UNIVERSITY OF CINCINNATI
Date: ______8th August 2008
I, ______Prashant Rajan ______, hereby submit this work as part of the requirements for the degree of: Master of Science in: Department of Chemical and Materials Engineering It is entitled : Understanding aggregate morphology in colloidal systems through small-angle scattering and reverse Monte Carlo (RMC) simulations
This work and its defense approved by:
Chair: ______Dale W. Schaefer ______Greg Beaucage ______Jude Iroh ______Steve Clarson ______
Understanding aggregate morphology in colloidal systems through small-
angle scattering and reverse Monte Carlo simulations
A thesis submitted to the
Division of Research and Advanced Studies
of the University of Cincinnati
in partial fulfillment of the
requirements for the degree of
Master of Science
in the Department of Chemical and Materials Engineering
of the College of Engineering
2008
by
Prashant Rajan
B.E. University of Pune, Pune, India
Committee Chair: Professor Dale W. Schaefer Abstract
We have studied the three dimensional structure of aggregated colloidal silica at sub- micron length scales. Our efforts are part of an integrated research project[1-3] focused on constructing a verifiable model for reinforcement of elastomers by colloidal fillers using small- angle scattering techniques. Our work is aimed at developing a simulation-based modeling tool that enables us to visualize the microstructure of reinforcing fillers in three dimensions by fitting small-angle scattering data. We have developed a program based on ‘ C’ programming language to describe scattering from a single aggregate. Our simulation creates the aggregate using a particle-cluster aggregation mechanism. The simulation then proceeds to make random moves on the aggregate surface while simultaneously fitting small angle scattering data on the system under study. Our results provide a real-space picture of aggregate structure that is consistent with scattering data and images obtained from electron microscopy. We estimate primary particle size accurately by generating aggregate morphology visualizations for different primary particle sizes. Visualizations for different primary particle size and size distributions provide a different qualitative picture of aggregate morphology for the same mass fractal dimension. We arrive at the correct value for primary particle size by matching the visualizations with the slopes of power-law regime in the mass fractal domain. We have also distinguished between the morphology of silica aggregates in powder and inside elastomer by applying our program to small-angle-scattering data on silica powder and silica-filled-elastomer systems.
ii iii ACKNOWLEDGMENTS:
A Masters thesis is not usually an incredibly hard effort but this thesis is special in that it is the cumulative result of two years worth of experiences, work and relationships. I came to this program a naïve adolescent struggling to become an adult and I leave a much-chastened youth struggling to remain responsible. For this transition I thank first and foremost my advisor Dr.
Dale Schaefer. To me he does and always will symbolize simplicity, hard work and the highest ethical and research standards. My gratitude to my parents and my sister who let a floundering idiot leave their care, doubtless with much fear and trepidation. My grateful thanks to Dr.
Beaucage for teaching me that action speaks much louder than words. He helped me arrive at what is probably my biggest epiphany till date – learning IS indeed an end in itself. Thank you
Dr. Vasudevan, for believing in me and giving me a chance when I thought I had none. I thank
Drs. Clarson and Iroh for their cheerful support through my time here in Cincinnati.
I thank Dr. T.K. Hatwar for giving me the opportunity to work with him through Spring. I thank Drs. Tom Blanton and Joel Shoreman for their enthusiasm in discussing my research. I also thank Dr. Jan Ilavsky and the Advanced Photon Source at Argonne. This thesis would not be possible without availing of the world-class facilities at Argonne.
A big thank you to my friends in Cincinnati and elsewhere – Dhruva you remain my guiding star! Sujit, Rohan, Johny, Sachin, Aarti, Amrita – with friends like you, who needs therapists? Thank you Aniruddha, Ashay, Ashish, Adi, Ashwin, Mangya and the rest of the gang at West Nixon. You guys got me through the tough times.
I reserve special thanks for Ram, who I am sure my elder brother would have been like.
He perhaps has played a more immediate role than anyone else in effecting a semblance of change in me. I would like to thank Doug Kohls for helping with USAXS in samples. Also,
iv many thanks to Doug, for showing me the difference between professional and personal trust.
You gave me an accurate estimate of the price of both.
Last but never the least, I thank Shweta. Words fail me when I think of the efforts you have taken to get me to this stage. This is just the beginning. And the fact that I am writing this acknowledgement is proof that love is enough.
v TABLE OF CONTENTS:
ABSTRACT ii
ACKNOWLEDGEMENTS iv
LIST OF FIGURES viii
LIST OF TABLES xii
1. INTRODUCTION 1
1.1 BACKGROUND AND MOTIVATION 1
1.2 FRACTAL CONCEPTS 4
1.3 SMALL-ANGLE SCATTERING 6
1.4 REVERSE MONTE CARLO (RMC) METHODS: A BRIEF REVIEW 8
2. MATERIALS AND METHODS 17
2.1 ULTRA SMALL-ANGLE X-RAY SCATTERING (USAXS) 19
2.2 STRUCTURAL MODELING 19
2.2.1 RMC MODELING 20
2.3 IMPLEMENTATION OF RMC ALGORITHM 20
2.3.1 GENERATING THE AGGREGATE 21
2.3.2 EXECUTING MONTE CARLO MOVEMENT OF BEADS 21
2.3.3 FITTING THE DATA 23
2.4 TRANSMISSION ELECTRON MICROSCOPY (TEM) 23
3. RESULTS AND DISCUSSION 38
3.1 USAXS ANALYSIS 39
3.2 RMC ANALYSIS 39
3.3 TEM RESULTS 41
3.4 ANALYSIS OF COLLOIDAL SILICA DISPERSED IN RUBBER 52
vi 3.5 RELEVANCE OF RMC AND SMALL ANGLE SCATTERING ANALYSIS TO 55 REINFORCEMENT STUDIES
3.6 UNIQUENESS OF SOLUTION 60
3.7 ACCOUNTING FOR POLYDISPERSITY BETWEEN AGGREGATES 63
4. CONCLUSIONS AND FUTURE WORK 64
REFERENCES 66
APPENDICES 68
72
vii
LIST OF FIGURES
Figure 1. Mass Fractal form of a Romanesco broccoli (Brassica oleracea)
Figure 2. Crumpled paper as surface fractal in 3 dimensions
Figure 3. Scattering intensity plotted against the scattering vector q for a sample of
precipitated silica exhibiting 2 levels of structure as observed through Ultra
Small Angle X-ray Scattering (USAXS)
Figure 4. Unified fit model applied to USAXS data on precipitated silica sample
dispersed in rubber. Rg1 and Rg2 correspond to the radius of gyration
calculated for silica primary particle and aggregate respectively
Figure 5. Graphical representation of initial aggregate structure for an aggregate
composed of 220 beads
Figure 6. Normalized analytical form factor for a solid sphere (R = 8.5 nm) having
uniform scattering length density
Figure 7. Effect of analytical form factor P(q) on intensity I(q) sim calculated by the
RMC program
Figure 8. RMC fit using level 1 Unified fit in form factor P(q) calculation
Figure 9. Optimizing the structure factor S(q) calculation at each Monte Carlo step by
calculating only the contribution of the move under consideration to the overall
structure factor S(q).
Figure 10. Structure factor evolution over the duration of a simulation
Figure 11. Effect of monodisperse spherical assumption on S(q) and I(q) calculated by
RMC modeling program
Figure 12. Polydisperse collection of beads
viii
Figure 13. Effect of polydispersity on RMC calculated structure factor S(q)
Figure 14. Unified fit intensity modeled with RMC for a uniform distribution of bead
radii (0.5 Ǻ < R < 175 Ǻ), Aggregation number, z = 107
Figure 15. Chi square ( χ2) with no fluctuations allowed
Figure 16. Chi square ( χ2) calculated with fluctuations allowed
Figure 17. USAXS curves of colloidal silica samples:(a) DSI series: DSI pH6 and DSI
1289, (b) Goodyear: GY5011, (c) Nissan Snowtex. Flat background was
subtracted to emphasize the power-law profiles at high q. Data have been
scaled vertically to match in the high q region for comparison
Figure 18. Evolution of chi square over the number of Monte Carlo moves for DSI pH6,
DSI 1289, GY 5011 and Nissan Snowtex
Figure 19. Graphical representatations of aggregate structures.
Figure 20. Probability distributions of primary particle radii as estimated by Irena
modeling of Unified fit to USAXS data
Figure 21. Fits obtained from RMC modeling of Unified intensity for DSI pH6, DSI
1289, GY 5011 and Nissan Snowtex®
Figure 22. Aggregate morphology visualization: Nissan Snowtex "String of pearls", slope
of mass fractal domain P = - 1.68
Figure 23. Aggregate morphology visualization: GY 5011, slope of mass fractal domain
P = -2.7
Figure 24. Visualizing aggregate morphology for DSI 1289, mass fractal dimension as
obtained from Unified fit is -2.53
ix
Figure 25. Visualizing aggregate morphology of DSI pH6 through RMC modeling. Mass
fractal dimension obtained from Unified fit is - 3.0 indicating dense compact
aggregates
Figure 26. Unified fit to DSI pH6 for smaller sized primary particles (Rg1 = 155 Å).
