# Modeling and simulation of nanoparticle aggregation in colloidal systems

iii T ABLE OF CONTENTS LIST OF TABLES...................................vi LIST OF FIGURES..................................viii ACKNOWLEDGEMENTS..............................xiv CHAPTER 1.INTRODUCTION.........................1 1.1 Simulation Approaches................................9 1.1.1 Molecular Dynamics.............................9 1.1.2 Monte Carlo Simulation Algorithms....................10 1.1.3 Mesoscale Methods..............................12 1.2 Research Objectives.................................14 1.3 Original Contributions of this Dissertation.....................15 1.4 Outline of the Dissertation..............................17 CHAPTER2.ONBROWNIANDYNAMICS SIMULATIONOF AGGRE- GATION.......................................18 2.1 Abstract........................................18 2.2 Introduction......................................18 2.3 Convergence of Brownian Dynamics Simulations.................24 2.4 Aggregation Regime.................................29 2.5 Simulation Accuracy.................................31 2.6 Summary and Discussion..............................37 2.7 Acknowledgement...................................41 Bibliography........................................42

iv C HAPTER3.COARSE-GRAININGAPPROACHTOINFERMESOSCALE INTERACTIONPOTENTIALS FROMATOMISTICINTERACTIONS FOR AGGREGATING SYSTEMS.......................45 3.1 ABSTRACT.....................................45 3.2 INTRODUCTION..................................46 3.3 METHODS......................................51 3.3.1 Molecular Dynamics.............................52 3.3.2 Langevin Dynamics.............................53 3.3.3 Aggregation Statistics............................55 3.4 PERFORMANCE OF LD MODEL.........................56 3.5 RELATIVE ACCELERATION...........................57 3.6 IMPROVED BD MODEL..............................60 3.7 RESULTS WITH IMPROVED BD MODEL...................64 3.8 DISCUSSION.....................................68 3.9 CONCLUSIONS...................................71 3.10 ACKNOWLEDGMENTS..............................72 BIBLIOGRAPHY.....................................73 CHAPTER 4.EFFECT OF SHEAR ON COLLOIDAL AGGREGATION OF A MODEL SYSTEM USING LANGEVIN DYNAMICS SIMULA- TION.........................................92 4.1 Introduction......................................92 4.2 Improved Langevin Dynamics Simulation.....................95 4.3 Analysis of Colloidal Aggregation Under Shear..................97 4.3.1 Scale–separated and Scale–overlap Regimes................98 4.4 Energy Balance in Sheared Aggregating System..................102 4.5 Eﬀect of Shear on Aggregation Structure......................106 4.5.1 Aggregation Without Shear.........................106 4.5.2 Shear–induced Aggregation Mode......................110

v 4 .5.3 Characterization of Local Structure....................112 4.6 Prediction of Maximum Size of Aggregates Under Shear.............117 4.7 A Regime Map for Aggregation Under Sheared..................121 4.8 Discussion.......................................125 4.9 Conclusions......................................126 4.10 Acknowledgments...................................127 Bibliography........................................128 CHAPTER 5.CONCLUSIONS AND FUTURE WORK...........131 5.1 Summary and conclusions..............................131 5.2 Secondary ﬁndings..................................132 5.3 Future work......................................133 APPENDIX A.BUCKINGHAM PI ANALYSIS................136 APPENDIX B.RELATIVE ACCELERATION CALCULATION......140 APPENDIX C.DERIVATION OF PAIR CORRELATION EXPRESSION FOR A BINARY MIXTURE..........................145 APPENDIX D.DERIVATION OF THE TRANSPORT EQUATION FOR THE TWO–PARTICLE DENSITY ρ (2) ....................148 APPENDIX E.LIGHT SCATTERING ANALYSIS..............152 APPENDIXF.EVOLUTIONOF THE SECOND-ORDERDENSITYFOR MD MODEL....................................157 APPENDIX G.HYDRODYNAMIC EFFECT..................159 G.1 Short-range lubrication forces............................159 G.2 Long-range hydrodynamic interactions.......................162 BIBLIOGRAPHY...................................164

