p (kPa) e 1.932257103T 1.644626106T 2 )]

1/ 3 l  0.239[1 0.3291388(1T /Tc)  -3 2 / 3 1 Liquid density ρl (g cm ) 7.364340(1T /Tc )  9.985096(1T /Tc )  4 / 3 5.283608(1T /Tc ) ]

-3 3 2 Solid density ρs (kmol m ) 6.897  2.4610 T

-1 5 1 Surface tension γl (N m ) 0.02573 9.19010 (T  273.15)

3 Hvap  R exp(2.96690 1.93257910 T  Heat of vaporization ΔHvap 1.644626106T)[447.269 T(T  447.269) (J mol-1) (1.932579103  2.1644626106T)] Heat of fusion ΔH fus 2 2.871107 (J kmol-1)

192

Table B.4 Thermophysical properties of Nitrogen

Thermophysical Ref Property Molecular Weight 28.01 1

Specific isobaric vapor heat 1 -1 -1 29.12 capacity Cp (J mol K )

Table B.5 Thermophysical properties of Argon

Thermophysical Ref Property Molecular Weight 39.95 1 Specific isobaric vapor heat 20.78 1 -1 -1 capacity Cp (J mol K )

193

References:

1. D. Ghosh, D. Bergmann, R. Schwering, J. Woelk, R. Strey, S. Tanimura and B. E.

Wyslouzil, Journal of Chemical Physics, 2010, 132, 024307-024301 - 024307-

024317.

2. T. E. Daubert and R. P. Danner, Hemisphere Pub. Corp New York1989.

194

Appendix C: Fortran code to calculate thermodynamic unknowns with p and g as input

ccc this program version includes the ability to update the latent heat as a c function of temperature for h2o and d2o based on clausius- clapeyron c approximations to liquid-vapor equations. !chhfeb2001 ccc this version of the program calculates a "wet" isentrope based on the c measured dry isentrope and corrected for the differences in gamma. it c also starts the wet condensing flow integration on the desired data c point rather than on the wet isentrope to avoid any extraneous extra c shifts. ccc smoothes all of the good density data first, then integrates from an c initial value using finer integration grid (up to 5x) ccc modified to take in pressure data instead of density data ....jul97, jlc ccc note: stein used to smooth the integrated values as well... may consider c doing this for rough data... not yet implemented but easy to do.... bew ccc this version has been modified for Nozzle H on train B with Velmex (PP) ccc RTD probe is calibrated and temperature calibration factor is included ccc Now nu.dat has "tempcal" and this program reads in the value and does c temperature calibration as "to(i)=to(i)+tempcal"...... jun02, PP ccc fc=g *wi/(w10+w20) was replaced by fc=g/(w10+w20) March04 Shinobu ccc Inert gas is a mixture of N2 and CH4 March04 Shinobu ccc tisd=pp0d(i)**c0*t0 was replaced by pp0d(i)**c3*t0 July05 Shinobu ccc Function fk has been corrected 3/31/2007 Shinobu, Hartawan ccc See Vol.6, p9 and Vol. 9, p68 ccc tempcal is not used from cal07. 10/17/2007 Shinobu ccc Gas constant was set to 8.3145. 10/18/2007 Shinobu ccc Changing for Nonane-D2O, new Properties for Nonane 17/06/2009

implicit real*8(a-h,o-z) real*8 fcon 195

real*8 msq,msqw,mssq real*8 rg, pi, avog real*8 dotm,dotncal,pc10,pc20,zc10 real*8 p0, t0, tempcal real*8 xstart, xthroat, xnum real*8 tt0(2000),fc(2000),g(2000),u(2000), *rr0(2000),pp0(2000),pp0d(2000) real*8 tt0_is(2000),t_is(2000),rr0_is(2000),pp0_is(2000) real*8 t_is_s(2000),pp0_is_s(2000) real*8 aratio(2000),wg(2000),t(2000),tisd(2000) real*8 xd(2000),xw(2000),x(2000) real*8 dry(2000),dryf(2000),sdry(2000) real*8 wet(2000),swet(2000),wetf(2000) real*8 po(400),p(400),deltapo(400),deltap(400),to(400) real*8 deltadry(2000),deltadryf(2000),deltato(2000) real*8 deltawet(2000),deltawetf(2000),dtemp(2000,20) real*8 m_1,mssq_is,m_0,m_2 real*8 mdry(2000) real*8 t_is_up(2000) c c ccccccccccccccccccccccccccccccc ! Shinobu ccccccccc real*8 mssq_TDL,cp_TDL,cpr_TDL real*8 fc_TDL(2000),tt0_TDL(2000),u_TDL(2000),rr0_TDL(2000) real*8 g_TDL(2000),g_TDL2(2000),g_TDL3(2000),sol_TDL(2000) real*8 sol_TDL2(2000),sol_TDL3(2000),fsol_TDL(2000) real*8 x_TDL(2000),t_TDL(2000),ar_TDL(2000) real*8 cpv_TDL,cpc_TDL,y1_TDL,y2_TDL,y3_TDL,cpsol_TDL cccccccccccccccccccccccccccccccccccccccccccccccccccccccc

character*30 dryfil,wetfil,a character*8 specie(2) character*60 progname c character*4 title(3,2) common /xval/ xs(2000)

*------nomenclature c dhc,fdhc(zc10,t(i)) latent heat of condensible vapor c pc10,pc20 condensible vapor pressure (read in 2*Torr, works in dyne/cm^2) c t(i) Temperature of inert in Kelvin c zc10 Initial molar fraction of condensible vapor1 (zc10+zc20=1) *------nomenclature

progname='nuetodd2o_irCH4MFC_DryCp2Up_cal07' open(5,file='nu_CH4_MFC_decane_nitrogen.dat',status='old') ! March04 Shinobu open(10,file='4pp.out',status='unknown') open(11,file='wilson.out',status='unknown') open(9,file='new4pp.out',status='unknown') open(13,file='dtemp.out',status='unknown') c open(14,file='legend3.bat',status='unknown') c open(15,file='legend4.bat',status='unknown') 196

open(7,file='upstream.out',status='unknown')

call echo

pi=3.14159d0 rg=8.3145d7 avog=6.022d23 c read two condensible species read(5,41,end=50)specie c print 1006, specie 1006 format (2a8) 41 format(2a8) c read stagnation conditions-temp, pressure, partial pressure of c condensible--pressures are in mm of hg--note t0 and p0 are calculated from data files later. read(5,*)tempcal !PP02 !RTD probe calibration added write(*,*)'tempcal = ',tempcal,' (not used)' c convert pressures to dyn/cm**2 pconv = 760.d0/1.01325d6 c read molecular weights of carrier (1), condensible (2,3) and CH4 (4) read(5,*)wmN,wm2,wm3,wm4 ! March04 Shinobu c read specific heats of gases read(5,*)cpN,cp2,cp3,cp4 ! March04 Shinobu c read correction factor for MKS flowmeter read(5,*)demf c read latent heat of fusion unbits in J/g read(5,*)fdhd c read starting value and the number of points in the output read(5,*) xstart2, ilast2 c read the integration end points (may be different than ifrst, ilast) c the number of integrations attempts, and the number of integration c sub-intervals. c istart >= 2, ifin < ilast, ni=1 (for useless roop, do k=1,ni) read(5,*)istart2, ifin2, ni, nint c read name of dry pressure data file read(5,1)dryfil 1 format(a30) c read smoothing parameters: m-order, n-number of points read(5,*)md,nd c read x values and all of the dry data p0, p(x), and the associated standard deviations. c correct the pressures i.e. baratron calibrations and pressure loss due to mesh. open(unit=4,file=dryfil,status='old') c read total number of values in dry pressure data file read(4,1)a 197

read(4,*)idend dotncal=0.d0 p0dry=0.d0 c t0dry=273.15d0 +tempcal t0dry=0.0d0

do i=1,idend read(4,*)xd(i),po(i),deltapo(i),p(i),deltap(i),to(i),deltato(i), * dummy1,dummy2,flowmain,deltaflowmain,flowsub,deltaflowsub, * pthroat,deltathroat,pexit,deltapexit, * dummy3,dummy4 ! xd(i) in 0.01mm, po in 2*torr c po(i)=(po(i)*0.49967 + 2.19)-poloss !Shinobu cal04 c po(i)=po(i)/1.9888 !Shinobu cal05 c po(i)=po(i)*0.5025d0+0.457d0 !Shinobu cal07 c to(i)=to(i)*0.99147d0+0.510d0 ! RuiYang and Shinobu cal07 to(i)=to(i)*0.99375d0+0.352d0 ! Shinobu cal08 Vol.11, p.33 (same as cal 13) ccc Following equation includes the effect of poloss ccc po(i)=po(i)*0.498313d0+2.36259d0 !Shinobu cal14 c p0dry=p0dry+po(i) t0dry=t0dry+to(i) end do p0dry=p0dry/idend t0dry=t0dry/idend+273.15d0 c do i=1,idend p(i)=0.0374041+1.00004*p(i) !Shinobu Cal 2014 dry(i)=p(i)/p0dry deltadry(i)=dry(i) * *((deltap(i)/p(i))**2.0+(deltapo(i)/p0dry)**2.0)**0.5 c write(*,*) i, xd(i), dry(i), deltadry(i) !debug dotncal=dotncal+(flowmain+flowsub)*demf/idend/22.41d0 c p0dry=p0dry+po(i) c t0dry=t0dry+to(i) end do close(unit=4) write(*,*)p0dry

c p0dry=p0dry/idend c t0dry=t0dry/idend+273.15d0

c read the number of wet data sets and flow rate of CH4 read(5,*)ndata, dotCH4

wm0dry=(dotncal*wmN+dotCH4*wm4)/(dotncal+dotCH4) ! March04 Shinobu 198

w40dry=dotCH4*wm4/(dotncal*wmN+dotCH4*wm4) y30dry=dotCH4/(dotncal+dotCH4) cp0dry=( dotncal*wmN*cpN+dotCH4*wm4*fcp4(t0dry,p0dry,y30dry) ) * / (dotncal*wmN+dotCH4*wm4) c write(*,*)'wm0dry,w40dry,y30dry,cp0dry=', c & wm0dry,w40dry,y30dry,cp0dry write(11,1302)dryfil

do kd = 1,ndata

read(5,*)ntype,entry1,entry2,entry3 ! March04 Shinobu if(ntype.eq.0)then !pressure input (torr) pc10=entry1 pc20=entry2 tCH4=entry3 ! March04 Shinobu else if(ntype.eq.1)then !massflow and weight fraction input dotm=entry1 wfc10=entry2 wfc20=1.0d0-wfc10 c write(7,*)wfc10 tCH4=entry3 ! March04 Shinobu else write(*,*)'need to specify pressure (0)' write(*,*)'or mass flow with first weight fraction input(1)' stop end if c read name of wet pressure data file read(5,1)wetfil open(unit=4,file=wetfil,status='old') c read total number of values in wet pressure data file read(4,1)a read(4,*)idenw c read x values and all of the wet data p0, p(x), and the associated standard deviations. c correct the pressures i.e. baratron calibrations and pressure loss due to mesh. tN=0.d0 t0set=0.d0 c do i=1,idenw read(4,*)xw(i),po(i),deltapo(i),p(i),deltap(i),to(i),deltato(i), * dummy1,dummy2,flowmain,deltaflowmain,flowsub,deltaflowsub, * pthroat,deltathroat,pexit,deltapexit, * dummy3,dummy4 !po in 2*torr 199

