RHEOLOGY of COLLOIDAL SUSPENSIONS: a COMPUTATIONAL STUDY by SEYEDSAFA JAMALI Submitted in Partial Fulfillment of the Requirement

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RHEOLOGY OF COLLOIDAL SUSPENSIONS:

A COMPUTATIONAL STUDY

SEYEDSAFA JAMALI

Submitted in partial fulfillment of the requirements

For the degree of Doctor of Philosophy

Thesis Advisor: Dr. Joao Maia

Department of Macromolecular Science and Engineering

Case Western Reserve University

August, 2015 CASE WESTERN RESERVE UNIVERSITY

SCHOOL OF GRADUATE STUDIES

We hereby approve the thesis/dissertation of

Seyedsafa Jamali

Candidate for the Ph.D. degree *.

(Signed) Prof. Joao Maia (Chair of the committee)

Prof. Gary Wnek

Prof. Michael Hore

Prof. Daniel Lacks

(date) 05-12-2015

*We also certify that written approval has been obtained for any proprietary material contained therein.

Dedication

To my wife, Shaghayegh, and to my parents, Saeed and Kobra.

III

Table of Contents Chapter 1 Introduction ...... 19

1.1. Computer Simulations ...... 19

1.2. Dissipative Particle Dynamics ...... 22

1.3. Rheology ...... 24

1.4. Rheology of Suspensions ...... 26

1.5. Organization and Scope of Dissertation ...... 28

1.6. References ...... 29

Chapter 2 Bridging Simulation to Experiment ...... 31

2.1. Introduction ...... 31

2.2. Formalism and Parametrization of DPD ...... 32

2.3. Formalism of MDPD ...... 36

2.4. Equation of State in MDPD ...... 39

2.5. Compressibility and Flory-Huggins χ parameter ...... 46

2.6. Interfacial Tension ...... 52

2.7. Validation on polymer solutions ...... 53

2.8. Transport properties and Dynamics ...... 56

2.9. Conclusions ...... 59

2.10. References ...... 61

Chapter 3 DPD Rheometry ...... 64

3.1. Introduction ...... 64

3.2. Boundary Conditions ...... 67

3.3. Steady Shear Flow ...... 71

3.4. Zero Shear Viscosity ...... 77

3.5. Transient Shear and Poiseulle Flow ...... 81

3.6. Conclusions ...... 83

3.7. References ...... 84

Chapter 4 Stabilizing Shear Flows in DPD...... 87

4.1. Introduction ...... 87

4.2. Temperature Measurements under Shear with DPD ...... 90

4.3. GIANT, a new thermostat for DPD ...... 95

4.4. Viscosity measurement using GIANT ...... 102

4.5. Conclusions ...... 104

4.6. References ...... 105

Chapter 5 Rheology of Colloidal Suspensions ...... 108

5.1. Theoretical Background ...... 108

5.2. Prior DPD simulations on suspensions ...... 116

5.3. Modified DPD model ...... 120

5.4. Parameterization of colloidal forces ...... 125

5.5. Suspensions at Rest (quasi-equilibrium conditions) ...... 128

5.6. Crystallization of mono-sized suspensions ...... 136

5.7. Viscosity measurements...... 138

5.7.1. Effect of volume fraction ...... 139

5.7.2. Effect of particle strength ...... 141

5.7.3. Effect of lubrication interactions ...... 146

5.8. Other rheological parameters ...... 148

5.9. Microstructure under flow ...... 154

5.10. Rheology of bimodal suspensions ...... 156

5.10.1. Effect of particle composition ...... 157

5.10.2. Effect of size ratio ...... 158

5.11. Proposed mechanism ...... 160

5.11.1. Force Analysis ...... 160

5.11.2. Hydro-cluster formation...... 163

5.11.3. Contact network formation ...... 167

5.12. Potential energy analysis...... 170

5.13. Colloidal Gels ...... 177

5.13.1. Introduction ...... 177

5.13.2. Equilibrium properties ...... 180

5.13.3. Rate-dependent properties ...... 188

5.13.4. Time-dependent properties ...... 190

5.14. Conclusions ...... 195

5.15. Refernces...... 197

Chapter 6 Bibliography ...... 204

Table of Figures

FIGURE 1.1. DIFFERENT TIME AND LENGTH SCALES OF COMPUTER SIMULATIONS...... 21

FIGURE 1.2. TYPICAL COARSE-GRAINING AT DIFFERENT LEVELS, FROM THE MOLECULAR

LEVEL TO BEADS...... 23

FIGURE 2.1. PRESSURE AS A FUNCTION OF: A) BIJ PARAMETER, AND B) PARTICLE DENSITY.40

FIGURE 2.2.PRESSURE DIVIDED BY REPULSIVE PARAMETER AS A FUNCTION OF: A) DENSITY,

AND B) CUBIC DENSITY...... 41

FIGURE 2.3.PRESSURE VS. DENSITY FOR: A) BIJ =10, B) BIJ =30 AND, C) BIJ =50...... 41

FIGURE 2.4. ABSOLUTE VALUE OF THE TOTAL PRESSURE SUBTRACTED BY THE REPULSIVE

PRESSURE AS A FUNCTION OF DENSITY FOR: A) BIJ =10, B) BIJ =30 AND, C) BIJ =50...... 42

FIGURE 2.5.ABSOLUTE VALUE OF THE ATTRACTIVE PRESSURE NORMALIZED BY AIJ

PARAMETER FOR: A) BIJ =10, B) BIJ =30 AND, C) BIJ =50. THE DASH LINE SHOWS THE

EXPRESSION PREDICTED FOR THE CONSERVATIVE PARAMETER DEPENDENCE IN STANDARD

DPD BY GROOT AND WARREN...... 43

FIGURE 2.6.MEASURED PRESSURE DIVIDED BY ITS PREDICTED VALUE FROM THE EQUATION

15 AS A FUNCTION OF DENSITY...... 44

FIGURE 2.7. PRESSURE AS A FUNCTION OF AIJ, FOR A FLUID WITH COMPRESSIBILITY OF

WATER...... 47

FIGURE 2.8. BIJ AS A FUNCTION OF AIJ BASED ON EQUATION 16 AND 17...... 48

FIGURE 2.9.FLORY-HUGGINS INTERACTION PARAMETER AS A FUNCTION OF ∆A...... 50

FIGURE 2.10. A) THE COLOR MAP OF THE INTERACTION PARAMETER FOR A RANGE OF AIJ

AND ∆B CHOICES, AND B) INTERACTION PARAMETER AS A FUNCTION OF ∆B AT CONSTANT

AIJ PARAMETER...... 51

FIGURE 2.11.A) THE DIFFERENCE BETWEEN THE NORMAL AND TANGENTIAL COMPONENTS

OF THE LOCAL PRESSURE TENSOR ALONG THE CALCULATION BOX, ASSOCIATED WITH THE

INTEGRAL IN EQUATION 22 ON THE RIGHT AXIS WHICH CORRESPONDS TO THE SURFACE

TENSION, FOR DENSITY OF 3 AND ∆A=22. AND B) NORMALIZED SURFACE TENSION BY THE

NUMBER DENSITY AS A FUNCTION OF FLORY-HUGGINS INTERACTION PARAMETER...... 53

FIGURE 2.12.RADIAL DISTRIBUTION FUNCTION OF DPD FLUID FOR DIFFERENT MDPD CUT-

OFF VALUES...... 54

FIGURE 2.13.RADIUS OF GYRATION AS A FUNCTION OF CHAIN LENGTH FOR DIFFERENT

SOLVENT QUALITIES...... 56

FIGURE 2.14.A) DIFFUSION COEFFICIENT AND, B) ZERO SHEAR VISCOSITY OF MDPD FLUID

AS A FUNCTION OF REPULSION PARAMETER, AIJ FOR A RANGE OF BIJ PARAMETERS AND

DENSITIES...... 57

FIGURE 2.15. A) DIFFUSION COEFFICIENT (SOLID SYMBOLS) AND ZERO SHEAR VISCOSITY

(OPEN SYMBOLS), AND B) SCHMIDT NUMBER OF THE MDPD FLUID AS A FUNCTION OF

REPULSIVE PARAMETER, BIJ...... 58

FIGURE 2.16.MEAN SQUARE DISPLACEMENT OF THE MDPD FLUID, FOR A) B=100 AND A

RANGE OF ATTRACTIVE TERMS, AND B) A=20 AND A RANGE OF REPULSIVE TERMS...... 59

FIGURE 3.1.SCHEMATIC VIEW OF LEES-EDWARDS BOUNDARY CONDITIONS ...... 71

FIGURE 3.2. VELOCITY PROFILES FOR A RANGE OF SHEAR RATES WITH DISSIPATIVE

PARAMETER OF: A) 4.5 AND, B)50...... 72

FIGURE 3.3. DIMENSIONLESS TEMPERATURE VERSUS THE SHEAR RATE FOR A RANGE OF

DISSIPATIVE PARAMETERS...... 73

FIGURE 3.4. VISCOSITY VERSUS THE SHEAR RATE FOR A RANGE OF DISSIPATIVE

PARAMETERS...... 75

FIGURE 3.5. VISCOSITY AS A FUNCTION OF DISSIPATIVE TERM, GAMMA FOR DIFFERENT

SHEAR RATES...... 77

FIGURE 3.6. ZERO SHEAR VISCOSITY VS. TEMPERATURE, FOR DIFFERENT DISSIPATIVE

TERMS (LEGEND), WITH CONSERVATIVE FORCE PARAMETER BASED ON: A) ADAPTIVE AND B)

NON-ADAPTIVE METHOD...... 78

FIGURE 3.7. ZERO SHEAR VISCOSITY VS. DISSIPATIVE TERM, FOR DIFFERENT

DIMENSIONLESS TEMPERATURES), WITH CONSERVATIVE FORCE PARAMETER BASED ON: A)

ADAPTIVE AND B) NON-ADAPTIVE METHOD...... 80

FIGURE 4.1.TEMPERATURE VERSUS SHEAR RATE USING DPD THERMOSTAT FOR A RANGE OF

DISSIPATIVE PARAMETERS...... 90

FIGURE 4.2. DISTRIBUTION OF PARTICLE VELOCITIES IN THE FLOW DIRECTION, FOR A RANGE

OF SHEAR RATES AND DISSIPATIVE PARAMETERS...... 93

FIGURE 4.3. MAXIMUM (FILLED SYMBOLS) AND MINIMUM (EMPTY SYMBOLS) VELOCITIES

OF EACH LAYER AS A FUNCTION OF POSITION IN THE VELOCITY GRADIENT DIRECTION, FOR A

RANGE OF SHEAR RATES...... 94

FIGURE 4.4.TEMPERATURE VERSUS SHEAR RATE USING: A) GIANT, AND B) LOWE-

ANDERSEN THERMOSTAT...... 97

FIGURE 4.5. VELOCITY (IN THE FLOW DIRECTION) DISTRIBUTION FOR A RANGE OF SHEAR

RATES, USING: A) STANDARD DPD, AND B) GIANT THERMOSTAT...... 99

FIGURE 4.6. MAXIMUM (FILLED SYMBOLS) AND MINIMUM (EMPTY SYMBOLS) VELOCITIES

OF EACH LAYER AS A FUNCTION OF POSITION IN THE VELOCITY GRADIENT DIRECTION, FOR A

RANGE OF SHEAR RATES USING GIANT THERMOSTAT...... 99

FIGURE 4.7.RELATIVE VELOCITY DISTRIBUTIONS FOR DIFFERENT SHEAR RATES IN

DIFFERENT DIRECTIONS...... 101

FIGURE 4.8.VELOCITY PROFILE AT DIFFERENT SIMULATION TIMES FOR SHEAR RATES OF: A,

D) 0.5, B, C) 2.0, C, F) 10.0, USING STANDARD DPD (TOP ROW) AND NEW THERMOSTAT

(BOTTOM ROW)...... 102

FIGURE 4.9.VISCOSITY VS. SHEAR RATE FOR A RANGE OF DISSIPATIVE PARAMETERS USING:

A) REGULAR DPD THERMOSTAT, B) OUR PROPOSED THERMOSTAT, AND C) LOWE-

ANDERSEN METHOD COUPLED WITH DPD...... 103

FIGURE 5.1.A SUSPENSION REPRODUCED BY DPD (LEF) COMPARED TO HOW A SUSPENSION

HAS TO BE REPRESENTED IN ORDER TO PRESERVE THE COMPLETE HYDRODYNAMICS...... 117

FIGURE 5.2. SCHEMATIC REPRESENTATION OF POLY DISPERSED SUSPENSION THROUGH

BUNDLING INDIVIDUAL DPD PARTICLES (BOTTOM) VERSUS THE IDEAL SINGLE-PARTICLE

DEFINITION (TOP)...... 119

FIGURE 5.3. CORE-MODIFIED DEFINITION OF COLLOIDAL PARTICLES AS PRESENTED BY

WHITTLE AND TRAVIS...... 120

FIGURE 5.4. SCHEMATIC DECAY FUNCTIONS FOR CONSERVATIVE FORCE AND CONTACT

FORCES OF DIFFERENT SOFT LAYER THICKNESSES...... 126

FIGURE 5.5. THE SCHEMATIC DECAY FUNCTIONS OF COLLOIDAL INTERACTIONS AT VERY

CLOSE SEPARATION DISTANCES...... 127

FIGURE 5.6. ZERO SHEAR VISCOSITY OF SUSPENSIONS MODELED AT DIFFERENT VOLUME

FRACTIONS COMPARED WITH EXPERIMENTAL AND COMPUTATIONAL DATA...... 131

FIGURE 5.7. MEAN SQUARED DISPLACEMENT OF COLLOIDAL PARTICLES IN DIFFERENT

VOLUME FRACTIONS AS A FUNCTION OF TIME...... 132

FIGURE 5.8. PAIR CORRELATION FUNCTION OF SUSPENSIONS AT EQUILIBRIUM FOR

DIFFERENT CONCENTRATIONS...... 134

FIGURE 5.9. ZERO SHEAR VISCOSITY AS A FUNCTION OF VOLUME FRACTION FOR

SUSPENSIONS OF DIFFERENT CHARACTERISTICS (EFFECTIVE RADIUS SIZE)...... 135

FIGURE 5.10. SNAPSHOTS OF CRYSTALLIZED SUSPENSIONS UNDER SHEAR (SHEAR RATE OF

1), IN DIFFERENT PLANES AND CORRESPONDING PAIR CORRELATION FUNCTION GRAPHS. 137

FIGURE 5.11. VISCOSITY VS. PECLET NUMBER FOR DIFFERENT COLLOIDAL FRACTIONS

(GIVEN AS INSERTS), AND NORMALIZED VISCOSITY BY THE VALUE AT ONSET OF SHEAR-

THICKENING FOR ALL VOLUME FRACTIONS (BOTTOM RIGHT CURVE)...... 140

FIGURE 5.12. VISCOSITY CURVE FOR 58% SUSPENSIONS AT DIFFERENT CLEARANCE GAP

VALUES IN THE LUBRICATION EQUATION...... 141

FIGURE 5.13. FLOW CURVES OF SUSPENSIONS OF VARYING COLLOIDAL MODULI WITH

DIFFERENT VOLUME FRACTIONS...... 144

FIGURE 5.14.RELATIVE VISCOSITY AGAINST FRACTION OF COLLOIDAL PARTICLES FOR:

LEFT) SOFT COLLOID WITH MODULUS OF 100, AND RIGHT) RIGID COLLOIDS OF THE

MODULUS 25000...... 144

FIGURE 5.15.MAXIMUM PACKING FRACTION CALCULATED BY DIFFERENT MODELS VERSUS

THE PECLET NUMBER FOR SOFT AND RIGID SUSPENSIONS...... 145

FIGURE 5.16. LEFT) RELATIVE VISCOSITY VERSUS THE COLLOIDAL FRACTION FOR A RANGE

OF CONTACT MODULUS VALUES AT HIGH SHEAR RATE, PE=320, AND RIGHT) MAXIMUM

PACKING FRACTION CALCULATED FROM THE RELATIVE VISCOSITY AS A FUNCTION OF

MODULUS...... 146

FIGURE 5.17. VISCOSITY VS. PECLET NUMBER (LEFT) AND STRESS (RIGHT) FOR 58%

SUSPENSIONS OF DIFFERENT STRENGTH, WITH (SOLID LINES AND SYMBOLS) AND WITHOUT

(DASHED LINES AND EMPTY SYMBOLS) LUBRICATION POTENTIAL...... 147

FIGURE 5.18. FIRST (LEFT) AND SECOND (RIGHT) NORMAL STRESS DIFFERENCES VERSUS

THE PECLET NUMBER FOR A RANGE OF VOLUME FRACTIONS AND CONTACT MODULUS

VALUES...... 151

FIGURE 5.19. FIRST (LEFT) AND SECOND (RIGHT) NORMAL STRESS DIFFERENCES VERSUS

THE SHEAR STRESS FOR DIFFERENT VOLUME FRACTIONS OF NEAR HARD-SPHERE

SUSPENSIONS...... 152

FIGURE 5.20. FIRST (LEFT) AND SECOND (RIGHT) NORMAL STRESS DIFFERENCE

COEFFICIENTS VERSUS THE VOLUME FRACTION OF COLLOIDAL PARTICLES FOR A RANGE OF

MODULUS VALUES AT HIGHEST SHEAR RATE (PE=320)...... 153

FIGURE 5.21. TOTAL PRESSURE OF: LEFT) 58% SUSPENSIONS VERSUS THE PECLET NUMBER,

AND RIGHT) SUSPENSIONS AT PE=320 VS. THE SOLID PARTICLE VOLUME FRACTION...... 154

FIGURE 5.22. PAIR CORRELATION FUNCTION IN VELOCITY-GRADIENT DIRECTION FOR SOFT

AND RIGID 58% SUSPENSIONS OVER A RANGE OF SHEAR RATES...... 155

FIGURE 5.23. PAIR CORRELATION FUNCTION OF THE 58% SUSPENSIONS WITH DIFFERENT

MODULI, AT PE=320...... 156

FIGURE 5.24. REDUCED VISCOSITY VS. REDUCED STRESS OF BIMODAL SYSTEMS WITH SMALL

PARTICLE- LARGE PARTICLE COMPOSITIONS OF: A)10-90, B)25-75, C)50-50 AND D)75-25.

EXPERIMENTAL RESULT FROM N. J. WAGNER ET AL...... 158

FIGURE 5.25. REDUCED VISCOSITY VERSUS SHEAR RATE FOR THE DIFFERENT SIZE RATIOS IN

BIMODAL SYSTEMS OF WITH THE COMBINATION OF 10-90 (A) AND 25-75 (B) FOR THE

SMALL-LARGE COLLOIDAL PARTICLES...... 159

FIGURE 5.26. REDUCED VISCOSITY (EXPERIMENTAL DATA FROM D’HAENE AND MEWIS)

VERSUS REDUCED STRESS FOR BIMODAL SYSTEMS OF WITH RATIO OF 6:1 AND

COMBINATION OF 10-90 (A) AND 50-50 (B) FOR THE SMALL-LARGE COLLOIDAL PARTICLES.

...... 160

FIGURE 5.27. CONTRIBUTION OF DIFFERENT FORCES (%) VERSUS PECLET NUMBER FOR

MONODISPERSE COLLOIDAL SYSTEM OF WITH RADIUS OF 1.5 (A) AND 3.0 (B) FOR THE

COLLOIDAL PARTICLES...... 162

FIGURE 5.28. CONTRIBUTION OF DIFFERENT FORCES (%) INTO TOTAL FORCE ACTING IN THE

SYSTEM VERSUS PECLET NUMBER, FOR BIMODAL SYSTEMS OF WITH RATIO OF 6:1 AND

COMBINATION OF 10-90 (A) AND 50-50 (B) FOR THE SMALL-LARGE PARTICLES...... 162

FIGURE 5.29. SNAPSHOTS OF THE SIMULATION SYSTEM FOR DIFFERENT PECLET NUMBERS.

THE PARTICLES CONTRIBUTING TO CLUSTERS ARE SHOWN IN RED...... 164

FIGURE 5.30. THE NUMBER OF HYDRO-CLUSTERS VERSUS THE PECLET NUMBER FOR

MONODISPERSE COLLOIDAL SYSTEM OF WITH RADIUS OF 1.5 (A) AND 3.0 (B) FOR THE

COLLOIDAL PARTICLES...... 165

FIGURE 5.31. PERCENTAGE OF PARTICLES CONTRIBUTING INTO CLUSTERS FOR

MONODISPERSE COLLOIDAL SYSTEM OF WITH RADIUS OF 1.5 (A) AND 3.0 (B) FOR THE

COLLOIDAL PARTICLES...... 166

FIGURE 5.32. CONTRIBUTION OF PARTICLES (%) IN FORMATION OF HYDRO-CLUSTERS

VERSUS PECLET NUMBER FOR BIMODAL SYSTEMS OF WITH RATIO OF 6:1 AND

COMBINATION OF 10-90 (A) AND 50-50 (B) FOR THE SMALL-LARGE COLLOIDAL PARTICLES.

...... 166

FIGURE 5.33. CONTACT NETWORKS FORMED AT DIFFERENT SHEAR RATES IN 58%

SUSPENSIONS AND MODULUS OF 25000...... 168

FIGURE 5.34. SNAPSHOTS OF CONTACT NETWORKS FORMED WITH (LEFT) AND WITHOUT

(RIGHT) LUBRICATION POTENTIALS...... 170

FIGURE 5.35. DIMENSIONLESS POTENTIAL ENERGIES OF SOLVENT AND COLLOIDAL

PARTICLES AS A FUNCTION OF SOLID CONCENTRATION...... 171

FIGURE 5.36. POTENTIAL ENERGY OF SUSPENSIONS WITH DIFFERENT VOLUME FRACTIONS

UNDER FLOW...... 172

FIGURE 5.37. POTENTIAL ENERGY OF 58% SUSPENSIONS WITH (SOLID LINE AND SYMBOLS)

AND WITHOUT (DASHED LINES AND EMPTY SYMBOLS) LUBRICATION POTENTIAL, FOR

VARIOUS CONTACT MODULI...... 173

FIGURE 5.38. THE POTENTIAL ENERGY OF THE COLLOIDS (BOTTOM) AND SOLVENT (TOP) AS

A FUNCTION OF PÉ FOR DIFFERENT VOLUME FRACTIONS (LEFT) AND CONTACT FORCES

(RIGHT), NORMALIZED BY ITS VALUE AT THE EQUILIBRIUM CONDITION WITH (SOLID) AND

WITHOUT (OPEN) LUBRICATION...... 175

FIGURE 5.39.POTENTIAL ENERGY MEASURED FROM DIFFERENT TYPES OF INTERACTIONS:

(LEFT) SOLVENT-SOLVENT, (MIDDLE) COLLOID-SOLVENT AND, (RIGHT) COLLOID-COLLOID,

WITH (SOLID) AND WITHOUT (OPEN) LUBRICATION POTENTIAL, FOR A RANGE OF VOLUME

FRACTIONS (TOP ROW) AND CONTACT POTENTIALS (BOTTOM ROW), NORMALIZED BY ITS

VALUE AT EQUILIBRIUM...... 176

FIGURE 5.40. THE FRACTION OF PARTICLES IN CONTACT NORMALIZED BY THE TOTAL

NUMBER OF COLLOIDS WITH (SOLID) AND WITHOUT (OPEN) LUBRICATION INTERACTIONS

FOR DIFFERENT VOLUME FRACTIONS (LEFT) AND CONTACT POTENTIALS (RIGHT)...... 177

FIGURE 5.41. SNAPSHOTS OF GEL FORMATION IN 15% COLLOIDAL SYSTEMS AT DIFFERENT

TIMES...... 181

FIGURE 5.42. VISCOSITY/MODULUS AT NO-FLOW CONDITIONS VERSUS THE VOLUME

FRACTION FOR COLLOIDAL GELS...... 182

FIGURE 5.43. MEAN SQUARED DISPLACEMENT GRAPHS OF COLLOIDAL GELS WITH

DIFFERENT VOLUME FRACTION OF SOLID PARTICLES IN LINEAR (TOP) AND LOGARITHMIC

(BOTTOM) SCALES VERSUS TIME...... 184

FIGURE 5.44. AVERAGE NUMBER OF COLLOIDAL BONDS AS A FUNCTION OF TIME FOR

DIFFERENT VOLUME FRACTION OF COLLOIDAL PARTICLES...... 185

FIGURE 5.45. DISTRIBUTION OF NUMBER OF BONDS IN A 15% COLLOIDAL GEL SYSTEM AT

STEADY STATE CONDITIONS...... 186

FIGURE 5.46. PAIR CORRELATION FUNCTION GRAPHS OF COLLOIDAL GELS WITH DIFFERENT

CONCENTRATIONS...... 187

FIGURE 5.47. PCF GRAPHS OF 15% GEL AT DIFFERENT CALCULATION TIMES...... 188

FIGURE 5.48. PAIR CORRELATION FUNCTION GRAPHS OF 15% COLLOIDAL GELS AT VARIOUS

SHEAR RATES IN DIFFERENT PLANES...... 189

FIGURE 5.49. SNAPSHOTS OF COLLOIDAL GEL AT DIFFERENT SHEAR RATES SHOWN FOR

DIFFERENT PLANES...... 190

FIGURE 5.50. AVERAGE NUMBER OF BONDS VERSUS STRAIN UNITS, FOR 15% COLLOIDAL

GEL SYSTEM UNDER DIMENSIONLESS SHEAR RATE OF 0.5...... 192

FIGURE 5.51. DISTRIBUTION OF BOND NUMBERS AT DIFFERENT STRAIN UNITS FOR 15%

COLLOIDAL GEL UNDER SHEAR RATE OF 0.5, BASED ON: LEFT) SHORT-RANGE AND, RIGHT)

LONG-RANGE DEFINITION FOR BOND FORMATION...... 193

FIGURE 5.52. SNAPSHOTS OF COLLOIDAL GEL UNDER 0.5 SHEAR RATE FLOW, IN DIFFERENT

PLANES AT DIFFERENT STRAIN UNITS...... 194

Acknowledgments

One of the most challenging and yet enjoyable chapters of my life is coming to an end and it is definitely not an easy task to thank everyone who helped me to get where I am today. Not only because every single piece of education and advice throughout my life has led to this point, but also because without the support of the people who believed in me, I would have never been able to complete this chapter. So, I want to devote this section to express my sincerest gratitude to the ones who helped and guided me along this unforgettable journey.