vi L IST OF TABLES Table 2.1 Comparison of MD and position-velocity Langevin BD simulation time for 31 non-aggregating solute particles.All solvent and solute parti- cle interactions were modeled using Lennard-Jones potentials with well depth ε and particle radius σ.The time increment in both types of simulations was ﬁxed at 5 ×10 −15 seconds.................23 Table 2.2 Characteristic length,time,and velocity scales in BD simulations.The parameter ε represents the intermolecular potential energy minimum, or well depth.................................30 Table 2.3 Simulation parameters used to produce Figure 2.4.Particle interactions were modeled using Lennard-Jones potentials and simulations were car- ried out using the LAMMPS[9] software package.............31 Table 2.4 Simulation parameters used to produce Figure 2.5.Particle interactions were modeled using Lennard-Jones potentials.MD simulations were carried out using the LAMMPS[9] software package...........34 Table 3.1 Parameters used in MD simulations....................53 Table 4.1 Micro,meso,amd macro scales.Where τ v is the characteristic time,R g is the aggregate radius of gyration,L is the box length,and G is the shear rate...................................97 Table 4.2 Parameters used in LD simulations to produce Figure 4.2........100 Table 4.3 Energy budget for non–sheared and representative sheared aggregation systems....................................105

vii T able 4.4 Maximum size of aggregates calculation..................120 Table 4.5 (d f ,R max g ,LPED) Values as a function of P`eclet number Pe and po- tential well–depth ˆε.............................123 Table E.1 Validation of LS code............................153 Table E.2 Simulation parameters used to produce Figure E.1.Particle interactions are modeled using Lennard-Jones potentials................155

viii L IST OF FIGURES Figure 1.1 Figures taken from Cerda et al.[5] for diﬀerent aggregating regimes: (a) system at equilibrium with dimensionless potential well–depth ˆ U = U/k B T = 3.125;(b) non–equilibrium aggregation with dimensionless potential well–depth ˆ U = U/k B T = 4.0;(c) gelation with dimensionless potential well–depth ˆ U = U/k B T = 7.0..................2 Figure 1.2 The focus area of this dissertation relative to other work on colloidal aggregating system under shear.......................6 Figure 1.3 Dependence of the maximum system size versus simulation method...7 Figure 2.1 Dependence of MD simulation CPU time on the number of Lennard- Jones particles,N,and the solute/solvent diameter ratio,R for non- aggregating particles.All other simulation parameters are identical in the two sets of simulations.Solute volume fraction was chosen to be 0.01.21 Figure 2.2 Illustration of the ramp-well potential...................25 Figure 2.3 Dependence of statistical error,S p on number of independent simula- tions,M for a time step σ v ∞ ∆t/σ = 0.002.The slope of the linear ﬁt is -0.52 for p a = 0.9,and is -0.54 for p a = 0.74...............28 Figure 2.4 Clustering index (see color legend) as a function of reduced interaction potential well depth,ˆε and reduced diﬀusivity, ˆ D ∞ .Each curve repre- sents constant ˆε ˆ D ∞ .The region bounded by ˆε ˆ D ∞ 1 represents the regime of validity of the position and velocity Langevin to PL reduction.32 Figure 2.5 Extent of aggregation as a function of dimensionless time, ˆ t = σt/σ v ∞ , for BD and MD simulations described in Table 2.4............35

ix F igure 2.6 Comparison of cluster size distributions obtained fromMDand BDsim- ulations of aggregation carried out under conditions speciﬁed in Table 2.4 and at the same extent of aggregation ξ = 0.89............36 Figure 2.7 Scaled cluster size distributions for MD simulations...........37 Figure 2.8 Scaled cluster size distributions for BD simulations............38 Figure 2.9 Number of monomers as a function of radius of gyration,R g for MD simulations described in Table 2.4.The slope of the linear ﬁt is the volume fractal dimension,d f ........................39 Figure 2.10 Number of monomers as a function of radius of gyration,R g for BD simulations described in Table 2.4.The slope of the linear ﬁt is the volume fractal dimension,d f ........................40 Figure 3.1 Comparison of g AA (ˆr) predicted by the LD model with MD simulation data at time ˆ t = 86.5:a) DLA regime:ε AA /ε BB = 8.0;b) RLA regime:ε AA /ε BB = 4.0...........................78 Figure 3.2 Comparison of the cluster size distribution predicted by the LD model with corresponding MD simulation data at time ˆ t = 86.5:a) DLA regime:ε AA /ε BB = 8.0;b) RLA regime:ε AA /ε BB = 4.0......79 Figure 3.3 Direct relative acceleration between solute particles (1) and (2) (solid arrow between particles (1) and (2)) occurs due to their direct interac- tion.Indirect relative acceleration between solute particles (1) and (2) (dashed arrow) occurs (a) in MD due to interaction of particles (1) and (2) with probe solute particle and probe solvent molecule p;(b) in LD due to interaction of particles (1) and (2) with probe solute particle only.80