c po(i)=po(i)/1.9888 !Shinobu cal05 c po(i)=po(i)*0.5025+0.457 !Shinobu cal07 ccc Following equation includes the effect of poloss ccc po(i)=po(i)*0.498313d0+2.36259d0 ! Shinobu Cal2014 c t0set=t0set+to(i)/idenw c to(i)=to(i)+tempcal !ppaci02! RTD probe calibration c to(i)=to(i)*0.99147d0+0.510d0 ! RuiYang and Shinobu cal07 to(i)=to(i)*0.99375d0+0.352d0 ! Shinobu cal08 Vol.11, p.33 p(i)=0.0374041+1.00004*p(i) ! Shinobu Cal2014 wet(i)=p(i)/po(i) deltawet(i)=wet(i) * *((deltap(i)/p(i))**2.0+(deltapo(i)/po(i))**2.0)**0.5 c write(*,*) xw(i), wet(i), deltawet(i) !debug tN=tN+(flowmain+flowsub)*demf/idenw/22.41d0 enddo c cccccccccccccccccccccccccccccc ! Shinobu cccc read(4,*) iden_TDL write(*,*)iden_TDL do i=1,iden_TDL read(4,*) x_TDL(i),g_TDL2(i),sol_TDL2(i) enddo ccccccccccccccccccccccccccccccccccccccccccccc close(unit=4)

c figure out the average stagnant pressure and temperature

p0=0.0 t0=0.0 do i=1,idenw p0=p0+po(i)

t0=t0+to(i) enddo p0=p0/idenw t0=t0/idenw+273.15 devp0=0.0 do i=1,idenw devp0=devp0+(po(i)-p0)**2.0 enddo devp0=(devp0/(idenw-1))**0.5 write(*,*) 'average p0 is ', p0,'torr' write(*,*) 'p0 std dev is ', devp0,'torr' write(*,*) 'average t0 is ',t0,'k'

allflux=tN+tCH4+dotm*wfc10/wm2+dotm*wfc20/wm3 200

wm1=(tN*wmN+tCH4*wm4)/(tN+tCH4) w40=tCH4*wm4/(tN*wmN+tCH4*wm4+dotm) y30=tCH4/allflux

c !chh99 c now figure out pcondensible from calibration and average properties if(ntype.eq.1)then !now calculate pcondensible pc10=p0*dotm*wfc10/wm2/allflux ! June05 Shinobu write(*,*)'pc10= ',pc10,' torr' pc20=p0*dotm*wfc20/wm3/allflux write(*,*)'pc20= ',pc20,' torr' endif c convert pressures to dyn/cm**2 p0=p0/pconv pct0=pc10+pc20 pc10=pc10/pconv pc20=pc20/pconv if((pc10+pc20).lt.1.d-18) then zc10=0.d0 else zc10=pc10/(pc10+pc20) !chh22.02.01 endif y10=pc10/p0 y20=pc20/p0 c calculate stagnation gas mass density and condensible monomer mass c density (g/cm**3) c w2,w3 are mass fraction of condensible vapor in gas wmav=(wm1*(p0-pc10-pc20)+wm2*pc10+wm3*pc20)/p0 w20=wm2*pc10/p0/wmav w30=wm3*pc20/p0/wmav wi=1.d0-w20-w30 wN0=wi-w40 c gw17-2-00 assuming vapor condenses at constant composition let's define c a fictitious mean condensible vapor molecular weight wmc if((pc10+pc20).lt.1.d-18) then wmc=0.d0 else wmc=(wm2*pc10+wm3*pc20)/(pc10+pc20) endif c also let's save the inital average molecular weight wmav0=wmav

cp0= wN0*cpN+w40*fcp4(t0,p0,y30) & +w20*fcp2(t0,p0,y10)+w30*fcp3(t0,p0,y20)

201

gamma=cp0dry/(cp0dry-rg*1.d-7/wm0dry) !n2 gamma gamma0=cp0/(cp0-rg*1.d-7/wmav) !initial mixture gamma rhog0=p0/rg/t0*wmav write(*,*)'wmav',' w20',' w30',' wi',' cp0',' gamma0' !chh061098 write(*,*)wmav, w20,w30,wi,cp0,gamma0 c calculate various exponents and constants involving gamma eai = 2.d0*(gamma-1.d0)/(gamma+1.d0) eai0 = 2.d0*(gamma0-1.d0)/(gamma0+1.d0) ep = -gamma/(gamma-1.d0) ep0 = -gamma0/(gamma0-1.d0) erho = -1.d0/(gamma0-1.d0) emrho = gamma-1.d0 emrho0 = gamma0-1.d0 eam2 = (gamma+1.d0)/(gamma-1.d0) eam20 = (gamma0+1.d0)/(gamma0-1.d0) c1 = 2.d0/(gamma-1.d0) c10 = 2.d0/(gamma0-1.d0) c2 = (gamma0+1.d0)/2.d0 c0 = (gamma0-1.d0)/gamma0 c3 = (gamma-1.d0)/gamma c figure out where the throat is for the dry data c first figure out the value of pstar/p0=pstp0

pstp0 = (1.d0+ 1.0d0/c1)**ep tstt0=pstp0**c3

*********** Values at throat under Dry condition, Shinobu *************

stepm=100.0 nstepm=100 dm_1=1.d0/stepm m_1=0.d0+dm_1

t_0=t0dry p_0=p0dry

t_1=t_0 p_1=p_0 g_1=gamma cp_1=cp0dry

do i=2,nstepm

fk_1=( dgdt(t_1,p_1,0.d0,0.d0,y30dry,0.d0,0.d0,w40dry,wm0dry) & *t_1 + p_1*g_1/(g_1-1.d0)* & dgdp(t_1,p_1,0.d0,0.d0,y30dry,0.d0,0.d0,w40dry,wm0dry) )/ 202

& ( 2.d0+(g_1-1.d0)*m_1*m_1 ) dg_1=-2.d0*fk_1*m_1*(g_1-1.d0)*2.d0*dm_1/ & (1.d0+fk_1*m_1*m_1*(g_1-1.d0)/g_1) dt_1=-t_1/(2.d0+(g_1-1.d0)*m_1*m_1)* & ( 2.d0*(g_1-1.d0)*m_1*2.d0*dm_1 + & m_1*m_1*(g_1-1.d0)/g_1*dg_1 ) dp_1=p_1/t_1*g_1/(g_1-1.d0)*dt_1 dr_1=r_1/t_1/(g_1-1.d0)*dt_1

t_2=t_0+dt_1 p_2=p_0+dp_1 r_2=r_0+dr_1 cp_2=(1.d0-w40dry)*cpN+w40dry*fcp4(t_2,p_2,y30dry) g_2=cp_2/(cp_2-rg*1.d-7/wm0dry)

t_0=t_1 p_0=p_1 t_1=t_2 p_1=p_2 cp_1=cp_2 g_1=g_2 m_1=m_1+dm_1 c write(*,*)'m_1,cp_1,g_1',m_1,cp_1,g_1 write(*,*)'dg_1, dt_1, dp_1, ',dt_1,dp_1 c write(*,*)'p_1/p0, t_1/t0',p_1/p0,t_1/t0,i

enddo

tstcpdry=t_1 pstcpdry=p_1

write(*,*) ' pstp0, tstt0 (for constant Cp) =', pstp0,tstt0 write(*,*) 'pstcpdry/p0dry, tstcpdry/t0dry = ', & pstcpdry/p0dry, tstcpdry/t0dry

******************************************************************** ***

pstp0dry=pstcpdry/p0dry do i=1,idend c write(*,*) i, xd(i), dry(i) !debug if((dry(i).gt. pstp0dry).and.(dry(i+1).le. pstp0dry))then c write(*,*) 'true' !debug xthroat=( pstp0dry-dry(i))/(dry(i+1)-dry(i))*(xd(i+1)-xd(i)) & +xd(i) go to 5001 endif enddo 5001 continue write(*,*) 'dry throat of ',dryfil,' is at ',xthroat c now shift all the x and scale so that x(i) is in units of cm. 203 c find number of unused points before xstart !chh110698 do i=1,idend x(i)=(xd(i)-xthroat)/1000.0 ! in units of cm !Shinobu for Velmex on Train B enddo c now do linear interpolation to get fixed x intervals ************************************************ ixstart=1 xstart= int(x(1)*10.d0)/10.d0 ilast=ilast2+int( (xstart2-xstart)/0.1+0.1 ) write(*,*) 'xstart= ',xstart ************************************************ c save steps in inner loop by beginning interp. where left off lasti=ixstart !chh110698 do j=1,ilast xs(j)=xstart+(j-1)*0.1 !in intervals of 1 mm

do i=lasti,idend c write(*,*)i, x(i) !chh110698 if((x(i).le.xs(j)).and.(x(i+1).gt.xs(j)))then dryf(j)=dry(i)+(xs(j)-x(i))*(dry(i+1)-dry(i))/(x(i+1)-x(i)) deltadryf(j)=deltadry(i) lasti=i !chh110698 goto 5 endif write(*,*)xs(j), dryf(j) enddo

write(*,*) 'can not interpolate for point', j

5 continue enddo

c we now have an array dryf(j) at fixed xs(j) intervals. now put c through smoothing routine. c c smooth dry density values c first do points at ends of good data range do j=1,(nd-1)/2 k0=1 i=j call smooth(md,nd,i,k0,sval,dryf) sdry(i)=sval k0=ilast-nd+1 i=ilast+1-j call smooth(md,nd,i,k0,sval,dryf) sdry(i)=sval 204

enddo c next do points in good data range do i=3,(ilast-2) k0=i-(nd-1)/2 call smooth(md,nd,i,k0,sval,dryf) sdry(i)=sval enddo

1201 format(f10.4,f10.4,g14.4,f10.4,f10.4,g14.4) c figure out where the throat is for the wet data c first figure out the value of pstarw/p0=pstp0w

pstp0w= (1.d0+ 1.0d0/c10)**ep0

*********** Values at throat under Wet condition, Shinobu *************

stepm=250.0 nstepm=250 dm_1=1.d0/stepm m_1=0.d0+dm_1

t_0=t0 p_0=p0 r_0=rhog0

t_1=t_0 p_1=p_0 r_1=r_0 g_1=gamma0 cp_1=cp0

do i=2,nstepm

fk_1=( dgdt(t_1,p_1,y10,y20,y30,w20,w30,w40,wmav0) & *t_1 + p_1*g_1/(g_1-1.d0)* & dgdp(t_1,p_1,y10,y20,y30,w20,w30,w40,wmav0) )/ & ( 2.d0+(g_1-1.d0)*m_1*m_1 ) dg_1=-2.d0*fk_1*m_1*(g_1-1.d0)*2.d0*dm_1/ & (1.d0+fk_1*m_1*m_1*(g_1-1.d0)/g_1) dt_1=-t_1/(2.d0+(g_1-1.d0)*m_1*m_1)* & ( 2.d0*(g_1-1.d0)*m_1*2.d0*dm_1 + & m_1*m_1*(g_1-1.d0)/g_1*dg_1 ) dp_1=p_1/t_1*g_1/(g_1-1.d0)*dt_1 dr_1=r_1/t_1/(g_1-1.d0)*dt_1

t_2=t_0+dt_1 p_2=p_0+dp_1 r_2=r_0+dr_1 cp_2= wN0*cpN+w40*fcp4(t_2,p_2,y30) & +w20*fcp2(t_2,p_2,y10)+w30*fcp3(t_2,p_2,y20) 205

g_2=cp_2/(cp_2-rg*1.d-7/wmav)

t_0=t_1 p_0=p_1 r_0=r_1 t_1=t_2 p_1=p_2 r_1=r_2 cp_1=cp_2 g_1=g_2 m_1=m_1+dm_1 c write(*,*)'m_1,cp_1,g_1',m_1,cp_1,g_1 c write(*,*)'dg_1, dt_1, dp_1, ',dg_1,dt_1,dp_1 c write(*,*)'p_1/p0, t_1/t0',p_1/p0,t_1/t0

enddo

gammam=g_1 tstarcp=t_1 pstarcp=p_1 rstarcp=r_1 ustarcp=dsqrt(gammam*rg*tstarcp/wmav)