First and foremost, I want to thank my advisor, Dr. Joao Maia, for giving me the chance to explore and extend my boundaries beyond what I believed possible. His mentorship and guidance throughout the course of my graduate career have taught me how to think and how to perform scientific research independently. I would also like to thank my co-advisor Dr. Daniel Lacks, for his continuous support and guidance during the past few years. I am grateful to other members of my thesis committee, Dr. Gary Wnek and Dr.

Michael Hore for their time and invaluable input to enhance the quality of this thesis. I want to extend my gratitude to Macromolecular Science and Engineering department faculty who taught me many aspects of the scientific research during the past years: Dr.

Alexander Jamiesson, Dr. Hatsuo Ishida, Dr. Liming Dai and Dr. Ica Manas-Zloczower.

Special thanks goes to current and past members of Maia research team for their friendship and collaboration, and for providing a scientifically challenging environment:

Dr. Mikio Yamanoi, Dr. Jorge Silva, Dr. Patrick Harris, Dr. Jia Liu, Dr. Ricardo Andrade,

Dr. Creusa Ferreira, Dr. Rongzhi Huang, Sidney Carson, Jessey Gadley, Pamela Garcia,

Tyler Sneider, Unique Luna, Chaitania Danda, Sangjin Lee, Brandy Grove. A very special

thank goes to two members of the computational rheology sub-group in Maia research

team: Shaghayegh Khani and Arman Boromand. The energy, the ideas and the

encouragement that you gave me every day in our discussions and conversations, made

everything possible for me. Thanks for putting up with me for the past few years.

I want to thank my family back home, in Iran and in Australia. My mom has

sacrificed so much for me to pursue what I love, and continues to do so every day with an

endless love and energy. My dad, a hardworking family-loving man, taught me to believe in myself and reach for my dreams, and it is his forever-lasting memory that keeps me going in the toughest times. My brothers Reza and Sina, thank you for being the best brothers anyone could ask for and for always being there for me. I want to thank the rest of my family: Narmin, Neda, Masoud, Farangis and Shafagh, and my niece and nephew

Niki and Rastin. I promise that I will always do everything in my power to make you proud.

As an international student far from home, my Cleveland friends have quickly become family to me. My Cleveland family, Arman, Afsoon, Bardia, Maryam, Nahal,

Ladan, Elham, Bardia, Pat, and Aly (and Grady, and Riley too), you mean the world to me and I always will love you with my whole heart.

Finally, Shaghayegh, my wife, my love and my best friend. You have given up so much for me, and you have been with me through thick and thin for the past 11 years. You have always raised me when I fall, and loved me unconditionally. I hope to one day be able to give you the life you want and you deserve. I would have never been where I am today without your love and support. I love you with my whole life.

Rheology of Colloidal Suspensions: A Computational Study

SAFA JAMALI

Abstract

Computational studies have emerged as a key class of scientific approached to solving different problems of interest in the past few decades. Dissipative Particle

Dynamics, DPD, a mesoscale simulation technique based on Molecular Dynamics has been established as a powerful technique in recovering a wide range of physical and chemical processes. Nevertheless, absence of robust bridge between the computational parameters to the physical characteristics of a system has limited applications of DPD. Thus in the second chapter of this dissertation (after a brief introduction and organizational guideline in chapter 1) a systematic study will be presented, providing several routes for setting the simulation parameters based on the real experimental measures.

Although computational and theoretical works have always been a crucial areas of research in the rheology society, DPD has not been employed in rheological studies. This is mainly due to the fact that a step-by-step guideline does not exist for rheological measurements in DPD. Another reason for this lack of success in rheological community is that the built-in thermostat in DPD is not capable of providing a stable control over the thermodynamics of the system under flow conditions. Thus, firstly in chapter 3 different methods of viscosity measurement and rheological studies will be discussed in detail, and consequently in chapter 4 a novel thermostat is presented to modify the natural shortcomings of DPD under flow.

For decades now, scientists across different disciplines have attempted at

identifying the nature of versatile rheological response of colloidal suspensions. Exhibiting

Newtonian behavior at very low, shear-thinning at intermediate, and shear-thickening at high flow rates in dense colloidal suspensions exemplifies a broad range of rheological regimes within a simple solid-liquid system. Despite numerous experimental and computational efforts in explaining the underlying mechanism of these behavior, there is still an ongoing debate in the scientific community on the subject. Hence, in final chapter a comprehensive study on rheology of colloidal suspensions (including a complete flow curve, normal stress measurements and microstructural evolutions) is presented, based on the results and foundations in prior chapters as well as in the literature.

CHAPTER 1 INTRODUCTION

1.1. Computer Simulations

Science is defined (in the Oxford dictionary) as: “The intellectual and practical

activity encompassing the systematic study of the structure and behavior of the physical

and natural world through observation and experiment”. Thus a scientist’s job is to “know”

and to be able to explain a physical phenomenon. The systematic study of comprehending

a particular observation or experiment usually starts with hypothesizing. Consequently

based on the hypotheses, theories are constructed. Since the language of theory is

mathematics, a crucial step for this process is to relate the mathematical theories to the real

life physical phenomenon subject to study. Theories are often associated with assumptions

and approximations; however, the importance of these assumptions and the extent of

compromised details are both unknown and uncontrollable, making the theoretical

predictions questionable. While an over-simplified theory is very likely to miss out on important details and thus fail to explain the experiment, adding these details to the theory can significantly increase the computational cost to provide quantitative results from those mathematical expressions. This is the birthplace of the computer simulations.

The first time that computer simulations were used in a scientific project was during the World War I, in a project called “Manhattan project”. The simulations were performed on 12 hard spheres to study detonation of an atomic bomb. One can clearly understand the importance of details in such areas of study and the complicated nature of calculations in order to provide quantitative predictions. By improving the computer technologies, the computer simulations have also grown substantially ever since. Nowadays, computer

simulations are employed extensively in almost every branch of science. These studies are

based on the foundations of theories where usually exact solution of a phenomenon is

known, without the approximations and assumptions associated with analytical and

numerical methods. In this regard, one can think of computer as a laboratory where

different theories can be examined and tested. This practice has made substantial

contribution to current understanding of different physical phenomenon.

Similar to experimental practice, in computer simulations one needs certain tools

to carry out the study. The mathematical models and methods are the tools required for

performing any simulation. In other words, the theoretical expressions are tested via these

mathematical practices in order to provide quantitative results. As in the experimental world one may seek information about the atomistic and molecular state of a matter, simulation methods have been proposed and evolved in order to provide insight to these scales. Nevertheless, the applicability of these methods is lost, when long time behavior or extreme length scales are of interest. Thus, various simulation methods have been developed trying to solve the problems in different time and length scales to match the physical problem in question. Figure 1.1 provides a general scheme of different time and length scales and examples of the physical phenomenon that can be studied at each level.

Figure 1.1. Different time and length scales of computer simulations. One can clearly distinguish between the different scales of simulation studies and their relevance to the physical phenomenon in figure 1.1. While fundamental atomistic level simulations are generally the subject of physics and electronics research and employ quantum mechanics tools to perform the calculations, on the other extreme end of the graph engineering scales attempt to predict very long time behavior of large engineering scales such as the operation units or the aerodynamics and fluid dynamics in extreme conditions.

Nonetheless most of the relevant observations in the field of material science occur in the intersection of intermediate scales, where the molecular level chemistry and physics are employed to explain microstructural state or a particular response of a material. Very often this response is in the experimentally measurable scales of few micro seconds to few

seconds in time, and few nanometers to several hundred micrometers in length. In order to

explain the molecular scale phenomenon, several numerical models have been developed

over the years: Monte Carlo [] simulations are perhaps the very first method of simulation

used in material science, Lattice-Boltzman [2] method as a discrete approach, and perhaps the most widely used method of Molecular Dynamics (MD) [3]. In MD, very fundamental and not so new theories such as the Newton’s laws of motion are employed to solve the many-body problems in different conditions. An all-atom approach is taken in MD simulations, enabling quantitative calculation of all the molecular properties. Although the computational capabilities have substantially improved over the past decades, still performing molecular level simulations in order to explain the experimentally relevant time/length scale phenomenon in macroscale is not possible. Thus, intermediate scale models have been developed trying to bridge the atomistic level theories to the macroscopic responses of materials. This class of models is referred to as “mesoscale” methods. In the mesoscale models, the fundamental theories and laws of motion are used in combination with continuum models. In other words, several assumptions and simplifications have to be made in order to reduce the computational cost of the molecular level models and make these methods capable of reaching larger time and length scales required at macroscale methods. One of the youngest and yet most successful mesoscale techniques is Dissipative

Particle Dynamics (DPD). The following section briefly explains the initial introduction and birth of Dissipative Particle Dynamics method in the scientific community.

1.2. Dissipative Particle Dynamics

The Molecular Dynamics simulations were employed for solving scientific

problems, shortly after the appearance of the first digital computers [4]. As it was

22 mentioned in the previous section, in MD, fundamental laws of motion are employed to explain the properties of matter at different conditions; however, the computational cost for molecular level techniques to provide information about the experimentally relevant time/length scales limits the application of MD. DPD as a mesoscale was initially introduced to the scientific community based on similar foundations as MD [5, 6]. In DPD, instead of a complete description of all the molecular components in a system, many atoms or many molecules assemblies, represented in the form of particulate beads are used for simulation of different materials. The process of grouping several atoms/molecules into one bead is referred to as “coarse-graining”. A schematic representation of the coarse- graining process is given in figure 1.2.

Figure 1.2. Typical coarse-graining at different levels, from the molecular level to beads.

One can clearly see that as a result of coarse graining, and by changing the coarse

graining level the number of calculations significantly decreases. Although this process

substantially changes the numerical efficiency of a model, one has to be careful when using the coarse grained methods. This is simply because at the coarse-grained level, the molecular properties and clear definition of each component is not present anymore, and is replaced by equations that are associated with many assumptions and approximations.

Consequently in the process of coarse-graining one can simply lose the physical meaning of the mathematical expressions. Over the past 25 years, and since its initial introduction to the scientific community, DPD has been used for several applications: From initial attempts to model colloidal suspensions under flow [5, 6], to simulation of blood flow [7,

8], to complex polymeric systems [9, 10] are only a few examples. Despite of the broad applications of DPD, the number of publications with fundamental research on the methodology of the DPD itself is growing continuously. Additionally, many reports show that DPD fails to capture the experimental phenomenon in question properly. This suggests that regardless of the extensive success of DPD in modeling different real life observations, there is still many aspects of this simulation technique that remain unexplored. One of the most important fields of research, which has been the subject of computational research from the early developments in the simulation techniques, is the fluid dynamics and rheology. Thus in the following section a brief introduction to the field of rheology is presented.

1.3. Rheology

The word “Rheology” was firstly used by Eugene Bingham and Marcus Reiner in

1920 inspired by the famous quote of Heraclitus: “Everything flows” [11, 12]. Originated

by Greek “rheo: flow and logia: study of”, rheology focuses on the behavior of matter at

different flow conditions. While the fluid mechanics research has dealt with the flow

properties and the response of liquid state matters and their response, solid mechanics is

the field of studying the behavior of well-defined structures (at rest) upon external forces.

As mentioned above, inspired by the concept that everything flows, rheologists study the response of matter in general, regardless of its state, to the external force/flow. This is best exemplified by the concept of Deborah number (= Time of a physical event / Time of observation) which was firstly introduced to the rheology community in 1964 [12]. Marcus

Reiner named the dimensionless number of Deborah, after a verse in the song of Deborah judges 5:5: “Mountains flowed before the Lord”, meaning that one can observe the flow of mountains, if he/she had the life of Lord. This (although in extreme situation) exemplifies the importance and the applicability of rheology in different fields of science and everyday life. Thus the ultimate goal of rheologist is not only to study the reaction of a fluid to particular flow, but to also understand the underlying mechanisms that lead to such response.

Perhaps the most important quantity of interest in the field of rheology is viscosity of a fluid. Qualitatively, viscosity is generally defined as the resistance of the fluid to the applied force or flow, which is raised from the collision of fluid particles. Quantitatively, the ratio between the stress and the deformation rate in a fluid is defined as its viscosity.

Consequently a fluid is rheologically characterized in regards to its viscosity in different flow conditions. For example, a fluid is referred to as Newtonian when the viscosity remains constant independent of the deformation rate at a given temperature or pressure.

Respectively, when the viscosity decreases with increasing the deformation rate the fluid

is shear-thinning. An opposing response is observed when the fluid becomes more resistant

to the flow by increasing the strain rate, corresponding to shear-thickening behavior.

In order to measure the rheological properties of a fluid, different rheometry

techniques have been developed. The main goal of these techniques is to mimic the relevant

real flow conditions such as pressure driven, extensional and drag flows. Regardless of the

type of flow and the response of the fluid, a general methodology is adapted in all

rheometry techniques: an external deformation/stress is applied on the fluid, and the

corresponding stress/deformation is measured. Similar to experiments, in computational studies, researchers have made efforts in reproducing the same rheometry techniques and to measure the rheological properties of fluids virtually. One of the fields of rheology research, which has been the subject of numerous studies is the rheology of suspensions.

Thus in the following section we briefly present the historical aspects and the significance of this field.

1.4. Rheology of Suspensions

The mixture of a solid-fluid is called a suspension in general terms. Considering

the generality of this definition, one can find examples of suspensions virtually

everywhere: from our everyday foods (milk, juices, soups and etc.), to cosmetics

(fragrances, creams and lotions, toothpaste and etc.), to industrial examples (in oil and

energy industry), to extreme examples (icebergs in the ocean). Not only because of the

practical importance of this class of materials, but also because of their versatile rheological

responses, have suspensions always received increasing enthusiasm from the fluid

dynamicists and physicists. So-called dense suspensions show Newtonian behavior while

at quasi-equilibrium conditions or at very low shear rates, followed by shear-thinning

response to intermediate shear rates. This is surprisingly further followed by shear- thickening behavior at elevated deformation rates. Such versatile rheological responses from a simple solid-liquid mixture has attracted many scientists to try to understand the underlying mechanisms.

The particulate nature of the suspensions, as well as the relevant time/length scales at which different rheological behavior of a suspension is observed, makes the field of suspension rheology one of the most attractive and successful fields of numerical simulations. The very first simulations performed by Molecular Dynamics simulations [4]

and Dissipative Particle Dynamics [5] have attempted to reproduce the flow properties of

suspensions. In addition to these methods, Stokesian Dynamics and Brownian Dynamics

simulations have significantly contributed to the current understanding of the suspension

rheology [13-15].

Despite the exceptional success of numerical studies and the versatility of reports

focusing on this subject, rheology of suspensions is still one of the most controversial fields

of active research in the physics and fluid dynamics society. This is due to a continuous

and evolving development of different theories that aim at explaining the underlying

physics of suspensions. In other words, while opposing theories have been able to

reproduce different particular macroscopic response of a suspension, none of these theories

is able to recover the complete rheological behavior of suspensions.

1.5. Organization and Scope of Dissertation

In the next chapter, the details of DPD technique and more specifically the Many-

body Dissipative Particle Dynamics (MDPD) method are presented. Efforts will be made

in order to map the simulation parameters into relevant experimental measures. This is

absolutely crucial for any computer simulation method, in order to confirm the reliability

of the produced results. The parametrization methods presented in this chapter ensure that a real fluid is being modeled in the equilibrium (no flow) conditions.

Considering the need for a thorough characterization of the Dissipative Particle

Dynamics technique under flow, in Chapter 3 a systematic study on different rheometry and viscosity measurement techniques via DPD is presented. The aim of this chapter is to provide the necessary tools for a rheological study to be performed by DPD. Namely, the effect of each parameter on the rheological response of a DPD fluid, and the physical consequences of the choice of rheometry technique will be discussed in detail.

One of the most important limitations of DPD in rheological studies arises from the numerical instabilities associated with the thermostatting effect of the DPD ensemble. Thus in Chapter 4, after careful review of the aforementioned issue and identifying the sources of instabilities, a novel algorithm is presented to address the problem of temperature instabilities under flow conditions. By the end of this chapter, the rheological tools will be examined and the verification of the method will be made.

The limitations of DPD in order to capture the rheological response of suspensions will be discussed in Chapter 5. Consequently, modifications to the DPD model will be made accordingly and the results on the rheology of colloidal suspensions under shear will

28 be presented. General flow curve of a suspension, in addition to other rheologically relevant properties of suspensions will be discussed, and efforts will be made in order to correlate the macroscopic behavior of suspension to the theoretical explanations. Furthermore, effect of several parameters such as particle size and composition, with special emphasis on the rheological behavior of bimodal suspensions will be studied.

Upon a careful study of the neutral hard-sphere suspensions and based on the results obtained, chapter 5 will be concluded by a brief discussion on the rheology of colloidal gels. Namely, the rate dependent and time-dependent behavior of attractive gels under flow will be studied in addition to distinctive behavior of the colloidal gels at equilibrium conditions as compared to neutral freely-moving suspensions.

1.6. References

1. Binder, K. and D. Heermann, Monte Carlo simulation in statistical physics: an introduction. 2010: Springer Science & Business Media. 2. Chen, S. and G.D. Doolen, Lattice Boltzmann Method For Fluid Flows. Annual Review of Fluid Mechanics, 1998. 30(1): p. 329-364. 3. Rapaport, D.C., The Art of Molecular Dynamic Simulation. Second ed. 2004, New York: Cambridge University Press. 4. Alder, B.J. and T. Wainwright, Studies in molecular dynamics. I. General method. The Journal of Chemical Physics, 1959. 31(2): p. 459-466. 5. Hoogerbrugge, P.J. and J.M.V.A. Koelman, Simulating Microscopic Hydrodynamic Phenomena with Dissipative Particle Dynamics. Europhysics Letters, 1992. 19(3): p. 155-160. 6. Hoogerbrugge, P.J. and J.M.V.A. Koelman, Dynamic Simulations of Hard-Sphere Suspensions Under Steady Shear. Europhysics Letters, 1993. 21(3): p. 363-368. 7. Pivkin, I.V. and G.E. Karniadakis, Accurate Coarse-Grained Modeling of Red Blood Cells. Physical Review Letters, 2008. 101(11): p. 118105. 8. Ye, T., et al., Dissipative particle dynamics simulations of deformation and aggregation of healthy and diseased red blood cells in a tube flow. Physics of Fluids (1994-present), 2014. 26(11): p. -.

CHAPTER 2 BRIDGING SIMULATION TO EXPERIMENT

This chapter is adapted from: “Generalized mapping of multi-body dissipative particle dynamics onto compressibility and the Flory-Huggins theory”, by Safa Jamali, et al., Journal of Chemical Physics, 2015, 142 (16), pp: 164906 (11).

2.1. Introduction

The computational capabilities have advanced tremendously over the past few

decades, making a significant contribution to our current understanding of the molecular

scale characteristics of matter. Yet, there are clear limitations for the microscopic and

atomistic level simulation techniques to capture the features of real phenomenon. For

example, Molecular Dynamics (MD), one of the most popular simulation techniques, is

limited to a few nanometers and milliseconds in length and time respectively. For the same

reason methods of this scale are frequently employed to study the microstructure and

physics of phenomena. On the other hand, macroscopic methods such as Finite Element

Method (FEM) are unable to look at the microscopic evolution and are usually used to

discuss the bulk behavior of the material subject to simulations from the continuum

mechanics perspective. Thus, there is a clear need for simulation techniques that lie in

between these two different scales, i.e., in order to access length and time scales larger than

those accessible to microscale methods, but smaller than the lower limit in macroscale

ones. These simulation techniques are usually called “mesoscale” methods. The mesoscale

time and length scales are accessible by effective Coarse-Graining (CG) of the micro

scales. Coarse-graining is a process in which many small particles are grouped together to

form a larger bead. During the coarse-graining, small particles lose their degree of freedom

to an extent which is the relevant degree of freedom for the study at hand. Perhaps the most

well-known example of this process is coarse-graining of MD particles. However, one would need to make a very careful choice of mesoscale parameters since the meaning of different parameters fundamentally change by coarse-graining. For example, instead of simulating a real material/phenomenon, one can still reproduce a behavior through parameter manipulation without a clear definition of parameters. Thus, although CG techniques provide an effective way to extend the time and length scales, a potential limitation of these methods is lack of a clear map to actual materials and their characteristics.

Coarse-Grained Molecular Dynamics (CGMD) as an extension to traditional MD simulations provides a larger time/length scale compared to atomistic scales. Nevertheless, since CGMD in general utilized hard Lennard-Jones potentials, the coarse graining level accessible to this method remains limited. This is particularly important when time- dependent fluids, e.g. polymer melts, are subject of study. Dissipative Particle Dynamics

(DPD) was firstly proposed as a coarse-grained molecular dynamics method to study suspensions [5] and has been improving continuously ever since. The past two decades has seen an explosion of simulation studies using DPD mainly because while it uses the same concepts as MD, it allows for time and length scales that are orders of magnitude higher than MD.

2.2. Formalism and Parametrization of DPD

DPD is governed by the Langevin equation, which is a coupled stochastic first order

differential equation that describes the change in position ri and velocity vi of the particles

th in terms of vi and the total forces Fi on the i particle. The total force on a DPD particle is subsequently sum of three different forces and can be written as equation 2. In DPD, the equation of motion of a particle is written based on three different forces (equation 1-2):

The Random force (equation 3), which includes the thermal fluctuations in the system, the

Dissipative force (equation 4), which acts against the motion of particle and acts as the

viscosity of a fluid and the Conservative force (equation 5), which gives the chemical

identity of each particle and the extent of pressure between the different components of a

system.

dri = v , (eq. 1) dt i

dvi CDR mi =∑FFFij ++ ij ij , (eq. 2) dt

RRΘij Feij= σ ijw ij(r ij )ij , (eq. 3) ∆t

DD Fij=−⋅γ ijw ij(r ij )(v ij ee ij ) ij , (eq. 4)

CC Feij= aw ij ij(r ij ) ij , (eq. 5)

rij 1;−≤rrij c ω (rij ) = rc , (eq. 6)  ≥  0; rrij c

In equation 3, σ controls the strength of the thermal fluctuations and θij is a δ- correlated Gaussian random variable. The parameter γ in equation 4 controls the strength

of the dissipative force and acts as the viscosity for the system. Finally, the conservative

force in equation 5 is controlled by species dependent interaction parameter, aij . All DPD

potentials are calculated via a weight function (equation 6) which starts at unity when two

particles overlap and goes to zero at a distance between center of two DPD particles, called

“cut-off” distance. The cut-off distance in DPD is typically equal to the separation distance between two particles (=1.0). The weight functions in dissipative and random force are constrained by the Fluctuation-Dissipation [16] relations and define the dimensionless temperature in the system (equation 7). To date, equation 7 is the only mathematical

expression given for choosing the random and dissipative parameters. Since eq. 7 only

relates these parameters based on the dimensionless temperature, unlimited choices can be

made for these two parameters as long as the equation 7 is satisfied. Detailed study on the

physical consequences of setting these two parameters by this method will be presented in

chapter 3.