x F igure 3.4 Indirect average relative acceleration between A-A pairs resulting solely fromother A-type particle interactions.MDsimulations (10,000 A-type solute particles and 813,218 B-type solvent particles) compared with LD model predictions (10,000 A-type solute particles) at time ˆ t = 86.5: a) DLA regime:ε AA /ε BB = 8.0;b) RLA regime:ε AA /ε BB = 4.0.In- direct average relative acceleration is scaled as ∆ ˆ A I | ˆ r = ∆A I |r σ A m A /ε AA .81 Figure 3.5 Indirect average relative acceleration between A-A pairs resulting solely from B particles.MD simulation of 10,000 A-type solute particles and 813,218 B-type solvent particles at time ˆ t = 86.5:a) DLA regime: ε AA /ε BB = 8.0;b) RLA regime:ε AA /ε BB = 4.0............82 Figure 3.6 Evolution of the pair correlation function g AB (ˆr

,t) from MD simula- tion:a) DLAregime:ε AA /ε BB = 8.0;b) RLAregime:ε AA /ε BB = 4.0. Scaled time ˆ t = t D ∞ /σ 2 ..........................83 Figure 3.7 Comparison of the improved LD potential U LD AA,eﬀ and the modeled solvation potential ˜ U 2 with the Lennard-Jones potential U LJ AA :a) DLA regime:ε AA /ε BB = 8.0,C 2 = 0.51;b) RLA regime:ε AA /ε BB = 4.0, C 2 =3.15....................................84 Figure 3.8 Comparison of g AA (ˆr) predicted by improved LD model with corre- sponding MD result at time ˆ t = 86.5:a) DLA regime:ε AA /ε BB = 8.0; b) RLA regime:ε AA /ε BB = 4.0......................85 Figure 3.9 Comparison of the cluster size distribution predicted by improved LD model with corresponding MD result at time ˆ t = 86.5:a) DLA regime: ε AA /ε BB = 8.0;b) RLA regime:ε AA /ε BB = 4.0............86

xi F igure 3.10 Normalized cluster size distributions in the RLA regime ˆε AA /ˆε BB = 4.0 [top panel (a),(b) and (c)],and the DLA regime ˆε AA /ˆε BB = 8.0 [bottom panel (d),(e) and (f)] at diﬀerent times ˆ t = tD ∞ /σ 2 for MD simula- tions with LJ potential [left column (a) & (d)];LD simulations with LJ potential [middle column (b) & (e)];LD with improved potential [right column (c) & (f)]...............................87 Figure 3.11 Comparison between MD and LD (unmodiﬁed LJ and improved poten- tial) of the extent of aggregation ξ for RLA regime:ˆε AA /ˆε BB = 4.0 [top panel:(a) and (b)],and DLA regime:ˆε AA /ˆε BB = 8.0 [bottom panel:(c) and (d)].LD simulations with unmodiﬁed LJ potential are compared with MD in the left column [(a) & (c)],while LD with the improved potential is compared with MD in the right column [(b) & (d)].88 Figure 3.12 Indirect average relative acceleration between A-A pairs resulting solely from other A particle interactions.Improved LD model (10,000 A- type particles) compared with MD simulation (10,000 A-type parti- cles and 813,218 B-type particles) at time ˆ t = 86.5:a) DLA regime: ε AA /ε BB = 8.0;b) RLA regime:ε AA /ε BB = 4.0............89 Figure 3.13 Schematic showing a “probe” solvent molecule p which can occupy any point in 3-d space except volumes of solute particles 1 and 2,thus deﬁning the domain of integration for the relative acceleration calculation.90 Figure 3.14 Comparison of computed indirect average relative acceleration with the analytical result at ˆn = 0.1:a) 1-d case,computations with 150,000 particles averaged over 3,000 multiple independent trials,b) 2-d case, computations with 823,000 particles averaged over 240 multiple inde- pendent trials.................................91