write(*,*)'p*/p0, t*/t0, r*/r0 (constant Cp) ', & pstp0w,1.d0/c2,c2**erho write(*,*)'p_1/p0, t_1/t0, r_1/r0 ',p_1/p0,t_1/t0,r_1/rhog0 write(*,*) 'gamma0, gammam',gamma0,gammam ******************************************************************** ***

do i=1,idenw c write(*,*) i, xw(i), wet(i) !debug if((wet(i).gt.pstarcp/p0).and.(wet(i+1).le.pstarcp/p0))then c write(*,*) 'true' !debug xthroat=xw(i)+(pstarcp/p0-wet(i))/ & (wet(i+1)-wet(i))*(xw(i+1)-xw(i)) go to 5002 endif enddo 5002 continue write(*,*) 'wet throat of ',wetfil,' is at ',xthroat c now shift all the x and scale so that x(i) is in units of cm. c find the number of unused points before xstart. !chh110698 ixstart=0 !chh110698 do i=1,idenw x(i)=(xw(i)-xthroat)/1000.0 ! in units of cm if(x(i).le.xstart)ixstart=i !chh110698 enddo 206

write(*,*)'break1' read(*,*)xnum c write(*,*) 'ixstart= ',ixstart !chh110698 write(*,*) 'throat shifted' !debug c now do linear interpolation to get fixed x intervals lasti=ixstart !chh110698 do j=1,ilast c xs values have already been assigned in dry data analysis c write(*,*) xs(j) !debug do i=lasti,idenw !chh110698 if((x(i).le.xs(j)).and.(x(i+1).gt.xs(j)))then wetf(j)=wet(i)+(xs(j)-x(i))*(wet(i+1)-wet(i))/(x(i+1)-x(i)) deltawetf(j)=deltawet(i) lasti=i !chh110698 goto 6 endif enddo

write(*,*) 'can not interpolate for point', j 6 continue enddo c cccccccccccccccccccccccccccccccccc ! Shinobu lasti=1 do j=1,ilast c write(*,*) xs(j) !debug do i=lasti,iden_TDL if((x_TDL(i).le.xs(j)).and.(x_TDL(i+1).gt.xs(j))) then g_TDL3(j)=g_TDL2(i)+(xs(j)-x_TDL(i))* & (g_TDL2(i+1)-g_TDL2(i))/(x_TDL(i+1)-x_TDL(i))

sol_TDL3(j)=sol_TDL2(i)+(xs(j)-x_TDL(i))* & (sol_TDL2(i+1)-sol_TDL2(i))/(x_TDL(i+1)- x_TDL(i)) lasti=i !chh110698 goto 62 endif enddo

write(*,*) 'can not interpolate for point', j 62 continue enddo ccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccc

c we now have an array wetf(j) at fixed xs(j) intervals. now put c through smoothing routine. write(*,*) 'put through smoothing' 207 c c smooth wet pressure values c first do points at ends of good data range do j=1,(nd-1)/2 k0=1 i=j call smooth(md,nd,i,k0,sval,wetf) swet(i)=sval k0=ilast-nd+1 i=ilast+1-j call smooth(md,nd,i,k0,sval,wetf) swet(i)=sval enddo c next do points in good data range do i=3,(ilast-2) k0=i-(nd-1)/2 call smooth(md,nd,i,k0,sval,wetf) swet(i)=sval enddo write(*,*) 'finished interpolating points' c use finer integration step size than measured point spacing c generate interior points by linear interpolation c nint is the number of subintervals between each pair of original x values write(*,*) 'nint= ', nint c calculated the finer grid, interpolating on the wet condensing and c wet isentrope data write(*,*) 'calculate the finer grid'

********************************************** ifin=ifin2+int( (xstart2-xstart)/0.1+0.1 ) istart0=1 **********************************************

npts=ifin-istart0+1 nnpts=(npts-1)*nint+1 jinit=nnpts+2*nint+istart0-1 do i=ifin+1,istart0,-1 delx=xs(i)-xs(i-1) delprd = sdry(i)-sdry(i-1) c delprwi = sweti(i)-sweti(i-1) delprw = swet(i)-swet(i-1) delg=g_TDL3(i) - g_TDL3(i-1) ! Shinobu delsol=sol_TDL3(i) - sol_TDL3(i-1) jinit=jinit-nint jp=0 write(*,*)jinit,jinit-nint+1 do j=jinit,jinit-nint+1,-1 fint=1.d0*dfloat(jp)/(1.d0*nint) xs(j)=xs(i)-delx*fint write(*,*)fint, xs(j) 208

if(dabs(xs(j)).LT.1.d-4) ithroat=j pp0d(j) = sdry(i)-delprd*fint c pp0i(j) = sweti(i)-delprwi*fint pp0(j) = swet(i)-delprw *fint g_TDL(j)=g_TDL3(i)-delg*fint ! Shinobu sol_TDL(j)=sol_TDL3(i)-delsol*fint jp=jp+1 enddo enddo ifin1=istart0+nnpts-1

***************************************************************** istart= istart0+ & int( (xstart2-xstart)/0.1+istart2-istart0+0.1 )*nint write(*,*)'ithroat =',ithroat *****************************************************************

*** Pressure and temperature upstream of the integration region **** t_0=tstarcp p_0=pstarcp

cp_0= wN0*cpN+w40*fcp4(t_0,p_0,y30) & +w20*fcp2(t_0,p_0,y10)+w30*fcp3(t_0,p_0,y20) g_0=cp_0/(cp_0-rg*1.d-7/wmav)

dp=( pp0(ithroat+1)-pp0(ithroat-1) )*p0 dt=t_0/p_0*(g_0-1.d0)/g_0*dp

t_is_up(ithroat)=tstarcp t_is_up(ithroat+1)=tstarcp+dt/2.d0 t_is_up(ithroat-1)=tstarcp-dt/2.d0

t_1=t_is_up(ithroat+1) p_1=pp0(ithroat+1)*p0

do i=ithroat+2,istart cp_1= wN0*cpN+w40*fcp4(t_1,p_1,y30) & +w20*fcp2(t_1,p_1,y10)+w30*fcp3(t_1,p_1,y20) g_1=cp_1/(cp_1-rg*1.d-7/wmav)

dp=( pp0(i)-pp0(i-2) )*p0 dt=t_1/p_1*(g_1-1.d0)/g_1*dp

t_is_up(i)=t_is_up(i-2)+dt

t_1=t_is_up(i) p_1=pp0(i)*p0 enddo

t_1=t_is_up(ithroat-1) p_1=pp0(ithroat-1)*p0

do i=ithroat-2,istart0,-1 209

cp_1= wN0*cpN+w40*fcp4(t_1,p_1,y30) & +w20*fcp2(t_1,p_1,y10)+w30*fcp3(t_1,p_1,y20) g_1=cp_1/(cp_1-rg*1.d-7/wmav)

dp=( pp0(i)-pp0(i+2) )*p0 dt=t_1/p_1*(g_1-1.d0)/g_1*dp

t_is_up(i)=t_is_up(i+2)+dt

t_1=t_is_up(i) p_1=pp0(i)*p0 enddo

write(7,700) 700 format(' x(cm) pp0 p(Torr) Tisw(K)')

do i=istart0,istart write(7,710) xs(i),pp0(i),pp0(i)*p0*pconv,t_is_up(i) 710 format(f7.2,f8.4,2f10.2) enddo close(unit=7) ******************************************************************** ***

do k = 1,ni c need to calculate at istart-1 so adjust if istart=1 c since there is no good data avaiable before 1 write(*,*) 'start' write(*,5000) istart 5000 format(3(I3,2x)) if(istart.eq.1)istart=istart+1

***********Values at the start point of integration for Dry, Shinobu **********

stepp=100.0 nstepp=100 dp_1=(pp0d(istart-1)*p0dry-pstcpdry)/stepp t_0=tstcpdry p_0=pstcpdry cp_0=(1.d0-w40dry)*cpN+w40dry*fcp4(t_0,p_0,y30dry) g_0=cp_0/(cp_0-rg*1.d-7/wm0dry) m_0=1.d0 fk_0=( dgdt(t_0,p_0,0.d0,0.d0,y30dry,0.d0,0.d0,w40dry,wm0dry) & *t_0 + p_0*g_0/(g_0-1.d0)* & dgdp(t_0,p_0,0.d0,0.d0,y30dry,0.d0,0.d0,w40dry,wm0dry) )/ & ( 2.d0+(g_0-1.d0)*m_0*m_0 ) a_0=1.d0

dt_0=t_0/p_0*(g_0-1.d0)/g_0*dp_1 dm_0= -(2.d0+(g_0-1.d0)*m_0**2) 210

& *( g_0+fk_0*(m_0**2)*(g_0-1.d0) ) & /(2.d0*(g_0**2)*m_0)/p_0*dp_1 da_0= -a_0*(m_0**2-1.d0)/(g_0*m_0**2)/p_0*dp_1

t_1=t_0+dt_0 m_1=m_0+dm_0 p_1=p_0+dp_1 a_1=a_0+da_0 cp_1=(1.d0-w40dry)*cpN+w40dry*fcp4(t_1,p_1,y30dry) g_1=cp_1/(cp_1-rg*1.d-7/wm0dry) fk_1=( dgdt(t_1,p_1,0.d0,0.d0,y30dry,0.d0,0.d0,w40dry,wm0dry) & *t_1 + p_1*g_1/(g_1-1.d0)* & dgdp(t_1,p_1,0.d0,0.d0,y30dry,0.d0,0.d0,w40dry,wm0dry) )/ & ( 2.d0+(g_1-1.d0)*m_1*m_1 )

do i=2, nstepp dt_1=t_1/p_1*(g_1-1.d0)/g_1*2.d0*dp_1 dm_1= -(2.d0+(g_1-1.d0)*m_1**2) & *( g_1+fk_1*(m_1**2)*(g_1-1.d0) ) & /(2.d0*(g_1**2)*m_1)/p_1*2.d0*dp_1 da_1= -a_1*(m_1**2-1.d0)/(g_1*m_1**2)/p_1*2.d0*dp_1

t_2=t_0+dt_1 m_2=m_0+dm_1 a_2=a_0+da_1 p_2=p_0+2.d0*dp_1

cp_2=(1.d0-w40dry)*cpN+w40dry*fcp4(t_2,p_2,y30dry) g_2=cp_2/(cp_2-rg*1.d-7/wm0dry) fk_2=( dgdt(t_2,p_2,0.d0,0.d0,y30dry,0.d0,0.d0,w40dry,wm0dry) & *t_2 + p_2*g_2/(g_2-1.d0)* & dgdp(t_2,p_2,0.d0,0.d0,y30dry,0.d0,0.d0,w40dry,wm0dry) )/ & ( 2.d0+(g_2-1.d0)*m_2*m_2 )

t_0=t_1 m_0=m_1 a_0=a_1 p_0=p_1 t_1=t_2 m_1=m_2 a_1=a_2 g_1=g_2 fk_1=fk_2 p_1=p_2 c write(*,*)'dt_1,dm_1,da_1 =',dt_1,dm_1,da_1

enddo

tisd(istart-1)=t_1 211

aratio(istart-1)=a_1 ar_TDL(istart-1)=a_1 !harshad mdry(istart-1)=m_1

dp_2=dp_1+(pp0d(istart)-pp0d(istart-1))*p0dry dt_1=t_1/p_1*(g_1-1.d0)/g_1*dp_2 dm_1= -(2.d0+(g_1-1.d0)*m_1**2) & *( g_1+fk_1*(m_1**2)*(g_1-1.d0) ) & /(2.d0*(g_1**2)*m_1)/p_1*dp_2 da_1= -a_1*(m_1**2-1.d0)/(g_1*m_1**2)/p_1*dp_2

t_2=t_0+dt_1 m_2=m_0+dm_1 a_2=a_0+da_1 p_2=p_0+dp_2

tisd(istart)=t_2 mdry(istart)=m_2 aratio(istart)=a_2 ar_TDL(istart)=a_2 !harshad ******************************************************************** ************* c note! start the wet condensing flow integration on the desired data c point (i.e. on the wet curve data) rather than on the wet isentrope c to avoid any extraneous extra shifts/offsets in t etc. c msqw = c10*((1.d0/pp0(istart-1))**c0-1.d0) c tt0(istart-1)=1.d0/(1.d0+msqw/c10) c rr0(istart-1)=(1.d0+msqw/c10)**erho c msqw = c10*((1.d0/pp0( istart))**c0-1.d0) c tt0(istart)=1.d0/(1.d0+msqw/c10) c rr0(istart)=(1.d0+msqw/c10)**erho