σ 2 = k T (eq. 7) 2γ b

2 ωω=CR = ωωD =  ω (rij ) ( rij ) ( rrij ), ( ij ) ( rij ) , (eq. 8)

The dissipative and random forces together form a thermodynamic ensemble that controls the temperature in the system and can be used to simulate non-equilibrium dynamics and incorporates Navier-Stokes hydrodynamics [17]. However, recent developments have shown that traditional forces in DPD are not able to capture short-range hydrodynamics and thus there is a need for a lubrication force [18]. Since the conservative force accounts for the particle-particle interactions and allows the surface tension and pressure to be calculated, it has to be included, otherwise only systems of non-interacting athermal particles can be modeled. Also, tunable compressibility is only possible with the

inclusion of the conservative force. DPD potentials are so soft that the cut-off distance is usually defined as the separation distance between two particles (one diameter). In fact, it is employing of very soft potentials that allows much higher scales to be achieved in DPD when compared to MD simulations.

As mentioned before, the choice of simulation parameters in DPD plays a crucial role in defining what is being represented by a particle. Since DPD parameters are dimensionless, this choice is arbitrary and not inspired by the physical characteristics of matter unless a mapping is defined and a relation between the simulation and real

parameters is established. The chemical potential of a DPD particle can be correlated to the

one of real matter by matching its surface tension properties. This can be done by

monitoring the effect of different conservative force parameters on the interfacial tension

and deriving the equation of state based on this potential. Groot and Warren [19] proposed a method to match different DPD terms to physical time and space parameters. They provided relations for the conservative force controlling parameter, a, of a DPD particle to

−1 be calculated based on its compressibility,κ (equation 9) and the Flory-Huggins χ

parameter (equation 10). This has significantly extended the range of phenomena accessible by DPD simulations. In particular, by including χ, the mapping enables DPD to study dynamics of polymer systems.

κ −1 −1 a≈ kT , (eq. 9) ij B 0.2ρ

0.2∆aρ χ = , (eq. 10) kTB

2.3. Formalism of MDPD

In 2001 Pagonabarraga and Frenkel [20] proposed the so-called Multi-body

Dissipative Particle Dynamics (MDPD) method, based on DPD with a modified

conservative force including a local density dependent term. Potentials such a hard

Lennard-Jones type inherently provide the complete van der Waals loop; however, in soft

DPD interactions the three-body interactions are mandatory for completion of a van der

Waals loop, and thus the addition of the density dependent term in the conservative force

makes the model capable of reproducing multiphase phenomena. MDPD’s novelty and

utility arises from the introduction of a density dependent attractive term to the

conservative force. Thus the equation 5 for the conservative force in DPD is replaced by

the equation 11:

CC D Frij=(ab ijω(rij ) ++ iji( ρ ρω j) (rij )) ˆij , (eq. 11)

Where bij is the density dependent interaction parameter that also depends on the

th d species involved, ρi is the average local density of the i phase and ω (rij ) is a weight

function that writes the same units as ω (rij ) with a distance rd for the cut-off. It should be

mentioned that aij ≤ 0 indicates the attractive nature of this term, in contrary to repulsive

ρ characteristic of bij ≥ 0 in the conservative force. The weight function, ω (rij ) , differs

D from ω (rij ) and is given by

2  15 r ρ  1− ; rr≤ ω = 3 ij d (rij ) 2π rrdd  , (eq. 12) ≥  0; rrij d

Where rd is the density dependent cut-off radius, which is usually set to 0.75 [21-

23] with rrcd> to prevent discontinuities at the cut-off. Other choices of the conservative force are equally valid [24], but the formulation in equation 12 is used for its simplicity.

Also, it is important to note that the matrices of force constants for aij and bij do not change during the course of a given simulation.

MDPD has been successfully employed for various studies mainly when a multiphase flow is of interest [21, 22, 25-28]. Trofimov et al. [29] verified the thermodynamic consistency of this formulation and extended it to describe a strongly non-ideal, multiphase system with a correction factor. In a series of publications, Ghoufi and coworkers used

MDPD to mimic various multiphase systems ranging from simple vapor–liquid mixture to polyelectrolytes and polymer brushes [23, 26, 27, 30]. Density dependence is necessary to describe even noble gases; however, there is no unique way to represent a pair potential that depends on density. Even though different multiphase systems can be described by the

MDPD method, so far there is no report providing a robust link between the physical parameters and the MDPD terms.

For DPD, perhaps the most well-known study has been done by Groot and Warren

[19] who found a quadratic equation of state (EOS) in terms of the density ρ with mean-

field valueα =0.101 ± 0.001. In case of MDPD, Merabia and Pagonabarraga [31, 32]

considered one-body density dependent energies and constructed several general EOSs in

terms of the structure factor and pair correlation functions using a saddle point expansion,

which predicts the introduction of a third order term for the MDPD EOS. Warren [28, 33]

developed an explicit EOS for MDPD and found a third order term, but compressibility

37 relationships and a relation to the Flory-Huggins χ parameter were not developed. This is in part due to the fact that the density dependent term in the conservative force has an additional dimension, which complicates the analysis. In a recent report, Warren [34] showed that the density dependent controlling parameter, bij, has to take a constant value for different components in order for the potential to remain conservative. Other approaches have assumed an EOS or attempted to derive conservative forces from their desired EOS. Fedosov et al. [35] assumed that the density dependence of b would add a cubic or third order term to the EOS. A more general fitting approach was adopted by

Arienti et al. [25], who described the EOS as a power-law. However in all of the above mentioned reports, a functional form of the EOS was assumed, which potentially limits the fitting capability of these studies. To date, the only report providing a bridge between the chemistry of a real fluid to MDPD parameters is the work by Ghoufi and Malfreyt [27], where a multi-scale approach is adapted based on the Flory-Huggins theory. Authors showed that by doing so, realistic representation of electrolytes with different characteristics can be obtained.

In the present chapter this issue is addressed by presenting a direct relationship between the simulation parameters in multi-body dissipative particle dynamics (MDPD) and experimental measures including the isothermal compressibility and the Flory-

Huggins χ parameter. The development of a χ relationship for DPD enables a wide range of studies regarding material behavior, particularly in co-polymers and polymer blends.

One would expect that compressibility and χ parameter relationships will allow for systematic and physically meaningful choices of conservative force parameters in MDPD and, more importantly, allow MDPD studies to use and contribute to the large literature

base of studies using the Flory-Huggins χ parameter. In addition, the effect of MDPD parameters on the transport properties of the DPD fluid, namely the viscosity and diffusion coefficient will be studied, and the dependency of Schmidt number on the a and b terms

will be discussed.

2.4. Equation of State in MDPD

In order to link the interaction parameters used in the MDPD formulation to real

parameters, one would need to start from derivating their effect on the excess pressure of

the system; however, this has to be done by considering the fact that having a negative aij

parameter does not allow calculations consisting of only aij parameters to be performed,

and thus the effect of this term has to be studied after being decoupled from the bij term (if

possible, which will be discussed in detail shortly). Thus, we first performed simulations

with only the repulsive (bij) term to derive proper bij -dependence of the pressure. It should be mentioned that the bij=0 case, which corresponds to the standard DPD model has been

studied extensively before, and there is a second order density dependence with aij, which

agrees with previous calculations. The pressure was measured using the virial expression

(with ρ = NV/ ):

NN−−11NN 1 CCρ P=+⊗+ ρρ kTbb∑∑rFij ij = kT ∑∑rFij⊗ ij , (eq. 13) 3VN i=11 ji>=3 i ji>

There are a number of other ways that the pressure tensor can be measured from

the interfacial tension as given by the Kirkwood-Buff [36] relation, the Irving-Kirkwood

[37] method, or the test area method [38]. Based on the comparative study of Biscay et al.

[39], all of the methods give essentially the same pressure tensor for DPD. We chose

equation 13 for consistency and because it gives a scalar result.

For the EOS, a simulation box with periodic boundary conditions containing 10,000 particles at densities from 3 to 10 particles per unit volume was used. The dissipative and random parameters were chosen to keep the dimensionless temperature at 1.0. The system equilibrated for 105 time steps of 0.01 time units before sampling. Data was then collected every time step for the next 106 steps. Averages with error bars for the pressure were obtained for each run. Figure 2.1 shows the total pressure as a function of bij and density.

Figure 2.1. Pressure as a function of: a) bij parameter, and b) particle density. The linear curves in figure 2.1.a show that in the absence of an attractive parameter, the pressure of a MDPD fluid depends linearly on the value of the repulsive parameter at any specific density. This suggests that one can obtain a mastercurve by simply dividing the measured pressure by the bij parameter. Figure 2.2 shows the pressure divided by the bij parameter for different densities.

Figure 2.2.Pressure divided by repulsive parameter as a function of: a) Density, and b) Cubic density. The results in figure 2.2.b show a cubic density dependence of the pressure, which is in agreement with previous studies. Based on the results in figure 2.2, we derive the repulsion dependence of the pressure as:

43 2 P=ρ kTB +2 α brij d ( ρρ −+ c d ρ), (eq. 14)

Where cd=4.69, = 7.55andα = 0.101. Now, with the proper correlation between

the pressure and the repulsive term at hand, one can seek the same relation for the attractive

term of the conservative force. To do this, we have performed the simulations of varying

attractive parameter with three different repulsive terms of 10, 30 and 50. The results of

these simulations are plotted in figure 2.3.

Figure 2.3.Pressure vs. Density for: a) bij =10, b) bij =30 and, c) bij =50.

It should be noted that the results in figure 2.3 include the calculations at which the measured pressure is a positive value. This is particularly important because as soon as a negative pressure is developed in the calculation box (which is an unphyisical phenomenon), the MDPD fluid begins to retract and the effective volume of the fluid becomes smaller than the set value by the NVT ensemble. Althought negative pressure values are physically possible in particular situations such as in metasbale materials, in order to provide a physically meaningful representation of an experimental fluid, one can only obtain positive pressures. Thus, regardless of the possibility of a negative pressure, the measurement of this quantity does not remain valid. In order to decouple the partial pressure terms originated by aij and bij, we now plot the total pressure subtracted by the repulsive contribution given in equation 14 (figure 2.4).

Figure 2.4. Absolute value of the total pressure subtracted by the repulsive pressure as a function of density for: a) bij =10, b) bij =30 and, c) bij =50. The discrepencies between the results in figure 2.4 suggests that even after excluding the pressure from the repulsive term, the residual pressure still varies for different bij parameter values; however, following the same procedure as for the repulsive term, we divide the entities in the figure 2.4 by the attractive parameter, aij, in order to obtain the mastercurve. The results in figure 2.5 shows these curves, being normalized by the extent of attractive parameter.

Figure 2.5.Absolute value of the attractive pressure normalized by aij parameter for: a) bij =10, b) bij =30 and, c) bij =50. The dash line shows the expression predicted for the conservative parameter dependence in standard DPD by Groot and Warren. Equation 15 gives the expression derived from DPD and eq. 14 as the complete

equation of state for MDPD (the second term in the righ hand side is the dashed line in

figure 2.5):

2 43 2 P=ρ kTB + αρ a ij +2 α brij d ( ρ −+ c ρ d ρ), (eq. 15)

The results in figure 2.5 reveal that although the second order dependence of aij for

MDPD previously reported by Warren predicts successfully predicts the pressure at low values of repulsive term, by increasing the bij parameter it understimates the contribution

of the aij term. This deviation is more pronounced at higher bij terms, and is also a reverse function of the aij parameter itself. In other words, high values of bij and small aij terms

lead to large discrepencies from the predicted value. This suggests that by increasing the

value of bij, and specifically at high densities, an aijbij term has to be introduced in the

system as well.

Figure 2.6.Measured pressure divided by its predicted value from the equation 15 as a function of density. The descrepencies in figure 2.6, in comparison with the pressure predicted by the equation 15 clearly shows that although at high densities, namely where the 3rd order density dependence of the repulsive term dominates the pressure, eq. 15 succesfully can describe the EOS of a MDPD fluid; however, at low and intermediate densities which are used in majority of MDPD studies, eq. 15 overpredicts the pressure values by up to two orders of differences at the most sever deviations. Thus, we derive the following equation which now includes the aijbij term, in equation 16.

4 2 43 2 αbrij d 2 P=ρ kTB + αρ a ij +2( α brij d ρ −+− c ρ d ρ )0.5 ρ , (eq. 16) aij

The EOS in eq. 16 now can predict the pressure of a MDPD fluid at different aij and bij parameters, with an extra term to explain the deviations from equation 15. Although

rd nd Warren reported 3 order dependence of bij accompanied by 2 order dependence of A for

MDPD, his equation of state cannot properly predict the pressure in this regime. Now, with

the generalized EOS at hand, the physical significance of each term in 16 can be discussed.

The first term on the right-hand side of eq. 16, ρkTb , is the term that corresponds to the

2 ideal gas. The second term,αρaij , originates from pairwise potentials in DPD and thus

explains the two-body interactions. The third term describes the three-body interactions originating from the introduction of a density dependent term in the conservative force, which interacts with the standard two-body interactions. In fact, it is this term that allows

MDPD to reflect the local densities and thus describe phase transitions as predicted. The last term on the right-hand side equation, with a 2nd order dependence on density is unexpected. A previous study suggests that in density dependent conservative potentials between soft particles, this term possibly represents formation of structure or phase separation at higher densities [32]. The emergence of this last term in the EOS can be explained as follows: the attractive term of the conservative force (aij term) has a weight

function with the cut-off distance of 1.0, similar to traditional DPD, as opposed to the

repulsive term (bij) which diminishes at the cut off distance of 0.75. Consequently, there

exists a region where two neighboring particles are interacting solely via the attractive

potentials. This pulls the particles to separation distances where the repulsive term becomes

effective and eventually dominates the attractive term. This competition between the two

opposing forces gives rise to a new term, which depends on both terms. In other words, the

extent of pressure originated by the attractive term interactively changes the contribution

of the pressure originated by the repulsive term as well. On the other hand, since DPD

particles represent very soft spheres, the excluded volume might be significant and

captured by the van der Waals. However, since this term includes a second order density

dependence and is negligible at low and intermediate densities, and at high densities the

pressure is completely dictated by the 3rd order term, it is safe to neglect this term for

compressibility and Flory-Huggins interaction parameters.

2.5. Compressibility and Flory-Huggins χ parameter

Compressibility can be used to help map the MDPD simulation parameters to

experimental fluids. As mentioned before, this mapping is available for DPD. For MDPD,

since both aij and bij parameters contribute to the compressibility, a line is specified for compressibility, density, and temperature. The compressibility equation is calculated and presented in equation 17. For these calculations, the compressibility,

−1 -1 κρ=1/ (kb T )( d / dP ) , was equated to the compressibility of water (κ =16) and solved

for either aij or bij.

1 ∂p 2αραa br42(6 ρ−+ 4 c ρ 2 d ) 2 α br4 ρ =++1 ij ij d −ij d (eq. 17) kT∂ρ kT kT 0.5 BB BkTaB ij

On the other hand, as mentioned before one should carefully consider the possibility

of building up a negative pressure in the calculation box. Now we have 2 equations

(compressibility, and also the EOS where P=0) and two unknowns. Thus, based on the

expression in 17, we substitute bij with an aij -dependent term in eq. 16, and plot the

pressure as a function of aij (figure 2.7).

Figure 2.7. Pressure as a function of aij, for a fluid with compressibility of water. The curve in figure 2.7 reveals that for a MDPD fluid to take the compressibility of

water (given in eq. 17) and have a total positive pressure, the aij parameter can not take values of less than 10 regardless of the fluid density. This is in contradiction with the majority of parameter sets reported in the literature. Now we can measure the bij term as a function of aij as well (figure 2.8). Once again, it should be noted that the aij and bij pairs

from the figure 2.8 are the ones to: first, yield a positive pressure and second, satisfy the

compressibility relationship. Figure 2.8 also shows that in the majority of the previous

studies performed via MDPD, either compressibility or pressure of the fluid was

compromised as the window of accessible range to each parameter is rather narrow.

Figure 2.8. bij as a function of aij based on equation 16 and 17. The Flory-Huggins theory is formulated in terms of the free energy per lattice site

F as presented in the equation 18.

F ϕϕAB =++lnϕA ln ϕB χ ϕϕ AB, (eq. 18) kTbA N N B

Where ϕA and ϕB are the volume fraction of the A and B components respectively,

and N A and NB are the number of segments for each A and B molecule. The free energy

is minimized at

1−ϕ ln A ϕA χN = , (eq. 19) 12− ϕA

Which is consequently a useful way to measure χ through phase separation, when

= = . For the phase separation and computational χ measurement, a similar

𝑁𝑁 𝑁𝑁𝐴𝐴 𝑁𝑁𝐵𝐵

48 simulation condition (as used for EOS derivations) was used except that the box was twice as large in the z-direction as it was in the other two directions with 6,000 particles, and the phases were initialized on different sides of the box in the z-direction. This was done to reduce the interface area and extend the single phase length in the z-direction to facilitate measurement of densities far away from the interface.

Consequently, one can derive equations to calculate the Flory-Huggins interaction parameter based off aij and bij. As was mentioned before, the repulsive parameter, bij, must

take a constant matrix value ( bbbAA= AB = BB ) to remain conservative [34], and on the other hand at low and intermediate densities, the aij - bij interaction term can be neglected. Thus, equation 16 can be reduced to its traditional DPD format which consequently yields the same expression for the Flory-Huggins interaction parameter, χ .

2αρ( +− ρ)(aa) χ,= A B AB AA (eq. 20) kTb

Now, in order to validate the expression for χ, one would need to: first, study the phase separation of two different components at varying values of Δa and second, measure the Flory-Huggins interaction parameter as a function of Δa to confirm the validity of equation 20. Although a non-zero Δb is a no-go rule for MDPD, we also perform the same type of study on this parameter at a single density, to better understand the significance of the bij matrix at the most widely used simulation condition ( ρ = 3). In practice, the phase separation phenomenon was studied by monitoring the density of each phase at equilibrium over the calculation box and the results for Δa are shown in figure 2.9.

Figure 2.9.Flory-Huggins interaction parameter as a function of ∆a. It should be noted that the mean value of x at a distance far enough from the

interface was used to estimate the Flory-Huggins interaction parameter. It should be noted that since the EOS and compressibility equations provide a guideline for choosing the aij

and bij parameters rather than unique solution, various pairs of parameters can be used to model the same pair of fluids. Thus at each density, several choices have been made for the particles for the interaction parameter measurements. The curve in figure 2.9 shows a clear phase separation between the two fluids based on the interaction parameter.

Regardless of the choice of parameter pair choice at each density, the Flory-

Huggins interaction parameter linearly increases by increasing the Δa at a constant slope.

The calculated slopes are given in equation 21-23.

χρ=(0.322 ± 0.003) ∆=a ;( 3), (21)

χρ=(0.739 ± 0.014) ∆=a ;( 5), (22)

χρ=(1.107 ± 0.023) ∆=a ;( 7), (23)

Similarly to results of Groot and Warren for DPD, the slopes measured for the interaction parameter and density deviate from the ones predicted by equation 20; however, this deviation for all three densities is the same (≅− 2α 0.3) . The exponents for χ here are based on the empirical measurement of this parameter from the phase separation of binary mixtures at the interface, rather than using the EOS-derived expressions for the Flory-

Huggins interaction calculations. As a result, this exponent of this parameter defers from the one suggested by Ghoufi and Malfreyt [27] where the authors employed a multi-scale approach and used the FH theory to obtain this exponent. Figure 2.10 shows the map of the interaction parameter as a function of Δb, for a range of aij choices for the density of 3.

Figure 2.10. a) The color map of the interaction parameter for a range of aij and ∆b choices, and b) Interaction parameter as a function of ∆b at constant aij parameter.

The map of χ for the whole range of Δb and aij reveals not only that having a non- zero Δb results in a non-conservative type of potential, but also for a large range it cannot effectively introduce any phase separation. The fit in the figure 2.10.b has a slope of 0.0025

which proves that changing the Δb does not significantly change the mixing behavior of

the two fluids at this density. This is expected at the density of 3, because the repulsive

term of the derivative of the EOS becomes negligible and thus, insensitive to the bij

parameter; however, one should note that this term becomes dominant at higher densities

as it includes a second order density term.

2.6. Interfacial Tension

One can subsequently measure the interfacial tension of binary mixtures with

different Flory-Huggins interaction parameters, in order to confirm the expressions given

in equation 21-23, and to study the stability of an interface as a function of this parameter.

The surface tension can be expressed as a function of the normal ( pyN ( ) = pyyy ( ) ) and

1 tangential ( py( ) =[] p( y) + p( y) ) components of the local pressure tensor, and T 2 xx zz

using the Irving-Kirkwood expression [37] as shown in the equation 24.

L /2 1 y σ = pNT( y) − p( y) . dy (eq. 24) 2 ∫ −Ly /2

Various expressions and calculation methods for the surface tension measurments in MDPD and DPD can be found elsewhere [23, 26], where authors reported the surface tension measurments of a liquid-vapor mixture at different conditions. In our simulations, we measured the surface tension of bindary mixtures with different densities and Flory-

Huggins interaction parameters, in order to validate confirm the validity of equations 21-

23. Figure 2.11.a shows the typical pressure components and surface tension of a binary mixture, whilte figure 2.11.b presents the value of surface tension as a function of χ parameter, for several densities. As theoretically predicted, the curve of the normalized

52 surface tension given in figure 2.11.b collapses on a single curve for different densities, confirming the validity of eq. 21-23.

Figure 2.11.a) The difference between the normal and tangential components of the local pressure tensor along the calculation box, associated with the integral in equation 22 on the right axis which corresponds to the surface tension, for density of 3 and ∆a=22. And b) Normalized surface tension by the number density as a function of Flory-Huggins interaction parameter. 2.7. Validation on polymer solutions

One limitation of phase separation measurements is that only positive Δa values can be evaluated since it is not easy to measure the difference between a perfectly mixed solution and a pair-aggregate solution from the density profiles. This motivates the use of

Rg to determine the effect of both negative and positive Δa values. During the past decade,

DPD has been used extensively to simulate polymer melts/solutions. Consequently different models have been developed in order to study different aspects of a polymer system. For example, a so called “FENE” potential can be used in order to explain the dynamics of a polymer chain in regards to tube model [9]. However, traditionally this is done by the well-known bead and spring model, where groups of polymer repeat units are represented by DPD particles linked with simple spring force that serves as the covalent

53 bond between the repeat units. The spring force usually takes the form presented in equation 25.

S F= kr( ij − r eq )rˆij (eq. 25)

In this, k is the spring constant, and req is the equilibrium bond length. In our simulations we use spring constant of k = 300 and the equilibrium bond length of 0.65.

This is because by using the MDPD cut-off distance of 0.75, the first neighbor peak in the radial distribution function shifts to 0.65. Figure 2.12 shows the dependency of g(r) on the cut-off distance in MDPD.

Figure 2.12.Radial Distribution Function of DPD fluid for different MDPD cut-off values. The radius of gyration of a polymer chain as a characteristic scale reflects the effect of solvent quality on polymer size. The gyration radius of a polymer chain scales with a scaling exponent, v, of the chain length. Flory predicted the value of 0.6, 0.5 and 0.33 for

good, theta and bad solvents respectively for the scaling exponent. The gyration radius can

be calculated using equation 26.