Figure 27. Visualizing aggregate morphology of DSI pH6 through RMC modeling. Mass
fractal dimension obtained from Unified fit is - 3.06 indicating dense
aggregates.
Figure 28. Aggregate visualization for incorrectly calculated radius of gyration (Rg = 458
Å) of primary particles - DSI pH6.Through RMC visualization, we are able to
observe an open fractal aggregate which is inconsistent with the the calculated
slope of the mass fractal domain (-3.06). We understand that the intensity
curve was incorrectly interpreted leading to an erroneus assumption of broad
particle size distribution along with large Rg for primary particles.The
aggregates are actually composed of smaller primary particles that form a
dense cluster that is more representative of the slope in the mas fractal
domain.
Figure 29. Representative TEM image of DSI 1289 at 50 nm resolution
Figure 30. Representative TEM image of Nissan Snowtex collodial silica at 20 nm
resolution. Relatively monodisperse primary particles are connected in a linear
fashion that is correctly visualized through RMC simulation.
Figure 31. RepresentativeTEM image of Goodyear colloidal silica sample GY 5011 at 50
nm resolution. Sample appears as a ramified structure corroborated through
simulation.
x
Figure 32. Representative TEM image of DSI pH6 silica.at 50 nm resolution
Figure 33. RMC fit to USAXS data for Zeosil 1165 powder and for Zeosil 1165
compounded in Styrene-Butadiene Rubber
Figure 34. RMC visualization of aggregate structure for Zeosil 1165 silica powder
sample.
Figure 35. Aggregate visualization for Zeosil 1165 dispersed at 10phr in elastomer.
xi
LIST OF TABLES
TABLE I Unified fit parameters obtained for Dimosil grades (DSI pH6, 38
® DSI 1289), Goodyear sample (GY5011) and Nissan Snowtex
colloidal silica
TABLE II Maximum end-end distance calculated for final aggregate for 40
two different simulation runs on each sample
TABLE III Size distribution parameters for primary particles 42
xii
1. Introduction:
1.1 Background and Motivation:
The mechanical performance of a composite depends on the structure of reinforcing fillers at the micron and sub-micron length scales.[4-7] Understanding the structure of reinforcing fillers is an important step toward understanding elastomer reinforcement and if possible, tuning mechanical properties of such composites.[8, 9]
Reinforcing fillers like carbon black and colloidal silica have a complex structure that may extend over six decades in length scale.[10]
Filler structure has been studied with imaging techniques such as electron microscopy or atomic force microscopy.[11, 12] Most imaging techniques are susceptible to operator bias and are hindered by their two dimensional rendering of a three dimensional composite. Cross-sectional area of the sample being evaluated during imaging experiments is also very small.
Small angle scattering of light and X-rays is a valuable tool to distinguish between complex morphologies since scattering techniques cover an enormous range of length scales - up to 7 orders of magnitude for the momentum transfer vector q.[13]
Scattering is a powerful method of accessing bulk structure within the composite. X-ray scattering occurs due to contrast in the electron density of constituents and contains information about the three-dimensional arrangement of scattering objects. X-ray scattering measured at small angles is sensitive to large-scale inhomogeneities in electron density. X-rays having a wavelength of ~1.54 Ǻ can be used to investigate structures of
the order of a few 100 nanometers.
1
The principal goal of our work is to elucidate the morphology of sub-micron level structures formed by reinforcing fillers. Numerous studies have been undertaken to study filler structure and the effect of filler structure on mechanical properties of nanocomposites has also been studied.[1, 14-16] Small angle scattering provides detailed information regarding filler morphology and allows for extensive modeling especially in the case of hierarchical systems.[2] Modeling tools allow filler attributes such as the mass, surface area, radius of gyration and fractal dimensions of hierarchically organized filler systems to be calculated from the intensity curve. For example, the unified approach of Beaucage[17, 18] is widely used for the purposes of fitting scattering data to obtain information at multiple structural levels for hierarchically organized systems.
One drawback of scattering methods is that the data are collected in reciprocal space (or q-space) and not in real space. We are limited then in being able to visualize
structures and morphologies studied through scattering. A lot of work has been done in
the field of inverting data in the reciprocal space to real space information.[14, 15, 19,
20] The scope of our work is to develop a freely available tool that enables researchers to
not only visualize but also confirm their models describing filler morphology. We have
studied different samples of colloidal silica.
Colloidal silica is an important reinforcing agent and is widely used in rubber-
based applications such as tires.[9] The structure of colloidal silica has an important
effect on its reinforcing properties and has been well studied in the literature.[21-23] Up
to three levels of hierarchical organization of colloidal silica structure have been
reported.[24] These three levels comprise primary particles, aggregates of primary
particles and larger agglomerates. Primary particles and aggregates are typically smaller
2
than 0.1 micron and are referred to as hard structures because they do not break down under shear. Aggregates are critical for reinforcement. Agglomerates are much larger and
“softer” than aggregates and should be weak so that they easily break down to aggregates during rubber mixing. Weak agglomerates significantly reduce the energy required to disperse filler in rubber. It is clear that the structure of colloidal silica at different length scales contributes differently to reinforcement of the polymer or elastomer matrix.
Section 1.2, 1.3 and 1.4 discuss the concepts and terms that enable us to study the morphology of structures formed by hierarchically organized systems like colloidal silica.
3
1.2 Fractal concepts
Colloidal structures are often disordered hierarchies. Such structures can be
described in a quantifiable manner using fractal geometry concepts.[18, 25, 26]
A fractal possesses dilation symmetry or self-similarity in that the structure being
observed looks the same even if the resolution is increased.[27] The radius r of a sphere drawn around a point in the fractal object is related to the mass M (r) inside the sphere as:
d M(r) ∝ r m
d m is known as the mass fractal dimension; it is a number between 1 and 3 and may be a
fraction. For a solid three-dimensional object, d m = 3 . As the object becomes more open,
the value of d m drops and falls to 1 for a linear object. A fractional value of d m indicates
a branched or aggregated structure. The trend in d m can be understood by relating the
volume V (r ) and density ρ(r ) of the sphere with its radius as:
V (r) ∝ r 3
∴ ρ(r) ∝ r d −3
The density of the object under study is no longer constant and varies inversely with the
volume under consideration.[27]
A surface fractal is an object that has a rough surface also possessing the dilation
properties of a fractal. The surface area of a fractal object goes as:
d S(r) ∝ r s
d s is the fractal dimension of the surface fractal in three dimensional space. Its value lies between 2 (smooth surface) and 3 (a folded surface that completely fills up the space).
4
Figure 1 and Figure 2 illustrate examples of a mass fractal and a surface fractal respectively.
Figure 1. Mass Fractal form of a Romanesco broccoli (Brassica oleracea)
Figure 2. Crumpled paper as surface fractal in 3 dimensions
5
1.3 Small Angle Scattering:
For a monodisperse system of spheres each having volume Vs , the differential
1 dΣ scattering cross-section per unit sample volume , for an infinitely diluted V dΩ
suspension of spheres; is the product of the square of the scattering contrast ∆ρ between the spheres and the dispersion medium, the volume fraction φ of the spheres, the particle form factor P(q ) and the structure factor S(q ) which represents scattering from some arbitrary spatial organization of these spheres: [13]
1 dΣ = ∆ρ 2φV S(q)P(q) (1) V dΩ s
The normalized analytical form factor P(q) for a solid sphere of radius R with uniform scattering length density ρ0 is a function of qR :
9(sin qR - qR cos qR )2 P(q) = 6 ()qR (2)
The geometry of a single sphere or a collection of monodisperse spheres is described by the structure factor S(q) :
N N 1 sin qr jk S(q) = 1 + ∑ ∑ N qr j k≠j jk where,
rjk = rj − rk (3) rj and rk are the co - ordinates of the centers of the spheres. Appendix A contains the derivation for S(q).
Figure 3 shows a plot of small angle scattering data on precipitated silica
dispersed in rubber. Information on the structure and morphology of the system under
6
study is gained by measuring the scattered intensity, I(q) , as a function of scattering vector, q.
1 dΣ I(q) = C ⋅ V dΩ
Where, C is a constant.