***********Values at the start point of integration for Wet, Shinobu **********

stepp=100.0 nstepp=100 dp_1=(pp0(istart-1)*p0-pstarcp)/stepp

t_0=tstarcp p_0=pstarcp r_0=rstarcp g_0=gammam

dt_0=t_0/p_0*(g_0-1.d0)/g_0*dp_1 dr_0=r_0/p_0/g_0*dp_1 t_1=t_0+dt_0 r_1=r_0+dr_0 212

p_1=p_0+dp_1 cp_1= wN0*cpN+w40*fcp4(t_1,p_1,y30) & +w20*fcp2(t_1,p_1,y10)+w30*fcp3(t_1,p_1,y20) g_1=cp_1/(cp_1-rg*1.d-7/wmav)

do i=2,nstepp

dt_1=t_1/p_1*(g_1-1.d0)/g_1*2.d0*dp_1 dr_1=r_1/p_1/g_1*2.d0*dp_1 t_2=t_0+dt_1 r_2=r_0+dr_1 p_2=p_0+2.d0*dp_1 cp_2= wN0*cpN+w40*fcp4(t_2,p_2,y30) & +w20*fcp2(t_2,p_2,y10)+w30*fcp3(t_2,p_2,y20) g_2=cp_2/(cp_2-rg*1.d-7/wmav)

t_0=t_1 r_0=r_1 p_0=p_1 t_1=t_2 r_1=r_2 cp_1=cp_2 g_1=g_2 p_1=p_2

enddo

tt0_is(istart-1)=t_1/t0 rr0_is(istart-1)=r_1/rhog0 pp0_is(istart-1)=p_1/p0

msqw = c10*((1.d0/pp0(istart-1))**c0-1.d0) write(*,*)'pp0(istart-1), tt0, rr0', & pp0(istart-1),1.d0/(1.d0+msqw/c10),(1.d0+msqw/c10)**erho write(*,*)'p_1/p0,t_1/t0,r_1/rhog0', & pp0_is(istart-1),tt0_is(istart-1),rr0_is(istart-1)

dp_2=dp_1+(pp0(istart)-pp0(istart-1))*p0 dt_1=t_1/p_1*(g_1-1.d0)/g_1*dp_2 dr_1=r_1/p_1/g_1*dp_2 t_2=t_0+dt_1 r_2=r_0+dr_1

tt0_is(istart)=t_2/t0 rr0_is(istart)=r_2/rhog0 pp0_is(istart)=(p_0+dp_2)/p0

msqw = c10*((1.d0/pp0( istart))**c0-1.d0) write(*,*)'pp0(istart), tt0, rr0 ', & pp0(istart),1.d0/(1.d0+msqw/c10),(1.d0+msqw/c10)**erho write(*,*)'p_2/p0,t_2/t0,r_2/rhog0', & pp0_is(istart),tt0_is(istart),rr0_is(istart)

213

tt0(istart-1)=tt0_is(istart-1) tt0_TDL(istart-1)=tt0_is(istart-1) !harshad rr0(istart-1)=rr0_is(istart-1) rr0_TDL(istart-1)=rr0_is(istart-1) !harshad tt0(istart)=tt0_is(istart) tt0_TDL(istart)=tt0_is(istart) !harshad rr0(istart)=rr0_is(istart) rr0_TDL(istart)=rr0_is(istart) !harshad ******************************************************************** ***

g(istart)=0.d0 g(istart-1)=0.d0 fc(istart)=0.d0 !fraction condensed fc_TDL(istart)=0.d0 !harshad fsol_TDL(istart)=0.d0 write(10,1024) progname 1024 format('Program: ',a60)

write(10,1011)p0*pconv,devp0,t0-273.15, t0set, rhog0*1.0d3 !4pp plots write(10,1010) dotm,specie(1),wfc10 !4pp plots 1010 format('Weight flux of condensable =',f6.2, & ' g/min , Fraction of ',a, '=',f7.3) !4pp plots 1011 format('p0= ',f6.2,'+/-',f4.2,' Torr T0=',f6.2, & ' C (set T0=',f6.2,') rho0=', e11.4,' kg/m3') !4pp plots write(10,1012)pc10*pconv,specie(1),pc20*pconv, +specie(2) !4pp plots 1012 format('@subtitle "',2(f7.4,'torr ',a),'"') !4pp plots write(10,1013) allflux, (dotncal+dotCH4) !4pp plots 1013 format( 'Total mole flux =', & f6.3,' mol/min (',f6.3,' in Dry )' ) !4pp plots write(10,1019) tCH4, dotCH4 !4pp plots 1019 format( ' CH4 mole flux =', & f6.3,' mol/min (',f6.3,' in Dry )' ) !4pp plots write(10,1014)wetfil,dryfil !4pp plots 1014 format('@subtitle \"',a30,'with dry trace ',a30,'\"') !4pp plots write(10,1015) !4pp plots 1015 format(' x(cm) u(m/s) T(K) p/p0 Tis p/p0_is', &' MoleFract. g g/g_inf A/A* r/r0', &' Tisd p/p0_isd') !4pp plots c write(12,1016)kd,pc10*pconv,specie(1),pc20*pconv, c +specie(2) !4pp plots

214 c 1016 format('legend string ',i2,' \"',2(f7.4,'torr ',a),'\"') !4pp plots c write(14,1017)kd, wetfil !4pp plots c 1017 format('legend string ',i2,' \"',a13,'\"') !4pp plots c write(15,1018)kd-1,p0*pconv,devp0,t0-273.15 !4pp plots c 1018 format('legend string ',i2,' \"',f6.2,'+/-',f4.2,'torr ', c & f6.2,'celsius"') !4pp plots write(9,1037) !4pp plots in new output file 1037 format(' x(cm) u_PTM T_PTM g_PTM T_is Tptm-Tis P/po_is', &' P/po g/g_inf sol/g_inf T_tdl A/A*_PTM A/A*_TDL gginf_TDL', &' r/r0_TDL r/r0 g_TDL u_TDL')

write(*,*) 'start integration' write(*,5000)istart,ifin1

do i=istart,ifin1 c calculate local value of effective area ratio, aratio c msq is local mach number squared, mssq = (u/u*)^2

*********** Integration of the isentropic curve for Dry, Shinobu **************

dp_dry=( pp0d(i+1)-pp0d(i-1) )*p0dry/2.d0

p_dry=pp0d(i)*p0dry cp_dry=(1.d0-w40dry)*cpN + w40dry*fcp4(tisd(i),p_dry,y30dry) g_dry=cp_dry/(cp_dry-rg*1.d-7/wm0dry) fk_dry=( dgdt(tisd(i),p_dry,0.d0,0.d0,y30dry, & 0.d0,0.d0,w40dry,wm0dry) & *tisd(i) + p_dry*g_dry/(g_dry-1.d0)* & dgdp(tisd(i),p_dry,0.d0,0.d0,y30dry, & 0.d0,0.d0,w40dry,wm0dry) )/ & ( 2.d0+(g_dry-1.d0)*mdry(i)**2 )

dt_dry=tisd(i)/p_dry*(g_dry-1.d0)/g_dry*dp_dry dm_dry= -( 2.d0+(g_dry-1.d0)*mdry(i)**2 ) & *( g_dry+fk_dry*(mdry(i)**2)*(g_dry-1.d0) ) & /(2.d0*(g_dry**2)*mdry(i))/p_dry*dp_dry da_dry= -aratio(i)*(mdry(i)**2-1.d0)/(g_dry*mdry(i)**2)/ & p_dry*dp_dry

tisd(i+1)=tisd(i-1)+2.d0*dt_dry mdry(i+1)=mdry(i-1)+2.d0*dm_dry aratio(i+1)=aratio(i-1)+2.d0*da_dry

dar=dlog(aratio(i+1)/aratio(i-1))/2.d0 215

c write(*,*) 'Integration of the isentropic curve for Dry OK' ******************************************************************** ************* ccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccc c Arearatio for constant Cp of CH4 c c msq = c1*((1.d0/pp0d(i))**c3-1.d0) c aratio(i)= dsqrt(((c1/eam2*(1.d0+msq/c1))**eam2)/msq) c msq = c1*((1.d0/pp0d(i+1))**c3-1.d0) c aratio(i+1) = dsqrt(((c1/eam2*(1.d0+msq/c1))**eam2)/msq) c dar=dlog(aratio(i+1)/aratio(i-1))/2.d0 c c cccccccccccccccccccccccccccccccccccccccccccccccccccccccccccc

*********** Integration of the isentropic curve for Wet, Shinobu **************

dp=(pp0(i+1)-pp0(i-1))/2.d0

t_is(i)=tt0_is(i)*t0

u_is=ustarcp*rstarcp/rhog0/rr0_is(i)/aratio(i) mssq_is=(u_is/ustarcp)**2 cp_is= wN0*cpN+w40*fcp4(t_is(i),pp0_is(i)*p0,y30) & +w20*fcp2(t_is(i),pp0_is(i)*p0,y10) & +w30*fcp3(t_is(i),pp0_is(i)*p0,y20) cpr_is=cp_is/cp0 hpara_is=1.d0-(gamma0-1.d0)/gamma0/cpr_is

tempA=(1.d0-t_is(i)/tstarcp/gammam/mssq_is)/rr0_is(i) tempG=t0/tstarcp/gammam/mssq_is tempJ=rr0_is(i)*tt0_is(i)/ & (hpara_is-t_is(i)/tstarcp/gammam/mssq_is)

dtt0_is=(tt0_is(i)-tempA*tempJ)*dar dpp0_is=-tempJ*dar drr0_is=-(rr0_is(i)+tempG*tempJ)*dar

tt0_is(i+1)=tt0_is(i-1)+2.d0*dtt0_is pp0_is(i+1)=pp0_is(i-1)+2.d0*dpp0_is rr0_is(i+1)=rr0_is(i-1)+2.d0*drr0_is c write(*,*) 'Integration of the isentropic curve for Wet OK' *************** Smoothing, Shinobu ************************************ t_is_s(i)=t0*( tt0_is(i-1)+2.d0*tt0_is(i)+tt0_is(i+1) )/4.d0 pp0_is_s(i)=( pp0_is(i-1)+2.d0*pp0_is(i)+pp0_is(i+1) )/4.d0 c t_is_s(i)=t0*tt0_is(i) c pp0_is_s(i)=pp0_is(i) ******************************************************************** *** 216

c write(*,*)'mssq,mssq_is',mssq,mssq_is c write(*,*)'cp,cp_is',cp,cp_is c write(*,*)'hpara,hpara_is',hpara,hpara_is c write(*,*)'gamma0,gammam',gamma0,gammam c write(*,*)'tempA, tempJ,tempG',tempA,tempJ,tempG c write(*,*) rr0_is(i) c write(*,*)'dar',dar c write(*,*)'dtt0,dpp0',dtt0_is,dpp0_is c write(*,*)'mdry(i), Mach ,fk_dry=', c & mdry(i), dsqrt(c1*((1.d0/pp0d(i))**c3- 1.d0)),fk_dry