N 1 2 Rg = ∑( rri − CM ) , (eq. 26) N i=1

In this, N is the number of polymer beads (chain length), rCM is the center of mass

th for the chain and ri is the position of i bead. Schlijper et al. [40] investigated polymers in

DPD using a bead-spring model in an athermal (theta) solvent and nearly obtained the correct scaling exponent for Rg of 0.50. Khani et al. [9] performed the same study and found the proper scaling laws for a polymer melt. Kong et al. [41] considered the effect of solvent quality on Rg and found good, theta, and bad solvents by changing the conservative force

parameter for the polymer-solvent interactions. Other DPD studies have been performed

to reproduce scaling laws for polymer melts/solutions [42, 43]. The solvent quality and the interaction between different components of a system is best explained by the concept of free energy.

For the calculation of the gyration radius, single polymer chains of varying length

and 100,000 solvent particles were modeled. For these simulations, a large number of

solvent particles is required for two reasons: first, the scaling laws are valid only in the

dilute regime and second, the calculation box has to be large enough to avoid any polymer-

polymer interaction by the imaginary neighboring cells introduced in the periodic boundary

conditions. The scaling relationship of the Rg was investigated as a function of Δa and the results are presented in figure 2.13. Simulations for three Δa values over a range of chain lengths are shown in figure 2.13. Since naturally a theta solvent is defined as the condition at which the polymer chain does not feel the effect of solvent, Δa=0 is used to mimic it.

Using the curves in figure 10, Δa=10 was used to reproduce an interaction parameter of

χ ≈ 3to represent a bad solvent mixture and a Δa=-10 was employed for the good solvent simulations at the density of 3. For both Δa, the expected scaling exponents are precisely captured (straight dash lines represent theoretical values) to for the good, theta, and bad solvent with slopes of 1.18, 1.0 and 0.66 respectively.

Figure 2.13.Radius of gyration as a function of chain length for different solvent qualities. 2.8. Transport properties and Dynamics

So far we have studied the equilibrium and phase separation properties of MDPD fluids. Nonetheless, one of the major application of DPD and MDPD simulations is the transport properties and the dynamics of fluids. Thus, we have performed simulations on

MDPD fluids with different aij and bij parameters and calculated the zero shear viscosity and diffusion coefficient. Detailed study of the methodology and different techniques for viscosity measurements in DPD will be presented in the next chapter. The diffusion

coefficient of a MDPD fluid can be measured using two different methods: first, by

calculating the slope of the mean squared displacement of the ensemble as a function of

time and second, by using the velocity autocorrelation function and following equation 27.

1 ∞ = D∫ dtvvii(0). ( t ) (eq. 27) 3 0

In the same manner, the zero shear viscosity of a fluid can be calculated from the pressure autocorrelation function using equation 28.

V ∞ η = ∫ dtPP(0) : ( t ) , (eq. 28) kTB 0

Figure 2.14 shows the results for the diffusion coefficient and the zero shear viscosity of MDPD fluid as a function of the aij parameter for a range of bij parameters at

different densities. The results in figure 2.14 reveal that over the studied range, these two

main transport properties of a fluid are not a function of the attractive parameter. It should

be noted that the results in these curves are limited to the simulations with positive total

pressure in the calculation box.

Figure 2.14.a) Diffusion coefficient and, b) Zero shear viscosity of MDPD fluid as a function of repulsion parameter, aij for a range of bij parameters and densities.

The same curves can be plotted for the bij parameter, and subsequently using the viscosity and diffusion coefficient one can calculate the Schmidt number as a function of bij. The results are shown in figure 2.15, where a strict dependence of all transport

properties on the repulsion parameter is observed. Increasing bij directly increases the

viscosity of a MDPD fluid and at the same time significantly reduces its diffusion. As a

result, the Schmidt number which theoretically is in the order of ~103 for water molecules, increases significantly as the repulsive parameter is increased. This is particularly pronounced at elevated densities as the predominance of repulsion parameter is larger.

Figure 2.15. a) Diffusion coefficient (solid symbols) and zero shear viscosity (open symbols), and b) Schmidt number of the MDPD fluid as a function of repulsive parameter, bij. The insensitivity of the dynamics to the attractive parameter, and at the same time

its strict dependence on the repulsive parameter can be also observed in the mean squared

displacement of the MDPD fluid plotted in figure 2.16. The results are from simulation at

elevated densities ( ρ = 7 ) and show clearly that while the MSD curves of the same bij

parameter and different aij terms are within the same range, keeping the attractive term

constant and changing bij significantly changes the dynamics of the system. It should be

mentioned that the same type of behavior is observed for all densities.

Figure 2.16.Mean square displacement of the MDPD fluid, for a) B=100 and a range of attractive terms, and b) A=20 and a range of repulsive terms. 2.9. Conclusions

In this chapter, relations to experimental parameters have been derived for MDPD based on the Flory-Huggins interaction parameter. In order to do this, the EOS for MDPD

was revisited and was found to have a third order density-dependent term for the bij

parameter in agreement to prior claims; however, an additional aijbij interaction term was

found to play a crucial role at elevated densities and at low attractive parameter values.

Based on this newly added term, the EOS was completed and provides a better fit at higher

densities and predicts pressure within measurement errors. Based on the EOS, an

expression for the compressibility of the MDPD fluid was derived and was employed in

addition to the pressure measurements to define an accessible window of simulation

parameters. The solution of compressibility and pressure equation at the same time for aij

and bij enabled the definition of a criterion at which the compressibility of water is satisfied

and at the same time the total pressure of the system remains positive and physically

meaningful. The Flory-Huggins relationship contains a Δa term that motivates phase

separation or miscibility of mixtures. The actual effect of Δa on the Flory-Huggins χ

parameter was measured by monomer phase separation and Rg scaling. Although the slopes

of the relationship between the Δa and χ were predicted closely by the derived expressions, the same deviations reported for DPD were observed for MDPD. Nevertheless, the derived expressions for the Flory-Huggins interaction parameter provide a very good estimate for matching the simulation parameters to the experimental characteristics of the fluid. The calculation of the gyration radius of a polymer chain, Rg, and its scaling relationships by

the chain length for different solvent qualities showed that good, theta, and bad solvents

could be reproduced precisely using the derived expressions.

Finally, the dynamics and transport properties of the MDPD fluid, namely the

diffusion coefficient and zero shear viscosity, were studied as a function of attractive and repulsive parameters. We found that while these entities are insensitive to the attractive term, aij, they strictly depend on the value of the repulsive parameter, bij. Thus the dynamics

of a fluid and consequently the Schmidt number can be tuned for orders of magnitude by

changing this parameter. Overall, this work presents systematic choices of the MDPD

conservative force parameters in order to match experimental compressibility and the

Flory-Huggins χ. The χ relationship now makes the vast experimental literature using χ

accessible to MDPD studies.

2.10. References

5. Hoogerbrugge, P.J. and J.M.V.A. Koelman, Simulating Microscopic Hydrodynamic Phenomena with Dissipative Particle Dynamics. Europhysics Letters, 1992. 19(3): p. 155-160. 9. Khani, S., M. Yamanoi, and J. Maia, The Lowe-Andersen thermostat as an alternative to the dissipative particle dynamics in the mesoscopic simulation of entangled polymers. Journal of Chemical Physics, 2013. 138(17): p. 174903(1-10). 16. Marsh, C.A., G. Backx, and M.H. Ernst, Fokker-Planck-Boltzmann equation for dissipative particle dynamics. EPL (Europhysics Letters), 1997. 38(6): p. 411. 17. Espanol, P. and P. Warren, Statistical Mechanics of Dissipative Particle Dynamics. Europhysics Letters, 1995. 30(4): p. 191-196. 18. Jamali, S., M. Yamanoi, and J. Maia, Bridging the gap between microstructure and macroscopic behavior of monodisperse and bimodal colloidal suspensions. Soft Matter, 2013. 9(5): p. 1506-1515. 19. Groot, R.D. and P.B. Warren, Dissipative particle dynamics: Bridging the gap between atomistic and mesoscopic simulation. Journal of Chemical Physics, 1997. 107(11): p. 4423-4435. 20. Pagonabarraga, I. and D. Frenkel, Dissipative particle dynamics for interacting systems. Journal of Chemical Physics, 2001. 115(11): p. 5015-5026. 21. Chen, C., et al., A Many-Body Dissipative Particle Dynamics Study of Spontaneous Capillary Imbibition and Drainage. Langmuir, 2010. 26(12): p. 9533-9538. 22. Chen, C., et al., A Many-Body Dissipative Particle Dynamics Study of Forced Water–Oil Displacement in Capillary. Langmuir, 2011. 28(2): p. 1330-1336. 23. Ghoufi, A. and P. Malfreyt, Calculation of the surface tension from multibody dissipative particle dynamics and Monte Carlo methods. Physical Review E, 2010. 82(1): p. 016706. 24. Louis, A.A., Beware of density dependent pair potentials. Journal of Physics: Condensed Matter, 2002. 14(40): p. 9187. 25. Arienti, M., et al., Many-body dissipative particle dynamics simulation of liquid/vapor and liquid/solid interactions. The Journal of Chemical Physics, 2011. 134(20): p. -. 26. Ghoufi, A., J. Emile, and P. Malfreyt, Recent advances in Many Body Dissipative Particles Dynamics simulations of liquid-vapor interfaces. The European Physical Journal E, 2013. 36(1): p. 1-12. 27. Ghoufi, A. and P. Malfreyt, Coarse Grained Simulations of the Electrolytes at the Water–Air Interface from Many Body Dissipative Particle Dynamics. Journal of Chemical Theory and Computation, 2012. 8(3): p. 787-791.

28. Warren, P.B., Vapor-liquid coexistence in many-body dissipative particle dynamics. Physical Review E, 2003. 68. 29. Trofimov, S.Y., E.L.F. Nies, and M.A.J. Michels, Thermodynamic consistency in dissipative particle dynamics simulations of strongly nonideal liquids and liquid mixtures. Journal of Chemical Physics, 2002. 117(20): p. 9383-9394. 30. Goujon, F., et al., Frictional forces in polyelectrolyte brushes: effects of sliding velocity, solvent quality and salt. Soft Matter, 2012. 8. 31. Merabia, S. and I. Pagonabarraga, A mesoscopic model for (de)wetting. The European Physical Journal E, 2006. 20(2): p. 209-214. 32. Merabia, S. and I. Pagonabarraga, Density dependent potentials: Structure and thermodynamics. The Journal of Chemical Physics, 2007. 127(5): p. -. 33. Warren, P.B., A manifesto for one-body terms: the simplest of all many-body interactions? Journal of Physics: Condensed Matter, 2003. 15: p. 3467-3473. 34. Warren, P.B., No-go theorem in many-body dissipative particle dynamics. Physical Review E, 2013. 87(4): p. 045303. 35. Fedosov, D.A., G.E. Karniadakis, and B. Caswell, Steady shear rheometry of dissipative particle dynamics models of polymer fluids in reverse Poiseuille flow. The Journal of Chemical Physics, 2010. 132(14). 36. Kirkwood, J.G. and F.P. Buff, The Statistical Mechanical Theory of Surface Tension. The Journal of Chemical Physics, 1949. 17(3): p. 338-343. 37. Irving, J.H. and J.G. Kirkwood, The Statistical Mechanical Theory of Transport Processes. IV. The Equations of Hydrodynamics. The Journal of Chemical Physics, 1950. 18(6): p. 817-829. 38. Gloor, G.J., et al., Test-area simulation method for the direct determination of the interfacial tension of systems with continuous or discontinuous potentials. The Journal of Chemical Physics, 2005. 123(13): p. -. 39. Biscay, F., et al., Calculation of the surface tension from Monte Carlo simulations: Does the model impact on the finite-size effects? The Journal of Chemical Physics, 2009. 130(18): p. -. 40. Schlijper, A.G., P.J. Hoogerbrugge, and C.W. Manke, Computer simulation of dilute polymer solutions with the dissipative particle dynamics method. Journal of Rheology (1978-present), 1995. 39(3): p. 567-579. 41. Kong, Y., et al., Simulation of a Confined Polymer in Solution Using the Dissipative Particle Dynamics Method. lnternatiomd Journal of Thermoplastics, 1994. 15(6): p. 1093-1101. 42. Zhao, T., et al., Dissipative particle dynamics simulation of dilute polymer solutions—Inertial effects and hydrodynamic interactions. Journal of Rheology (1978-present), 2014. 58(4): p. 1039-1058. 43. Spenley, N.A., Scaling laws for polymers in dissipative particle dynamics. EPL (Europhysics Letters), 2000. 49(4): p. 534.

CHAPTER 3 DPD RHEOMETRY

This chapter is adapted from: “Viscosity measurement in Dissipative Particle

Dynamics”, by Safa Jamali, Armand Boromand and Joao Maia, Journal of Computer

Physics Communications, 2015, submitted.

3.1. Introduction

A DPD particle interacts with its neighbors through three effective forces

(explained in detail in previous chapter) called conservative, random and dissipative, with

each force being measured via a controlling parameter. In other words, these three

parameters define the system being modeled. Interestingly, there are only two expressions

at hand to set these parameters: (I) the conservative force controlling parameter can be

decided using the well-known Groot-Warren [44] formulation based on compressibility of a component which was discussed and derived in the Chapter 2, and (II) the dissipative and random force controlling parameters together define the dimensionless temperature in the system and thus can be chosen by solving the fluctuation-dissipation relationship [17].

Despite the crucial role of dissipative and random parameters in defining the dynamics of

a DPD system, they can take any arbitrary value as long as the dissipation-fluctuation relationship is satisfied. Due to the lack of an expression to provide unique values for these parameters, and perhaps due to the initial introduction in the method of values of 3.0 and

4.5 for random and dissipative coefficients respectively, the majority of the works reported in the literature have used this same parameter set. Regardless of the parameter choices and the material subject to study, one can categorize any simulation into two different conditions:

i) Equilibrium simulations, where material properties are monitored at rest (and

by definition replicates the behavior of the fluid in the linear regime);

ii) Non-equilibrium conditions, in which DPD particles are subject to external

fields and rate-dependent properties of a fluid can be reproduced.

The former involves simple periodic boundary conditions and yields consistent and thermally stable results, even though they sometimes suffer from statistical issues due to very poor signal-to-noise ratio, especially in the measurement of stress and thermodynamic quantities. On the other hand, non-equilibrium simulations in general lead to more stable and consistent calculations, but suffer from two sources of artifacts. Firstly, the traditional

Lees-Edwards boundary condition [45] is not capable of maintaining stable profiles at the boundaries in velocity direction, and therefore additional modifications are required.

Secondly, since the dissipative and random forces together serve as a built-in thermostat for DPD and contribute to dynamics of the system at the same time, out of equilibrium simulations can affect the temperature controlling effect of the dissipative force and hence the thermal stability of the calculations.

Different boundary conditions (BCs) including wall-bounded and unbounded, periodic Poiseulle flow (PPF), use of ghost particles and bounce back, etc. have been proposed as modifications for non-equilibrium DPD simulations. For example, Pivkin et al. [46, 47] proposed a wall-bounded BC which induces the shear profile in the DPD domain by moving the solid wall particles. Backer and coworkers [48] proposed a periodic

Poiseulle flow BC that results in stable velocity profiles and does not involve steady shear flow issues. However, for many applications/systems the method of choice is still steady shear since it imposes a constant shear rate along the calculation domain and using PPF

can lead to secondary flows e.g. migration of internal constituents in the case of multiphasic

systems or even polymer solutions or melts. Chatterjee [49], in an effort to modify the

Lees-Edwards boundary conditions, introduced ghost particles in limiting boundaries in

the velocity direction and obtained stable velocity profiles over a wide range of strains.

Nevertheless, using such particles makes reliability of the ensemble averages and

periodicity of the system questionable.

Different additional thermostats and built-in alternatives to DPD thermostat have

been proposed in order to compensate for the shortcoming of the dissipative-random

ensemble in temperature stabilization. Whittle and Travis [50] studied different external

(MD-based) thermostats and their performance on controlling the temperature of a

colloidal suspension under shear. Khani et al. [9] studied the temperature controlling effect of so-called Lowe-Anderson thermostat as an alternative to traditional DPD forces in polymer melts and found better stability of both temperature and dynamics using this method when compared to the simple dissipative-random ensemble. However, they have shortcomings at predicting the transport properties.

Perhaps one of the most interesting quantities of a DPD fluid, one that reveals information about the dynamics of a system, is its viscosity. Many efforts have been made to study the rheology of a DPD fluid at different conditions. Marsh et. al. [16] derived an expression for the effect of dissipative force parameter on the viscosity of a DPD fluid when there is no conservative force applied on the DPD particle. Different groups have studied rheological behavior of suspensions using both traditional and modified DPD [18,

51-54]. Polymer melts and solutions, as well as multiphase systems have also been studied with special emphasis on their rheological behavior using DPD [9, 10, 55]. However, one

can conclude from a careful review of the prior publications that, even for the simplest

system (water, first studied by Groot and Warren [44]), values reported for the viscosity

vary from one report to another, depending on the type of flow and method used for

calculation of viscosity, boundary conditions and the choice of force parameters [42, 55-

57]. In particular, different viscosity values from Green-Kubo [58] expression for stress autocorrelation function (zero-shear viscosity), stress tensor for steady shear simulations,

Poiseuille flow and transient start-up shear flow have been reported. Furthermore, theoretical predictions for temperature and dissipative parameter dependence of the viscosity are contradictory within the existing literature.

In this chapter a comprehensive study on different viscosity measurement methods and the effect of simulation parameters/conditions on rheological properties obtained by

DPD simulations is presented. The aim is to provide guidance on the suitability of each method to simulate the non-linear rheological behavior of complex fluids.

3.2. Boundary Conditions

Perhaps one of the most widely-used boundary condition methods for simulation

of fluids under non-equilibrium condition (mainly shear) is the well-known Lees-Edwards

boundary condition [45]. The method was initially proposed (and always been successful)

for molecular dynamics simulation of sheared fluids. The Lees-Edwards boundary

conditions (L-E BC) introduces the desired velocity profile in any given calculation box

by applying a constant velocity to the surrounding imaginary cells (with different signs for

upper and lower neighboring cells). However, it has been shown that use of the original

periodic L-E BC results in distorted velocity profiles which originates from the fact that

67 the dissipative force in DPD uses relative velocity of a particle pair to calculate the force between them [49]. This is more pronounced when greater values of γ, the dissipative force controlling parameter, is used to reproduce higher Schmidt numbers. Different solutions have been proposed to address this issue for DPD simulations of sheared fluids. Chatterjee

[49] proposed using so-called ghost particles in an additional layer at the borders of the central calculation cell. Some authors have been using a simple modification of the original scheme of LE BC that corrects the dissipative force by taking into account the velocity of the parent cells [59, 60]. This scheme is discussed in detail in a separate chapter and is called the Eulerian approach for the rest of this chapter. This method will induce high deviations of imposed shear rate from measured value at low shear rates, which have been observed and reported in other literatures as well [60-62]. Other methods that have been used resort to frozen particles and wall-bounded flows; e.g. Pivkin et al. [46] and Chen et al. [63] introduced the velocity profile by bounding the upper/lower cell borders with a moving wall in order to avoid direct interaction between two neighbor particles in opposite sides of the calculation box. However, using wall-bounded simulation cells results in additional effects, such as slip and density fluctuations near the wall. Although authors have been trying to address the density fluctuation issue [47, 64] it should be considered that utilizing the approach of using frozen particles should be avoided unless the physics of confinement geometry needs to be addressed. In general there are three reasons for not including the frozen particles: i) Calculation times increase; ii) It violates the momentum conservation equation, since it does not hold for the interactions between solvent and wall particles; iii) Unlike MD, using frozen DPD particles as a coarse-grained level of MD does not provide any physical explanation unless meaningful coarse-graining procedures are

followed, which is not the case for any of the proposed wall models. It should also be noted

that using frozen particles has been the choice for most of steady shear studies via DPD. In

the present chapter, we introduce a modification approach to the traditional L-E BC, and

study the resulting velocity profiles, temperature control and pressure/stress measurements

for each system.

One way to look at the DPD particles under flow is looking at them as individual

momentum carriers that interact with each other in a reference coordinate moving with the flow field. To analyze the flow properties with DPD, one can look at the fluid parcels rather that a specific position in the flow field. We proposed both ways to look at the flow properties of the DPD fluid. The Lagrangian approach, in which we analyze the flow in the reference coordinate attached to each fluid parcels, and Eulerian approach in which flow is

analyzed in a reference coordinate system attached to the calculation cell will be discussed

in the next section. Detailed explanation of the Lagrangian approach is to appear in another

Dissertation by Arman Boromand of the same group; however here we briefly explain the

concept of this method. DPD momentum carriers can start either from a stationary position

or possessing an imposed initial velocity related to the set temperature (which reduces the

time necessary to reach equilibrium state). After reaching the equilibrium state, each DPD

carrier has a peculiar velocity and the ensemble has a velocity distribution. When any flow

field is imposed on the calculation cell, e.g. shear, extension, or mixed state of shear and

extension flow, DPD carriers will follow the field and it can be assumed that this velocity

and the peculiar velocities of the DPD momentum carries can be superimposed. Generally

solving DPD equations involves using velocity-Verlet, VV, algorithm, which is described in detail in [44]. Due to the dependency of the dissipative force (DPD particle’s

acceleration) on its velocity and the dependency of the DPD particle’s velocity on its

acceleration, VV should be divided into two parts. In VV-Part-I particles’ position and

velocities are updated from the forces on the DPD particles and then forces on the DPD

particles will be calculated based on updated velocities. In VV-Part-II the DPD velocities are corrected by the updated force values. In Lagrangian scheme, the flow field (shear) will

be screened after VV-Part-I and the DPD particles will interact by their peculiar velocities;

after updating the values for the peculiar velocities, the flow field will be superimposed.

The simplest modification to the traditional L-E BC can be applied by modifying the dissipative force between two neighboring particles located on opposite sides of the calculation box. One would need to move the upper and lower neighboring cells with the

γL velocity of VVUpper=−= Lower y , to obtain the shear rate of γ for a calculation box of Cell Cell 2

(,LLLxyz ,) size. A schematic view of the imposed velocity is shown in figure 3.1.

Figure 3.1.Schematic view of Lees-Edwards boundary conditions While it should be noted that the calculation cell is fixed in this approach, we focus

D on the calculation of dissipative force between particles i and j, Fij , and more specifically

the relative velocity of these particles, vij , interacting from opposite sides of the calculation

box. In order to take into account the effect of continuous velocity profile, in addition to

the periodicity of the system, we now change the dissipative force to the following form:

DD Fij=−γ ijw ij(r ij )[ev ij .( i−− v j 2 V Cell )] e ij (eq. 1)

3.3. Steady Shear Flow

Figure 3.2Error! Reference source not found. shows the steady velocity profiles at different shear rates developed using abovementioned procedure for Eulerian method.

The velocity profiles reveal that a wide range of shear rates can be obtained using this

approach.

Figure 3.2. Velocity profiles for a range of shear rates with dissipative parameter of: a) 4.5 and, b)50. It should be noted that results in Figure 3.2 are measured for a dissipative force

parameters of γ = 4.5,50 averaged over 100 time steps. Same results can be obtained for

Lagrangian approach that are not shown here. One can calculate temperature of a system by measuring ensemble average kinetic energy of the particles, which can be determined by

1 N T=∑ mriiiii(vu − ( )) ⋅− ( vu ( r )) (eq. 2) 3N i=1

The term ur()i corresponds to the value of the velocity (from the velocity profile)

at the position of i-th particle in the velocity gradient direction, where vii− ur() is the

peculiar velocity of the particle i-th, mi is the mass of a DPD particle and N is the total

number of particles. Figure 3.3 shows the temperature controlling effect of the modified

dissipative force for the Eulerian approach, at given dissipative terms. Detailed study of

the thermostatting capabilities in DPD and the origins of the instabilities will be presented

in Chapter 4.

Figure 3.3. Dimensionless temperature versus the shear rate for a range of dissipative parameters. An increase of the temperature under strong non-equilibrium conditions is a well-

known phenomenon that has been studied extensively and different thermostats have been

proposed to address the issue [9, 50]. As one can clearly conclude from the figure 3.3, the dissipative force with higher strengths can control the temperature up to higher shear rates.

