AGGREGATION IN COLLOIDS AND AEROSOLS by FLINT G. PIERCE B.S., Kansas State University, 1994 B.S., Kansas State University, 1994 AN ABSTRACT OF A DISSERTATION submitted in partial fulfillment of the requirements for the degree DOCTOR OF PHILOSOPHY Department of Physics College of Arts and Sciences KANSAS STATE UNIVERSITY Manhattan, Kansas 2007 Abstract This work is the result of a wide range of computer simulation research into the aggregation behavior of dispersed colloidal and aerosol particles in a number of different environments from the continuum to the free-molecular. The goal of this research has been to provide a bridge between experimental and theoretical researchers in this field by simulating the aggregation process within a known model. To this end, a variety of inter- particle interactions has been studied in the course of this research, focusing on the effect of these interactions on the aggregation mechanism and resulting aggregate structures. Both Monte Carlo and Brownian Dynamics codes have been used to achieve this goal. The morphologies of clusters that result from aggregation events in these systems have been thoroughly analyzed with a range of diverse techniques, and excellent agreement has been found with other researchers in this field. Morphologies of these clusters include fractal, gel, and crystalline forms, sometimes within the same structure at different length scales. This research has contributed to the fundamental understanding of aggregation rates and size distributions in many physical system, having allowed for the development of improved models of the aggregation and gelation process. Systems studied include DLCA and BLCA in two and three dimension, free-molecular diffusional (Epstein) system, selective aggregation in binary colloids, ssDNA mediated aggregation in colloidal systems, and several others. AGGREGATION IN COLLOIDS AND AEROSOLS by FLINT G. PIERCE B.S., Kansas State University, 1994 B.S., Kansas State University, 1994 A DISSERTATION submitted in partial fulfillment of the requirements for the degree DOCTOR OF PHILOSOPHY Department of Physics College of Arts and Sciences KANSAS STATE UNIVERSITY Manhattan, Kansas 2007 Approved by: Major Professor Amit Chakrabarti Abstract This work is the result of a wide range of computer simulation research into the aggregation behavior of dispersed colloidal and aerosol particles in a number of different environments from the continuum to the free-molecular. The goal of this research has been to provide a bridge between experimental and theoretical researchers in this field by simulating the aggregation process within a known model. To this end, a variety of inter- particle interactions has been studied in the course of this research, focusing on the effect of these interactions on the aggregation mechanism and resulting aggregate structures. Both Monte Carlo and Brownian Dynamics codes have been used to achieve this goal. The morphologies of clusters that result from aggregation events in these systems have been thoroughly analyzed with a range of diverse techniques, and excellent agreement has been found with other researchers in this field. Morphologies of these clusters include fractal, gel, and crystalline forms, sometimes within the same structure at different length scales. This research has contributed to the fundamental understanding of aggregation rates and size distributions in many physical system, having allowed for the development of improved models of the aggregation and gelation process. Systems studied include DLCA and BLCA in two and three dimension, free-molecular diffusional (Epstein) system, selective aggregation in binary colloids, ssDNA mediated aggregation in colloidal systems, and several others. Table of Contents Page - Section xi List of Figures xxiii List of Tables xxiv Acknowledgements 001 1. Introduction 001 1.1. Aggregation Defined 001 1.2. Scope of Thesis 003 1.3. Organization of Thesis 004 1.4. Interactions 006 1.5. Current Work 007 2. Regimes of Particle Motion 007 2.1. Drag 008 2.2. Continuum Regime 008 2.3. Free-Molecular Regime 009 2.4. Knudsen Number 009 2.5. Diffusive vs. Ballistic Motion 013 2.6. Calculating Drag 014 2.7. Regime Crossover 016 2.8. Special Considerations 017 2.9. Nearest Neighbor Distance (R nn ) 019 2.10. Diffusion Limited Cluster Aggregation (DLCA) 020 2.11. Ballistic Limited Cluster Aggregation (BLCA) 