******************************************************************** *** t(i)=tt0(i)*t0 t_TDL(i)=tt0_TDL(i)*t0 !harshad u(i)=ustarcp*rstarcp/rhog0/rr0(i)/aratio(i) u_TDL(i)=ustarcp*rstarcp/rhog0/rr0_TDL(i)/ar_TDL(i) ! harshad mssq=(u(i)/ustarcp)**2 mssq_TDL=(u_TDL(i)/ustarcp)**2

fcon=dotm * (wfc10/wm2 + (1-wfc10)/wm3) c if((pc10+pc20).lt.1.d-18) then c y1=0.d0 c y2=0.d0 c else c y1=(1-fc(i))*fcon/(allflux-fc(i)*fcon)*y10/(y10+y20) c y2=(1-fc(i))*fcon/(allflux-fc(i)*fcon)*y20/(y10+y20) c endif c y3=y30*allflux/(allflux-fc(i)*fcon) c write(*,*)'(y1+y2+y3+tN/allflux*y3/y30), (wN0+w20+w30+w40)', c & y1+y2+y3+tN/allflux*y3/y30, wN0+w20+w30+w40 c harshad-y values required only in case of clustering y1=0 y1_TDL=0 y2=0 y2_TDL=0 y3=0 y3_TDL=0 c gw2-17-00 update specific heat if((pc10+pc20).lt.1.d-18) then cpv=0.d0 cpc=0.d0 else cpv=( w20*fcp2(t(i),pp0(i)*p0,y1) + & w30*fcp3(t(i),pp0(i)*p0,y2) )/(w20+w30) cpc=( w20*fcpl2(t(i))+w30*fcpl3(t(i)) )/(w20+w30) endif 217

cp= wN0*cpN+w40*fcp4(t(i),pp0(i)*p0,y3) & +(w20+w30-g(i))*cpv+g(i)*cpc cpr=cp/cp0 c c gw2-17-00 update specific heat for TDL if((pc10+pc20).lt.1.d-18) then cpv_TDL=0.d0 cpc_TDL=0.d0 cpsol_TDL=0.d0 else cpv_TDL=( w20*fcp2(t_TDL(i),pp0(i)*p0,y1_TDL) + & w30*fcp3(t_TDL(i),pp0(i)*p0,y2_TDL) )/(w20+w30) cpc_TDL=( w20*fcpl2(t_TDL(i))+w30*fcpl3(t_TDL(i)) )/(w20+w30) cpsol_TDL=( w20*fcpsol2(t_TDL(i)) + & w30*fcpsol3(t_TDL(i)))/(w20+w30) endif cp_TDL= wN0*cpN+w40*fcp4(t_TDL(i),pp0(i)*p0,y3_TDL) & +(w20+w30-g_TDL(i))*cpv_TDL+g_TDL(i)*cpc_TDL cpr_TDL=cp_TDL/cp0

c gw2-17-00 update "mu/(1-g)" = wmu, and related factors if((pc10+pc20).lt.1.d-18) then wmu=wm1 wg(i)=0.d0 else wmu=wm1*wmc/(wi*wmc+(w20+w30-g(i))*wm1) wg(i)=wmu/(wmc) endif wmuu0=wmu/wmav0 c c gw2-17-00 update "mu/(1-g)" = wmu_TDL, and related factors if((pc10+pc20).lt.1.d-18) then wmu_TDL=wm1 wg_TDL=0.d0 else wmu_TDL=wm1*wmc/(wi*wmc+(w20+w30-g_TDL(i))*wm1) wg_TDL=wmu_TDL/(wmc) endif wmuu0_TDL=wmu_TDL/wmav0

hpara=wmuu0-(gamma0-1.d0)/gamma0/cpr dr=dp/tstarcp*t0/gammam/mssq-rr0(i)*dar dgp=(hpara-t(i)/tstarcp/mssq/gammam)/rr0(i)*dp+tt0(i)*dar if((pc10+pc20).lt.1.d-18) then dg=0.d0 else dg=dgp*cp*t0/(fdhc(wfc10,t(i))-cp*t(i)*wg(i)) ! Shinobu endif c gw2-17-00 update dtt0 218

dtt0=(wmuu0-t(i)/tstarcp/gammam/mssq)/rr0(i)*dp+ & tt0(i)*(dar+wg(i)*dg)

tt0(i+1)=tt0(i-1)+2.0d0*dtt0 rr0(i+1)=rr0(i-1)+2.0d0*dr g(i+1)=g(i-1)+2.0d0*dg

if((w20+w30).gt.0.0)then fc(i+1)=g(i+1)/(w20+w30) ! March04 Shinobu else fc(i+1)=0.0 end if c c write(*,*) 'Integration of the Wet trace OK' cc ***********************correction for the code when g is input******** ap_TDL=wmuu0_TDL*gamma0-(gamma0-1.d0)/cpr_TDL ! Shinobu h_TDL=ap_TDL/gamma0 ! Shinobu cc ccccccccccccccccccc ! Shinobu cccccccccccccccccccccccc

dg_TDL=(g_TDL(i+1)-g_TDL(i-1))/2.0d0 dsol_TDL=(sol_TDL(i+1)-sol_TDL(i-1))/2.0d0 tempA=(wmuu0_TDL-t_TDL(i)/tstarcp/gammam/mssq_TDL)/rr0_TDL(i) tempB=tt0_TDL(i) tempC=tt0_TDL(i)*wg_TDL tempF=fdhc(wfc10,t_TDL(i))/cp_TDL/t0-tt0_TDL(i)*wg_TDL tempD=(h_TDL - t_TDL(i)/tstarcp/gammam/mssq_TDL)/rr0_TDL(i)/tempF tempE=tt0_TDL(i)/tempF tempS=fdhd/cp_TDL/t0 dlar_TDL=dg_TDL/tempE-tempD/tempE*dp+tempS/tt0_TDL(i)*dsol_TDL dtt0_TDL=tempA*dp+tempB*dlar_TDL+tempC*dg_TDL dr_TDL=dp/tstarcp*t0/gammam/mssq_TDL-rr0_TDL(i)*dlar_TDL

ar_TDL(i+1)=ar_TDL(i-1)*dexp(2.0d0*dlar_TDL) rr0_TDL(i+1)=rr0_TDL(i-1)+2.0d0*dr_TDL tt0_TDL(i+1)=tt0_TDL(i-1)+2.0d0*dtt0_TDL wm_TDL=wmav0*dotncal/ & (dotncal-(dotncal-tN-tCH4)*fc_TDL(i)) ! Molecular weight for c c sound velocity ga_TDL=cp_TDL/(cp_TDL-8.3145/wm_TDL) ! Specific heat ratio for c sound velocity a_TDL=(ga_TDL*8.3145*t_TDL(i)/wm_TDL*1000.0)**0.5

if((w20+w30).gt.0.0)then fc_TDL(i+1)=g_TDL(i+1)/(w20+w30) ! Shinobu fsol_TDL(i+1) = sol_TDL(i+1)/(w20+w30) else fc_TDL(i+1)=0.0 fsol_TDL(i+1)=0.0 end if 219

write(9,1105)xs(i),u(i)/100,t(i),g(i),t_is_s(i),t(i)- t_is_s(i), & pp0_is_s(i),pp0(i),fc(i), fsol_TDL(i), & t_TDL(i),aratio(i),ar_TDL(i),fc_TDL(i),rr0_TDL(i), & rr0(i),g_TDL(i),u_TDL(i)/100 1105 format(3f8.2,f8.4,2f8.2,4f8.4,f8.2, & 5f8.4,f8.6,f8.2) cc ***********************correction for the code when g is input over******** write(10,1020)xs(i),u(i)/100,t(i),pp0(i),t_is_s(i),pp0_is_s(i), ! June05 Shinobu * (1-fc(i))*fcon/(allflux-fc(i)*fcon),g(i),fc(i),aratio(i), * rr0(i),tisd(i),pp0d(i)

1020 format(f8.3,f10.2,f8.2,f8.4,f8.2,f8.4,2e13.4,f8.4,f8.4,f8.4, & f8.2,f8.4) !4pp plots 1000 format(e12.3,e12.4,f8.2,e12.3,f8.2,f7.3,2e12.4,e12.4) 1100 format(i5,e12.3,5e12.4) 1110 format(e12.3,e13.5) enddo

write(*,*)'start search' c now search for the onset conditions using both t(i)-t_is_s(i) and t(i)-tisd

do i = istart,ifin1-1 dtemp(i,kd) = t(i)-t_is_s(i) dt1 = t(i) - t_is_s(i) dt2 = t(i+1) -t_is_s(i+1) if(dt1.le.0.5d0.and.dt2.gt.0.5d0)then xon = xs(i)+(0.5-dt1)/(dt2-dt1)*(xs(i+1)-xs(i)) pp0on = pp0(i)+(0.5-dt1)/(dt2-dt1)*(pp0(i+1)-pp0(i)) pp0ion = pp0_is_s(i)+(0.5-dt1)/(dt2-dt1)*(pp0_is_s(i+1)- pp0(i)) ton = t(i)+(0.5-dt1)/(dt2-dt1)*(t(i+1)-t(i)) tiswon = t_is_s(i)+(0.5-dt1)/(dt2-dt1)*(t_is_s(i+1)- t_is_s(i)) else endif enddo write(*,*)'using the t-t_is_s = 0.5 k' write(*,1300)xon,pp0on*pct0,ton, & pp0ion*pct0,tiswon 1300 format('onset occurs at x =',f8.4,f8.4,f7.2,f8.4,f7.2) write(11,1301)t0,p0*pconv,pct0,ton,pp0on*pct0,

& pp0on*pc10*pconv,pp0on*pc20*pconv,wetfil,xon 1301 format(f8.2,f8.2,f8.4,f8.2,f8.4,f8.4,f8.4,2x,a13,f7.1) 1302 format('@\"t0 p0 pct ton pon p1on p2on', & 6x,a13,'\"') xon=0.0 220

pp0on=0.0 ton=0.0 do i = istart,ifin1-1 dt1 = t(i) - tisd(i) dt2 = t(i+1) - tisd(i+1) if(dt1.le.0.5d0.and.dt2.gt.0.5d0)then xon = xs(i)+(0.5-dt1)/(dt2-dt1)*(xs(i+1)-xs(i)) pp0on = pp0(i)+(0.5-dt1)/(dt2-dt1)*(pp0(i+1)-pp0(i)) pp0don = pp0d(i)+(0.5-dt1)/(dt2-dt1)*(pp0d(i+1)-pp0(i)) ton = t(i)+(0.5-dt1)/(dt2-dt1)*(t(i+1)-t(i)) tisdon = tisd(i)+(0.5-dt1)/(dt2-dt1)*(tisd(i+1)-tisd(i)) else endif enddo write(*,*)'using the criterion t-tisd = 0.5 k' write(*,1300)xon,pp0on*(pc10+pc20)*pconv,ton, & pp0don*(pc10+pc20)*pconv,tisdon

write(*,*) 'finished integration' enddo enddo c now write out the dtemp files to dtemp.out

do i = istart,ifin1 write(13,1313)xs(i),(dtemp(i,j),j=1,ndata) enddo 1313 format(f8.4,20(f8.2))