Although it is generally accepted that the use of large friction parameters, requires smaller time steps for better temperature stability, we have run simulations with a wide range of

(0.0001-0.1) and found that the time step does not affect the temperature stability while the∆𝑡𝑡 noise to signal ratio increases by decreasing the time step (will be presented in the

Chapter 4). In contrast to a better temperature controlling effect, it was reported [49] that

using higher dissipative parameters will result in a distorted velocity profile due to

limitations of the L-E boundary conditions. However, the velocity profiles for the

dissipative coefficient of γ = 50.0 plotted in Figure 3.2.b shows that the Eulerian approach

constructs the correct velocity profile and consequently correct shear rate. Given the forces

between each pair of particles, and the velocity of individual particles, one can calculate

the pressure tensor for a DPD system by equation 3.

N NN−1 1 T P=∑mri( vu i − ( i )) ⊗− ( vui ( r i )) +∑∑ rij ⊗ F ij , (eq. 3) V i=11ji>= i

where ⊗ is the dyadic product of the two vectors. The stress tensor would be the negative of the pressure tensor (S = -P), and thus the shear viscosity of a DPD fluid can be

calculated from the xy component of the stress tensor (equation 4).

S η = xv , (eq. 4) γ

The bracket in (eq. 4) represents the time average value of the shear stress. By

careful review of the Lagrangian approach, one can see that the force calculation is done

in the stage that there is no flow present and stress of the flow field should be calculated

separately and be added to the stress calculated form particles’ interactions. For this reason

we calculated the flow stress though the dissipative tensor proposed by Marsh et al. [65] in

this approach. Viscosity measurement data for Eulerian and Lagrangian approaches are

depicted in figure 3.4. As it can be seen from the graphs both method predict the same

values and dependency of the viscosity to the dissipative parameter and shear rate. It should

be noted that the minimum accessible shear rate for Lagrangian is lower and signal to noise

ratios are higher compare to Eulerian approach.

Figure 3.4. Viscosity versus the Shear Rate for a range of dissipative parameters. The limitation of Eulerian approach can be explained through the physical concept behind the LE BC. Owing to the fact that velocity gradient in Eulerian method requires diffusion throughout the calculation cell through imaginary cells at the top and bottom of the calculation cell, this method is always limited by the lower values of strain rates. This phenomena can be illustrated in a numerical example: Consider a calculation cell of a size used in all simulations of this study; 10 [rc] in each direction. If one aims at studying the

viscosity of the system by applying a shear rate of 0.01, knowing that the maximum

velocities are occurring at the top and bottom of the calculation cell (5 x 0.01 = 0.05) and

the fact that all the simulations are running at the temperature (KbT = 1.0) we can calculate

the share of the shear field compare to the mean velocity of the particle due to its

temperature; which is 5%. This small share will hinder the diffusion of momentum into the calculation cell. In fact our data showed that while Eulerian approach is limited to shear

rate 0.05 velocity profile can be reproduced for shear rate of 0.005 for Lagrangian

approach. This discrepancy between the applied shear rate and the measured shear rate is

the reason for the increase in the viscosity at the low shear rates (result are not shown here).

This is a very important shortcoming of Eulerian approach that is neglected or has not been

studied in detailed in some literature using DPD at low shear rate studies [55, 60].

It should be noted that an unphysical shear-thickening behavior is observed at high

shear rates for both Eulerian and Lagrangian approaches when a small value of dissipative

parameter is employed. This can be explained through equation 3. The first term on the

right hand side of the pressure tensor, i.e., the kinetic part, is a function of particle velocity.

As explained before, the deviation of particle velocities at high shear rates (which gives

rise to temperature deviation) is reflected in this part of the equation. Thus, an increased

velocity (meaning increased temperature) results in increased viscosity. It should be also

noted that if one tends to measure viscosity or other material functions e.g. first normal- stress coefficients, with the LE- BC (Eulerian approach) at low shear rates (less than 0.05), the shear flow field is not stably developed even after 106 steps (results are not shown here)

and thus, calculating the viscosity based on the applied shear rates (not the measured one)

will lead to erroneous high values.

Now with the viscosity values at hand, one can look at the effect of dissipative

parameter on the dynamics of DPD fluid. Figure 3.5 presents the viscosity of DPD fluid as

a function of dissipative term, γ for a range of shear rates. It should be mentioned that in

this graph, only the results that are measured in thermally stable simulations are plotted. In

another word, the results of the simulations where the temperature deviates from its set

value are excluded. One can argue that while at very low dissipative terms, the viscosity is

76 rather independent of this parameter, by increasing the friction term, this dependency becomes very strong.

Figure 3.5. Viscosity as a function of dissipative term, Gamma for different shear rates. 3.4. Zero Shear Viscosity

With the pressure tensor at hand, one can calculate the zero-shear viscosity of a given system, using Green-Kubo expression of pressure auto-correlation function:

V ∞ η= αβ≠ ∫ dt Pαβ( t ) : P αβ (0) ;( ), (eq. 5) 3kTB 0

In practice the simulations were ran for 3.1× 107 time steps were the first 106 steps were not included in the results, assuming that the system reaches its equilibrium state during this time. Pressure autocorrelation function was calculated over 40,000 time origins, and the result were integrated out for 2000 time steps, t∗ = 20 . According to the

parametrization method of Groot-Warren and as derived and explained in the previous

section, there is a direct relationship between the conservative force parameter and

temperature of a DPD system ( aij= 25 kT B , considering a constant compressibility and number density of 3.0). However, based on the expression for the calculation of the pressure tensor, increasing the temperature will increase the first term on the right hand side of the equation 3 (kinetic part), and adapting the conservative force will increase the

second term as well (virial part). Increase in shear viscosity by increasing the maximum

repulsion parameter has been also observed in other literature [42]. These two effects

together will increase the viscosity of a DPD system as a function of temperature, which is

in contradiction with the standard temperature dependence of the shear viscosity. Figure

3.6 shows this behavior for a range of dissipative parameters and two different conservative

forces: one calculated by an adaptive approach, where the conservative parameter has been adapted following the Groot-Warren expression, and one by a non-adaptive approach, where the conservative force has been kept constant, for different temperatures.

Figure 3.6. Zero shear viscosity vs. Temperature, for different dissipative terms (legend), with conservative force parameter based on: a) Adaptive and b) Non-adaptive method.

The results in figure 3.6 proves that when an adaptive scheme is followed for the conservative force parameter, the zero shear viscosity increases by the kinetic temperature.

On the other hand, the non-adaptive approach reproduces physically meaningful results in regards to temperature dependence of the viscosity. It is important to note that using the non-adaptive approach does not violate any physical laws. Although one can argue that the systems with different temperatures and the same conservative force differ in their chemical identity, presence of a minimum required conservative force prevents the system to become compressible. As it is stated in the recent publication by Pan et al. [56] DPD fluids are susceptible to compressibility issues. They have shown that DPD fluids can be considered (nearly) incompressible under 3 conditions: 1- small mass of DPD particles, 2- high temperature in systems with no conservative interactions, and finally increasing the repulsive force higher than the value predicted by GW scaling relation. Backer et al. [48] performed similar study to compare different methods for viscosity measurements, however they did not use the conservative force at the temperature of 0.5 which results in a compressible fluid. Marsh et.al. [65] suggested different dissipative parameter dependence for viscosity of a DPD fluid as presented in equation 6.

45kT 2πρ2 γ η =B + , (eq. 6) 4πγ 1575

Equation 6 predicts a rather steep decrease in the viscosity as the dissipative parameter increases, followed by an increase; ultimately, the viscosity becomes independent of temperature at high dissipative parameters. Considering the fact that equation 6 is written for a DPD fluid without conservative force, this behavior is a direct consequence of having a compressible fluid. Although the viscosity increases by increasing

79 the dissipative parameter at all temperatures for the adaptive method as expected by the abovementioned expression, our simulations show different trends at different temperature regimes when the non-adaptive approach is used (figure 3.7).

Figure 3.7. Zero shear viscosity vs. Dissipative term, for different dimensionless temperatures), with conservative force parameter based on: a) Adaptive and b) Non-adaptive method. At low temperatures (<1.0), the viscosity increases almost linearly with the

dissipative parameter. At unity ( kTB =1.0 ), which is the temperature of the choice for majority of DPD simulations, the viscosity is in agreement with the analytical predictions.

At high temperatures (>1.0), and where a low dissipative parameter is used, the viscosity starts to rise and eventually at very high temperatures, the viscosity of a DPD fluid with dissipative parameter of 1 becomes larger than the one withγ =100 . This can also be explained by equation 6, as at the low dissipative parameter values, the first term on the right hand side of the equation is the dominant factor (diffusive), which predicts a linear increase of the viscosity by the temperature. However, when the friction term is effectively increased, the second term on the right hand side dictates the viscosity of the fluid

(dissipative term). Needless to mention that introduction of the conservative force based

on the expression given by Groot-Warren changes this scheme. Although in our case there is a conservative force the effect of conservative force can be interpreted through diffusion

term. At low dissipative strength increase the maximum repulsion parameter will

significantly decrease the diffusion term and consequently the compressibility of the liquid

(non-adaptive approach) that is why the trend at low viscosity and low temperature is

totally opposite. Also at high temperature and low dissipative parameter non-adaptive

approach will lead to more diffusion dominated state and it can be seen through a sharp

erroneous increase in the viscosity. On the other hand at low temperature and low

dissipative force increase in the repulsive force will lead to the correct behavior in the case

of non-adaptive and erroneous results from adaptive state that is due to the compressibility

or high diffusion in the adaptive case.

3.5. Transient Shear and Poiseulle Flow

In this section two other methods of viscosity measurement in DPD is briefly

discussed; however, the detailed explanations and results are to appear in Doctoral

Dissertation by Arman Boromand and also in an article published in Computer Physics

Communications journal. While measuring microscopic stresses is common in the

literature as a way to measure viscosity in MD simulations and generally in particle-based

simulation methods, numerical fitting is a powerful method to obtain viscosity of a medium

both experimentally e.g. particle image velocimetry or Doppler optical coherence

tomography, and by simulation methods. Two types of well-understood and characterized

flow types for this purpose are Poiseuille flow, PF, and start-up shear transient flow. There are two well-established methods to generate PF in DPD. The first method is the so-called periodic Poiseuille flow method (PPFM) [48, 66], in which the calculation cell is divided

81 into two symmetric regions and a specific body force is applied to each DPD particle located in the top half of the calculation cell, while an equal force in the opposite direction is applied to the DPD particles in the other half of the calculation cell. The second method is wall-bound PF in which the fluid domain is confined by presence of the wall particles located at its boundaries, and a body force is applied to all solvent particles bound between the two walls. Different boundary conditions can be imposed to obtain no-slip at solid- liquid boundaries, e.g. bounce-back, specular reflection, and Maxwellian [46, 64, 67-69].

Despite all the efforts, wall-bounded flows remains as one of the most controversial and unsettled issues in DPD. Solving the Navier-Stokes equation for a rectangular channel

∂P under pressure gradient equal to −=ρ f yields [48]: ∂x b

y VV(y)= (1 − ( )2 ) , (eq. 7) x center d

Vcenter is proportional to channel depth and inversly proportional to the viscosity of the fluid.

Ly ∂P V =() − , (eq. 8) center 2µ ∂x

The values of viscosity and channel depth can be calculated from velocity fitting.

The channel depth value is obtained to verify the fitting procedures. The viscosity measurement can also be performed by another fitting technique based on start-up transient

2 shear flow. Solving the momentum diffusion equation in 1D ( ∂=∂txVVν yx), one can provide a methodology to obtain the kinetic viscosity and respectively dynamic viscosity of the DPD fluid. Start-up transient shear flow has been employed by other authors to

82 measure viscosity of particle-based simulation techniques and the procedures used to obtain the viscosity is discussed elsewhere in detail [70-74].

3.6. Conclusions

Since DPD is a relatively recent simulation method and the literature on DPD simulations out of equilibrium is rather scattered, a guideline for viscosity measurement using DPD is non-existent. Furthermore, the need for correct boundary conditions and a wider range of shear rates accessible to DPD simulations cannot be neglected. Thus, in this chapter a series of systematic studies on different methods of viscosity measurements were performed.

In this chapter, a comprehensive study was conducted to compare different techniques to measure the viscosity by Dissipative Particle Dynamics, and using two methods. First, using the pressure tensor and the microscopic definition of stress in the non- equilibrium system; second, using the autocorrelation function and the pressure tensor at no-flow condition. It was showed that the temperature instabilities and artifacts at high shear rates are functions of the strength of the dissipative force and the fact that using smaller time steps cannot alleviate the problem. To understand the origin of this problem, the velocity distribution at each shear rate for different values of the dissipative forces will be analyzed in the next chapter; however, it was found that higher values of the dissipative force control the dissemination of the thermal instabilities in the system better. This fact reveals one of the shortcomings in the DPD scheme, the fact that the dissipative force controls the dynamics of the system while it is also part of the thermostat. To decouple the thermostatting role and solely study the dynamics of the system, stress autocorrelation

function was employed in order to measure the zero-shear viscosity at different temperature

and dissipative forces. The result of zero shear viscosity measurements revealed that the

Groot-Warren scaling law for compressibility out of equilibrium will lead to erroneous

results at low dissipative part and is due to the diffusion that is resembling gas like

behavior.

3.7. References

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48. Backer, J.A., et al., Poiseuille flow to measure the viscosity of particle model fluids. The Journal of Chemical Physics, 2005. 122(15): p. 154503. 49. Chatterjee, A., Modification to Lees–Edwards periodic boundary condition for dissipative particle dynamics simulation with high dissipation rates. Molecular Simulation, 2007. 33(15): p. 1233-1236. 50. Whittle, M. and K.P. Travis, Dynamic Simulations of Colloids by Core-Modified Dissipative Particle Dynamics. Journal of Chemical Physics, 2010. 132: p. 124906(1)-124906(16). 51. Jamali, S., A. Boromand, and J. Maia, Dissipative Particle Dynamics simulation of colloidal suspensions. Bulletin of the American Physical Society, 2014. 52. Chatterjee, A. and L.-M. Wu, Predicting rheology of suspensions of spherical and non-spherical particles using dissipative particle dynamics (DPD): methodology and experimental validation. Molecular Simulation, 2008. 34(3): p. 243-250. 53. Padding, J.T. and A.A. Louis, Hydrodynamic interactions and Brownian forces in colloidal suspensions: Coarse-graining over time and length scales. Physical Reviews, 2006. 74: p. 031402. 54. Martys, N.S., Study of a dissipative particle dynamics based approach for modeling suspensions. Journal of Rheology (1978-present), 2005. 49(2): p. 401-424. 55. Pan, D., N. Phan-Thien, and B.C. Khoo, Dissipative particle dynamics simulation of droplet suspension in shear flow at low Capillary number. Journal of Non- Newtonian Fluid Mechanics, 2014. 212(0): p. 63-72. 56. Pana, D., et al., Numerical investigations on the compressibility of a DPD ﬂuid. Journal of Computational Physics 2013. 242 p. 196–210. 57. Clark, A.T., et al., Mesoscopic Simulation of Drops in Gravitational and Shear Fields. Langmuir, 2000. 16(15): p. 6342-6350. 58. Kubo, R., Statistical-mechanical theory of irreversible processes. I. General theory and simple applications to magnetic and conduction problems. Journal of the Physical Societ of Japan, 1957. 12(6): p. 570-586. 59. Fedosov, D.A., I.V. Pivkin, and G.E. Karniadakis, Velocity limit in DPD simulations of wall-bounded flows. Journal of Computational Physics, 2008. 227(4): p. 2540-2559. 60. Fedosov, D.A., G.E. Karniadakis, and B. Caswell, Steady shear rheometry of dissipative particle dynamics models of polymer fluids in reverse Poiseuille flow. The Journal of Chemical Physics, 2010. 132(14): p. 144103. 61. Boek, E.S., P.V. Coveney, and H.N.W. Lekkerkerker, Computer simulation of rheological phenomena in dense colloidal suspensions with dissipative particle dynamics. Journal of Physics: Condensed Matter, 1996. 8(47): p. 9509. 62. Boek, E.S., et al., Simulating the rheology of dense colloidal suspensions using dissipative particle dynamics. Physical Review E, 1997. 55(3): p. 3124-3133.

63. Chen, S., et al., Flow around spheres by dissipative particle dynamics. Physics of Fluids (1994-present), 2006. 18(10): p. 103605. 64. Ranjith, S.K., B.S.V. Patnaik, and S. Vedantam, No-slip boundary condition in finite-size dissipative particle dynamics. Journal of Computational Physics, 2013. 232(1): p. 174-188. 65. Marsh, C.A., G. Backx, and M.H. Ernst, Static and dynamic properties of dissipative particle dynamics. Physical Review E, 1997. 56(2): p. 1676-1691. 66. Visser, D.C., H.C.J. Hoefsloot, and P.D. Iedema, Modelling multi-viscosity systems with dissipative particle dynamics. Journal of Computational Physics, 2006. 214: p. 491–504. 67. Revenga, M., et al., Boundary Models in DPD. International Journal of Modern Physics C, 1998. 09(08): p. 1319-1328. 68. Revenga, M., I. Zuniga, and P. Espanol, Boundary conditions in dissipative particle dynamics. Comput Phys Commun, 1999. 121: p. 309-311. 69. Visser, D.C., H.C.J. Hoefsloot, and P.D. Iedema, Comprehensive boundary method for solid walls in dissipative particle dynamics. J. Comput. Phys., 2005. 205(2): p. 626-639. 70. Haber, S., et al., Dissipative particle dynamics simulation of flow generated by two rotating concentric cylinders: Boundary conditions. Physical Review E, 2006. 74(4): p. 046701. 71. Filipovic, N., et al., Dissipative particle dynamics simulation of flow generated by two rotating concentric cylinders: II. Lateral dissipative and random forces. Journal of Physics D: Applied Physics, 2008. 41(3): p. 035504. 72. Willemsen, S., H. Hoefsloot, and P. Iedema, No-slip boundary condition in dissipative particle dynamics. Int J Mod Phys, 2000. 11: p. 881-890. 73. Lei, H., D.A. Fedosov, and G.E. Karniadakis, Time-dependent and outflow boundary conditions for Dissipative Particle Dynamics. Journal of Computational Physics, 2011. 230(10): p. 3765-3779. 74. Wagner, A.J. and I. Pagonabarraga, Lees–Edwards Boundary Conditions for Lattice Boltzmann. Journal of Statistical Physics, 2002. 107(1-2): p. 521-537.

CHAPTER 4 STABILIZING SHEAR FLOWS IN DPD

This chapter is adapted from: “A Gaussian-Inspired auxiliary thermostat for non- equilibrium Dissipative Particle Dynamics simulations”, by Safa Jamali, Armand

Boromand and Joao Maia, Journal of Computer Physics Communications, 2015, submitted.

4.1. Introduction

Dissipative Particle Dynamics (DPD) as a mesoscale method was first proposed in

1992 [5] based on coarse-grained molecular dynamics model. Substitution of hard

Lennard-Jones (L-J) potentials in MD by very soft potentials acting on many-atoms ensemble represented by a bead rather than a single atom enables DPD to access length/time scales orders of magnitude higher than the ones approachable by MD.

Numerous studies have been performed in order to map the DPD simulation parameters to physically meaningful characteristics [17, 19, 75-77]. Simple formulation and versatility of DPD have attracted many researchers to employ DPD for a wide range of different studies [8, 42, 55, 78-83]. One of the major fields of research in which DPD has been widely adapted is fluid dynamics, where rheological features of a fluid are studied at different (equilibrium or under flow) conditions [35, 52, 84-88]. To study non-equilibrium phenomena one would need to assure complete computational stability over the conditions under which the simulation is performed. This includes (but is not limited to) boundary conditions, flow profiles and thermal stability of a system.

In order to study the dynamics of a fluid in response to a specific flow condition, different types of flows can be imposed in the simulation. Similarly to the experimental means of fluid dynamics studies, these flow types include extensional, pressure-driven and drag shear flows. Perhaps the most relevant flow type to real-life phenomena is the drag shear flow, which in its simplest form consists of a steady shear profile of a constant shear rate over the calculation box. Regardless of the flow type, reproducing these flow conditions in the simulation of a particulate system requires use of relevant boundary conditions and often is associated with numerical artifacts. For example, the most widely accepted boundary conditions for steady shear simulation is the well-known Lees-Edwards

[45] boundary conditions (LE-BCs). Employing the LE-BCs, one takes into account a series of imaginary cells surrounding the calculation box and imposes the shear profile in the system by simply dragging the upper and lower imaginary cells. However, since DPD utilizes a built-in thermostat that controls the dynamics of a system as well, using LE-BCs in DPD simulations gives rise to two main issues: i) The relative velocity of neighboring particles in the imaginary cells results in a distorted shear profile [49]; ii) At high shear rates the DPD thermostat loses its ability to control the temperature properly. Thus, the temperature deviates from its ideal value, which consequently introduces unphysical behavior in the dynamics of a system [89]. In the previous section, the former issue was addressed by introducing two different boundary conditions: i) An Eulerian approach which is simply a modified LE-BCs to fit in DPD simulations; ii) A Lagrangian approach that disassembles the shear profile and interaction potentials in one step, and superimposes the shear profile on the calculation box in the second step. The proposed boundary conditions have successfully reproduced imposed shear profiles for a wide range of shear

rates and corresponding viscosities were studied by different methods of viscosity

measurement.

Different thermostats and alternatives have been proposed in efforts to fix the

temperature controlling effect in DPD at rest (equilibrium conditions) [9, 90]. The reason

for this is that increasing the time-step in simulation of viscoelastic fluids with DPD even at rest (no-flow) can be problematic [91]. There also have been reports where different measurement techniques or external thermostats have been proposed for DPD simulations under flow conditions [16, 50]. Kremer et al. [92, 93] performed a thorough study on the

use of DPD thermostat in non-equilibrium MD simulations, namely on the hard particles rather than usual soft DPD ones, and found DPD as perhaps the ideal thermostat for NEMD simulations as it preserves the hydrodynamics under the flow conditions and it stabilizes the simulations effectively. Nevertheless, the issue of temperature deviation under high shear rate conditions remains unresolved, as the shear rates subject to previous studies mainly correlate to ones commonly used in MD simulations. The level of shear rates required for simulation of a fluid varies from one to another based on the fluid characteristics and the subject of the study. For instance, in order to study the dynamics of polymer melts under flow, one needs low/intermediate shear rates as this value is strictly dictated by the length scales appropriate for polymer chains [9]. On the other hand, much higher values of shear rates are required when rheology of sheared colloidal suspensions are studied. In fact, since the flow-dependent properties of suspensions such as shear-

thickening occur only at shear rates where hydrodynamic interactions dominate the

Brownian forces, stable access to a wide range of shear rates is a prerequisite for any

simulation technique to capture the flow properties of colloidal dispersions [18]. Thus, in

89 this chapter firstly the origins of temperature instability in out of equilibrium DPD simulations will be studied, and secondly a Gaussian inspired auxiliary non-equilibrium thermostat (GIANT) for DPD simulations will be presented, enabling of precise control over the temperature. Comparison between the newly proposed method’s results and one of the most widely used thermostats, the Lowe-Andersen [90], will be made.

4.2. Temperature Measurements under Shear with DPD

Figure 4.1 shows the temperature as a function of shear rate for: a) a range of dissipative parameters using standard DPD and the time step of 0.01, and b) different time steps. One can argue three main conclusions from the figure 4.1: i) The temperature

deviates from its set value of kTB =1.0 in a continuously increasing manner as the shear rate increases; ii) The built-in DPD thermostat loses its capability of holding the temperature as the dissipative parameter decreases, and iii) At the time steps smaller than

0.01, changing the time step size does not affect the temperature of the system as the data points in the figure 4.1.b fall on the same curve for over two decades of time step sizes.

Figure 4.1.Temperature versus shear rate using DPD thermostat for a range of Dissipative parameters.

The first two conclusions are expected, because of the nature of the dissipative force. As explained in the previous chapters, the dissipative force acts against the relative motion of particles and thus increasing the strength of this potential retains the velocity of interacting particles in a narrower range. On the other hand, the Fluctuation-Dissipation

relationship has to be satisfied and thus, the larger dissipative parameter, γ mn , the larger

the random force controlling parameter, σ mn ; hence increasing the dissipative strength is limited by the noise parameter as well. In contradiction with our results in figure 4.1.b, one may expect a better thermostatting effect as the time step size is decreased; however, one should keep in mind that decreasing the time step size directly increases the strength of the random force as it appears in the denominator in the formalism of the random potential.