The relation between q and scattering angle ( θ) is governed by the equation:
4π θ q = sin λ 2
λ is the wavelength of the radiation in the medium.
q has the units of reciprocal length. For a given q, scattering is affected by real space inhomogeneities on the scale q-1. Large q is then related to small length scales or sizes and vice-versa.
In Figure 3, the length scale ( q-1) extends over at least three decades and can be
divided into two distinct regimes with evidence of a third power law regime at low q.
Each distinct regime consists of a power law and a knee or a crossover region where two power laws meet.
7
Figure 3. Scattering intensity plotted against the scattering vector q for a sample of precipitated silica exhibiting 2 levels of structure as observed through ultra small angle X-ray scattering (USAXS)
In the high q region ( q > 0.002 Ǻ) the power-law scattering is representative of
the surface scattering from primary particles. Silica primary particles are dense structures formed by covalent bonding between silica monomers and oligomers.[28] S(q) is constant in this region.
8
The power observed in the high q regime depends on the smoothness of particle surface.
The relation between intensity and q is then obtained as:
2π I(q) ∝ q ds −6 for q > R g1 (4)
R is the radius of gyration of the primary particles. The radius of gyration g1
The value of the q-exponent varies between 3 and 4 depending on ds (2 < ds < 3).
For the case of a uniformly smooth particle surface with ds = 2,
−4 I(q) ∝ q (5)
This result is the well known Porod law[29] of scattering. The value of the exponent may be less than -4 for diffuse interfaces.
In the low q regime, the form factor P(q) is constant and,
2π 2π I(q) ∝ q −dm for < q < R R g2 g1 (6)
dm is the mass fractal dimension. R g2 is the radius of gyration of the aggregates formed by
primary particles.
In the crossover regime between the two power laws the intensity is proportional
to the exponential of the square of the radius of gyration of the structure associated with
the particular level.
− R 2 q 2 I(q) ∝ exp g for qRg << 1 3 (7)
This region is known as the Guinier regime.[30] The Guinier regime represents the size of the structure being observed at a particular level of resolution.
9
The size and morphology at each structural level represented by the scattering data in Figure 4 can be quantified by fitting the plot of scattering intensity against q using the unified model of Beaucage.[18, 24]
Figure 4. Unified fit model applied to USAXS data on precipitated silica sample dispersed in rubber. Rg1 and Rg2 correspond to the radius of gyration calculated for silica primary particle and aggregate respectively
Figure 4 shows the parameters that are calculated by the unified fit for each structural level. R g1 and R g2 are the radii of gyration for primary particles (level 1) and aggregates (level 2) respectively. P1 and P2 are the slopes of the local power-laws.
Porod’s law (P1 = -4) is observed for primary particles with smooth surface at high q (q >
.02 Ǻ):
−4 2 I(q) = Bq where B = 2 πz(∆ρ) Sv (8)
Surface Area S = v Scattering Volume
10
z is the aggregation number of primary particles in the scattering volume, ∆ρ is the scattering length density difference between the scattering material and the background
(air, rubber, solvent), and S is the surface area of a primary particle. The Porod regime at high q is adjacent to a knee-like decay in scattering that represents the Guinier regime for primary particles. We observe Guinier’s law for both primary particles and their aggregates:
2 2 q Rgi I(q) = G exp − i 3 Gi is the Guinier prefactor for level i, 2 2 Gi = zi ()∆ρ Vi (9) i = 1 for primary particles i = 2 for aggregates
zi is the number density and V i is the volume of scattering entities at level i.
The two Guinier regimes are separated by a second power law. Its slope is indicated by P2. Scattering in this regime obeys a fractal dependence: [23, 24, 31]
−d f I(q) = B2q (10) df is the mass-fractal dimension (P 2 = -df) and B 2 is the power law prefactor for the aggregates. The power law prefactor, B 2 for a fractal cluster is calculated as:
G2d f d f B = Γ 2 d f R 2 g2 (11)
G2 is the Guinier prefactor defined in and Γ is the mathematical gamma function.[18, 24]
11
Level 1 of the unified fit describes the primary particle size in terms of the radius of gyration, R g1 , where,
1/2 Rsphere = (5/3) Rg1 (12)
The primary particle diameter can be calculated from the ratio of particle volume to
surface area as:
6V 6Q d v / s = = (13) S πB1
d v/ s is called the Sauter mean diameter. Q is the Porod invariant calculated for primary
particles and is given by: Q ∫ ∞q2I q q 2N 2V = 0 ( )d = 2π (∆ρ) (14)
Particle diameter in (13) can be written as:
6Q 6 ⋅ 2π2 N(∆ρ)V 6V d = = = v / s πB N 2 S S 1 ⋅π 2π (∆ρ) (15)
Equation 15 assumes primary particles to be spheres and expresses the particle diameter as the volume/surface ratio of the particles. d v/s is the ratio of the third moment to the second moment of a given particle size distribution, d v/s = M 3/M 2.[32, 33]
Aggregates can be described by the degree of aggregation, z. z is the number
density which is simply the number of primary particles that constitute the aggregate.
The degree of aggregation can be calculated from the unified fit parameters:[34]
G2 z = (16) G1
12
G1 and G 2 are the Guinier prefactors for level 1 primary particles and level 2
aggregates. The degree of aggregation has been used to relate silica structure with the
mechanical properties of silica-filled elastomer compounds.[3]
1.4 Reverse Monte Carlo methods: A brief review
Studies of the physical properties of materials are complemented by forward and inverse modeling techniques.[35] In forward modeling, numerical quantities are calculated from atomic/molecular configurations that are generated using basic equations.
The calculated numerical quantities are then compared with experimental data.[35] For example, the starting expression could be a numerical approximation of the forces that exist between atoms/molecules. By contrast, inverse modeling methods use the experimental data as the starting point. The aim of an inverse modeling process is to generate atomic or molecular configurations that give the best possible agreement with the experimental data within the experimental error when the configurations are subject to some constraints.[36] As an immediate consequence, there is no requirement for representation of inter-atomic forces since the expectation of the inverse modeling process is to obtain good agreement with experimental data.[36]
Reverse Monte Carlo (RMC) modeling techniques are variants of the standard
Metropolis Monte Carlo method (MMC).[37] RMC modeling is a general method of structural modeling based on experimental data.[38] The fundamental RMC algorithm has been described previously.[39-41] RMC methods have historically been used to complement data obtained from scattering and diffraction studies of ordered and disordered systems.[36] RMC methods when applied to scattering techniques are inverse
13
modeling techniques based on inverting the Fourier transform of the intensity of scattered radiation from the system under study.[27, 42] Atomic/molecular configurations are generated and their co-ordinates are varied in a random manner to obtain the best possible agreement with experimental data.[36, 38, 43] RMC methods are based on the principles of statistical mechanics.[35] Over time, an RMC simulation will tend to maximize the entropy in the configuration of atoms/molecules of the system being modeled. The results of an RMC simulation correspond to the most disordered atomic/molecular configurations that are consistent with experimental data. There may be a range of different configurations that maintain the same order of consistency with the data over varying degrees of disorder. It is important to choose as wide a range of data as possible in order to arrive at solutions that are appropriate not only in the quality of the fit to data but also in the amount of disorder in the atomic/molecular configuration.[35]
It cannot be overemphasized that the aim of RMC modeling is to simply produce structural models that are consistent with experimental data. There are no correct or unique solutions with RMC modeling.[36] The value of RMC modeling is in its ability to improve our understanding of the relationship between structure and a specific physical property of the system being studied. RMC modeling is complementary to experimental work. RMC modeling results may throw light on possible future experiments with respect to a particular problem.[36]
14
2. Materials and Methods:
2.1 Sample preparation and ultra small angle x-ray scattering (USAXS):
Aqueous colloidal silica suspensions from Nissan Chemical (SNOWTEX ®) and powder samples of silica from Dimona Silica Industries (DSI 1289 powder) and Rhodia
(Zeosil 1165 powder) were studied using ultra small angle x-ray scattering (USAXS).
The USAXS measurements of the silica samples were carried out in ambient conditions using a Bonse-Hart camera over a q range of 0.0002 - 0.4 Å -1 (Beam line ID-33,
Advanced Photon Source, Argonne National Laboratory). The x-ray beam dimensions were 0.4-mm x 2-mm. A special liquid sample cell supplied by Argonne National Lab was used for holding aqueous suspensions of Nissan SNOWTEX ®. Powder samples of
DSI 1289 and Rhodia Zeosil were spread between 2 pieces of Scotch tape and pressed in
a uniform film. Two pieces of tape were run as background. Corrections were applied to
the scattering data for sample transmission and for background scattering from the
tape/liquid cell and air. Indra software (available online at www.uni.aps.anl.gov) was
used to desmear the slit-smeared data. The desmeared data was analyzed with Irena
software provided by Argonne National Laboratory.