50 stop end c subroutine smooth(m,n,k,k0,sval,y) c this subroutine produces smoothed values of a tabulated function y c based on technique described in ralston, "a first course in num. anal." c y values do not have to be equally spaced, but x values must be supplied c regardless of the spacing c c m - order of the highest polynomial used in smoothing c n - number of y points in interval over which smoothing is performed c k - point whose smoothed value is desired c k0 - first point in set of n c sval - smoothed value returned to calling program c real*8 p(-2:5,1:200),b(0:5),omega(0:5),gamma(0:5),beta(-1:5) *,alpha(0:5),y(200),sval,x common /xval/ x(1000) beta(-1)=0. beta(0)=0. gamma(0)=n omega(0)=0. 221

alpha(1)=0. do i=k0,(n+k0-1) omega(0)=omega(0)+y(i) alpha(1)=alpha(1)+x(i) p(-2,i)=0. p(-1,i)=0. p(0,i)=1. enddo b(0)=omega(0)/gamma(0) alpha(1)=alpha(1)/gamma(0) sval=b(0) do j=1,m gamma(j)=0. omega(j)=0. alpha(j+1)=0. do i=k0,(n+k0-1) p(j,i)=(x(i)-alpha(j))*p(j-1,i) - beta(j-1)*p(j-2,i) gamma(j)=gamma(j)+p(j,i)*p(j,i) alpha(j+1)=alpha(j+1)+x(i)*p(j,i)*p(j,i) omega(j)=omega(j)+y(i)*p(j,i) enddo alpha(j+1)=alpha(j+1)/gamma(j) beta(j)=gamma(j)/gamma(j-1) b(j)=omega(j)/gamma(j) sval=sval+b(j)*p(j,k) enddo return end

subroutine echo character*100 a write(9,3) 15 read(5,1,end=99)a write(9,2)a goto 15 99 continue rewind 5 return 1 format(a100) 2 format(1x,a100) 3 format(1h1,20x,'input file',//) end *------chh22.02.01---* * c real function fdhc(dhc) c fdhc = dhc c return * real*8 function fdhc(wfc10,tk) double precision zc10,wfc10,tk,rg

222

*------general nomenclature-* c rg universal gas constant in units of c tk temperature of vapor condensing in kelvin c zc10 molar fraction of condensible 1 in vapor (zc10+zc20=1.0) *------condensible nomenclature-* c a2h2o - a4h2o h2o vapor pressure constants, wagner correlation c a1d2o - a6d2o d2o vapor pressure constants, c mwd2o d2o molecular weight c z d2o intermediate variable *------* double precision a1d2o,a2d2o,a3d2o,a4d2o,a5d2o,a6d2o,z,mwd2o !d2o pve constants double precision bbu,cbu,mwetod,dhcetod

rg=8.3145d0

*------* *-d2o clausius-clapeyron relation applied to equilibrium vapor pressure *-d2o valid for temperature range of 275-823K *-d2o hill, mcmillan, and lee, j. phys chem ref data, vol 11, no.1, p1-14 (1982) a1d2o= -7.81583d0 a2d2o= 17.6012d0 a3d2o=-18.1747d0 a4d2o= -3.92488d0 a5d2o= 4.19174d0 a6d2o=643.89d0 mwd2o=20.03d0

z=1-tk/a6d2o d2oa=a1d2o*z+a2d2o*z**1.9+a3d2o*z**2+a4d2o*z**5.5+a5d2o*z**10. d2ob=a1d2o+1.9d0*a2d2o*z**0.9+2.d0*a3d2o*z+5.5d0*a4d2o*z**4.5 &+10.d0*a5d2o*z**9. dhcd2o=-rg*(a6d2o*d2oa+tk*d2ob)/mwd2o

*------* *-BuOH clausius-clapeyron relation applied to equilibrium vapor pressure *-BuOH valid for temperature range of 243.2-303.2 K * And T. Schmeling and R. Strey, Ber. Bunsenges. Phys. Chem., vol 87, p871-874 (1983) c bbu= 9412.61d0 c cbu= 10.54d0 223 c mwbuOH=74.12d0 c dhcbuOH=rg*(bbu-cbu*tk)/mwbuOH c Use corrected equation 20 of Ruzicka and Majer J physical chem ref data 23, 1994 p 1-39 c Note the T is missing from the a1 term!Original units are J/mol a0 = 2.96690d0 a1 = -1.93225794d-3 a2 = 1.644426d-6 Tb = 447.269 mwetod=142.280d0

Term1 = rg*exp(a0 + a1*tk + a2*tk*tk)

Term2 = Tb + tk*(tk - Tb)*(a1+2.0d0*a2*tk) ccc dhcetod= Hvap von Octane!!!

dhcetod = (Term1*Term2)/mwetod

*------* fdhc = (wfc10*dhcetod)+(1.d0-wfc10)*dhcd2o c write(28,*)'dhc debug: fdhc= ',fdhc,' wfc10= ',wfc10,' tk= ',tk,' K' !debug dhc return

end

*------*

***************** Functions for Cp **********************************

real*8 function fcp2(tk,p,y1) double precision tk,p,y1 double precision mw,a0,a1,a2 c** Cp of BuOH ****** p: Total static pressure ****** y1: Mole fraction of condensable 1 in vapor phase c mw=74.12 c a0=30.941d0 c a1=0.10037d0 c a2=7.322d-5 c fcp2=(a0+a1*tk+a2*tk*tk)/mw (data for EtOD) c fcp2=1.473 used a const cp for BuOh?? c [J/g*K]

224

*----Cp octane--Ruzicka & Majer J.Phys.Chem.Ref.Data.Vol.23.No.1,1994 ------***NEU machen * taken from Bures et al, Chem Eng Sci, 36, 529-537 (1981) c------Einheit J/molK------

A = 163.73837d0 B1 = 320.24325d0 B2 = 191.23849d0 C1 = 1379.9706d0 C2 = 3024.7636d0

Term1 = (C1/tk)*(C1/tk)*dexp(-C1/tk)/(1.0d0 - dexp(- C1/tk))**2

Term2 = (C2/tk)*(C2/tk)*dexp(-C2/tk)/(1.0d0 - dexp(- C2/tk))**2

cpnonane = A + B1*Term1 + B2*Term2

mw=142.28d0

fcp2 = cpnonane/mw

return end C Functions for Cp solid (J/g-K) real*8 function fcpsol3(tk) fcpsol3= ((-34900+2528.0*tk-12.65*tk**2+0.0284*tk**3)) & /142280 return end real*8 function fcpsol2(tk) fcpsol3= ((-34900+2528.0*tk-12.65*tk**2+0.0284*tk**3)) & /142280 return end c unit for the code should be J/g*K

real*8 function fcp3(tk,p,y2) double precision tk,p,y2 double precision rg,mw,a0,a1,a2,a3 c** Cp of D2O: Fitted to the data in JCP 22, 2051 (1954) at T= 160- 340 K rg=8.3145 mw=20.03 a0=4.1712 225

a1=-2.2388d-3 a2= 8.6096d-6 a3=-5.6304d-9 fcp3=rg*(a0+a1*tk+a2*tk**2+a3*tk**3)/mw c fcp3=1.710d0

return end

real*8 function fcp4(tk,p,y3) double precision tk,p,y3 double precision rg,mw,a0,a1,a2,a3 c** Cp of CH4: Fitted to the data in J. Chem. Eng. Data 8, 547 (1963) at T= 160-340 K rg=8.3145 mw=16.04 a0=4.337 a1=-3.7677d-3 a2= 9.1107d-6 a3= 1.0178d-9 fcp4=rg*(a0+a1*tk+a2*tk**2+a3*tk**3)/mw c fcp4=2.226d0 return end

************** Temperature derivative of Cp ****************

real*8 function fdcp2dt(tk,p,y1) double precision tk,p,y1,mw,a1,a2,c1,c2 c mw=74.12 c a1=0.10037d0 c a2=7.322d-5 c fdcp2dt=(a1+2.d0*a2*tk)/mw c----Cp Nonane--Ruzicka & Majer J.Phys.Chem.Ref.Data.Vol.23.No.1,1994 ------***NEU machen * taken from Bures et al, Chem Eng Sci, 36, 529-537 (1981) c------Einheit J/molK------

A = 163.73837d0 B1 = 320.24325d0 B2 = 191.23849d0 C1 = 1379.9706d0 C2 = 3024.7636d0

mwnonane=142.28d0 ccc Näherung für cp/dt

tk1 = tk+ 0.01d0 tk2 = tk - 0.01d0

Term1 = (C1/tk1)*(C1/tk1)*dexp(-C1/tk1)/ 226

& (1.0d0 - dexp(-c1/tk1))**2 Term2 = (C2/tk1)*(C2/tk1)*dexp(-C2/tk1)/ & (1.0d0 - dexp(-c2/tk1))**2

cpnonane1 = (A + B1*Term1 + B2*Term2)/mwnonane

Term1 = (C1/tk2)*(C1/tk2)*dexp(-C1/tk2)/ & (1.0d0 - dexp(-c1/tk2))**2 Term2 = (C2/tk2)*(C2/tk2)*dexp(-C2/tk2)/ & (1.0d0 - dexp(-c2/tk2))**2

cpnonane2 = (A + B1*Term1 + B2*Term2)/mwnonane

fdcp2dt = (cpnonane1-cpnonane2)/(0.02d0)

return end

real*8 function fdcp3dt(tk,p,y2) double precision tk,p,y2 double precision rg,mw,a1,a2,a3 c** Cp of D2O: Fitted to the data in JCP 22, 2051 (1954) at T= 160- 340 K rg=8.3145 mw=20.03 a1=-2.2388d-3 a2= 8.6096d-6 a3=-5.6304d-9 fdcp3dt=rg*(a1+2.d0*a2*tk+3.d0*a3*tk**2)/mw c fdcp3dt=0.0 return end

real*8 function fdcp4dt(tk,p,y3) double precision tk,p,y3 double precision rg,mw,a1,a2,a3 c** Cp of CH4: Fitted to the data in J. Chem. Eng. Data 8, 547 (1963) at T= 160-340 K rg=8.3145 mw=16.04 a1=-3.7677d-3 a2= 9.1107d-6 a3= 1.0178d-9 fdcp4dt=rg*(a1+2.d0*a2*tk+3.d0*a3*tk**2)/mw c fdcp4dt=0.0 return end

*************** Pressure derivative of Cp *********************

real*8 function fdcp2dp(tk,p,y1) 227

double precision tk,p,y1 fdcp2dp=0.d0 return end

real*8 function fdcp3dp(tk,p,y2) double precision tk,p,y2 fdcp3dp=0.d0 return end

real*8 function fdcp4dp(tk,p,y3) double precision tk,p,y3 fdcp4dp=0.d0 return end

***************** Functions for Cpl ********************************** c** Cpl of Nonane real*8 function fcpl2(tk) double precision tk fcpl2=2.217 return end c** Cpl of D2O real*8 function fcpl3(tk) double precision tk fcpl3=4.205d0 return end

******** Temperature derivative of gamma of gas mixture ***

real*8 function dgdt(tk,p,y1,y2,y3,w20,w30,w40,wmav0) double precision tk,p,y1,y2,y3,w20,w30,w40,wmav0 double precision rg,cpN,cp,gamma,dcpdt

rg=8.3145 cpN=1.0397

cp= (1.d0-w20-w30-w40)*cpN+w40*fcp4(tk,p,y3) & +w20*fcp2(tk,p,y1)+w30*fcp3(tk,p,y2) gamma=cp/(cp-rg/wmav0)

dcpdt= w20*fdcp2dt(tk,p,y1)+w30*fdcp3dt(tk,p,y2) & +w40*fdcp4dt(tk,p,y3)

228

dgdt=gamma*(1.d0-gamma)/cp*dcpdt

return end

******** Pressure derivative of gamma of gas mixture ******

real*8 function dgdp(tk,p,y1,y2,y3,w20,w30,w40,wmav0) double precision tk,p,y1,y2,y3,w20,w30,w40,wmav0 double precision rg,cpN,cp,gamma,dcpdp

rg=8.3145 cpN=1.0397

cp= (1.d0-w20-w30-w40)*cpN+w40*fcp4(tk,p,y3) & +w20*fcp2(tk,p,y1)+w30*fcp3(tk,p,y2) gamma=cp/(cp-rg/wmav0)

dcpdp= w20*fdcp2dp(tk,p,y1)+w30*fdcp3dp(tk,p,y2) & +w40*fdcp4dp(tk,p,y3)

dgdp=gamma*(1.d0-gamma)/cp*dcpdp

return end

229

Appendix D: PTM and SAXS results for n-nonane pv0 = 489 Pa

(a)