021 3. Interactions 021 3.1. van der Waals 021 3.1.1. Origin of vdw 022 3.1.2. vdW for Particles 023 3.1.3. vdW Dependence on Material v 025 3.1.4. vdW for the Medium 025 3.1.5. Size Scaling of vdW 026 3.2. Repulsive Potentials 026 3.2.1. Need for Repulsion 027 3.2.2. Lennard-Jones Potential 027 3.2.2.1. LJ Form 028 3.2.2.2. Origin of the Repulsive Term 029 3.2.2.3. Utility 029 3.2.3. Chemical Considerations and the DLVO potential 029 3.2.3.1. Surface Charging in a Liquid Medium 030 3.2.3.2. DLVO potential 032 3.2.3.3. Reaction Limited Cluster Aggregation (RLCA) 034 4. Simulation Methods 034 4.1. Code 034 4.2. Random Variable Assignment 037 4.3. Monte Carlo Simulations 037 4.3.1. Advantages and Disadvantages 038 4.3.2. Standard MC Aggregation Algorithm 038 4.3.2.1. Initial Placement 038 4.3.2.2. Periodic Boundary Conditions (PBC) 041 4.3.2.3. Link-Cell method 046 4.3.2.4. Cluster Lists 047 4.3.2.5. Unfolding 049 4.3.2.6. Cluster Motion 050 4.3.2.6.1. Ballistic 053 4.3.2.6.2. Diffusional 054 4.3.3. Metropolis MC 056 4.4. Brownian Dynamics (BD) 056 4.4.1 Molecular Dynamics 057 4.4.2. Initial Particle Velocities 058 4.4.3. Langevin Equation vi 059 4.4.4. Method and Implementation 064 5. Aggregation Theory 064 5.1. Smoluchowski (Coagulation) Equation 065 5.1.1. Kernels 067 5.1.2. Cluster Fragmentation 072 5.1.2.1. Examples 074 5.2. Fractal Aggregates 075 5.2.1. Fractal Dimension and Mass Scaling 078 5.2.2. Determination of Fractal Dimension 078 5.2.2.1. Ensemble Method 079 5.2.2.2. Onion-shell Method 080 5.2.2.3. Structure Factor Method 093 5.3. Gelation 095 5.3.1. Conditions for Gelation 095 5.3.1.1. Case 1 096 5.3.1.2. Case 2 096 5.3.1.3. Case 3 097 5.3.1.4. Case 4 098 5.4. Aggregation Kinetics 098 5.4.1. Kinetic Exponent from Homogeneity 099 5.4.2. Kinetics for Various Regimes 103 5.4.3. Kinetics for Fractal Aggregates 106 5.5. Cluster Size Distributions (CSD) 109 6. Results: Aggregation in Aerosols 110 6.1. Diffusion Limited Cluster Aggregation in Two-Dimensional Systems - Dilute to Dense 110 6.1.1. Introduction 110 6.1.2. Kinetics 113 6.1.3. CSD 114 6.1.4. Scaling form 116 6.1.5. Conclusion vii 117 6.2. Kinetic and Morphological Studies of DLCA in Three Dimensions Leading to Gelation - Simulation and Experiment 117 6.2.1. Introduction 117 6.2.2. Kinetics and Gelation 119 6.2.3. Df (Ensemble Method) 122 6.2.4. Df (Structure Factor Method) 122 6.2.5. Df Crossover in Gelled Clusters 124 6.2.5.1. Low fv (0.001) 124 6.2.5.2. Intermediate fv (0.010) 124 6.2.5.3. High fv (0.200) 125 6.2.6. Df (Perimeter Method) 125 6.2.6.1. Current Study 127 6.2.6.2. Experimental Study 129 6.2.7. Conclusions 130 6.3. Diffusion Limited Cluster Aggregation in Aerosols Systems with Epstein Drag 130 6.3.1. Introduction 130 6.3.2. Cluster-Dilute, Cluster-Dense, and Intermediate Systems 131 6.3.3. Regimes of Motion 133 6.3.4. Mobility Radius 134 6.3.5. Scaling of the Aggregation Kernel 135 6.3.5.1. Ballistic Regime 135 6.3.5.2. Diffusion Regime 141 6.3.6. Simulation Model and Methods 142 6.3.7. Cluster Projection and Mobility Radii Exponents 144 6.3.8. Simulation Results in the Epstein Regime 144 6.3.8.1. Fractal Dimension 147 6.3.8.2. Kinetics of Aggregation 149 6.3.8.3. Cluster Size Distributions 150 6.3.9. Conclusions viii 151 6.4. Motional Crossover from Ballistic to Epstein Diffusional in the Free- Molecular Regime 151 6.4.1. Introduction 151 6.4.2. Persistence Length and characteristic time 154 6.4.2.1. Spherical Particles 155 6.4.2.2. Fractal Aggregates 157 6.4.3. Rnn and Rnns 164 6.4.4. Kinetics 164 6.4.5. Simulation 164 6.4.5.1. Procedure 167 6.4.5.2. Morphologies/Fractal Dimension 171 6.4.5.3. Kinetics and Gelation 175 6.4.5.4. CSD 177 6.4.6. Conclusions 179 7. Results: Aggregation in Colloidal Systems 180 7.1. Selective Aggregation in Binary Colloids 180 7.1.1. Introduction 181 7.1.2. Model and Numerical Procedure 182 7.1.3. Results 182 7.1.3.1. Morphologies and Df 189 7.1.3.2. Kinetics 192 7.1.3.3. CSD 193 7.1.4. Conclusions 195 7.2. Aggregation-Fragmentation in a Model of DNA-Mediated Colloidal Assembly 195 7.2.1. Introduction 197 7.2.2. Model and Numerical Procedure 200 7.2.3. Results 200 7.2.3.1. Morphologies 208 7.2.3.2. Kinetics of Growth 216 7.2.3.3. Scaling Behavior at the Steady State ix 218 7.2.4. Conclusions 221 7.3. A Short-Range Model for Protein Aggregation in Highly Salted Solutions 221 7.3.1. Introduction 222 7.3.2. Simulation 222 7.3.3.