xii F igure 4.1 Space of dimensionless parameters in which we scale to characterize aggregation outcomes for Gτ (1) v as a function of the dimensionless po- tential well depth ˆε,dimensionless diﬀusion coeﬃcient ˆ D ∞ ,and P´eclet number Pe..................................100 Figure 4.2 Gτ (1) v as a function of the dimensionless potential well depth ˆε and P´eclet number Pe for LD simulations....................101 Figure 4.3 Evolution of kinetic energy in mean velocity E mean and kinetic energy in ﬂuctuating particle velocity E fluct in k B T ref units for system with ˆε = 8 and Pe = 2.1..............................103 Figure 4.4 Evolution of the trace of each component in Eq.4.11 in σ 3 v ∞ /σ units for system with ˆε = 8 and Pe = 2.1.Inset represents the same values at longer time when system reaches a steady–state............104 Figure 4.5 The fractal dimension D f from the LD with eﬀective potential U LD eff at time ˆ t = 3244:a) simulations are done with ˆε = 8.0;b) simulations are done with ˆε = 50.0..............................107 Figure 4.6 Snapshots for two typical aggregates for ˆε = 50.0 at time ˆ t = 3244: a) aggregate containing 150 monomers with the radius of gyration R g = 2.8 σ;b) aggregate containing 966 monomers with the radius of gyration R g = 7.1 σ..................................108 Figure 4.7 The fractal dimension D f from the LD with eﬀective potential U LD eff under shear ﬂow with Pe = 2.1 at time ˆ t = 113..............111 Figure 4.8 Snapshots for typical aggregate for ˆε = 50.0 at time ˆ t = 113 for aggre- gate containing 7144 monomers with the radius of gyration R g = 15 σ.111 Figure 4.9 For aggregation without shear the dimensionless local volumetric poten- tial energy density ˆ U(r)/ ˆ V cl in ε/σ 3 units as a function of the anisotropy A ij ,where i,j = 1,...,3 for ˆε = 8.0 and ˆε = 50.0.Color legend repre- sents the number of monomers in each cluster...............114

xiii F igure 4.10 Local volumetric potential energy density ˆ U(r)/ ˆ V cl in ε/σ 3 units as a function of the anisotropy A ij ,where i,j = 1,...,3 for sheared abbre- gating systems with Pe = 2.1.The color legend represents the number of monomers in each cluster.Dashed line represents average LPED value for non-sheared case..........................116 Figure 4.11 Sum of the relative accelerations due to potential interaction and the relative acceleration due to shear at time ˆ t = 113:a) system with ˆε = 8.0;b) system with ˆε = 50.0.Arrays show prediction of the maximum size of aggregates R max g .....................120 Figure 4.12 Relative forcef pot,sh as a function of the dimensionless interparticle po- tential well–depth ˆε,and P`eclet number Pe.The dashed line represents the boundary between non–aggregating and aggregating systems,and dotted line identiﬁes region when a compactness of the local structure is observed.Values in brackets for selected systems represent D f ,R max g , and LPED correspondently.........................124 Figure E.1 Structure of the largest cluster for 3-D BD simulation..........156 Figure G.1 Truncated dimensionless lubrication force ˆ F h TL and correction force ˆ F 2 as function of ε for equisize particles....................161 Figure G.2 The radial distribution function for the state point N = 256 and φ = 0.3403 using the three equations of motion/algorithms:I,no many-body hydrodynamics;II,with many-body hydrodynamics;and III,many- body hydrodynamics with an incomplete algorithm.Figure is taken from Heyes work [49].............................163

xiv A CKNOWLEDGEMENTS I would like to take this opportunity to express my sincere thanks and deep appreciation to those who helped me with various aspects of conducting research and the writing of this dissertation.First and foremost,I would like to thank Dr.Shankar Subramaniam for his constant guidance,encouragement,patience and support throughout this research and the writing of this dissertation.His scientiﬁc knowledge,technical skills and enlightened views on the research process has left a deep impression on me.How fortunate I am to have been a graduate student in his laboratory. I would also like to thank my POS committee members for their eﬀorts and contributions to this work:Dr.Rodney Fox,Dr.Monica Lamm,Dr.Pranav Shrotriya,Dr.Sriram Sundararajan,and Dr.Dennis Vigil. My special appreciation extends to all my lab-mates:Ragul Garg,Madhu Pai,Xu Ying, Vidyapati,Tenneti Sudheer,Ravi Kolakaluri,Karthikeyan Devendran,Mohammad Mehrabadi, Bo Sun,and Christopher Schmitz.Thank you for your friendship and essential support.You made my life in the lab comfortable and enjoyable. My thanks to the Department of Mechanical Engineering and its professional and adminis- trative staﬀ,especially Amy Carver,Jannet Huqqard,Sherrie Nystrom,Janelle Miranda,Hap Steed,and Nate Jensen for their ﬂawless and cheerful assistance during the past years. Finally,I would like to extend love and thanks to my family,especially to my wife Svitlana for her love,patience and enduring support during my graduate studies at ISU.I am grateful for my mother and my mother-in-law for their help with caring for our daughter Alina,whose entry into our lives gave us extra motivation to pursue our dreams.