In order to better understand the nature of this temperature deviation under high shear rates, one needs to monitor the distribution of particle velocities at each simulation condition. Statistical mechanics dictates that at any given instance the velocity of individual particles of a fluid should form a Gaussian distribution with the maximum at zero and the full width at half-maximum (FWHM) indicating the temperature [94, 95]. Needless to mention that this has served as the physical foundation of the majority of thermostats used in mesoscale and molecular level simulation techniques [90, 96-98]. This applies to each individual layer of particles subject to shear profile as well, thus one can measure the velocity distribution in the whole calculation box by subtracting the corresponding layer velocities from the particle velocities.

Figure 4.2 shows the distribution of particle velocities in the flow direction for a range of shear rates and dissipative parameters using the standard DPD thermostat. The velocity distribution curves in figure 4.2 reveal that regardless of the dissipative parameter

91 used for the simulation, increasing the shear rate broadens the velocity distribution, which shifts the FWHM to higher values and thus higher temperatures in the system are observed; however, increasing the dissipative parameter reduces this effect to some extent. As can be observed in figure 4.2, it is only at γ = 50 that the DPD thermostat is capable of even keeping the general form of the distribution curves.

Figure 4.2. Distribution of particle velocities in the flow direction, for a range of shear rates and dissipative parameters. It should be mentioned that despite the increase in temperature and deviation of the velocities from the expected distribution, the velocity profile in the calculation box remains constant and yields the correct shear rate. This is clear in figure 4.3, where the maximum

and minimum velocity of each layer for γ = 4.5 is plotted for a range of different shear rates. The graph shows that as the shear rate increases, the difference between the maximum and minimum velocity of a given layer is increased, meaning that the velocity distribution is broader; however, both maximum and minimum velocities follow the same slope (applied shear rate). In other words, the deviation of the velocity distribution is only a function of shear rate and not the value of the velocity itself, as at the center of the calculation box where velocity is the lowest this deviation is similar to the highest velocity layers.

Figure 4.3. Maximum (filled symbols) and minimum (empty symbols) velocities of each layer as a function of position in the velocity gradient direction, for a range of shear rates.

4.3. GIANT, a new thermostat for DPD

In order to address this issue, we introduced here an external thermostat that acts based on the real-time calculation of the particle velocity distributions in the system. To do this, we have modified the time integration scheme and introduced an additional step where the following occurs:

1- The velocity profile is monitored by dividing the calculation cell into layers,

2- The velocity distribution is formed using the particle velocities in regards to

average velocity of the parent layer,

3- FWHM is calculated from distribution curve and,

4- If the kinetic temperature of the fluid has deviated more than 5% from the set

temperature, all the particle velocities are divided by the calculated FWHM in

kT order to make sure that FWHM is corrected to the set value ( vvNew = B ). iiFWHM

One should note that the calculation of temperature in any DPD simulation, and more pronouncedly in non-equilibrium conditions, is associated with fluctuations in the measured temperature. This fluctuation is normally in the range of less than 3%, meaning that for a system that retains its set temperature of 1.0, at each individual time step, the temperature might vary between 0.97-1.03 which also has been reported previously [99,

100]. By imposing an applied shear rate in the calculation box the value of this fluctuation can increase by some extent, while the time-average temperature is kept unchanged. It should be noted that although this slight deviation in the temperature is present, the dynamics of the system is not affected by it, and thus this small deviation can be ignored.

The reason for this error is that in order to calculate the temperature, the very initial step is

95 to divide the calculation box into number of layers in the velocity gradient direction and

measure the average velocity of each layer, u()ri . Each of these particle layers take a finite thickness which cannot be too small or too large. A slim choice of layer thickness results in a very few number of particles to calculate the average layer velocity from, while an overestimated layer thickness includes an inherent velocity gradient within itself that gives rise to unrealistic temperature/distribution measurement. In other words, the small deviations in the temperature, particularly under shear, are not considered as physical deviations as they are originated by the numerical means of measurement. Hence in applying the thermostat (the 4th step above), two entities (the FWHM and the temperature) being compared are subject to intrinsic numerical errors of the range <5% that are of the same nature. The value of 5% error was employed to ensure that the thermostat is only applied in the system when deviations measured at each time step are larger than the so- called normal deviations measurable in any DPD simulation, and are originated by a physically meaningful changes in the velocity distribution function, rather than a common numerical error associated with these types of simulations.

We performed similar simulation studies using the well-known “Lowe-Andersen”

[90] scheme coupled with DPD thermostat. This is simply because we aim at studying the effect of dissipative parameter on the viscosity of a simple DPD fluid, thus instead of employing a stand-alone LA scheme which excludes the dissipative and random potentials completely, we have coupled this time integration method as an external heat sink with the

DPD thermostat. The Lowe-Andersen scheme utilizes the relative velocity if interacting particle pairs as opposed to individual particle velocities. Equation 1-3 show different steps

′ of the L-A thermostat. In the first step, a relative velocity ( vij ) is taken from the

Maxwellian distribution by using the random number of ξij with unit variance and zero

mean value. Subsequently, particle velocities are corrected using product of the

′ Maxwellian relative velocity ( vij ) and the real relative velocity ( vij ) of particles in the last

step. It should be mentioned that this scheme eliminates the random and dissipative forces

from the equation of motion, and replaces them by the equations 1-3, and by doing this

changes the dynamics of a system abruptly.

′ vij= ξ ij2,kT B (eq. 1)

2∆=ijrv ij (′ ij − v ij ). r ij , (eq. 2)

vv′′i= i +∆ ij;, v j = v j −∆ ij (eq. 3)

Previously our group reported the effect of this scheme on the linear and non-linear viscoelastic properties of the polymer melts at equilibrium and low shear rate out of equilibrium conditions [9].

Figure 4.4.Temperature versus shear rate using: a) GIANT, and b) Lowe-Andersen thermostat.

Figure 4.4 shows the temperature as a function of shear rate for a wide range of

dissipative parameters using: a) our thermostat, and b) the Lowe-Andersen thermostat.

Comparing the results in figures 4.1 and 4.4 shows that introducing the thermostats provides an excellent control over the temperature at the complete range of shear rates even at the lowest dissipative parameter values of 1. In the case of the new thermostat, it should be noted that the temperature is stable at the value of 1.06 as opposed to set temperature of

1.0; this constant deviation is attributed to the fact that real-time calculation of the velocity

distribution curve and consequently FWHM from limited number of velocities (number of

particles) is always associated with numerical errors. Also, since finite thickness of each

layer is being utilized in order to form the velocity distributions, the FWHM is always

calculated for particles belonging to different shear rate regions but same parent layers.

This can be decreased by increasing the number of particles, which increases the

calculation time. Nonetheless, the introduced thermostat effectively controls the

temperature without significant deviations at high shear rates. On the other hand, the LA

thermostat shows a contradictory results. Although the temperature remains constant for a

wide range of shear rates, it starts to deviate as the shear rate is increased, and more

importantly increasing the dissipative parameter results in increased temperatures at high

shear rates. As it was explained before, higher dissipative parameters yield narrower

velocity distributions, and since the LA scheme is applied based on the relative velocity of

interacting particle pairs as opposed to individual particle velocities, increasing the

dissipative term reduces the efficiency of the LA thermostat. Figure 4.5 shows the velocity

distribution of the DPD particles at different shear rates using the proposed thermostat and

the dissipative parameter of 4.5. Compared to the one obtained from the regular DPD

(figure 4.2), one can conclude that the thermostat successfully retains the velocity distributions intact even at the highest shear rates that are not accessible to regular DPD.

Figure 4.5. Velocity (in the flow direction) distribution for a range of shear rates, using: a) standard DPD, and b) GIANT thermostat.

Figure 4.6. Maximum (filled symbols) and minimum (empty symbols) velocities of each layer as a function of position in the velocity gradient direction, for a range of shear rates using GIANT thermostat.

Also, the velocity profiles presented in figure 4.6 show that regardless of the shear rate, the maximum and minimum velocity of each layer are in a close vicinity, preventing the distribution to deviate from its proper form.

One of the main features of DPD simulations is that the momentum is conserved both locally and globally in DPD and thus the hydrodynamic is naturally preserved. The

DPD formalism does this via the dissipative and random forces as it continues to do so without the conservative interactions; however, as the distribution curves of the figure 4.2 suggest, at high shear rates the DPD particles do not represent a physically meaningful fluid anymore and although the momentum is conserved at all conditions, the hydrodynamic is not preserved anymore.

Using the new thermostat, one can retain the interacting particles in a realistic velocity neighborhood and thus ensure that hydrodynamic is preserved even at high shear rates. This is more pronounces when the relative velocity of interacting particle pairs is studied. Figure 4.7 shows the relative velocity distribution of interacting particle pairs, for standard DPD and the new thermostat (GIANT), for dissipative parameter of 4.5 and at three different shear rates in different directions.

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Figure 4.7.Relative velocity distributions for different shear rates in different directions. Theoretically, the relative velocity of interacting particles should form a

Maxwellian distribution (this in fact serves as the foundation of the LA scheme). At low

shear rate of 0.5, where the standard DPD thermostat successfully controls the temperature,

both distribution curves (black line and symbols) coincide which proves that at low shear

rates the dynamics of DPD remains unchanged upon introduction of the thermostat.

Nonetheless, at high shear rates the DPD thermostat shows a broadened distribution which

results in abrupt increase of the dissipative stress in the system. At this stage, since the

interacting particles hold velocities that largely vary, unrealistic dynamics are recovered.

The thermostat however retains the general distribution of relative velocities intact and thus

provides proper description of the fluid even at extremely large shear rates.

The physical consequence of the anomalies in standard DPD is clearly observed

when transient flow properties of a fluid is considered. Using the L-E boundary condition,

at the start-up of the flow particles in the boundary layer are the first ones to adapt the

velocities induced by the imposed shear rate. Shortly, the velocity profile is developed in

the calculation box by natural diffusion of the momentum through interaction of

neighboring particles (dissipative force). This transient flow behavior is characteristic of

the fluid and can be studied by monitoring the velocity profile in the flow direction at

different simulation times as the system reaches the steady state. Figure 4.8 shows the velocity profiles of the standard DPD and the ones from thermostat-coupled simulations

for the dissipative term of 4.5 and at different shear rates.

Comparing the velocity profiles in figure 4.7.a and d proves that at low shear rates

the same transient properties are observed with/out the thermostat. Nevertheless, one

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clearly observes a change of behavior as the shear rate is increased. The DPD fluid at high

shear rates (figure 4.8.b and c) start to develop the imposed shear profile similar to the ones

observed at low shear rates, followed by an unrealistically fast development of the velocity

profiles. This change of transient properties is due to an accelerated diffusion of momentum

in the system caused by widened relative velocity distributions in figure 4.7. The particles

that are now in a much wider velocity neighborhood interact through dissipative term, and

since the dissipative force spreads the momentum in the system, transient properties of the

DPD fluid is significantly affected (unphysical internal momentum source). In contrary, the proposed thermostat provides the proper velocity distributions and relative velocities required for the dissipative force to naturally carry the momentum in the calculation box.

Figure 4.8.Velocity profile at different simulation times for shear rates of: a, d) 0.5, b, c) 2.0, c, f) 10.0, using standard DPD (top row) and new thermostat (bottom row). 4.4. Viscosity measurement using GIANT

Now with the proper thermostat at hand, we can investigate the rheological

response of a DPD fluid to an extended range of shear rates. Figure 4.9 shows the viscosity

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as a function of shear rate for a range of dissipative parameters. The viscosity of the DPD

fluid shows a Newtonian plateau at low shear rate regime followed by an unphysical shear-

thickening at higher shear rates which is pronounced when low friction parameters are

employed. This is a direct consequence of the temperature instability in the simulation, as

the kinetic contribution of the stress starts to substantially increase by increasing the

temperature.

Figure 4.9.Viscosity vs. shear rate for a range of dissipative parameters using: a) regular DPD thermostat, b) our proposed thermostat, and c) Lowe-Andersen method coupled with DPD. Figure 4.9.b on the other hand shows that introducing the thermostat effectively

increases the range of shear rates by more than a decade. The slight shear-thinning of the viscosity at low γ values can be explained by the fact that since the contribution of the

conservative force is continuously decreasing (by shear rate), at low dissipative parameter

values, the contribution from the random and dissipative potential in stress tensor is not strong enough to compensate for this effect. However, shear-thinning quickly vanishes when γ is increased to 25. The results in figure 4.9.c clearly shows that coupling the DPD

thermostat with LA method significantly changes the dynamics of the system. Thus

although LA scheme can effectively control the temperature, at the same time it makes the

results incomparable to the ones obtain by regular DPD. Obviously, in the case of stand-

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alone LA time integration scheme, which does not include the dissipative and random

force, there is no friction parameter dependence to be studied and the fluid dynamics is

solely controlled by the bath probability function introduced in the LA scheme which is

studied in depth elsewhere [9].

4.5. Conclusions

In this chapter, a thorough discussion on the sources of temperature instabilities in

DPD simulations of out-of-equilibrium conditions was presented. Consequently, a novel

auxiliary thermostat for non-equilibrium Dissipative Particle Dynamics simulations that

provides accurate control of the temperature over a wide range of shear rates and dissipative

parameters was proposed. This was achieved by real-time calculation of the distribution of

the particle velocities in regards to their imposed shear profile. Subsequently, the velocity

of individual particles was corrected to adapt into the Gaussian distribution of the desired

temperature.

In addition to temperature controlling effect of the proposed thermostat, we have

investigated the rheological response of the DPD fluid over a range of shear rates in regards

to the stress tensor. Stabilizing the temperature revealed that the shear rate dependence of

the viscosity is virtually the same for all the dissipative parameters and only showed slight

shear-thinning behavior at low γ values and at very high shear rates due to insufficient contribution from the dissipative parameters incapable of compensating for the decrease in the conservative stress. Thus, it can be concluded that using the proposed approach, the window of shear profiles accessible to DPD simulation can be significantly extended by controlling the temperature without compromising the physical nature of the phenomenon.

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This can be used in situations where high shear rate conditions are required in order to

study a specific phenomenon such as shear-thickening behavior in colloidal suspensions, droplet break-up under flow, and many more.

4.6. References

5. Hoogerbrugge, P.J. and J.M.V.A. Koelman, Simulating Microscopic Hydrodynamic Phenomena with Dissipative Particle Dynamics. Europhysics Letters, 1992. 19(3): p. 155-160. 8. Ye, T., et al., Dissipative particle dynamics simulations of deformation and aggregation of healthy and diseased red blood cells in a tube flow. Physics of Fluids (1994-present), 2014. 26(11): p. -. 9. Khani, S., M. Yamanoi, and J. Maia, The Lowe-Andersen thermostat as an alternative to the dissipative particle dynamics in the mesoscopic simulation of entangled polymers. Journal of Chemical Physics, 2013. 138(17): p. 174903(1-10). 16. Marsh, C.A., G. Backx, and M.H. Ernst, Fokker-Planck-Boltzmann equation for dissipative particle dynamics. EPL (Europhysics Letters), 1997. 38(6): p. 411. 17. Espanol, P. and P. Warren, Statistical Mechanics of Dissipative Particle Dynamics. Europhysics Letters, 1995. 30(4): p. 191-196. 18. Jamali, S., M. Yamanoi, and J. Maia, Bridging the gap between microstructure and macroscopic behavior of monodisperse and bimodal colloidal suspensions. Soft Matter, 2013. 9(5): p. 1506-1515. 19. Groot, R.D. and P.B. Warren, Dissipative particle dynamics: Bridging the gap between atomistic and mesoscopic simulation. Journal of Chemical Physics, 1997. 107(11): p. 4423-4435. 35. Fedosov, D.A., G.E. Karniadakis, and B. Caswell, Steady shear rheometry of dissipative particle dynamics models of polymer fluids in reverse Poiseuille flow. The Journal of Chemical Physics, 2010. 132(14). 42. Zhao, T., et al., Dissipative particle dynamics simulation of dilute polymer solutions—Inertial effects and hydrodynamic interactions. Journal of Rheology (1978-present), 2014. 58(4): p. 1039-1058. 45. Lees, A.W. and S.F. Edwards, The computer study of transport processes under extreme conditions. Journal of Physics C: Solid State Physics, 1972. 5: p. 1921- 1929. 49. Chatterjee, A., Modification to Lees–Edwards periodic boundary condition for dissipative particle dynamics simulation with high dissipation rates. Molecular Simulation, 2007. 33(15): p. 1233-1236.

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50. Whittle, M. and K.P. Travis, Dynamic Simulations of Colloids by Core-Modified Dissipative Particle Dynamics. Journal of Chemical Physics, 2010. 132: p. 124906(1)-124906(16). 52. Chatterjee, A. and L.-M. Wu, Predicting rheology of suspensions of spherical and non-spherical particles using dissipative particle dynamics (DPD): methodology and experimental validation. Molecular Simulation, 2008. 34(3): p. 243-250. 55. Pan, D., N. Phan-Thien, and B.C. Khoo, Dissipative particle dynamics simulation of droplet suspension in shear flow at low Capillary number. Journal of Non- Newtonian Fluid Mechanics, 2014. 212(0): p. 63-72. 75. Espanol, P., Fluid particle model. Physical Review E, 1998. 57(3): p. 2930-2948. 76. Espanol, P., Dissipative particle dynamics for a harmonic chain: A first principle derivation. Physical Review E, 1996. 53(2): p. 1572-1578. 77. Ghoufi, A. and P. Malfreyt, Coarse grained simulations of the electrolytes at the water−air interface from many body Dissipative Particle Dynamics. Journal of Chemical Theory and Computation, 2012. 8: p. 787-791. 78. Ibergay, C., P. Malfreyt, and D.J. Tildesley, Mesoscale modeling of polyelectrolyte brushes with salt. Journal of Physics Chemistry B, 2010. 114: p. 7274-7285. 79. Malfreyt, P. and D.J. Tildesley, Dissipative Particle Dynamics simulations of grafted polymer chains between two walls. Langmuir, 2000. 16: p. 4732-4740. 80. Schnell, G.M.B., et al., Multiscale modeling approach toward the prediction of viscoelastic properties of polymers. Journal of Chemical Theory and Computation, 2012. 8(11): p. 4570–4579. 81. Mai-Duy, N., N. Phan-Thien, and B.C. Khoo, A numerical study of strongly overdamped Dissipative Particle Dynamics (DPD) systems. Journal of Computational Physics, 2013. 245(0): p. 150-159. 82. Pan, D., et al., Numerical investigations on the compressibility of a DPD fluid. Journal of Computational Physics, 2013. 242(0): p. 196-210. 83. Phan-Thien, N., et al., Exponential-time differencing schemes for low-mass DPD systems. Computer Physics Communications, 2014. 185(1): p. 229-235. 84. Fedosov, D.A., G. Em Karniadakis, and B. Caswell, Dissipative particle dynamics simulation of depletion layer and polymer migration in micro- and nanochannels for dilute polymer solutions. The Journal of Chemical Physics, 2008. 128(14): p. -. 85. Fedosov, D.A., I.V. Pivkin, and G.E. Karniadakis, Velocity limit in DPD simulations of wall-bounded flows. Journal of Computational Physics, 2008. 227: p. 2540-2559. 86. Pan, W., B. Caswell, and G.E. Karniadakis, Rheology, microstructure and migration in Brownian colloidal suspensions. Langmuir, 2010. 26(1): p. 133-142. 87. Pivkin, I.V. and G.E. Karniadakis, A new method to impose no-slip boundary conditions in dissipative particle dynamics. Journal of Computational Physics, 2005. 207: p. 114-128.

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88. Pivkin, I.V. and G.E. Karniadakis, Coarse-graining limits in open and wall- bounded dissipative particle dynamics systems. Journal of Chemical Physics, 2006. 124. 89. Pastorino, C., et al., Comparison of dissipative particle dynamics and Langevin thermostats for out-of-equilibrium simulations of polymeric systems. Physical Review E, 2007. 76(2): p. 026706. 90. Lowe, C.P., An alternative approach to dissipative particle dynamics. Europhysics Letters, 1999. 47(2): p. 145-151. 91. Khani, S., M. Yamanoi, and J. Maia, Mesoscale simulation of entangled polymers: Part II. Lowe-Andersen thermostat. Bulletin of the American Physical Society, 2014. 92. Praprotnik, M., L. Delle Site, and K. Kremer, Adaptive resolution scheme for efficient hybrid atomistic-mesoscale molecular dynamics simulations of dense liquids. Physical Review E, 2006. 73(6): p. 066701. 93. Soddemann, T., B. Dünweg, and K. Kremer, Dissipative particle dynamics: A useful thermostat for equilibrium and nonequilibrium molecular dynamics simulations. Physical Review E, 2003. 68(4): p. 046702. 94. Kramers, H.A., Brownian motion in a field of force and the diffusion model of chemical reactions. Physica, 1940. 7(4): p. 284-304. 95. Uhlenbeck, G.E. and L.S. Ornstein, On the Theory of the Brownian Motion. Physical Review, 1930. 36(5): p. 823-841. 96. Berendsen, H.J.C., et al., Molecular dynamics with coupling to an external bath. The Journal of Chemical Physics, 1984. 81(8): p. 3684-3690. 97. Nosé, S., A unified formulation of the constant temperature molecular dynamics methods. The Journal of Chemical Physics, 1984. 81(1): p. 511-519. 98. Andersen, H.C., Molecular dynamics simulations at constant pressure and/or temperature. The Journal of Chemical Physics, 1980. 72(4): p. 2384-2393. 99. Travis, K.P., et al., New parametrization method for dissipative particle dynamics. Journal of Chemical Physics, 2007. 127. 100. Travis, K.P., P.J. Daivis, and D.J. Evans, Thermostats for molecular fluids undergoing shear flow: Application to liquid chlorine. Journal of Chemical Physics, 1995. 103(24): p. 10638-10651.

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CHAPTER 5 RHEOLOGY OF COLLOIDAL SUSPENSIONS

This chapter is adapted from:

“Bridging the gap between microstructure and macroscopic behavior of

monodisperse and bimodal colloidal suspensions”, by Safa Jamali, Mikio Yamanoi and

Joao Maia, Soft Matter, 2013, 9 (5), pp-1506-1515.

“Microstructure-rheology relationship in colloidal suspensions studied by potential energy analysis”, by Safa Jamali, Armand Boromand, Shaghayegh Khani and Joao Maia,

Journal of Colloid and Interface Science Communications, 2015, submitted.

“Normal stress measurements in shear-thickening suspensions”, by Safa Jamali,

Armand Boromand, Norman Wagner and Joao Maia, Journal of Rheology, 2015, to be submitted.

“Time and rate dependent properties of colloidal gels under steady shear flows”, by

Safa Jamali, Armand Boromand and Joao Maia, Physics Review Letters, 2015, to be submitted.

5.1. Theoretical Background

The non-Newtonian rheological response of colloidal suspensions at different flow

regimes is perhaps one of the most discussed, and yet controversial subjects in the field of

fluid dynamics and physics. Conceivably flow properties of suspensions is also the most

demonstrative of the rheological responses, considering even a non-expert can understand

the concept of shear-thickening/thinning behavior exemplified by running over a pool of

cornstarch and water, and sinking in while standing still. Nevertheless for several decades

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researchers have been trying to explain the nature of this versatile behavior, which is still

an ongoing discussion in the physics and rheology community. Regardless of the

differences in the details of models and theories suggested to explain the rate-dependent

rheology of suspensions, there is a consensus that as these macroscopic measures change

at different flow conditions, the microstructure of the suspension undergoes

transitions/changes correlatively [101].

Recent advances in the experimental capabilities, such as fast confocal microscopy

[102] and neutron scattering measurements [103] of the sheared suspensions have brought

insight into microstructural state of a suspension at different flow regimes. At the same

time, accurate measurements of the rheological properties are made possible by advanced normal stress transducers and controlled stress rheometry, enabling reliable characterization of the suspensions at different flow regimes [104]. However,

computational studies have contributed significantly to the current understanding of the

microstructure-rheology relationship in colloidal suspensions. Recently, series of

simulations and theoretical reports [105-107] based on the frictional contacts have

successfully reproduced the so-called Discontinuous Shear-Thickening (DST), with

remarkable agreement compared to experimental results. On the other hand, the hydrocluster formation and lubrication driven shear-thickening at high shear rates have been able to reproduce the proper microstructure and rheological properties with semi-quantitative compared to experimental measurements [15, 18, 108]. In the following we briefly discuss the opposing theories and some of the experimental studies on the rheological characterization and material properties of a colloidal suspension at shear-thickened state.