(b)

230

Appendix E: Mathematica code to generate an n-alkane crystal

a={.470,0,0}; b={.470 Cos[Pi/3],.470 Sin[Pi/3],0}; c = {0,0,2.678};

Na=15;Nb=18;Nc=6; th2=ArcCos[-1/3]/2; offset=a/2+b/2+.05c; l=.153; rho=l Cos[th2]; t1a={ rho Cos[Pi/6], rho Sin[Pi/6],l Sin[th2]}; t1b={-rho Cos[Pi/6],-rho Sin[Pi/6],l Sin[th2]}; t2a={-rho Cos[Pi/6], rho Sin[Pi/6],l Sin[th2]}; t2b={ rho Cos[Pi/6],-rho Sin[Pi/6],l Sin[th2]}; imol=0; iatom=0; atomname={" CAA"," CAB"," CAC"," CAD"," CAE"," CAF"," CAG"," CAH"," CAI"," CAJ"," CAK"," CAL"," CAM"," CAN"," CAO"," CAP"," CAQ"," CAR"," CAS"}; stream=OpenWrite["C:\\Users\\modak.CHBMENG\\Documents\\Viraj_Project_2\\Projec t_Main_June2015\\MD Simulations\chains_NND.gro"]; WriteString[stream,"initial octane config\n"]; WriteString[stream,ToString[15 18 6 19],"\n"]; Do[ r0=offset+na a+(b-a/2)nb+If[OddQ[nb],a/2,0]+nc c; If[EvenQ[nb],ta=t1a;tb=t1b, ta=t2a;tb=t2b];

ia=0;ib=0;imol+=1; Do[ 231

iatom+=1; t=r0+ia ta + ib tb; If[OddQ[i],ia+=1,ib+=1]; WriteString[stream,IntegerString[imol,10,5]," NND ",atomname[[i]],IntegerString[iatom,10,5], PaddedForm[t[[1]],{6,3}], PaddedForm[t[[2]],{6,3}], PaddedForm[t[[3]],{6,3}], "\n"],{i,19}]; (* Print[na," ",nb," ",nc]; *)

{nc,0,Nc-1},{nb,0,Nb-1},{na,0,Na-1}];1

WriteString[stream, PaddedForm[Na a[[1]],{12,6}], PaddedForm[Nb b[[2]],{12,6}], PaddedForm[Nc c[[3]],{12,6}]]; WriteString[stream,"\n"]; Close[stream];

232

Appendix F: GROMACS topology file for n-octane

; ; This file was generated by PRODRG version 071121.0636 ; PRODRG written/copyrighted by Daan van Aalten ; and Alexander Schuettelkopf ; ; Questions/comments to [email protected] ; ; When using this software in a publication, cite: ; A. W. Schuettelkopf and D. M. F. van Aalten (2004). ; PRODRG - a tool for high-throughput crystallography ; of protein-ligand complexes. ; Acta Crystallogr. D60, 1355--1363. ; ; [ defaults ] ; nbfunc comb-rule gen-pairs fudgeLJ fudgeQQ 1 2 no 1.0 1.0 [ atomtypes ] ;name at.num mass charge ptype sigma eps CH2 6 14.02700 0.000 A 0.401 0.469 CH3 6 15.03500 0.000 A 0.401 0.469 ;[ bondtypes ] ;[ pairtypes ] ;[ angletypes ] ;[ dihedraltypes ] [ nonbond_params ] ; i j func sigma eps CH3 CH3 1 0.401 0.469 CH3 CH2 1 0.401 0.469 233

CH2 CH2 1 0.401 0.469 [ moleculetype ] ; Name nrexcl OCT 3 [ atoms ] ; nr type resnr resid atom cgnr charge mass 1 CH3 1 OCT CAA 0 0.000 15.0350 2 CH2 1 OCT CAB 1 0.000 14.0270 3 CH2 1 OCT CAC 2 0.000 14.0270 4 CH2 1 OCT CAD 3 0.000 14.0270 5 CH2 1 OCT CAE 4 0.000 14.0270 6 CH2 1 OCT CAF 5 0.000 14.0270 7 CH2 1 OCT CAG 6 0.000 14.0270 8 CH3 1 OCT CAH 7 0.000 15.0350 [ bonds ] ; ai aj fu c0, c1, ... 1 2 1 0.153 292000.0 ; CAA CAB 2 3 1 0.153 292000.0 ; CAB CAC 3 4 1 0.153 292000.0 ; CAC CAD 4 5 1 0.153 292000.0 ; CAD CAE 5 6 1 0.153 292000.0 ; CAE CAF 6 7 1 0.153 292000.0 ; CAF CAG 7 8 1 0.153 292000.0 ; CAG CAH [ pairs ] ; ai aj fu sigma eps 1 4 1 0.401 0.00 ; CAA CAD 2 5 1 0.401 0.00 ; CAB CAE 3 6 1 0.401 0.00 ; CAC CAF 4 7 1 0.401 0.00 ; CAD CAG 5 8 1 0.401 0.00 ; CAE CAH [ angles ] 234

; ai aj ak fu theta0 k_theta 1 2 3 1 109.526 502.08 ; CAA CAB CAC 2 3 4 1 109.526 502.08 ; CAB CAC CAD 3 4 5 1 109.526 502.08 ; CAC CAD CAE 4 5 6 1 109.526 502.08 ; CAD CAE CAF 5 6 7 1 109.526 502.08 ; CAE CAF CAG 6 7 8 1 109.526 502.08 ; CAF CAG CAH [ exclusions ] 1 2 3 4 2 1 3 4 5 3 1 2 4 5 6 4 1 2 3 5 6 7 5 2 3 4 6 7 8 6 3 4 5 7 8 7 4 5 6 8 8 5 6 7 [ dihedrals ] ; ai aj ak al fu c0, c1, c2, c3, c4, c5 4 3 2 1 3 6.505 16.995 3.62 -27.12 0.00 0.00 ; dih CAD CAC CAB CAA 5 4 3 2 3 6.505 16.995 3.62 -27.12 0.00 0.00 ; dih CAE CAD CAC CAB 6 5 4 3 3 6.505 16.995 3.62 -27.12 0.00 0.00 ; dih CAF CAE CAD CAC 7 6 5 4 3 6.505 16.995 3.62 -27.12 0.00 0.00 ; dih CAG CAF CAE CAD 8 7 6 5 3 6.505 16.995 3.62 -27.12 0.00 0.00 ; dih CAH CAG CAF CAE ; system name [ system ] octane solid or liquid [ molecules ] OCT 480

235

Appendix G: Sample GROMACS simulations parameter file

; define = -DPOSRES constraints = none integrator = md emstep = 0.001 emtol = 0.01 ; ; Center of mass removal ------comm_mode = linear ; Change for no pbc comm_grps = System nstcomm = 10 ; freq of center of mass removal ; ; Time steps ------dt = 0.001 ; ps ! nsteps = 50000000 ; ; ; Output control ------nstxout = 5000 ; freq of coord output to trr nstvout = 5000 ; freq of velocity output to trr nstfout = 0 ; freq of force output to trr nstlog = 5000 ; freq of output to log file nstenergy = 100 ; default of -1 sets equal to nstlist ; ; Neighbor list and pbc's ------nstlist = 5 ; default is 10 -- try more frequent ns_type = grid ; simple (for no pbc) or grid ; pbc = xyz ; CHANGE periodic_molecules = no ; 236

; Interaction parameters ------rlist = 1 rcoulomb = 1 ; coulombtype = cut-off epsilon_r = 1 ; vdwtype = cut-off rvdw = 2.4 DispCorr = no ; no or EnerPres ; energygrps = System ;freezegrps = freeze ;freezedim = Y Y Y ;energygrp_excl = freeze freeze ; ; temperature coupling is on ------Tcoupl = v-rescale ;or v-rescale berendsen nose-hoover tau_t = 0.3 tc-grps = System ref_t = 205.0 ; Pressure coupling is not on ------Pcoupl = no ; no berendsen Parrinello-Rahman tau_p = 4 pcoupltype = anisotropic ; isotropic anisotropic compressibility = 10.e-5 10.e-5 10.e-5 0.0 0.0 0.0 ref_p = 1.0 1.0 1.0 0.0 0.0 0.0 ; Generate velocites ------gen_vel = no gen_temp = 215.0 gen_seed = 173529 237

Appendix H: Fortran code to calculate the orientational order in a liquid slab from the trajectory file

implicit none integer maxAtom, maxzBin, maxAngBin parameter (maxAtom=6000,maxzBin=200, maxAngBin=50) character*10 title character*5 molName, atomName integer nAtom, nMol, jMol, m, k, i1, i2, ctr integer nframe, nzbin, nAngbin, izbin, iAngbin integer dens(0:maxzBin, 0:maxAngBin) real*8 r(3,8,720),v(3),L(3),u(3, 720),comz(720),mag(720) real*8 vec(720),ang(720),proj(6),avgvec(6),unorm(6),t real*8 t1a,t1b real*8 dz,dAng, dummy c nzbin=200 nAngbin=50 nframe=0 c do m=0,maxzBin do k=0,maxAngBin dens(m,k) = 0 end do end do c 100 continue c c read frame --- read(*,*, end=200) title read(*,*) nAtom nMol=nAtom/8 do jMol=1,nMol do m=1,8 read(*,'(i5,2a5,i5,3f8.3,3f8.4)') i1,molName,atomName,i2, > (r(k,m,jMol),k=1,3),(v(k),k=1,3) end do end do

238

read(*,*) (L(k),k=1,3) dz=L(3)/dfloat(nzbin) dAng=1.0d0/dfloat(nAngbin) nframe=nframe+1 c

do jMol=1,nMol unorm=0.d0 do k=1,3 do ctr=1,6 t1a=r(k,ctr,jMol)-r(k,ctr+2,jMol) if (k==1 .or. k==2) then t1a=t1a-nint(t1a/L(k))*L(k) end if unorm(ctr)=unorm(ctr)+t1a**2 t1b=(r(k,ctr,jMol)+r(k,ctr+2,jMol)) if (k==1 .or. k==2) then t1b=t1b-nint(t1b/L(k))*L(k) end if t1b=t1b/2 if(k==3) then proj(ctr)=abs(t1a)/(unorm(ctr)**0.5d0) izbin=t1b/dz iAngBin=proj(ctr)/dAng dens(izbin,iAngbin)=dens(izbin,iAngbin)+1 end if end do c end do end do c go to 100 c 200 continue c open(unit=4,file='OrientationDist.txt') do izbin = 0, nzbin do iAngbin = 0,nAngbin t=1.d0/(dfloat(nFrame)*dz*dAng*L(1)*L(2)) write(4,*) dz*(dfloat(izbin)+0.d0), 239

> dAng*(dfloat(iAngbin)+0.d0), t*dens(izbin, iAngbin) end do end do c close(4) write(*,*) nFrame, ' frames read.' stop end