15
2.2 Structural modeling:
One of our primary goals was to measure and verify the degree of aggregation for the samples under study. We believe the degree of aggregation is a defining parameter that can help distinguish between different grades of colloidal silica. Filler-reinforced composites are usually crowded systems and their structural analysis is complicated. We propose a simple method for structural analysis that will enable us to draw a clearer picture of structure-property relationships in the future.
We analyze the scattering intensity in two ways. First, the scattering intensity is fit using the unified model of Beaucage.[17] The parameters obtained from the unified fit are then used to model aggregate structure in direct space by implementing reverse
Monte Carlo techniques.
2.2.1 Reverse Monte Carlo modeling:
We analyze the scattering signal from colloidal silica samples into its component structure factor and form factor. We extract real-space information from the so-called intra-aggregate structure factor [16] by fitting it using a reverse Monte Carlo (RMC) technique based on dynamic feedback between the structure studied in real and reciprocal space. The RMC algorithm was developed and coded on a computer running on an Intel®
Core™ 2 Duo – 1.66 GHz; dual core CPU. Our program has been developed using open source programming tools that are freely available online.[44-46] Our algorithm involves generating representative forms for aggregates that are subjected to modification in shape through random movements of primary particles. The structure factor after each move is calculated and compared with the experimental structure factor for the sample. Moves
16
that improve the quality of fit between the experimental and simulated structure factor are accepted. The fitting procedure is essentially automatic. Once the model parameters are defined, the simulation proceeds with the Monte Carlo steps indefinitely. If the algorithm converges the result is a structure that is compatible with the experimentally measured intensity.
2.3 Implementation of the RMC algorithm:
2.3.1 Generating the aggregate:
The first step of the RMC algorithm is to generate an aggregate of beads representing silica primary particles from the sample in terms of size distribution and number. The size of primary particles is calculated from their radii of gyration estimated by the Unified fit applied to the scattering curve. Particle size distributions can be obtained by fitting a range of bead sizes to a function describing the distribution of primary particle radii. From the values of G2 and G1 calculated from the unified fit
(equations 9 and 16) we obtain an estimate of the number of primary particles ( z)
constituting a representative aggregate.
The initial aggregate is created by adding particles one-by-one to a seed particle located at the origin. Each new particle is added at a random angle to a particle chosen randomly from those already present in the aggregate. For an aggregate comprised of i beads, our program performs a check at each addition to ensure that there are no collisions between the added ( i+1) th particle and the other i particles. Additions that pass this check are accepted and the process repeated till we obtain an aggregate composed of
17
z beads or primary particles. An illustration of initial aggregate structure created in the manner outlined above is shown in Figure 5.
Figure 5. Graphical representation of initial aggregate structure for an aggregate composed of 220 beads. The red beads are on the surface.
2.3.2 Executing Monte Carlo movement of beads:
The RMC algorithm modifies the aggregate shape by moving individual beads (or particles) to randomly chosen positions in the aggregate. The algorithm then compares the intensity of scattering from the modified aggregate with the measured experimental scattering intensity. It is important to maintain the structural integrity of the aggregate while moving the beads to different positions on or within the aggregate. Moving a bead should not cause the aggregate to break up into two separated parts. Therefore, only
18
beads situated on the aggregate surface (red beads in Fig. 5) can be selected for a move.
A surface bead is defined as a bead having only one adjacent neighbor. Our program identifies and records the number and indices of all surface beads. Each Monte Carlo step
(or move) consists of choosing a surface bead at random and placing it in contact with another randomly chosen bead at a random angle. Again, the check for collision with other particles in the aggregate has to be performed for the Monte Carlo step to be deemed physically acceptable. A prospective move becomes successful when it alters aggregate structure in such a way that it improves the fit between the intensity of scattering from the modified aggregate and the scattering data.
2.3.3 Fitting the data:
Each physically acceptable Monte Carlo step is evaluated by calculating the product of the spherical form factor P(q) for monodisperse spheres, the aggregate structure factor S(q) and the constant term in equation 1 and comparing it with the experimental intensity. Since the focus of our efforts is on determining aggregate morphology, we have fit the experimentally determined structure factor with S(q) for the
aggregate modified by a prospective move. We have developed the RMC program for
fitting data on the basis of equation 1 as follows:
19
I] Calculating form factor P(q) :
The normalized analytical form factor for scattering from a single solid sphere is described by equation (2) and depicted in Figure 6:
Figure 6. Normalized analytical form factor for a solid sphere (R = 8.5 nm) having uniform scattering length density .
20
Figure 6 shows the periodic variation in P(q) with the minima occurring for values of q that satisfy,
qR = tan qR ≅ (2k + 1)π 2
Maxima occur at,
qR ≅ 2π
The analytical form factor P(q) produces oscillations observed at high q in the intensity of scattering I(q) sim computed by the RMC program for the aggregate. In Figure
7, I(q) exp is the intensity being fit by the RMC program. I(q) exp is obtained from a 2 level
Unified fit applied to the experimental scattering data.
21
Figure 7. Effect of analytical form factor P(q) on intensity I(q) sim calculated by the RMC program .
We use the a two-level description of primary particles and aggregates following
Beaucage [24]:
22
d f 2 2 3 − q R erf ()qR 6 2 2 g [ g ] −q R I(q) = G1 exp + B1 exp exp + 3 q 3
p 3 − q 2 R 2 erf ()qR 6 G exp + B exp [] 2 2 3 q
A level 1 Unified fit to experimental data in the high q region (recall Figure 4) reproduces the experimental form factor scattering devoid of any oscillations observed in the high q region. The single level Unified fit also accounts for the constant term in equation 1. We modified the RMC routine to accept P(q) values from a level 1 Unified fit to the data. It is then no longer necessary to compute the analytical form factor of the sphere.
Oscillations produced by the analytical form factor are also eliminated. A representative plot is shown in Figure 8.
23
Figure 8. RMC fit using level 1 Unified fit in form factor P(q) calculation.
24
II] Structure factor S(q) :
The structure factor accounts for the correlations between the different primary particles in the aggregate. We use the Debye formula[15, 47] to calculate the structure factor for a single aggregate (intra-aggregate structure factor[15]):
1 N N sin qr S(q) = 1+ ∑∑ jk N j k≠ j qr jk where,
rjk = rj − rk r and r are the co - ordinates of the centers of the particles j k
From the above equation it can be seen that calculating the structure factor
involves calculating the distance between each particle in the simulated aggregate with
every other particle. The double summation translates into a nested double loop in the
program code. The program was considerably slowed down by the structure factor calculation, since S(q) had to be calculated for every prospective (or physically acceptable) move. To avoid this time consuming step, we calculated the contribution of the particle under consideration to the structure factor both before and after the move. It is the change in the position of the particle to be moved that changes the structure factor.
All the other particles in the aggregate retain their respective positions. Therefore, their distances with respect to each other remain the same as they were before the move is made. By factoring in only the effect of the change in position of a single particle to the structure factor we bypass the time-consuming double summation and reduce computing time significantly. (Figure 9)
25
Figure 9. Optimizing the structure factor S(q) calculation at each Monte Carlo step by calculating only the contribution of the move under consideration to the overall structure factor S(q).
All the beads other than the one to be moved retain their respective positions. Only the distances of each bead with respect to the bead that is moved ( r6 in the figure) do change.
At this stage, we are focused on understanding the aggregate morphology. The
RMC fitting routine calculates the structure factor for the aggregates and fits it to the
26
experimentally calculated structure factor as represented by an example shown in Figure
10.
Figure 10. Structure factor evolution over the duration of a simulation.
The interparticle structure factor S(q) simply describes how the differential scattering cross-section is affected by the interference effects in radiation scattered by different scattering entities. Aggregation leads to an increase in S(q) in the low q
27
regime[15] – from equation 3,
S(q → )0 = z where, is z is the aggregatio n number or the number of primary particles in the aggregate
At high q, S(q) tracks interactions between adjacent particles and S(q) tends to unity. Working under the monodisperse spherical assumption means two spherical beads/primary particles can only contact each other at one point. In Figure 10, the dip in
S(q) calculated by the RMC program is because of the initial assumption of
monodispersity in primary particle size. This assumption leads to poor data fitting in the
crossover regime as shown in Figure 11.
28
Figure 11. Effect of monodisperse spherical assumption on S(q) and I(q) calculated by RMC modeling program
III] Polydispersity:
We have accounted for the dip in the RMC calculated S(q) by introducing a polydispersity function in the program code for primary particle size. In physical terms we can attach smaller sized beads/primary particles in the space between beads of the same size.(Figure 12)
29
Figure 12. Polydisperse collection of beads
The polydispersity function has been coded to accept values from any probability distribution describing the distribution of primary particle size. The RMC program converts such a probability distribution to a frequency distribution. The frequency distribution of particles sizes is divided into an appropriate number of bins that allow for particles corresponding to the range of particle sizes in each bin to be represented in the
30
simulation. A representative plot of the experimental structure factor calculated from the
Unified fit intensity is shown in along with the RMC model fit for monodisperse beads and a uniform distribution of bead sizes in Figure 13 .