1 C HAPTER 1.INTRODUCTION Nanoparticles are widely used as building blocks in nanotechnology research and they oﬀer the promise of creating new materials and new applications in the nanoscale range.Moreover, properties of such materials diﬀers from bulk material properties [1].These novel properties are observable only at the nano-scale dimensions have already found their ﬁrst commercial applications [2].For example,latex nanoparticles are used for a variety of biological applica- tions [3]. Two high-rate synthesis methods are commonly used in the industry:aerosol reactors in a gaseous environment and colloidal reactors in a ﬂuid environment [1,4].In both methods the synthesis of the particles occurs in turbulent reactors due to the reaction of chemical precursors and the formation of nuclei,which rapidly grow due to surface addition and/or aggregation. Such a synthesis process subjects nanoparticle aggregates to a spatially homogeneous,time- varying shear ﬂow and is characterized by the variety of time–scales and length–scales from size of single particle to the size of particles aggregate.The next generation of applications will require improvement in the quality of the monodispersity,purity,and uniform surface chemistry of nanoparticles [4].Since the aggregation of the sheared colloidal nanoparticles is the important part of this process,a better understanding of the sheared aggregation phenomenon will help to improve synthesis methods. Aggregating systems that are studied in the literature can be classiﬁed by:(a) their com- positions:for example solute particles in solvent (latex particles in solvent),polymer chains in colloidal systems,etc;(b) concentration of substance;(c) presence (or absence) of external force:gravitational force,shear ﬂow,etc.In aggregating systems a rich variety of phenomena is observed.Competition between the physical mechanisms of interparticle attraction,intensity

2 (a) (b) (c) F igure 1.1:Figures taken fromCerda et al.[5] for diﬀerent aggregating regimes:(a) system at equilibriumwith dimensionless potential well–depth ˆ U = U/k B T = 3.125;(b) non–equilibrium aggregation with dimensionless potential well–depth ˆ U = U/k B T = 4.0;(c) gelation with dimensionless potential well–depth ˆ U = U/k B T = 7.0.

3 o f external shear ﬂow,and thermal energy determines whether a system will evolve reversibly or irreversibly as shown,for example,on Figure 1.1.These diﬀerent regimes are observed by varying the strength of interparticle interaction related to thermal energy ( ˆ U = U/k B T). When interparticle interaction is weak relative to thermal energy and is lesser than some criti- cal value then the formation of only small aggregates is observed (Figure 1.1a).As interparticle interaction become stronger,larger aggregates are formed (Figure 1.1b,c). Application of external shear force to the nanoparticle aggregation system is important because in large–scale reactors,the ﬂow is turbulent and aggregating nanoparticles will be subjected to time varying shear ﬂow at the Kolmogorov scale.In such a system the wide range of time and length scales are present.The wide range of time scales is introduced by the presence of short-time Brownian motion and the long-time hydrodynamic behavior of solvent.The wide range of length scales occurs due to the size separation of clusters of colloidal nanoparticles and solvent molecules.However,because of the scale separation between nanoparticle clusters and Kolmogorov scale,for a ﬁrst approximation the ﬂow can be treated as locally uniform time-varying shear.Once formed,aggregates do not break apart, and to introduce breakage some external forces such as shear ﬂow must be introduced into the system.When the thermal energy and kinetic energy associated with external shear ﬂow is able to overcome the interparticle interactions then a reversible change is expected.If interparticle attraction dominates,then an irreversible change is expected.In sheared colloidal systems,aggregation may occur due to particle–cluster (monomer addition) and cluster–cluster aggregation.The breakage and restructuring of these clusters is promoted by shear ﬂow,and all these processes are related to irreversible changes.Recent experiments observe restructuring of clusters in the presence of external shear force [6].However,there is no complete explanation of such behavior,which is a good reason to use computational approach for such a case. Before discussing the characterization of aggregation outcomes,it is useful to clarify some terminology speciﬁc to aggregation.Colloidal aggregation is sometimes classiﬁed as reversible or irreversible depending on the system’s characteristics (Figure 1.1a and Figure 1.1b,c cor- respondingly).However,the thermodynamic deﬁnition of a reversible process and reversible