240

Appendix I: Fortran code to calculate the fluctuations along the molecular axis from the trajectory file

implicit none integer maxAtom, maxPosBin parameter (maxAtom=6000,maxPosBin=1000) character*10 title character*5 molName, atomName integer nAtom, nMol, jMol, m, k, i1, i2, ctr integer nframe, nPosBin, izbin,ihbin real*8 r(3,8,720),v(3,8,720),L(3), dz,dh real*8 Posz(0:maxPosBin),Posh(0:maxPosBin) real*8 rcom(3,720),rcomz(720),rcomh(720)

do m=0,maxPosBin Posz(m)=0.0d0 Posh(m)=0.0d0 end do c nPosBin=500 nFrame=0

100 continue c read(*,*, end=200) title read(*,*) nAtom nMol=nAtom/8 do jMol=1,nMol do m=1,8 read(*,'(i5,2a5,i5,3f8.3,3f8.4)') i1,molName,atomName,i2, > (r(k,m,jMol),k=1,3),(v(k,m,jMol),k=1,3) end do end do c write(*,*) 'Loop executed' c read(*,*) (L(k),k=1,3) dz = 1/dfloat(nPosbin) dh = 1/dfloat(nPosbin) nframe=nframe+1 rcom=0.0d0 rcomz=0.0d0 rcomh=0.0d0 c do jMol=1,720 241

do k=1,2 do ctr=1,8 rcom(k,jMol)=rcom(k,jMol)+r(k,ctr,jMol) rcom(k,jMol)=rcom(k,jMol)-nint(rcom(k,jMol)/L(k))*L(k) end do end do do ctr=1,8 rcom(3,jMol)=rcom(3,jMol)+r(3,ctr,jMol) end do rcomz(jMol)=rcom(3,jMol)/8 rcomh(jMol)=((rcom(1,jMol)**2+rcom(2,jMol)**2)**0.5)/8 end do c write(*,*)rcom

dz = 15/dfloat(nPosbin) dh = 5/dfloat(nPosbin) c do jMol=1,720 izbin=rcomz(jMol)/dz ihbin=rcomh(jMol)/dh Posz(izbin)=Posz(izbin)+1 Posh(ihbin)=Posh(ihbin)+1 end do c go to 100 c 200 continue close(4) c write(*,*) nFrame, ' frames read.' open(unit=3,file='PositionZ.dat') do izbin=0,nPosBin write(3,'(f10.4,1x,f10.3)') dz*(dfloat(izbin)+0.d0),Posz(izbin) end do close(3) stop end

242

Appendix J: Fortran code to calculate the energy densities of either a solid, liquid or a surface frozen slab from the trajectory file as a function of slab length

implicit none integer maxAtom,maxMol,maxBin parameter (maxAtom=6000,maxMol=750,maxBin=800) character*10 title character*5 molName,atomName integer nAtom,nMol,jAtom,jMol,m,k, > nFrame, > i1,i2,jMol1,jMol2,m1,m2 real*8 r(3,maxAtom),rr(3,8,maxMol),v(3),L(3), > rij(3),rij1(3),rij2(3),rsq,rsq1,rsq2,rCut,rCutsq c integer nbin,ibin,dens(0:maxBin) real*8 dz,edensLJ(0:maxBin),edensLJ0(0:maxBin), > edensBend(0:maxBin), edensDih(0:maxBin), edensBond(0:maxBin), > sigma,epsilon,sigmasq,eps2, > ktheta,ktheta3,theta0,Pi, dBond,kBond,kBond4 real*8 t,t1,t2, u32(3),r12(3),r43(3) c c parameter definitions sigma=0.401d0 epsilon=0.469d0 ktheta=251.04d0 theta0=1.23d0 dBond=0.153d0 kBond=292000.d0 rCut=2.4d0 c rCutsq=rCut*rCut sigmasq=sigma*sigma eps2=2.d0*epsilon ktheta3=ktheta/3.d0 kBond4=0.25d0*kBond Pi=4.d0*datan(1.d0) c nbin=800 c zero arrays do m=0,maxBin dens(m)=0 edensLJ(m) =0.d0 edensLJ0(m) =0.d0 edensBend(m)=0.d0 243

edensDih(m) =0.d0 edensBond(m)=0.d0 end do c open(unit=14,file='edensInverseTrig.dat') c c read configurations and accumulate averages nFrame=0 100 continue read(*,*,end=200) title read(*,*) nAtom nMol=nAtom/8 jAtom=0 do jMol=1,nMol do m=1,8 jAtom=jAtom+1 read(*,'(i5,2a5,i5,3f8.3,3f8.4)') i1,molName,atomName,i2, > (r(k,jAtom),k=1,3),(v(k),k=1,3) do k=1,3 rr(k,m,jMol)=r(k,jAtom) end do end do end do read(*,*) (L(k),k=1,3) c nFrame=nFrame+1 c c accumulate densities dz=L(3)/dfloat(nbin) c c number density do jAtom=1,nAtom ibin=r(3,jAtom)/dz dens(ibin)=dens(ibin)+1 end do c c LJ energy density - intermolecular do jMol1=2,nMol do m1=1,8 do jMol2=1,jMol1-1 do m2=1,8 rsq=0.d0 do k=1,3 rij(k)=rr(k,m1,jMol1)-rr(k,m2,jMol2) rij(k)=rij(k)-nint(rij(k)/L(k))*L(k) rsq=rsq+rij(k)*rij(k) 244

end do if(rsq.le.rCutsq) then t=sigmasq/rsq t=eps2*( t**6-t**3) ibin=rr(3,m1,jMol1)/dz edensLJ(ibin)=edensLJ(ibin)+t ibin=rr(3,m2,jMol2)/dz edensLJ(ibin)=edensLJ(ibin)+t end if end do end do end do end do c c LJ energy density - intramolecular do jMol=1,nMol do m1=1,4 do m2=m1+4,8 rsq=0.d0 do k=1,3 rij(k)=rr(k,m1,jMol)-rr(k,m2,jMol) rij(k)=rij(k)-nint(rij(k)/L(k))*L(k) rsq=rsq+rij(k)*rij(k) end do if(rsq.le.rCutsq) then t=sigmasq/rsq t=eps2*( t**6-t**3) ibin=rr(3,m1,jMol)/dz edensLJ0(ibin)=edensLJ0(ibin)+t ibin=rr(3,m2,jMol)/dz edensLJ0(ibin)=edensLJ0(ibin)+t end if end do end do end do c c bending energy density do jMol=1,nMol do m=2,7 rsq1=0.d0 do k=1,3 rij1(k)=rr(k,m-1,jMol)-rr(k,m,jMol) rij1(k)=rij1(k)-nint(rij1(k)/L(k))*L(k) rsq1=rsq1+rij1(k)*rij1(k) end do rsq2=0.d0 245

do k=1,3 rij2(k)=rr(k,m+1,jMol)-rr(k,m,jMol) rij2(k)=rij2(k)-nint(rij2(k)/L(k))*L(k) rsq2=rsq2+rij2(k)*rij2(k) end do t=0.d0 do k=1,3 t=t+rij1(k)*rij2(k) end do t=t/dsqrt(rsq1*rsq2) if(dabs(t).gt.1.d0) then write(14,'("frame:",i8," molecule:",i6, > " bend:",3i2)') nFrame,jMol,m-1,m,m+1 write(14,'("old:",d25.18)') t t=dsign(1.d0,t) write(14,'("new:",d25.18)') t end if t=Pi-dacos(t) t=ktheta3*(t-theta0)**2 ibin=rr(3,m-1,jMol)/dz edensBend(ibin)=edensBend(ibin)+t ibin=rr(3,m ,jMol)/dz edensBend(ibin)=edensBend(ibin)+t ibin=rr(3,m+1,jMol)/dz edensBend(ibin)=edensBend(ibin)+t end do end do c c bond energy density open(unit=4,file='dummy2.dat') open(unit=3,file='dummy1.dat') do jMol=1,nMol do m=1,7 rsq1=0.d0 do k=1,3 rij1(k)=rr(k,m+1,jMol)-rr(k,m,jMol) rij1(k)=rij1(k)-nint(rij1(k)/L(k))*L(k) rsq1=rsq1+rij1(k)*rij1(k) end do t=dsqrt(rsq1) write(3, '(f10.4)')t t=kBond4*(t-dBond)**2 write(4, '(f10.4, f10.4)')kBond4, t c ibin=rr(3, m,jMol)/dz edensBond(ibin)=edensBond(ibin)+t 246

ibin=rr(3,m+1,jMol)/dz edensBond(ibin)=edensBond(ibin)+t end do end do close(4) c c dihedral energy density 1--2--3--4 c u32=r3-r2 (normalized) r12=r1-r2 r43=r4-43 c do jMol=1,nMol do m=2,6 t=0.d0 do k=1,3 u32(k)=rr(k,m+1,jMol)-rr(k,m,jMol) u32(k)=u32(k)-nint(u32(k)/L(k))*L(k) t=t+u32(k)*u32(k) end do t=1.d0/dsqrt(t) do k=1,3 u32(k)=t*u32(k) end do c t1=0.d0 t2=0.d0 do k=1,3 r12(k)=rr(k,m-1,jMol)-rr(k, m,jMol) r12(k)=r12(k)-nint(r12(k)/L(k))*L(k) t1=t1+r12(k)*u32(k) r43(k)=rr(k,m+2,jMol)-rr(k,m+1,jMol) r43(k)=r43(k)-nint(r43(k)/L(k))*L(k) t2=t2+r43(k)*u32(k) end do c t=0.d0 do k=1,3 r12(k)=r12(k)-t1*u32(k) r43(k)=r43(k)-t2*u32(k) t=t+r12(k)*r43(k) end do c t1=0.d0 t2=0.d0 do k=1,3 t1=t1+r12(k)*r12(k) t2=t2+r43(k)*r43(k) end do 247

t=t/dsqrt(t1*t2) if(dabs(t).gt.1.d0) then write(14,'("frame:",i8," molecule:",i6, > " dihedral:",4i2)') nFrame,jMol,m-1,m,m+1,m+2 write(14,'("old:",d25.18)') t t=dsign(1.d0,t) write(14,'("new:",d25.18)') t end if t=dcos(Pi-dacos(t)) t=6.505d0+t*(16.995d0+t*(3.62d0-t*27.12d0)) t=0.25d0*t c ibin=rr(3,m-1,jMol)/dz edensDih(ibin)=edensDih(ibin)+t ibin=rr(3,m ,jMol)/dz edensDih(ibin)=edensDih(ibin)+t ibin=rr(3,m+1,jMol)/dz edensDih(ibin)=edensDih(ibin)+t ibin=rr(3,m+2,jMol)/dz edensDih(ibin)=edensDih(ibin)+t end do end do c c go to 100 c 200 continue c c write output c write(*,*) nFrame, ' frames read.' c open(unit=3,file='numberdens.dat') do ibin=0,nbin t=1.d0/(dfloat(nFrame) *dz*L(1)*L(2)) write(3,'(f10.4,1x,f10.5)') dz*(dfloat(ibin)+0.d0),t*dens(ibin) end do close(3) c open(unit=3,file='eLJdens.dat') do ibin=0,nbin t=1.d0/(dfloat(nFrame) *dz*L(1)*L(2)) write(3,'(f10.4,1x,f10.5)') dz*(dfloat(ibin)+0.d0), > t*edensLJ(ibin) end do close(3) c 248

open(unit=3,file='eLJintradens.dat') do ibin=0,nbin t=1.d0/(dfloat(nFrame) *dz*L(1)*L(2)) write(3,'(f10.4,1x,f10.5)') dz*(dfloat(ibin)+0.d0), > t*edensLJ0(ibin) end do close(3) c open(unit=3,file='eBend.dat') do ibin=0,nbin t=1.d0/(dfloat(nFrame) *dz*L(1)*L(2)) write(3,'(f10.4,1x,f10.5)') dz*(dfloat(ibin)+0.d0), > t*edensBend(ibin) end do c open(unit=3,file='eBond.dat') do ibin=0,nbin t=1.d0/(dfloat(nFrame) *dz*L(1)*L(2)) write(3,'(f10.4,1x,f10.5)') dz*(dfloat(ibin)+0.d0), > t*edensBond(ibin) end do c open(unit=3,file='eDih.dat') do ibin=0,nbin t=1.d0/(dfloat(nFrame) *dz*L(1)*L(2)) write(3,'(f10.4,1x,f10.5)') dz*(dfloat(ibin)+0.d0), > t*edensDih(ibin) end do close(3) c close(14) c stop end

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