Figure 13. Effect of polydispersity on RMC calculated structure factor S(q).
The intensity calculated from the structure factor for polydisperse spherical beads/primary particles fits the Unified intensity well in the crossover region (0.001 Ǻ < q
< 0.01 Ǻ) as shown in Figure 14.
31
Figure 14. Unified fit intensity modeled with RMC for a uniform distribution of bead radii (0.5 Ǻ < R < 175 Ǻ), Aggregation number, z = 107.
32
IV] Chi square ( χ2) calculation:
The intensity calculated by RMC modeling is compared with the experimental intensity (the Unified fit intensity in our case) at each Monte Carlo step. The comparison between the RMC calculated intensity and experimental intensity is made by calculating
χ2 as:
2 I(q) − I(q) 2 1 exp sim χ = ∑ where, N i I(q)exp
I(q)exp is the experiment al intensity
I(q) sim is the RMC calculated intensity
Moves that decrease χ2 are accepted. We have developed a criterion that allows χ2 to fluctuate within a user-specified limit and probability. This strategy allows the RMC program to search for the global minimum during simulation without getting stuck in local minima. Figure 15 and Figure 16 show the chi square ( χ2) calculated for an
experimental data-set on Nissan silica particles. A static χ2 calculation restricts the simulation’s search for the best fit and the simulation stagnates after just 500 moves.
Allowing for fluctuation in the χ2 term lets the simulation run longer and ultimately a better fit is obtained.
33
Figure 15. Chi square ( χ2) with no fluctuations allowed.
Figure 16. Chi square ( χ2) calculated with fluctuations allowed.
34
V] Exporting data:
The primary objective of our project is to develop a three dimensional (3D) visualization of small-angle scattering data. We have developed a simple program using
Visual Python[44] to develop 3D visualizations of silica aggregates. Our Visual Python program takes as input the aggregation number of beads/primary particles (N), the co- ordinates of each bead, bead diameter, and number of neighbors for each bead along with surface bead information. The input for visualizing the RMC results is actually exported by the RMC program as a text (.txt) file. In addition, the intensity curves calculated by the RMC routine are exported in comma separated value (.csv) format for plotting purposes. Lastly, parameters of interest such as the radius of gyration (R g) of the
aggregate and the maximum end to end distance within the aggregate are exported along
with χ2 values for each move as another .csv file.
All the information described above can be exported at user-specified intervals.
2.4 Transmission Electron Microscopy (TEM):
A Phillips CM20 transmission electron microscope operating in high tension at 200 kV was used to image the samples. Images were collected at different magnifications ranging from 175 kx to 275 kx up to a resolution of 20 nm.
35
3. Results and discussion:
3.1 USAXS Analysis:
USAXS curves of the samples from Dimona Silica Industries - DSI (DSI pH6,
DSI 1289), Goodyear (GY 5011) and Nissan Chemicals (Snowtex ® ST-PS-SO) are
shown in Figure 17. Solid lines are the fits obtained by applying the Unified model. The
parameters obtained from the Unified model are listed in Table I.
Figure 17. USAXS curves of colloidal silica samples:(a) DSI series: DSI pH6 and DSI 1289, (b) Goodyear: GY5011, (c) Nissan Snowtex. Flat background was subtracted to emphasize the power-law profiles at high q. Data have been scaled vertically to match in the high q region for comparison.
36
Table I: Unified fit parameters obtained for Dimosil grades (DSI pH6, DSI 1289),
® Goodyear sample (GY5011) and Nissan Snowtex colloidal silica.
Level 1 DSI pH6 DSI 1289 GY 5011 Nissan Snowtex ST-PS-SO Rg (Å) 156 180 147 138 P -4 -4 -4 -4 G 534 2985 2771 1080 B 2.47E-06 1.08E-05 2.16E-05 1.20E-05
Level 2 DSI pH6 DSI 1289 GY 5011 Nissan Snowtex ST-PS-SO Rg (Å) 2149 2818 3753 812 P -3 -2.5 -2.44 -1.68 G 8.67E+05 2.78E+06 5000130 15788 B 0.00028438 0.016012 0.020368 0.19998 Z 1625 933 1805 15
Primary particle size across all four samples is 155 Å on average within an error of 15%. Aggregate sizes are similar for the Dimosil and Goodyear grades. Nissan
Snowtex sample has smaller aggregates. Slope of the mass fractal domain varies from -
3.0 to -1.68 for the four samples. This observation means that we have diverse aggregate morphologies ranging from dense compact structures (DSI pH6) to a more open aggregate morphology indicated by the level 2 slope of -1.68 for Nissan Snowtex. The aggregation number, z is calculated from Equation 16. The four samples show wide ranging z values ranging from 1805 primary particles for GY 5011 aggregate to 15
primary particles per aggregate for Nissan Snowtex ®. From Table I, it can be seen that for
37
the similar sized primary particles there are different sized aggregates with differing degree of aggregation.
3.2 RMC Analysis:
USAXS data on samples that possess higher aggregate mass or degree of
aggregation usually require longer time to be fit by the RMC algorithm. Figure 18 shows
the trend in the average value of chi square for two different runs on the four sample
data-sets. Nissan Snowtex ® aggregates are made of just 11 to 15 primary particles. Thus, the RMC algorithm fits USAXS data on the Nissan sample relatively fast. Dimosil and
Goodyear grades have higher aggregation numbers and require significantly longer time to be fit by the program. The RMC simulation constructs aggregates from a different random seed for each run. Figure 19 shows representative visualizations for an aggregate at the beginning and the end of a simulation run.
38
Figure 18. Evolution of chi square over the number of Monte Carlo moves for DSI pH6, DSI 1289, GY 5011 and Nissan Snowtex.
Figure 19. Graphical representatations of aggregate structures.
Initial aggregate shown on the left hand side. Structure obtained after the RMC program fits the data is on the right
39
While the final aggregate structures will appear different depending on the initial condition and angle of view, their measured properties are similar across different runs.
Table II lists the maximum end-end distance calculated for the final aggregate
representing a good RMC fit to USAXS data on each of the samples.
Table II: Maximum end-end distance calculated for final aggregate for two different simulation runs on each sample
Maximum end-end distance run 1 Maximum end-end distance run Sample name (Å) 2 (Å) GY 5011 3314 2721
DSI 1289 2625 2620
DSI pH6 2107 2114
Nissan Snowtex 609 723
Gaussian size distributions for primary particle size corresponding to the level 1
Unified intensity of each sample were generated using Irena modeling tools developed at
Argonne National Laboratory. The RMC program chooses the diameters of primary
particles used to generate the aggregate from the particle size distributions measured
through Irena modeling software. Figure 20 shows a plot of probability distributions for
primary particle size for the four samples. The mean and width of the distributions are
listed in Table III .
40
Figure 20. Probability distributions of primary particle radii as estimated by Irena modeling of Unified fit to USAXS data.
41
Table III: Size distribution parameters for primary particles:
Sample name mean(Å) width(Å) GY 5011 119 45
DSI 1289 155 40
DSI pH6 107 55
Nissan Snowtex 75 54
We compare the mean intensity calculated through RMC modeling for each of the
samples with respective Unified fit intensities in Figure 21.
42
Figure 21. Fits obtained from RMC modeling of Unified intensity for DSI pH6, DSI 1289, GY 5011 and Nissan Snowtex®
Figure 21 shows excellent agreement between the intensity calculated through
RMC modeling and the Unified fit intensity. Thus, the degree of aggregation calculated
from equation 16 is shown to be a reasonable estimate of the number of primary particles
in an aggregate.
43
The aggregate morphology for the different samples can be visualized from the aggregate snapshots shown in Figure 22, Figure 23, Figure 24 and Figure 25. Red beads indicate primary particles on the surface of the aggregate. Images are not scaled.
Figure 22. Aggregate morphology visualization: Nissan Snowtex "String of pearls", slope of mass fractal domain P = - 1.68.
Figure 23. Aggregate morphology visualization: GY 5011, slope of mass fractal domain P = -2.7.
44
Figure 24. Visualizing aggregate morphology for DSI 1289, mass fractal dimension as obtained from Unified fit is -2.53.