4 a ggregation phenomenon are diﬀerent.Thermodynamically a reversible change is one that is performed quasi-statically such that the system remains inﬁnitesimally close to thermody- namic equilibrium.Such changes can always be reversed and the system brought back to its original thermodynamic state without causing any changes in the thermodynamic state of the universe [7].But when we are talking about a reversible aggregation process we mean a sys- temat non-equilibriumsteady state (NESS).In aggregating systems,once aggregation starts it continues irreversibly.Reversibility (due to aggregate breakage) may occur only through shear ﬂow or increase in thermal energy.Thus,we can conclude that aggregating systems cannot be in thermodynamic equilibrium,instead,aggregating systems are in non–equilibrium steady states (NESS).An example of aggregating irreversible system is represented on Figure 1.1b from simulations performed by Cerda et al.[5] for 2D systems where large and dense aggre- gates are formed together with the presence of single particles.After an irreversible change the system cannot be brought back to its original thermodynamic state without causing a change in the thermodynamic state of the universe [7].Based on this deﬁnition we can conclude that an irreversible colloidal nanoparticle aggregation leads to such non–equilibrium steady state as gelation which is a ﬁrst order phase transition (as in a ﬁrst order phase transition a system either absorbs or releases a ﬁxed amount of energy).An aggregate structure that corresponds to the gelation stage is represented on Figure 1.1c from Cerda et al.[5] work.In this case large aggregates with a ramiﬁed structure are formed that occupy all the system’s volume.At this stage no single particles are observed. There also have been eﬀorts to classify this phase–change behavior of aggregating systems on a phase diagram.Anderson and Lekkerkerker [8] described all these regimes with the phase diagram for the colloid–polymer systems.In these systems polymer is added to colloidal systems to produce an attraction between the particles.By varying the relative size of polymer and the colloid;the polymer concentration and colloid volume fraction the range of particle– particle interaction can be tuned and a variety of phase diagrams can be realized.Anderson and Lekkerkerker [8] reported that the aggregating outcome depends on the initial conditions and slight change in one condition may signiﬁcantly change the outcome.They conclude that it

5 i s diﬃcult to reliably predict the transition mechanisms of colloid and colloid–polymer systems. The processes of aggregate formation and aggregate breakage have been investigated from an experimental and computational perspective [9,10,11,12].Researchers agree that colloidal particle aggregating phenomena is very complex and multiscale problem where aggregation outcome depends signiﬁcantly on the initial conditions.However,a single uniﬁed aggregation map that would determine diﬀerent aggregating outcomes based on the initial parameters of the sheared aggregating systemis not available.Such an aggregation map would be very useful when designing eﬃcient turbulent reactors used for synthesis of the particles with good size control of product. In principle,such an aggregation regime map could be generated based on a purely the- oretical description of aggregation;or using experimental approach;or using a computation approach.Colloidal particles aggregation phenomena is not completely described yet therefore pure theoretical approach for describing aggregating phenomena is not appropriate.Experi- mental approach allows to measure aggregation in real systems.However,it is not feasible to control and measure all the parameters that determine aggregation phenomena.Therefore,in this dissertation a computational approach is adopted to develop a fundamental understanding of colloidal aggregation. The focus area of this dissertation with respect to the work of other researchers is shown in Figure 1.2.The system complexity axis on this map represents model approaches used to study aggregation processes beginning from the simplest model LJ systems to more detailed and complicated systems such as protein molecules.The solute–concentration axis represents the range of solute densities,while the shear axis represents increase in shear ﬂow intensity in the system.Dark–gray areas represent work of other researchers,such as Hobbie [13] who had performed experimental studies of depletion–driven phase separation for dilute polystyrene spheres.Aggregation processes under shear ﬂow for dilute latex nanoparticles were studied by Chakrabarti,Sorensen,et al.[14].Aggregation in systems with dense polymeric spherical nanoparticles are performed by Lekkerkerker [9] as well as Shepherd [10] for systems with and without shear.On this map the focus of the present work is represented with light–gray