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At the quasi-equilibrium condition or at rest, colloidal particles take a random

morphology and thus the microstructure of the suspension stays isotropic. This is mainly

because in lack of shear forces, or external pressures to push the colloidal particles in

specific directions, the total force on a colloid is dominated by the Brownian motion and

thermal fluctuations. As the suspension is subjected to shear flows, the microstructure and

the rheological response changes. At low and intermediate shear rates, the Brownian

motions are not strong enough to keep up with the external forces, and the suspension shear-thins. In other words, at these shear rates, the particles tend to follow each other’s

path in the flow direction which makes the flow of the suspension easier. This in

macroscopic scale results in shear-thinning of the suspension, as the Brownian contribution to the force is constant while the external forces are increased at higher shear rates.

Generally, shear-thinning in these shear rate regime has been suggested to be either

originated by the ordering of suspending particles under shear, or lack of resistance from

the fluid to keep up with the flow as the nature of thinning.

By further increasing the shear rate (flow force), the suspensions become

increasingly more resistant to the flow, which in rheological terminology is referred to as

shear-thickening. Different theories have been suggested to explain the genesis of the

shear-thickening behavior. Traditionally order-to-disorder transition at the critical shear

rate was thought as the cause of shear-thickening behavior [109]. However, experimental

studies have shown that shear-thickening can be observed without an order-to-disorder

transition [110, 111]. For the first time, Brady showed that short-range hydrodynamic

interactions, also known as lubrication, at high shear rates effectively holds the colloidal

particles in close vicinity and form the so-called hydroclusters which stand in the way of

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the flow and originate shear-thickening behavior [112]. The essence of the lubrication

theory is that high shear rate flows can effectively push the particles to the point where the

distance between the colloidal particles is small enough for the hydrodynamic interactions

to take over and give rise to lubrication stresses. Since the lubrication potential strictly

depends on the surface separation distance of two interacting colloids, this effect is more

pronounced for dense suspensions. At high shear rates, the external shear forces reduce the

distance between the colloidal particles. Since the lubrication forces are of dissipative

nature and tend to defy the relative motion of particles, as the particles get in close

proximity interacting colloids become fixated in small gaps forming larger so-called

hydroclusters. These hydroclusters can resist to the flow and give rise to shear-thickening

behavior. At these shear rates the net force on a colloidal particle and consequently the

microstructure as well as the rheological response of a suspension is dictated by the

hydrodynamic interactions. Since this versatile rate dependent response of a suspension is

governed by the competition between the hydrodynamic and Brownian forces, these

distinct regimes are frequently presented as a function of a dimensionless number, Péclet,

6πη γR3 Pe = 0 . Where the shear rate dependent numerator is the strength of the kTB

hydrodynamic forces and the Brownian motions are represented as the temperature in the

denominator.

In contrary to the theory of lubrication driven shear-thickening, during the past few

years several studies have brought back the theory of dilatancy and frictional contacts [106,

107, 113-115]. Recently, Jaeger and coworkers [113, 116-118] have reported studies, bringing back the idea that was firstly introduced by Metzner and Whitlock [119], where

111 dilatancy and frictional forces at high shear rates are used to explain discontinuous shear- thickening. In other studies by Seto and coworkers [105, 106], frictional contact forces were added to a Stokesian Dynamics (SD) simulation system and discontinuous shear- thickening was successfully recovered. However, the proposed model does not agree with the experimental measurements of normal stress differences [104]. In a separate study

Wyart and Cates [114] theoretically showed the possibility of observing DST as a physical consequence of the frictional contacts. Fernandez et al. [107] proposed a transition from hydrodynamically lubricated to boundary lubricated mechanism as the microscopic origin of shear-thickening and performed experimental and simulation measurements confirming the theory. In a separate study, formation of correlating clusters due to frictional forces was suggested to give rise to shear-thickening in non-Brownian suspensions [115]. Regardless of the details of these studies, a general methodology is utilized: at very close colloidal contacts, a slip-stick combination scenario similar to friction laws of granular physics governs the rheological response of the fluid, and the net force exerted on a particle is dominated by the contact forces that take over the lubrication-driven stresses. As opposed to the role of hydrodynamic stresses, these reports suggest that at the very small separation distances the lubrication layer breaks and the contact between the two colloidal particles occurs. As soon as this contact point is formed, the microstructure and the motion of colloidal particle is governed by the competition between the normal repulsive contact potentials and the tangential friction forces. Thus, a stick-slip mechanism based on the

Coulomb’s law of friction is adapted to explain the frustrated motion of colloidal particles.

In several reports based on this methodology, DST and S-shaped stress curves of

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suspensions were reproduced successfully by tuning the friction coefficient of colloidal

particles.

While viscosity is perhaps the most indicative rheological landmark of a

suspension’s microstructure, drawing a comprehensive picture of the micro-macro

relationship is only possible by considering the complete tensorial form of the stress,

including the normal stresses. In other words, one can claim proper dyanmics and the

underlying physics only if complete rheological characterization of a suspension is

reproduced. One of these rheological properties, which directly reflects the microstructural

changes of a suspension under flow is the normal stress difference. However, first and

second normal stress differences, N1 and N2, are generally of smaller magnitudes compared

to shear stresses and also are associated with various instabilities in practical

measurements. Hence, the number of experimental reports where these two parameters are

accurately measured are very limited; however, recent advances in the rheometry

techniques has allowed reliable measurements of normal stress differences for dense

suspensions at high shear rates [104]. Regardless of the experimental evidences (which will be briefly discussed later), one can predict the microstructure and the normal stresses of a

suspension in different regimes based on the theoretical approaches explained before. The

hydrocluster formation as a result of large lubrication stresses in the center-center line of the colloidal particles suggests that the colloids will be concentrated in the compression axis of the shear plane [15]. Subsequently, particles will form a highly anisotropic structure where the probability of finding a neighboring colloid in a close vicinity is much larger than other coordinates. At this anisotropic states, the hydrodynamic and the Brownian contributions to the normal stress differences suggest a negative N1 and N2. A detailed

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explanation of the effect of microstructure on the sign and magnitude of normal stress

differences (based on the hydrodynamics theory) can be found in the work by [15]. On the other hand, the frictional contact model is based on the completion between the tangential and normal forces. At high friction coefficients, the tangential forces dominate the type of interaction between the two overlapping colloids and their frustrated movement gives rise to an effective transient contact network to be formed at high shear rates. This contact network is able to bear large amounts of stresses and thus DST can be reproduced by this method. In other words, the shear thickening is recovered only at the conditions where the tangential friction dominates the net force on a single colloidal particle. Obviously, at these conditions one can expect positive N1 and negative N2 to be measured due to the nature of

frictional forces [105].

Regardless of the theoretical predictions on the sign and magnitude of normal stress

differences, several experimental studies have reported measurements of these properties.

One should be cautious when referring to these data as in many cases, the authors state that the results are associated with intrinsic instrumental limitations. Perhaps the first quantitative measurements of normal stress differences are done by [120]. He reported the

N1 of the same magnitude as the shear stress with a negative sign, and N2 with half of this

magnitude and a positive sign, for a 58.7 vol% suspensions of styrene/ethyl acrylate

copolymer particles. Aral and Kalyon [121] reported increasing negative first normal

stresses differences by increasing the volume fraction of non-colloidal particles and the shear rate. Another example of the negative N1 in the shear-thickening regime was reported

by Lee [122] for near hard-sphere suspensions. Although the scattered data sets measured

for N2 did not let the authors to report reliable second normal stress differences, it was

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stated that the magnitude of N2 is larger than the N1. Arguably the most accurate normal

stress measurements of colloidal suspensions were reported in a recent study by Cwalina

and Wagner [104], where negative normal stress differences were reported for a range of

different volume fractions at the shear-thickening regime. N1 and N2 were found to be linearly increasing functions of shear rate in the shear-thickened state, which enabled

authors to define the shear-thickened state and consequently calculate the material

properties such as first and second normal stress different coefficients and the maximum

packing fraction at the shear-thickened state.

Theoretically, at the very close distances between the colloidal particles, and at high

shear rates where the shear stresses are extremely large, the particles start to deform elasto- hydrodynamically [123]. To explain this phenomenon, one has to take into account the fact that colloidal particles in real experiments have finite hardness or modulus, and thus will begin to deform as soon as the normal stresses exerted on them exceeds this modulus.

Kalman [124] showed that by taking this deformation into account one can explain the

second shear-thinning regime which is usually observed for soft suspensions. Nevertheless,

the computational models to date have not been able to study and examine this theory. This

is because in numerical approaches, a hard short-ranged repulsive force is usually employed to define the hard-sphere identity of a colloidal particle.

Computational studies in general have successfully predicted the local dynamics of colloidal particles, but have been restricted to small scale systems and failed in explaining large time and/or length scales [13-15, 50, 112, 125-127]. Similarly, most numerical studies on suspensions have focused on either near-equilibrium flow conditions or on dilute suspensions because of the difficulties in dealing with multi-body hydrodynamic

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interactions and their crucial role in such systems [128]. For example, Furukawa and

Tanaka [129] studied the role of multi-body hydrodynamic interactions on the aggregation and gelation process by fluid particle dynamics (FPD) method under equilibrium conditions; however, the authors did not study nor explained if the method is capable of predicting the flow behavior of such system, so its potential to capture the flow behavior of dense suspensions still unknown.

5.2. Prior DPD simulations on suspensions

The history of simulations on the dynamics of suspensions in DPD begins with the

very first reports using the method [5, 6]. In these initial reports, colloidal particles were represented by frozen DPD particles in a bath of flowing solvent particles. DPD inherently preserves the long-range hydrodynamics through dissipative and random ensemble, as well as multi-body interactions through simulation of solvent particles. These properties of DPD makes this method superior to most simulation techniques, and more specifically the ones traditionally used to model the colloidal suspensions, such as Brownian Dynamics.

Nevertheless the first attempts in employing DPD to model the flow properties of suspensions had only limited success. The authors in those reports successfully calculated the drag force on a colloidal particle, as well as the general dependency of the suspension viscosity on the colloidal content. The main reason for this lack of success can be explained by two inherent shortcomings of DPD which will be discussed in following.

In real suspensions, the solvent is present everywhere in between the colloidal particles. This is of utmost importance in capturing the proper physics and dynamics of suspensions as explained in the previous section; however, in DPD simulations, when the

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distance between two colloidal particles is smaller than a single solvent particle the solvent

is pushed outward. Thus instead of a suspension where the hydrodynamics is captured

properly throughout the calculation box, one will lose the so-called short-ranged

hydrodynamics in instances where the colloidal particles are in close proximity. This is

schematically depicted in figure 5.1.

Figure 5.1.A suspension reproduced by DPD (lef) compared to how a suspension has to be represented in order to preserve the complete hydrodynamics. In figure 5.1, one can argue that when the distance between the red colloidal

particles is smaller than the size of solvent in DPD (shown in smaller rectangles), the effect

of solvent is lost and thus instead of representation of a continuous solvent through

particulate representation, hydrodynamic breaks down. A remedy to this issue, that has

been adapted in several reports is to represent the solid particles as an assembly of DPD

particles linked together to represent a single colloid [54, 61, 130]. In real suspensions, generally the colloidal particles are much larger in size and mass when compared to solvent species, thus this approach provides a physically more meaningful definition to the colloidal particles. Nevertheless, there are two main issues associated with this method:

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firstly, because many DPD particles are required to form a single colloid, the computational

cost of simulating a dense suspensions is very high. Secondly, in the process of coarse-

graining through bundling the single DPD particles into colloids, the smooth surface

definition of interacting components is lost. In other words, since large forces are used in order to keep the colloidal particles together, the integrity of the colloidal surface is compromised. This is schematically depicted in figure 5.2. While a suspension in real life is consisted of colloidal particles with various sizes (fig. 5.2.top), in this approach one will need to bundle many DPD particles to reproduce such a system (fig. 5.2.bottom). It is clear that using this method the separation distance between two interacting colloids is vaguely

defined, which gives rise to several unphysical consequences. Ideally, in order to model

the colloidal suspensions properly without compromising these crucial details, one requires

a clear definition of particle size that can be tuned to desired values as well as a formalism

that ensures preservation of hydrodynamics in all conditions.

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Figure 5.2. Schematic representation of poly dispersed suspension through bundling individual DPD particles (bottom) versus the ideal single-particle definition (top). This definition was firstly introduced by Whittle and Travis [50], where they introduced a core-modified DPD method. In their method, colloidal particles were defined as the single rigid-cored particles that were coated with a soft layer. The core-modified definition by Whittle and Travis is presented in figure 5.3. There are two main advantages to this definition of solid particles in DPD: i) the integrity of particle surface is intact as the surface-surface distance between two colloidal particles, hij is clearly defined, and ii) one can reproduce particles of different sizes and thus suspensions of varying dispersities by changing the arbitrary size of the colloidal radius, Ri. Based on the definition by Whittle and Travis in the next section we propose a modified DPD method that is capable of capturing the proper dynamics of colloidal systems.

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Figure 5.3. Core-modified definition of colloidal particles as presented by Whittle and Travis. In a series of reports, Phan-Thienn and coworkers [55, 131, 132] have proposed a

similar approach of defining colloidal particles as rigid-cored components; however, the

details of the formulations were significantly different compared to the proposed method

by Whittle and Travis. Namely, the numerical method of providing the rigid nature to the

colloidal particle is of significant importance. One can incorporate hard-potentials similar to Lennard-Jones generally used in MD simulations. Another method proposed by Phan-

Thien and coworkers provides a rigid definition to colloids by changing the exponents of the weight functions in the formalism of DPD and by changing the cut-off distance for

specific colloid-colloid and colloid-solvent interactions.

5.3. Modified DPD model

The explicit solution of the equation of motion for solvent particles, and a built-in

thermostat that forms the canonical ensemble, and a series of pairwise interactions that

ensure conservation of momentum in a system, enables the long range hydrodynamics to

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be naturally preserved in DPD formalism. However, the early attempts on shear rheology

of suspensions using DPD had a limited success as the hydrodynamics breaks down at the

close separation distances [127]. In our modified DPD model, we use an arbitrary-sized

rigid core to represent the colloidal particle, with additional lubrication potentials to

compensate for the short-ranges lubrication interactions. Thus the equation of motion for a

colloidal particle in this formalism is written based on 5 main pairwise interaction

potentials.

dvi C D R H Contact mi =∑FFFFFij ++++ ij ij ij ij (eq. 1) dt

This equation reduces to the first 3 forces for the solvent particles, where only conservative, random and dissipative interactions are solved. The first force, conservative, is the extent of pressure between the interacting species and for the solvents is parametrized

κ −1 −1 based on the compressibility of water at ambient temperature, a≈ kT [19], ij B 0.2ρ

where ρ is the number density of DPD particles. Equation 2 shows the expression for the

conservative force, where aij is the conservative parameter, ωij()r ij is the weight function

based on the separation distance between the interacting particle pair, and eij is the unit

rij vector as eij = . rij

CC Feij= a ijω ij(r ij ) ij (eq. 2)

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The random force generates the thermal fluctuations in the system using a random

function, Θij , of zero mean value and unit variance. ∆t in equation 2 is the time step used

in the simulation, and σ ij is the strength of the thermal fluctuations in the system.

RRΘij Feij= σω ij ij(r ij )ij , (eq. 3) ∆t

Dissipative force acts against the relative velocity of particles, vij= vv i − j as the

heat sink and dissipates the generated heat in the random force and ensures the proper

thermodynamics to be reproduced. Hence, the dissipative parameter (frequently referred as

friction parameter),γ ij has to be coupled to the random parameter in equation 3 [17]. This

is done via the so-called fluctuation-dissipation theorem and consequently the

2 σ ij dimensionless temperature in the system is defined as = kTB . Additionally, the weight 2γ ij

functions used in these forces are inter-related via the equation 5. In fact, all of the DPD potentials are calculated via this weight function which starts at unity and decays to zero at a distance called the cut off. In standard DPD method, the interparticle potentials are so soft that often the cut off distance is assumed as the diameter of a DPD particle, as the particles are allowed to overlap and pass through one another.

DD Fij=−⋅γω ij ij(r ij )(v ij ee ij ) ij , (eq. 4)

r −≤ij CR D0.5 1;rrij c ωωij(r ij )= ij (r ij ) = [ ω ij (r ij )] = ωij (r ij ) = rc (eq. 5)  ≥  0; rrij c

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One of the hallmarks of DPD is that although the random force is the main source

of Brownian motion in a DPD fluid, since together with the dissipative force they conserve

the momentum both locally and globally they also preserve the proper long-range and

multi-body hydrodynamics. In addition to these forces, which are used for solving the

equation of motion for both solvent and the colloidal particles, the short-range lubrication

H Contact potential, Fij , and the contact forces, Fij , are also introduced for the colloidal particles.

H The former is based on the pair drag term, µij , in squeeze mode hydrodynamics [133],

which diverges at the surface-surface contact point, hij = 0 (equation 6). R is the radius of

the colloidal particle and η0 is the viscosity of the suspending fluid. Since the lubrication

potential is singular at the contact, a small gap, δ , is introduced to the pair drag term to regularize this force. At these distances the lubrication potential becomes independent of the surface-surface distance between the two interacting colloids.

 2 3πη0 R ,0<≤hij δ 2δ HH H Fij=−=µµ ij( ve ij.; ij) e ij ij  , (eq. 6) 3πη R2  0 > δ , hij 2hij

The lubrication force tries to reduce the relative velocity of particles and therefore

contributes to the kinetic energy of the system. Thus the random force has to be corrected

to satisfy the fluctuation-dissipation theorem (eq. 7).

R 2 HRθij Fij =(σ mn + 2,kT B f ij) w ij e ij (eq. 7) ∆t

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Contact The contact force, Fij , defines the rigid nature of the colloidal particles and

prevents the overlap between the two cores, and vanishes cubically at a particular

separation distance, ∆ , comparable to the properties of a colloidal particle in real

experiments. Additionally, since the real particles have finite modulus the softness and hardness of colloids can be adjusted by tuning the magnitude of the contact modulus,

f Contact .

 fContact (1+≤ hh ), 0  ij ij Contact Contact hij 3 Fij = fh(1 − ) , 0 ∆  ij

One should thus note that by changing the soft layer thickness, ∆, at which the

contact force goes to zero, one can reproduce colloidal particles with different

characteristics. Namely, if one chooses to model electrostatically charged suspensions with

significantly large double layer thicknesses, this parameter has to be correspondingly large.

In fact one can define a characteristic ratio based on this thickness and the radius of the

hard-sphere as presented in equation 9.

∆ RReff= HS 1, + (eq. 9) RHS

In contrary one can model so-called hard-sphere suspensions by simply reducing this thickness value to zero; however, in real experiments, perfec hard-spheres do not exist as all colloidal particles are associated with surface roughness values in ranges of 10-3 times

the colloidal radius, and most rigid particles are coated with very thin layers of polymers.

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It should be mentioned that in our methodology, we assume elasto-hydrodynamic

deformation of colloidal particles at close gaps. In other words, instead of assuming a

contact that breaks the lubrication layer and sticks the colloidal particles to one another,

we assume that as the particles get closer to each other the hydrodynamic stresses continue

to increase and eventually will elastically deform the particles. Thus, at the event of overlap between the colloidal cores, the contact force linearly increases. It has been previously shown that the stress level between the particles at this condition, depends directly on the value of the shear modulus of colloids. This is in contrary to the model adapted in recently developed frictional contact models, where a stick-slip scenario is considered based on the frictional contact models of granular physics. In our model the contact forces are only applied in the normal direction and tangential forces are neglected (both in contacts and the lubrication). In the next section we present the physical consequences of the choice of parameters for each individual force in our model.

5.4. Parameterization of colloidal forces

As it was mentioned in the previous section, the choice of interaction potential

between the solvent particles can be made based on the Groot-Warren expression for

compressibility of water at ambient temperature. Also, in Chapters 2-4 we have explained

in detail the methodology of choosing dissipative and random parameters for each simulation. Thus one in the first step has to establish the method to choose the interaction parameter between the colloidal particles, namely the conservative and the contact potentials. Figure 5.4 provides the schematic view of the conservative force and contact

forces of different soft layer thickness values.

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Figure 5.4. Schematic decay functions for conservative force and contact forces of different soft layer thicknesses. The contact force is of a semi-hard nature, and is much larger when compared to conservative force, at distances smaller than the soft layer thickness, where this force is effectively active. Thus at these distances the net force on the colloidal particles is dominated by the extent of this potential; however, at larger distances the conservative force becomes larger and plays the major role in defining the equilibrium distance between the colloidal particles.

Regardless of the long range interaction behavior of colloidal particles and the effect of each force on the physical meaning of the particles, as it was thoroughly discussed in the background section, the rheological response and macroscopic behavior of colloidal suspensions at high shear rates and specially in dense suspension are controlled by the

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nature of interactions at very small gaps. Figure 5.5 depicts the schematic view of decay

functions applied on a near hard-sphere suspension with the layer thickness of 10-3R to

mimic the roughness of real particles with different values of contact moduli.

Figure 5.5. The schematic decay functions of colloidal interactions at very close separation distances. It should be noted that although the lubrication potential is clearly dominant to other

types of interactions at small gaps, this potential strictly depends on the relative velocity of

interacting colloids and thus becomes significant at high shear rates. Nonetheless, one can simply conclude that the short-ranged behavior of suspensions are greatly governed by the extent of contact potentials, while long-range interactions are affected by value of conservative force. As it will be shown later, a completely different class of colloidal system can be simulated by simply eliminating the conservative force from the interaction

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scheme; however, in this chapter the same conservative parameter of 25.0 is used in our

simulation for simulation of colloidal particles.

5.5. Suspensions at Rest (quasi-equilibrium conditions)

The first step of the validation of the new model has to be made on the rheology

and dynamics of colloidal systems at equilibrium conditions. In order to do so, here we

define the mass and size of colloidal particles, which is used for calculation of the colloidal

volume fractions. Equation 10 defines the mass and volume of colloidal particles in respect

to the number density of the calculation system, in order to keep the density of solvent and

colloids matched for each simulation.

4 MV=ρ = ρπ R3 , (eq. 10) 3

Using the volume of each particle one can simply calculate the fraction of solid particles as sum of the total colloidal volume divided by the volume of the calculation box; however, this definition is only valid for the near hard-sphere colloids and has to be

modified for the colloidal and stabilized suspensions based on the effective radius of

particles (eq. 11).

3 ∆ φφeff= HS 1, + (eq. 11) RHS

Figure 5.6 shows the zero shear viscosities measured for near hard-sphere suspensions of different volume fractions compared with the experimental and theoretical results in the literature [15, 134-137]. As one would expect the viscosity of a suspension diverges as the volume fraction of the solid content reaches the maximum packing

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fractions. Theoretical predictions for volume fraction dependency of a suspension can be

explained as the following.

As it was discussed in the introduction section, the viscosity of a suspension is

strictly dependent on the viscous, and Brownian forces as well as the excluded volume of

the colloidal particles [101, 138]. The thermal fluctuations introduced in our model by the random potential ensure that the Brownian motion is always present for small enough particles, as generally colloidal particles of sizes larger than 1 μm are assumed as non-

Brownian particles. On the other hand, the multi-body hydrodynamic effects play a major role in defining the viscosity of a suspension. The dependence of the viscosity of a suspension on the fraction of colloidal particles was firstly discussed by Einstein [139,

140], where a theoretical expression was derived for this relationship in dilute regime (low particle concentrations:

ηφ=1 + 2.5 , (eq. 12)

The 2.5 prefactor of the equation 12 is derived for the spherical particles, and differs for different shapes and geometries. By increasing the number of colloidal particles, the multi-body interactions and hydrodynamics effects due to crowding of colloidal species becomes increasingly more dominant which suggests stronger dependencies on the volume fraction for the viscosity. Perhaps the most widely accepted relation providing this dependency is given by Krieger and Dougherty [141] presented in equation 13.