Figure 25. Visualizing aggregate morphology of DSI pH6 through RMC modeling. Mass fractal dimension obtained from Unified fit is - 3.0 indicating dense compact aggregates
From the aggregate snapshots, we confirm that RMC modeling captures the essential features of aggregate morphology in terms of the aggregation number ( z) and
the slope of the mass fractal domain as calculated from the Unified fit.
45
Another important result obtained from RMC modeling is that it gives a qualitative confirmation of the correct radius of gyration for primary particles. In the case of DSI pH6, initial attempts at applying the Unified fit to USAXS intensity in the high q and cross-over region suggested a broad distribution of primary particle sizes with a relatively large radius of gyration of 458 Å. Fitting for a broad distribution and large radius of gyration yielded low aggregation number (z = 96). RMC simulations fit the
Unified intensity for the low aggregation number. However, the visualization showed a much more open aggregate morphology than would be expected for mass fractal domain with slope -3.0 (see
Figure 28). Fitting the USAXS intensity with a smaller radius of gyration for primary particles gave a much higher aggregation number (z = 1624) (see Figure 26).
Resulting visualizations gave a much denser cluster of primary particles in accordance with a slope of -3.0 (Figure 27).
46
Figure 26. Unified fit to DSI pH6 for smaller sized primary particles (Rg1 = 155 Å).
Figure 27. Visualizing aggregate morphology of DSI pH6 through RMC modeling. Mass fractal dimension obtained from Unified fit is - 3.0 indicating dense aggregates.
47
Figure 28. Aggregate visualization for incorrectly calculated radius of gyration (Rg = 458 Å) of primary particles - DSI pH6.Through RMC visualization, we are able to observe an open fractal aggregate which is inconsistent with the calculated slope of the mass fractal domain (-3.06). We understand that the intensity curve was incorrectly interpreted leading to an erroneous assumption of broad particle size distribution along with large Rg for primary particles.The aggregates are actually composed of smaller primary particles that form a dense cluster that is more representative of the slope in the mass fractal domain.
It is interesting that a similar model of closely packed dummy atoms/ beads has been used to study the structure of biological macromolecules in solution at low resolution. [48] Svergun et al have developed a simulation method wherein a multiphase model for a particle composed of dummy atoms is generated by assigning individual atoms to a specific phase or the solvent.[48, 49] Their method uses a simulated annealing approach to search for those atomic configurations that fit the data while minimizing the interfacial area. The protein-RNA distribution in the ribosome of the E. coli bacterium
48
has been evaluated using ab-initio methods of shape determination using small-angle
scattering data.[50]
49
3.3 Transmission Electron Microscopy Results:
The aggregate morphology as visualized through RMC simulation has been verified through Transmission Electron Microscopy (TEM). (Figure 29, Figure 30, Figure
31 and Figure 32)
Figure 29. Representative TEM image of DSI 1289 at 50 nm resolution
Figure 30. Representative TEM image of Nissan Snowtex collodial silica at 20 nm resolution. Relatively monodisperse primary particles are connected in a linear fashion that is correctly visualized through RMC simulation.
50
Figure 31. RepresentativeTEM image of Goodyear colloidal silica sample GY 5011 at 50 nm resolution. Sample appears as a ramified structure corroborated through simulation.
Figure 32. Representative TEM image of DSI pH6 silica at 50 nm resolution.
51
3.4 Analysis of colloidal silica dispersed in rubber:
We performed RMC analysis on USAXS data collected on Zeosil 1165, a
commercial available grade of silica from Rhodia Silicea. USAXS measurements were
made on powders spread between Scotch Tape as well as silica compounded with
Styrene-Butadiene Rubber (SBR) samples. Radius of gyration of the primary particles
was 104 Å. Radius of gyration for aggregates in powder sample is 1540 Å. Radius of
gyration of silica aggregates dispersed in elastomer is relatively smaller (1244 Å). Slope
of the mass fractal domain changes from -2.01 for powder sample to -2.51 for the silica
aggregates inside SBR.
Figure 33 shows the fit obtained from RMC modeling of the USAXS data.
Aggregation number ( z) as determined from Unified model is confirmed through RMC simulation.
52
Figure 33. RMC fit to USAXS data for Zeosil 1165 powder and for Zeosil 1165 compounded in Styrene-Butadiene Rubber.
53
Figure 34 and Figure 35 show visualizations of Zeosil silica aggregates in powder and in SBR. The increase in slope of the mass fractal domain is reflected in the relatively dense aggregate visualizations for the silica dispersed in rubber.
Figure 34. RMC visualization of aggregate structure for Zeosil 1165 silica powder sample.
Slope of mass fractal domain, P = -2.1 indicating ramified aggregates with relatively open morphology.
54
Figure 35. Aggregate visualization for Zeosil 1165 dispersed at 10phr in elastomer.
We have shown that small angle scattering based RMC analysis is a useful tool toward understanding the process of dispersion of filler compounds in the elastomer/polymer matrix. Recent work on the morphological characterization of carbon black (CB) in different polymer matrices raises important questions about the three dimensional organization of the elemental structural unit in hierarchically organized fillers.[51] Hashimoto, Koga et al[51] show that the smallest structural unit of CB filler obtained after sonication in toluene is itself a spherical aggregate of about 9 primary CB particles that are fused together. Spherical aggregates of primary particles form higher order mass fractal ( D = 2.6 in toluene) structures. The aggregates described above form dispersible units that are bound by polymer chains in the matrix. The dispersible units of
CB filler in PI and SBR matrices were modeled as ellipsoids of different size and shape for the respective polymer matrices. The size of the dispersible unit for CB filler compounded in polyisoprene (PI) was found to be higher than the size for the dispersible
55
unit in styrene-butadiene rubber (SBR). The larger size of dispersible unit in PI raises the upper cutoff length of the mass fractal structure in PI while retaining the mass fractal dimension of CB filler dispersible units in both PI and SBR matrices. The authors’ observations were a result of replacing the Guinier function in the Unified model with form factors of structural units corresponding to a particular level. The results discussed above point to a new approach toward understanding filler structure beyond the traditional model of primary particles, hard aggregates and soft, weakly bonded agglomerates. We propose to include similar models of elementary structural units in future analysis and study.
56
3.5 Relevance of RMC and small angle scattering analysis to reinforcement studies:
The reinforcement of an elastomer by fillers such as colloidal silica or carbon black depends on the extent to which the filler disperses in the elastomer. The filler dispersion is a function of filler size and structure and the interaction between the filler and elastomer.
Filler size and structure:
The presence of filler at large length scales in the matrix should be avoided as the
larger filler units will acts as point of stress concentration and reduce the mechanical
strength of the composite. At sizes exceeding 10 microns, fillers tend to have a negative
effect on polymer/elastomer reinforcement. In order to ensure effective dispersion, the
filler surface area should be maximized to facilitate interaction between the matrix and
the filler. Smaller sized particles have greater surface area and particles in the range of 10
to 100 nm are found to contribute positively to reinforcement of the matrix. Witten,
Rubenstein and Colby (WRC), first studied the reinforcement of elastomers by filler
aggregates on the basis of fractal concepts.[8] They reported that the strength of the
elastomer-filler composite is determined by the deformability of the filler aggregates.
57
They found that the bending modulus of an aggregate decreases with the aggregate size as:
a 3+C G ~ R where, G is the bending modulus of the aggregate a is the size of primary particles R is the aggregate size C is the connectivi ty exponent
The critical size ζ of an aggregate is obtained by equating the aggregate modulus with the
modulus of the elastomer matrix.
()3+C −1 a ζ ~ G0
Aggregates at size scales greater than ζ do not contribute to reinforcement as they are softer than the elastomer matrix. Typical values of ζ are between 100 to 200 nm.
Beyond a critical concentration, the filler phase in the composite becomes continuous and is the primary stress bearing component. The modulus of the composite above the critical concentration is governed by silica structure.
ζ D−3 φ * = a
D is the aggregate fractal dimension. Typical values of φ * are ~0.2.
Extremely fine particles may present problems during processing as the particles at very small length scales will increase the viscosity during compounding operations.
58
Agglomerate strength:
The surface energy of the filler particles determines the strength of the structures at different length scales. Agglomerates exist at higher length scales and usually break down easily as compared to smaller aggregates that do not break down into smaller sizes for the same input energy. It is important to minimize the strength of the agglomerates in the system in order to ensure effective dispersion.
Primary particle size distribution:
The particle size distribution should be as monodisperse as possible so as to
obtain reinforcing aggregates. Greater number of particles that are larger in size will
impede dispersion and reduce reinforcement.
Relating structure and surface area:
As mentioned above, higher particle surface areas are suitable for reinforcing filler applications within limits. Surface area is dependent on the size of the primary particles and the porosity or the morphology of the aggregates and agglomerates. Fillers that contain aggregate and agglomerate structures that are highly porous (high structure) have lower bulk density, higher surface area and are suitable for elastomer reinforcement.