6 Model LJ systems AO (depletion colloids) Latex nanoparticles Composites/ Proteins Shear No shear Dense Dilute Hobbie [13] Chakrabarti & Sorensen et al.[14] Lekkerkerker group [8] Focus of present work Solute concentration Shepherd et al.[10] Systemcomplexity Figure 1.2:The focus area of this dissertation relative to other work on colloidal aggregating system under shear. area which represents dilute systems of solute particles in solvent,with and without shear ﬂow,for LJ model systems and depletion colloids.Since this work is a computational study, corresponding model systems are used instead of physical ones due to feasibility limitations of numerical methods.Therefore,the focus of this dissertation is on simple LJ model systems and depletion colloids.Dilute systems are chosen to compare our aggregating results with experimental results obtained by Mokhtari et al.[14] for latex nanoparticles.And shear ﬂow is applied to aggregating systems for the reasons given before. A fundamental understanding of changes in aggregate structure due to presence of the external shear ﬂow is required to correctly describe aggregation growth and breakage processes. Therefore,an eﬃcient numerical model that would accurately predict aggregation phenomenon in colloidal nanoparticles systems must be chosen. Currently,the following simulation approaches for aggregation are commonly used [11], [15]-[18] 1.Molecular dynamics (MD),which is a microscale method (described in Chapter 2). 2.Mesoscale methods,such as Langevin dynamics (LD) and Brownian dynamics (BD)

7 (described in Chapter 2),stochastic rotational dynamics (SRD),and dissipative particle dynamics (DPD). 3.Monte Carlo methods,such as lattice Monte Carlo (LMC) method and oﬀ-lattice Monte Carlo (OLMC) method. However,requirement for a signiﬁcant aggregation statistics leads to consideration of large simulation systems that can signiﬁcantly increase computational costs and decrease simula- tion eﬃciency.Ideally the model which is chosen to predict sheared colloidal nanoparticle aggregation should accurately describe physico-chemical interactions of relatively large physi- cal systems,and at the same time,simulate at a low computational cost.In reality this is hard to achieve.In many cases if the model is very accurate it is usually not eﬃcient and cannot be used to simulate a physical problem.On the other hand,more eﬃcient models usually are not very accurate in terms of representing the physics,thereby limiting their applicability. Thus,a computational model which is chosen to predict a sheared aggregation of colloidal nanoparticles should maintain the balance between the level of accuracy and computational eﬃciency. Systemsize Ο(10 0 ) Fichthorn [40,47,48,51] Chakrabarti & Sorensen et al.[35,36,39,50] Present work Simulation method Ο(10 6 ) Ο(10 5 ) MD Mesoscale MC Accuracy Less cost model solvent no solvent solvent Others [11,16,19-21,23,27] Figure 1.3:Dependence of the maximum system size versus simulation method.

8 T o study sheared aggregation we focus on coarse-grained (or mesoscale) simulation methods such as a Langevin dynamics.These methods are computationally eﬃcient when compared with microscale methods such as molecular dynamics,and they have the ability to accurately represent the aggregate structure when compared with the Monte Carlo methods.The hierar- chy of these methods is represented on Figure 1.3.From this Figure we can observe decrease in the maximum system size (represents the number of solute particles in a system) for models with more detailed solvent representation such as MD.And increase in the system size when solvent eﬀect is removed (Monte Carlo methods).The mesoscale methods still include solvent eﬀect through solvent modeling that allows to decrease computational cost and increase system size to get good aggregation statistics. As we show in Chapter 2,using MD to simulate aggregation phenomenon for realistic sys- tems is too expensive.At the same time the oﬀ-lattice Monte Carlo (OLMC) simulation has limitations in simulating aggregate restructuring,because it is not capable of representing re- structuring of the cluster after the cluster is formed.On the other hand,the mesoscale methods such as BD,LD,DPD,and SRD have the promise of low cost and accurate representation of aggregation structure on today’s computers,but it applicability for simulating non-equilibrium systems should be established.The development of LD and BD methods for solving sheared aggregation problems requires consideration of the following points: 1.The current LD and BD models are not adequate for aggregation. 2.Numerical accuracy and approach not well established in context of aggregation.This leads to Chapter 2. 3.Model accuracy of LD and BD is not satisfactory for aggregating systems when compared with established MD approach.This motivates the need for improved BD model with potential mean force (PMF) that accounts for solvent interaction in non-equilibrium aggregating systems which leads to Chapter 3. 4.The minimum set of characteristics and metrics required for complete description of sheared aggregation phenomenon is not established and the correspondent aggregation