We extract size and morphology parameters described above by applying the
Unified and RMC modeling techniques to small angle scattering data. The reinforcing
mechanism of fillers at nano-scale can be explained on the basis of the fractal nature of
filler aggregates using models such as the WRC model discussed above.
59
3.6 Uniqueness of solution:
It is evident from the RMC procedure outlined above that many equivalent
realizations of aggregate structure can be created for a particular USAXS data-set. The
aggregate structure is different based on the initial condition and the number of Monte
Carlo moves made during the fitting routine. However, given that the final aggregate
forms have measurable properties (maximum end-end distance, eccentricity etc.) that lie
within reasonable error from a global mean, the RMC-generated aggregate morphologies
for a particular USXAS data-set can be considered as representative of that particular
class of aggregates. However, as mentioned before, there no unique solution exists.
60
3.7 Accounting for polydispersity among aggregates:
Our code accounts for polydispersity in the sizes of the primary particles that
constitute a single aggregate. Any scattering experiment on materials exhibiting
hierarchical/fractal organization will involve an ensemble of polydisperse aggregates, i.e.,
the sample under study will comprise of aggregates or clusters of different sizes.
Polydispersity is observed in aggregate size because the actual process of aggregation in
colloids and aerosols gives a finite distribution of cluster sizes.[52] This polydispersity
assumes significance for analysis involving use of a single - cluster structure factor. The shape of the observed structure factor changes with polydispersity in aggregate size.[52]
The structure factor for polydisperse aggregates is then different than that of the single cluster structure factor that we compute in RMC analysis. The single cluster structure factor S(q) in the aggregate power law regime is given by:[52]
−D S(q) = C(qR g ) , qRg >> 1
For a single aggregate, the high qR g power law regime contains information on the fractal
dimension. The value of coefficient C has been estimated by Sorensen and Wang [52]to be 1.0 ± 0.5 to describe the power-law regime for the single aggregate structure factor in range of 1.7 < D < 2.1. The dependence of the structure factor on the power-law negative fractal dimension is retained in polydisperse systems of aggregates within limits of polydispersity.[53] The value of the coefficient C is modified by introducing
polydispersity in an ensemble of aggregates. When the value of C is significantly different from unity the use of a single aggregate structure factor is erroneous.
Calculating the correct C-value provides a measure of the width of aggregate size
distribution.[52] While our simulations do not account for polydispersity in the aggregate
61
ensemble, proper analysis must include effects of polydispersity and not just rely on the single cluster structure factor.
62
4. Conclusions and future work:
We have demonstrated a simple technique to fit the Unified intensity of scattering
from aggregated structures with our reverse Monte Carlo program. Our program is
reasonably fast, accurate and capable of producing structures that are similar to each
other for a given data-set. We have analyzed the Unified intensity on four different
sample sets that represent a wide range of aggregation numbers, aggregate sizes and
aggregate morphologies. RMC fitting of sample data-sets confirms the aggregation
number as determined through Unified fit analysis for all the samples. Our program fits
the Unified intensity of these different sample data-sets well and produces visualizations
that are compatible with the slope of the mass fractal domain as observed through the
Unified fit. The complementary nature of Unified modeling and RMC modeling allow us
to take advantage of the generality and simplicity of the Unified model while adapting
our simulations for each specific case. Representative TEM images corroborate the
structures seen through simulation
We have shown how RMC visualization can be more than just a qualitative tool
for estimating correct primary particle size and aggregation number from the Unified fit.
RMC visualizations can be extended to aggregates dispersed in a matrix as shown by our
work on simulating silica aggregate structure in elastomer.
While our eventual goal is to use RMC-Unified based analysis to construct a real
space picture of aggregate reinforcement in rubber, an exact explanation of the effect of
filler morphology on elastomer reinforcement is beyond the scope of this thesis. The
differences in structure of the samples studied over different length scales make it
63
difficult to correlate the mechanical properties of rubber with a particular model of reinforcement.
Light scattering analysis is a promising avenue for future research. In the future we intend to construct a model that can account for interacting aggregates by performing multi-body simulations. It is of interest to verify if simulation based modeling will lead to a solution of monodisperse aggregate or if it will require an account of polydispersity.
64
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APPENDICES:
Appendix A: Structure factor calculation:
N -i q⋅r A( q) =∑ A( q)e j j N N N N iq⋅r -i q⋅r I( q) = A( q)∑ e-i q⋅rk A( q)∑ e j = A2(q) ∑∑ e jk k j j k
A(q) is the amplitude of scattering from one particle A2(q) = v2∆ρ2P(q) scattering cross section from 1 bead we want cross section per unit volume for scattering from N- bead cluster
2 2 N N I (q) v ∆ρ P(q) -i q⋅r N = ∑ ∑e jk scattering from 1 aggregate with N beads V V j k N N N 1 -i q⋅r = v2∆ρ2P(q) ∑∑e jk V N j k N N I (q) nNv 1 -i q⋅r Nn = v∆ρ2P(q) ∑∑e jk n aggregates of N beads V V N j k = averaging over many realizations N N 1 -i q⋅r = φv∆ρ2P(q) ∑∑e jk N j k N N 1 -i q⋅r φv∆ρ2P(q)1 + ∑ ∑e jk N j k≠ j 1 N N sin qr = φv∆ρ2P(q)1 + ∑ ∑ jk N qr j k≠ j jk
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Appendix B: Code for Reverse Monte Carlo simulation
#include
time(&t); srand((long)t); //seeding random number generator with time since epoch. n_agg=220; num_bins=10; read_sizes(); build_sizes(); read_curve();
constant=(phi)*(4/3)*(PI)*(MAX_SIZE/2)*(MAX_SIZE/2)*(MAX_SIZE/2); constant/=pow(10,24); constant*=drho2; constant=1; printf("%E\n",constant); getchar();
init(); printf("\nDONE WITH INITIAL STRUCTURE. CALLING DO_SIM\n"); num_moves=do_sim(); printf("\nDONE WITH DO_SIM AFTER %d MOVES\n",num_moves); getchar();
return 0; }
/*Function do_sim * takes nothing, returns number of moves after which simulation done. * calls do_move()
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*/ int do_sim() { /* some mechanism by which we can check whether to stop or not
*/ FILE *fp,*fq; int i=0,last_ret_value; int tot_retry=0; char filename[256];
printf("Printng '#' per 1 moves...\n");
for(i=1;i printf("\nTook total %d retries!!!\n",tot_retry); return MAX_MOVES; //FIXME } /*Function do_move * returns fit-value after the move * alters beadsinfo. * */ int do_move(int move) { int count_surface_beads=0,i,j=0,collided; int index_tomove; int index_surface_beads[MAX_BEADS]; int stick_to_index; 70 int retry=0; beadinfo tempbead,tempbead2; double angxy,angz; curvedata expcurve; for(i=0;i angxy=(long double)deg_to_rad(rand()%(_RES_ANGLE)); angz =(long double)deg_to_rad(rand()%(_RES_ANGLE)); tempbead.pos.x = beads[stick_to_index].pos.x + ( ( (beads[stick_to_index].diameter/2) + (tempbead.diameter/2) )*sin(angz)*cos(angxy)); tempbead.pos.y = beads[stick_to_index].pos.y + ( ( (beads[stick_to_index].diameter/2) + (tempbead.diameter/2) )*sin(angz)*sin(angxy)); tempbead.pos.z = beads[stick_to_index].pos.z + ( ( (beads[stick_to_index].diameter/2) + (tempbead.diameter/2) )*cos(angz)); collided=0; //set flag saying bead is 'not collided state'. for(i=0;i 71 } } if(collided==0) { //successful move as far as positioning new particle is concerned. // add check here. // build simcurve here build_simulation_data(move,index_tomove,beads[index_tomove],tempbead); if(chisq[(move%2)] < min_chisq) { min_chisq=chisq[(move%2)]; } if( (chisq[(move%2)] for(i=0;i break; //from while 1 } else { // 72 retry++; } } else { retry++; printf("\nBAD MOVE %d!!! BEAD COLLIDED!!!!!!!!!\nTrying again!\n",move); } } return retry; } /*Function init * generates aggregate */ void init() { int n,i,stick_to_index,collided,num_moves,beads_diameter_index; double angxy,angz; beadinfo tempbead; char filename[256]; n_agg = 220; beads[0].pos.x=beads[0].pos.y=beads[0].pos.z=0; beads_diameter_index=rand()%n_agg; beads[0].diameter=beads_diameter[beads_diameter_index]; beads_diameter[beads_diameter_index]=-1; beads[0].isSurface=1; beads[0].count=0; beads[0].stuck_to_index=0; n=1; while(n //we add to the beads array here, bead by bead. //1. choose particle randomly to attach new particle to. stick_to_index=(rand()%n); //2. choose direction (from center of stick_to_index bead ) and size of new bead. angxy=(long double)deg_to_rad(rand()%(_RES_ANGLE)); angz =(long double)deg_to_rad(rand()%(_RES_ANGLE)); do { beads_diameter_index=rand()%(n_agg); tempbead.diameter = beads_diameter[beads_diameter_index]; 73 }while(tempbead.diameter==-1); // from this calculate center of tempbead. tempbead.pos.x = beads[stick_to_index].pos.x + ( ( (beads[stick_to_index].diameter/2) + (tempbead.diameter/2) )*sin(angz)*cos(angxy)); tempbead.pos.y = beads[stick_to_index].pos.y + ( ( (beads[stick_to_index].diameter/2) + (tempbead.diameter/2) )*sin(angz)*sin(angxy)); tempbead.pos.z = beads[stick_to_index].pos.z + ( ( (beads[stick_to_index].diameter/2) + (tempbead.diameter/2) )*cos(angz)); //3. now check whether this 'tempbead' collides with any 'other' existing bead. collided=0; //set flag saying bead is 'not collided state'. for(i=0;i tempbead.isSurface=1; beads[stick_to_index].count++; tempbead.count=1; tempbead.stuck_to_index=stick_to_index; if(beads[stick_to_index].count>1) beads[stick_to_index].isSurface=0; printf("\nAttaching new particle number %d to particle number %d\n,angxy=%lf,angz=%lf ",n,stick_to_index,angxy,angz); print_bead(tempbead); 74 // break; //4. Put tempbead into beads array. beads[n]=tempbead; //5. Invalidate the chosen diameter in the beads_diameter array beads_diameter[beads_diameter_index]=-1; n++; // WE have one more bead!! } // build_simulation_data(0); // populate the simcurve structure. // get chi_sq_value for move 0 // set_Pq(); for(i=0;i export_info(0,0); export_info(0,1); export_info(0,2); for(i=0;i void read_curve() 75 { FILE *fp; int i,j; char line[1024]; for(i=0;i i=0; fp=fopen("C:\\zeosil silica\\data-prashant-sim3-unified.csv","r"); // fp=fopen("C:\\data-prashant-toluene-not-unified.csv","r"); if(fp==NULL) { printf("\nError Opening File to Read Curve \n"); getchar(); exit(1); } while(!feof(fp)) { fscanf(fp,"%lf,%lf,%lf\r",&curve[i].intensity,&curve[i].q,&curve[i].Pq); printf("%lf\t%lf\t%lf\t\n",curve[i].intensity,curve[i].q,curve[i].Pq); i++; if(i>=MAX_POINTS) break; } fclose(fp); i=0; while((curve[i].intensity!=-1)&&(i 76 for(i=0;i i=0; fp=fopen("C:\\zeosil silica\\size distr.csv","r"); if(fp==NULL) { printf("\nError Opening File to Read Curve \n"); getchar(); exit(1); } while(!feof(fp)) { fscanf(fp,"%lf,%lf,%lf\r",&a,&b,&c); sizecurve[i].diameter=b*2; sizecurve[i].pdfval=c; sizecurve[i].fdval=c*n_agg; i++; if(i>=MAX_POINTS) break; } fclose(fp); i=0; while((sizecurve[i].diameter!=-1)&&(i double get_max_distance() { int i,j; double dist_max=0,dist; for(i=0;i 77 { // if(i!=j) // { dist=dist_between_beads(beads[i],beads[j]); if(dist_max==0) dist_max=dist; else if(dist>dist_max) { dist_max=dist; } // } } } return dist_max; } void export_info(int move,int export_type) { //Add switchfor different types of export here. char filename[256]; FILE *fp; int i; switch(export_type) { case 0: // Creating NEW File export_beads_move_move-number.txt, used for Python Visualization. sprintf(filename,"%sexport_beads_move_%d.txt",dir,move); fp=fopen(filename,"w"); fprintf(fp,"%d\n",n_agg); for(i=0;i 78 fprintf(fp,"%.15f,%.15f,%.15f,%.15f\n",simcurve[i].q,simcurve[i].intensity,simcurve[i].S q,simcurve[i].Pq); fclose(fp); break; case 2: // Appending to Existing FILE export_simdata.csv, used for exporting Simulation Data // file format : // for each exported move : move_number, chi_sq_value, max_distance, radius_of_gyration \n for the simulated structure. // Keeping this as seperate case as this distances info is not necessary for running the simulation, so we can not call for // ths export if we want to speed up things set_distances(); sprintf(filename,"%sexport_simdata.csv",dir); fp=fopen(filename,"a"); if(fp==NULL) { printf("XXXXXXXXXXXXXXXXX Unable to open file %s!! Mybe open already?? Exiting for now.\n",filename); getchar(); exit(1); } fprintf(fp,"%d,%.15f,%.15f,%.15f\n",move,chisq[move%2],distances.dist_max,distances. Rg); fclose(fp); break; case 3: // Debug Sq Verify Export sprintf(filename,"%scheck_Sq_move_%d.csv",dir,move); fp=fopen(filename,"w"); fprintf(fp,"q,Sq_fast(q),Sq_slow(q)\n"); for(i=0;i fprintf(fp,"%.15f,%.15f,%.15f\n",curve[i].q,tmpcurve[i].Sq,chkcurve[i].Sq); fclose(fp); break; default: break; } } 79 double get_radius_of_gyration() { // Rg is rms of the distance between all beads. int i,j; double Rg,dist_sum=0,dist; for(i=0;i double _get_Sq_fast_(double q, int index_bead_under_move, beadinfo bead_before_move, beadinfo bead_after_move, double Sq_before_move) { int i; double contribution_of_bead_before_move=0,contribution_of_bead_after_move=0; double x,dist,sum_before_move,sum_after_move,Sq; sum_before_move = (Sq_before_move-1)*n_agg; for(i=0;i dist=(dist_between_beads(beads[i],bead_after_move)); x=q*dist; contribution_of_bead_after_move+=(double)(sin(x)/x); } } sum_after_move=(sum_before_move- 2*contribution_of_bead_before_move+2*contribution_of_bead_after_move); Sq=1+(sum_after_move/n_agg); return Sq; 80 } double _get_Sq_(double q) { int i,j; double Sq,sum=0,curr,dist,x; for(i=0;i sum+=(double)(sin(x)/(x)); } } } Sq=1+(sum/n_agg); return Sq; } double _get_Pq_(double q) { int i; for(i=0;i return i; } double get_chi_sq() 81 { double chi_sq=0; int i; for(i=0;i void build_simulation_data(int move, int index_bead_under_move, beadinfo bead_before_move, beadinfo bead_after_move) { // populate the tmpcurve structure. // get chi_sq_value for this move and store in the chisq array at move%2 int i; // populate the tmpcurve structure. for(i=0;i // populate the distances structure for(i=0;i 82 dist=dist_between_beads(beads[i],beads[j]); dist_sum+=dist*dist; if(dist_max==0) dist_max=dist; else if(dist>dist_max) { dist_max=dist; } } } dist_sum=sqrt(dist_sum); distances.dist_max=dist_max; distances.Rg=dist_sum/n_agg; } void debug_fn_verify_Sq(int move,int index_bead_under_move, beadinfo bead_before_move,beadinfo bead_after_move) { int i,j,k; beads[index_bead_under_move]=bead_after_move; for(i=0;i } beads[index_bead_under_move]=bead_before_move; if(move%EXPORT_INFO_AFTER_MOVES==0) export_info(move,3); } void build_sizes() { int i,j,start_bead_index=0; int beads_in_bin,total_beads_in_bins; int bin_start_index,bin_end_index; double bin_sum_fdval; int range; int bin_size_in_points= (int)(sizedata_points/num_bins); 83 if(num_bins>sizedata_points) num_bins=sizedata_points; //initialize beads_diameter, todifferentiate when used actually. for(j=0;j //code for num_bins for(i=0;i for(j=start_bead_index;j if(sizecurve[bin_start_index].diameter beads_diameter[j]+=(rand()%range); } start_bead_index+=beads_in_bin; } //code for num_bins ends. //code for remainder bin bin_sum_fdval=0; 84 for(i=bin_size_in_points*num_bins;i for(j=start_bead_index;j //code for remainder beads mean_diameter=0; for(i=0;i for(j=0;j getchar(); 85 } double get_diameter() { int sizepoint; double probability_check; sizepoint=rand()%n_agg; return beads_diameter[sizepoint]; } void set_Pq() { int i,j; double ans=0,x; for(i=0;i 86