9 m ap is not deﬁned.This leads to development of new characteristics and metrics fully described in Chapter 4. Also accurate modeling of physico-chemical interactions is required.This can be achieved by developing the coarse–grained particle interaction potentials derived fromquantummechan- ics calculations,that are suitable for large scale nanoparticle aggregation simulations.In this case,the atomic models for surface molecules of polystyrene nanoparticles can be developed to calculate surface-molecule,surface-surface,and molecule-molecule interaction forces.These results can be validated by atomic force microscopy (AFM) measurements of polystyrene- polystyrene nanoparticles,and used in simulation of nanoparticles aggregation as a physical potential. 1.1 Simulation Approaches In this subsection the various approaches used to simulate nanoparticle aggregation are brieﬂy reviewed and their advantages and disadvantages are considered. 1.1.1 Molecular Dynamics Molecular dynamics (MD) simulation is an established technique that can simulate col- loidal nanoparticle aggregation [15].In the MD approach,solute and solvent particles interact through a modeled,intermolecular potential,and the positions and velocities of these parti- cles evolve in time according to Newton’s equations of motion.In most MD simulations,the intermolecular potential energy is taken to be the sum of isolated pair interactions,which is called the pairwise additivity assumption.The main diﬃculty with such an approach is that it cannot be used to model aggregation of a realistic system of colloidal nanoparticles.The requirements of large size separation between nanoparticles and solvent molecules (d NP ∼ 40 nm,d solv ∼ 0.3 nm and d NP /d solv ∼ 100 at solvent molecules volume fraction λ solv ∼ 0.45 and very low nanoparticle volume fraction λ solute ∼ 0.005),and the large number of nanoparticles that are modeled to have good statistics of aggregated clusters lead to an enormous number of solvent molecules in the system (on the order of 10 10 ).Moreover,calculation of intermolecular

10 f orces between solvent molecules in MD would require resolving time scales on the order of ˆτ F = 0.125.However,the time scale of evolution of cluster statistics is much larger and is on the order of ˆτ cl ∼ 40,000.00.Therefore,simulating any colloidal system even far from realistic physical parameters is a challenging and sometimes even impossible task.Alterna- tive approaches are needed to resolve this problem.One alternative is to use Monte Carlo approaches for nanoparticle aggregation simulation. 1.1.2 Monte Carlo Simulation Algorithms Based on the oﬀ-lattice Monte Carlo (OLMC) simulation,several methods are frequently used to model nanoparticle aggregation.These models include diﬀusion-limited aggrega- tion (DLA),diﬀusion-limited cluster aggregation (DLCA),ballistic-limited aggregation (BLA), ballistic-limited cluster aggregation (BLCA),reaction-limited aggregation (RLA),and reaction- limited cluster aggregation (RLCA). In DLA models,particles diﬀuse through a random-walk from distant points and ﬁnally arrive and stick to the surface of the growing aggregate [16,19].In the DLCA model,the particles and clusters move in random-walk trajectories,which represent the Brownian motion of the particles and clusters in a dense ﬂuid [20,21].According to this model,particles and aggregates are moved randomly,and when the distance between centers of two particles approach “cluster distance” r cl (the maximum distance between two neighbor particles which belong to the same cluster) they irreversibly link.After this linking if the distance between any pair of particles in two diﬀerent clusters appears to be less than r cl ,two clusters move apart along their approach path until the separation is equal to the cluster distance.Thus,stickiness probability p stick for these two approaches is unity.DLCA is the more appropriate model when simulating colloidal aggregation because in reality,aggregates grow not only due to cluster- monomer interaction but also due to the cluster-cluster interaction.Both DLA and DLCA models allow simulating the aggregation of systems with more than a million nanoparticles, which gives good statistics of aggregates.However,these approaches can only be applied if the interparticle interactions are smaller than k B T ref ,where k B is the Boltzmann constant and