−[ηφ] φ Max η =1, + (eq. 13) φMax

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In equation 13 ϕMax is the maximum packing fraction of colloidal particles,

corresponding to the maximum possible solid content at each condition. For instance, FCC

lattice structures allow solid contents of up to 74% while in randomly packed structures

this value does not exceed ~ 60-64% [142]. As the volume fraction of colloidal particles is

increased to these maximum allowed concentrations, the relative motion of particles

becomes increasingly more restricted up to a point where particles become fixated in well-

defined structures and form so-called colloidal glasses. In these situations, the viscosity

rises to infinity. Several experimental measurements have shown that the product of the

intrinsic viscosity and the maximum packing fraction of a suspension is arguably constant

at 2, which reduces the Krieger–Dougherty's equation to a equation with a single tunable parameter in equation 14 [143, 144].

−2 φ η =1, + (eq. 14) φMax

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Figure 5.6. Zero shear viscosity of suspensions modeled at different volume fractions compared with experimental and computational data. Now with the proper viscosity measurements at the equilibrium condition, one can look at the dynamics of colloidal particles in absence of flow for different volume fractions.

Figure 5.7 shows the mean squared displacement as a function of simulation time for different volume fraction of colloidal particles.

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Figure 5.7. Mean squared displacement of colloidal particles in different volume fractions as a function of time. There are three main observations to be made based on the results in figure 5.7: i)

regardless of the volume fraction of colloidal particles, the very first slope in the MSD

curve in logarithmic scale is larger than 1, which corresponds to super diffusive motion of

these particles at very short times. This is expected as all species show the same type of

behavior due to thermal fluctuations. ii) By increasing the volume fraction in the diffusive

regime (where slope is 1), the motion of colloidal particles becomes increasingly restricted

as the MSD systematically decreases at each given time for higher concentrations. And iii)

as the concentration is increased to so-called glassy regime, particles become arrested in glassy structures. This is evident in the plateau regimes observed for the highest volume fractions in fig 5.7; however, eventually different microscopic phenomenon such as cage

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hopping and particle rearrangements lead to increase in the MSD curves at longer time

periods. This will be explained in the next chapter in detail.

Another important study to be made on the colloidal suspensions is the

microstructural analysis. This is usually done by plotting the pair correlation function, g(r),

plots of suspensions in different planes. PCF in 3-D is usually defined as the probability of

finding a colloidal particle at a particular location in space (x,y,z) in respect to any given

center of particle. This quantity is most commonly presented by the value of this distance

as gr()= x2 ++ y 22 z . However, one can obtain more detailed information about the

microstructural changes of a fluid by separating the g(r) in different 2-D planes. This is done by dividing the sample subject to study into several layers in a specific direction, and performing the same statistical analysis in the other two directions. For instance, in order to study the microstructure of a fluid in x-y plane, one has to divide the fluid into several layers in the z-direction and provide the probabilities in respect to the relative coordinate of particles as opposed to the distance between them, g(x,y). In our simulations, this has been done in different directions; however, it should be noted that in order to provide the planar projections of PCF the thickness of each sample layer in the reference direction must be less than the diameter of a single particle, so that only particles in the same plane are

included in the statistical analysis. In lack of flow, the structure of near hard-sphere suspensions remains isotropic and thus presentation of pair correlation function (PCF) graphs in any particular plane is indicative of the microstructure in all directions. Figure

5.8 shows the PCF graphs of suspensions with different volume fractions of solid particles at no-flow conditions.

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Figure 5.8. Pair Correlation Function of suspensions at equilibrium for different concentrations. The dark sphere in the center of PCF graphs of figure 5.8 shows arises from the fact that there is no overlap between the rigid cores of particles at equilibrium condition.

Nonetheless, the scattering patterns suggest different microstructures for the colloidal suspensions at different volume fractions. While the PCF graphs at dilute and semi-dense fractions show no particular structure formation and is indicative of a rather dispersed microstructure, dense suspensions in the glassy regime (bottom row in figure 5.8) clearly show evidences of layered structure formation.

As it was explained in the previous section, one can reproduce suspensions of different characteristics by changing the thickness of the soft layer in colloidal particles.

Theoretically by substituting the fraction of mono-dispersed near hard-sphere particles in

Krieger-Dougherty expression (equation 14) by the effective volume fraction of the

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colloidal particles in equation 11, the volume fraction dependency of the viscosity for

different types of particles can be calculated by equation 15.

−2 φeff η =1, + (eq. 15) φMax

In order to validate the capability of our DPD model to capture the viscosity of different suspensions, simulations on suspensions of different effective radius values have been performed, and the results are presented in figure 5.9.

Figure 5.9. Zero shear viscosity as a function of volume fraction for suspensions of different characteristics (effective radius size). Expectedly, as the effective radius of the colloidal particles is increased, the zero

shear viscosity diverges at much lower fractions. This is because by increasing the effective

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radius, the effective volume fraction increases cubically, reaching the maximum packing

fractions at lower hard-sphere contents. We have shown that rheological behavior of

colloidal suspensions with different characteristics and natures, as well as their

microstructures and dynamics at equilibrium can be recovered by our model. Now we study

the behavior of colloidal systems under shear.

5.6. Crystallization of mono-sized suspensions

Prior to presenting the results on the rheological behavior of suspensions under

shear, here in this section we present a common numerical observation in particulate

suspension simulations which has a direct experimental relevance. One can reproduce

suspensions of different mono-sized colloidal concentrations and subject them to shear

flows of varying rates; however, very careful consideration of ordering in high shear rates

should be taken into account as dense suspensions tend to crystallize very quickly under

high shear forces. Although examples of this behavior has been observed experimentally,

as the particles crystalize in super-lattice structures the rheological response of the fluid cannot be generalized to other suspensions. Needless to mention that at low and intermediate volume fractions this problem does not affect the quality of results, as the

excluded volume of colloidal particles is large enough to prevent crystallization. Examples

of the snapshots of crystalized structures of 50% monodispersed and bimodal (95:5 ratio

of small:large particles) suspensions with the corresponding pair correlation function graphs in different planes are presented in figure 5.10.

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Figure 5.10. Snapshots of crystallized suspensions under shear (Shear rate of 1), in different planes and corresponding pair correlation function graphs.

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As one can clearly see in the PCF graphs presented in figure 5.10, super-lattice crystal structures are formed under shear for dense suspensions, even in the case of small bimodality. Our preliminary results (not shown here) prove that bimodal suspensions of at least 85:15 ratio for small:large particles are required in order to avoid strong ordering and crystallization of suspensions under shear. The effect of this ratio on the macroscopic behavior and rheology of suspensions will be discussed in detail later in this chapter.

5.7. Viscosity measurements

In practice, we have performed simulations on suspensions with different volume

fractions of the total number density of 3.0 and dimensionless temperature of kTB =1.0 in

the calculation box of LRi= xyz,, = 25 . Colloidal particles with R =1.0 were reproduced with

4 the mass of mR= ρπ 3 in order to ensure the density matching between the solvent and 3

the colloidal particles. It should be mentioned that in order to avoid strong ordering under

shear for monodispersed suspensions, a volumetric ratio of 1:1 of larger colloidal particles

with R =1.4 were added to the mixture. The dissipative parameter was set at γ ij = 50.0

and correspondingly giving the random parameter ofσ ij =10.0, with conservative

parameter of aij = 25.0 for all interacting species. The clearance gap used for the

regularization of the lubrication potential was set at δ =10−6 R (the effect of this parameter is also discussed in the next section) and the surface roughness of the colloidal particles at which the contact potential is neutralized is set to ∆=10−3 R in order to reproduce the most realistic representation of a colloid. Although the contact force prevents the overlap between the rigid cores of interacting colloids to occur, since finite time steps are used in

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discrete particulate simulation method, the event of overlap cannot be completely avoided.

However, in order to reduce these events and the numerical instabilities in the calculations

very small time steps of ∆=t 0.000005 were used in our simulations. All of the simulations

were performed over 10 million time steps to ensure steady and stable statistical results.

5.7.1. Effect of volume fraction

The general flow curve of suspensions with different volume fractions and contact modulus of 2500 is presented in figure 5.11. Regardless of the volume fractions presented in the flow curves of figure 5.11, all suspensions exhibit a slight shear-thinning behavior in the low and intermediate shear rates and at high Peclet numbers, this behavior is followed by a shear-thickening regime. One can clearly observe a magnified behavior in both thinning and thickening regimes as the concentration of colloidal particles increases. This is clear in the normalized viscosity curves where all the viscosities are normalized to their value at the onset of shear-thickening. As the volume fraction of the colloidal particles is increased the hydrodynamic forces can form the hydroclusters more effectively at high shear rates, which gives rise to pronounced shear-thickening behavior.

An interesting observation to be made in the result of figure 5.11 is that at very high shear rates/Peclet numbers the suspension with the highest fraction of colloidal particles, after a strong shear-thickening regime, the suspension exhibits a second shear-thinning regime. It should be mentioned that to date, the most significant shear-thickening behavior reported in computational works based on the hydrodynamic theory is in the range of viscosity ratios of 3 or less. In fact, this is the main reason for new developments in the frictional contact theory. In the next section, we show that these properties (second shear-

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thinning regime, and the quality of shear-thickening) is significantly controlled by the rigidity of the colloidal particles.

Figure 5.11. Viscosity vs. Peclet number for different colloidal fractions (given as inserts), and Normalized viscosity by the value at onset of shear-thickening for all volume fractions (bottom right curve). In order to ensure that the value of the clearance gap used in the lubrication potential

does not change the viscosities measured, we have performed simulations of the dense

suspensions with different gap values and the results are presented in figure 5.12. The

choice of dense suspensions was made to magnify the differences as the lubrication

140 potential and the close gaps between the colloidal particles are essential in these regimes.

Our results show that changing the gap to smaller values than the surface roughness of the colloidal particle (10-3R) does not change the macroscopic behavior of the suspension, which is in agreement with the previous reports by Seto and coworkers [105, 106].

Figure 5.12. Viscosity curve for 58% suspensions at different clearance gap values in the lubrication equation. Nonetheless, in all of our simulations the gap was kept at its lowest value of 10-5 to minimize the effect of the finite time steps and the discrete calculations.

5.7.2. Effect of particle strength

The general flow curve for suspensions with different volume fractions are given in figure 5.13. The suspensions regardless of the contact modulus and the volume fraction, show a general shear-thinning followed by shear-thickening at higher shear rates. At low volume fractions viscosity is rather independent of the contact modulus, as all different

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suspensions exhibit weak thickening. Also, one can observe the same behavior at low and

intermediate shear rates, namely the shear-thinning regime and the onset of shear- thickening, for different colloidal modulus values; however, the dense suspensions of small contact modulus show a second shear-thinning regime at elevated Peclet numbers, which vanishes for the rigid particles. Interestingly, one can argue that the onset of second shear- thinning is observed when the total stress exceeds the contact modulus of colloidal particles. In other words, the hydrodynamic stresses that give rise to shear-thickening of the suspension at high shear rates, continue to increase to a point where they eventually become larger than the strength of the colloidal particles. At this state, the hydrodynamic stresses begin to elastically deform the soft colloids. In contrary, near hard-sphere particles only exhibit shear-thickening which becomes stronger by increasing the shear rate.

Now that the effect of volume fraction and the effect of particle rigidity are individually discussed on the flow curve of suspensions, one can combine the two and discuss the capability of the empirical models in the literature in explaining the effect of particle rigidity and the shear flows on material properties. Figure 5.14 shows the relative

σ viscosity of the suspensions, ηr = , for different shear rates as a function of colloidal ηγ0 

volume fraction. The results in figure 5.14 clearly shows the difference between the soft

and rigid particles. At low and intermediate shear rates the curves of viscosity against the

volume fraction of colloidal particles is very similar for the soft and rigid suspensions;

however, by increasing the shear rate the behavior of the dense regime is significantly

different. Based on the empirical models to fit the viscosity curves in the figure 5.14 one

can calculate the maximum packing fraction of each system at different shear rates.

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143

Figure 5.13. Flow curves of suspensions of varying colloidal moduli with different volume fractions.

Figure 5.14.Relative viscosity against fraction of colloidal particles for: left) Soft colloid with modulus of 100, and Right) Rigid colloids of the modulus 25000. Figure 5.15 shows the maximum packing fractions calculated based on two different models of Maron-Pierce (equation 16) [144] and Eilers (equation 17) [143]:

−2 φ ηr =1, − (eq. 16) φMax

−1 2 φ ηφ=+−1 1.5 1 , (eq. 17) r φ Max

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Figure 5.15.Maximum packing fraction calculated by different models versus the Peclet number for soft and rigid suspensions. One can clearly correlate the calculated maximum packing fractions in the figure

5.15 with the general flow curves in figure 5.13: in the shear-thinning regime the maximum

packing slightly increases as the suspensions are dominated by the Brownian and flow

forces, however at the onset of shear-thickening, the maximum packing starts to decrease

as the viscosity is mainly hydrodynamically driven in this regime. For the rigid particles

that continue to thicken by increasing the shear rate, this maximum packing fraction is

reduced steadily. This in agreement with findings of Cwalina [104], where they calculated a constant maximum packing for the shear-thickened-state of suspensions, at lower fractions compared to the high shear maximum packing fractions for near hard-sphere particles. It should be mentioned that in our results there is no clear second plateau to define

a clear shear-thickened state which explains the continuously decreasing curve of φMax as

opposed to a constant value at high Peclet numbers. On the other hand, the maximum

packing of the soft suspensions start to grow as the second shear-thinning regime is

145 observed. Needless to mention that the quality of fitting is substantially affected in this regime.

One can compare the properties of the soft to rigid suspensions by comparing the relative viscosities of suspensions with different moduli at a constant shear rate. Thus in figure 5.16, we plotted the relative viscosities at the highest Peclet number in our simulations, Pe=320, and the maximum packing fractions calculated from these curves using the Eilers model (equation 10) against the hardness of colloidal particles.

Figure 5.16. Left) Relative viscosity versus the colloidal fraction for a range of contact modulus values at high shear rate, Pe=320, and Right) Maximum packing fraction calculated from the relative viscosity as a function of modulus. The relative viscosity at high Peclet numbers (figure 5.16) reflects the effect of particle softness, considering that the maximum packing fraction is significantly decreased by increasing the modulus of colloidal particles.

5.7.3. Effect of lubrication interactions

As it was discussed in the introduction section, the newly-developed theories based on the frictional contact forces suggest that the behavior of colloidal suspensions at high shear rates is dominated by the tangential contacts between the hard-spheres and that one

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does not need the lubrication interactions to reproduce the proper dynamics. In fact, there

always has been a debate on whether or not lubrication interactions are the source of

versatile rheological behavior in colloidal suspensions. Thus in this section we aim at studying the effect of these interactions by simply excluding them from the equation of motion.

We have performed the same series of simulations presented in the previous sections including the range of volume fractions and the contact modulus, with the same

H equation of motion as equation 1, without Fij . It should be mentioned however that since

DPD simulations inherently reproduce long-range and multi-body hydrodynamics through

the explicit calculations on the solvent particles and its thermostat, the effect of these

interactions cannot be masked in our model. Thus in order to magnify the effect of

lubrication potentials, figure 5.17 presents the flow curves with and without the lubrication

potentials for the highest volume fraction studied (58%).

Figure 5.17. Viscosity vs. Peclet number (left) and Stress (right) for 58% suspensions of different strength, with (solid lines and symbols) and without (dashed lines and empty symbols) lubrication potential.

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Results presented in figure 5.17 confirm the theory of lubrication-originated

hydroclusters as the suspensions without this type of interactions exhibit very slight or no

shear-thickening over a wide range of shear rates and contact moduli. One can argue that

the suspensions without lubrication forces, and high enough contact moduli exhibit shear- thickening behavior to some extent; however, it should be noted that at these particle strength values a very rigid hard-sphere is being modeled which in real experiments does not show a second shear-thinning behavior. Thus although weak shear-thickening followed by second thinning region for some suspensions is recovered without the lubrication potentials, the reproduced flow curve does not represent typical hard sphere suspensions under shear. We would like to stress that the viscosity-shear stress curves in figure 5.17 suggest that in all of the suspensions, simulation including the lubrication potential lead to shear stress values that are ~10 times larger than the ones calculated without lubrication forces. This validates the crucial role of hydrodynamic stresses in a dense suspension and reproducing the proper dynamics.

5.8. Other rheological parameters

As it was discussed in the introduction, a complete rheological analysis including

the normal stress measurements is crucial in understanding the underlying mechanisms.

Thus figure 5.18 shows the measured N1 and N2 values for a range of volume fractions and

contact modulus values. There are several observations to be made from the first and

second normal stress difference measurements in figure 5.18.

First of all, N1 stays negative throughout the whole range of Peclet numbers regardless of the volume fraction and contact modulus of the colloidal particles. The only

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exception for this is at very low shear rates where positive values are measured but remain

negligible compared to the total pressure of the system. At high shear rates and in the shear-

thickening regime N1 increases linearly with the Peclet number (shear rate).

The second normal stress differences are also negative in the range of Peclet numbers examined; however, there is a clear change of behavior for soft and rigid colloidal particles at high shear rates. As the second shear-thinning occurs for the soft suspensions, the magnitude of N2 begins to decrease and eventually the sign of N2 reverses to positive

values at very high shear rates for the soft colloids. By comparing the curves of the figure

4 for different volume fractions, one can clearly conclude that the magnitude of both

quantities, N1 and N2 , increases by increasing the volume fraction of solid contents. Also one can argue that while at low volume fractions first normal stresses are substantially

larger than the second normal stress differences, then N2 increases faster by increasing

the volume fraction and eventually becomes larger than the N1 at elevated fractions.

Another important conclusion from the figure 5.18 is that virtually in all the

suspensions, regardless of the colloidal fraction or the contact modulus, N1 and N2

remain small and independent of these parameters when lubrication potentials are excluded

in the simulation. This confirms the hydrodynamically dominated nature of the rheological

properties at these shear rates. As it was explained in the background section presence of

the anisotropic microstructure and the lubrication stresses are majorly responsible for the

sign and magnitude of normal stress differences. This is particularly important considering

the fact that the reports based on the frictional contact theory have successfully recovered

a range of continuous to discontinuous shear-thickening by simply tuning the friction

149 coefficient, but failed to recover the proper normal stress differences. This suggests that one cannot claim a proper physically meaningful without providing a complete rheological analysis, even if the shear viscosity as the most important rheological parameter is being properly reproduced. Furthermore, it should be noted that since experimental measurement of normal stress differences is very challenging and associated with several sources of error, these parameters have not been reported for the soft suspensions. Another reason for lack of literature data on these parameters is that these quantities are generally of much lower magnitudes compared to the shear stress of the suspensions.

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Figure 5.18. First (left) and second (right) normal stress differences versus the Peclet number for a range of volume fractions and contact modulus values.

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In order to compare these results with prior reports on the near hard-sphere

suspensions we plotted the N1 and N2 versus the shear stress in figure 5.19 for the highest

modulus in our simulations, f Contact = 25000 .

Figure 5.19. First (left) and second (right) normal stress differences versus the shear stress for different volume fractions of near hard-sphere suspensions. One can define the first and second normal stress coefficients for suspensions based

on the shear stress of the suspending fluid and the measured N1 and N2:

−N1,2 ψ1,2 = , (eq. 18) ηγ0 

The first and second normal stress difference coefficients for near hard-sphere

colloids, ψ1 and ψ2 , show the same dependencies on the volume fraction of the solid content as the relative viscosity of the suspensions given in figure 5.16; however, since N2

exhibits an unexpected change of magnitude for the soft suspensions in the second shear-

thinning regime, the typical divergence near the maximum packing fraction is not observed

for ψ2 of soft suspensions. In order to show this behavior, we have plotted the first and second normal stress difference coefficients of the suspensions with different modulus

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values, at the highest shear rate examined in our simulations (figure 5.20). One should note

that as the contact modulus is increased, these data reveal the material properties for suspensions in the second shear-thinning (soft) to strong shear-thickening (hard) regime.

Once again we have plotted the same data for the simulations without the lubrication interactions.

Figure 5.20. First (left) and second (right) normal stress difference coefficients versus the volume fraction of colloidal particles for a range of modulus values at highest shear rate (Pe=320). Theoretically, total pressure of the suspension should diverge near the maximum

packing fraction, similar to the relative viscosity and normal stress difference coefficients.

The total pressure of the dense suspensions (58%) for different particle modulus values

(figure 5.21.left) shows that while in the shear-thinning regime the pressure is rather

unchanged, it begins to exponentially increase at the onset of shear-thickening. Nonetheless in the case of soft particles, pressure becomes steady again in the second shear-thinning regime. The pressure at high shear rates (Pe=320) similar behavior when compared to the relative viscosities in figure 5.16.

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Figure 5.21. Total pressure of: Left) 58% suspensions versus the Peclet number, and Right) suspensions at Pe=320 vs. the solid particle volume fraction. 5.9. Microstructure under flow

As it was explained in the introduction section, the rheological response of a

suspension to different flow conditions is associated with changes in the microstructural

configuration of colloidal particles. These microstructural changes are usually presented in form of pair correlation function, g(r), in different planes. Figure 5.22 plots the pair correlation functions of the soft (modulus of 100) and rigid (modulus of 25000) at different shear rates in the velocity-gradient plane.

The pair correlation functions in figure 9 show different rheological regimes of each suspension. At the lowest Pe, both suspensions are in the shear-thinning regime where identical isotropic structures are observed. At the onset of shear-thickening, Pe=16, the microstructures start to become anisotropic which continue to grow to be more evident at higher Peclet of 32. One can clearly observe the same microstructure for the soft and rigid suspensions in these regimes; however, at even higher shear rates, where the soft suspension exhibits a second thinning behavior, the pair correlation functions are

154 significantly different. The rigid suspensions continue to form highly anisotropic structure where the colloidal particles are concentrated in the compressional axis. On the other hand, the soft particles are deformed in the flow direction at high shear rates and the microstructural anisotropy is substantially smaller compared to the g(x,y) of the rigid colloids.

Figure 5.22. Pair correlation function in velocity-gradient direction for soft and rigid 58% suspensions over a range of shear rates. Figure 5.23 shows the pair correlation functions of the suspensions with different moduli, at the highest Peclet numbers of 320 in different planes. The pair correlation function graphs given in figure 5.23 clearly show the pronounced anisotropy as the colloidal particles become more rigid. Also one can explain the positive N2 for the soft particles based on the pancake-like microstructures at these shear rates. Additionally, layered structures evident in vorticity planes for the rigid particles are absent for the lower moduli, where particle deformation is the dominant mechanism in defining the microstructure of the suspension. On should note that intact structures observed at high contact moduli confirms that near hard-sphere suspensions are being represented by those parameters, as opposed to softer suspensions with lower modulus. This in experimental

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term is similar to comparison of PMMA particles (soft) as opposed to Silica colloidal

particles of very high shear modulus strengths. A comprehensive study on the rheological

response of these materials can be find in the work by Kalman and coworkers [124].

Figure 5.23. Pair correlation function of the 58% suspensions with different moduli, at Pe=320. 5.10. Rheology of bimodal suspensions

Although monodisperse suspensions have been the system of preference in most

prior works in the area (especially theoretical works), this type of system is not the norm

in nature or industry, where a distribution of different particle sizes is present and yields

profoundly different properties from those of monodisperse suspensions. With the addition

of small fractions of fine particles to coarse ones in bimodal systems, the viscosity drops

significantly and the dynamics of cluster formation in the shear-thickening regime changes

[145-147]. The role of size ratio between the large-small particles is so crucial that under

specific conditions the shear-thickening behavior even disappears [145, 148]. As far as we

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are aware, there is no report explaining all the different parameters affecting the behavior

of the bimodal systems.

5.10.1. Effect of particle composition

The effect of the fine particle fraction in bimodal suspensions was studied by

simulating systems with different combinations of fine-coarse particle fractions, for the

size ratio of 2:1 and ϕeff = 64vol % . These conditions were particularly chosen because we

aim at comparing our simulation data with one from the reliable experimental report on

similar particle fraction and size ratio [146].

The simulation results were compared with the experimental data [146] and the

results which show very good agreement are shown in figure 5.24. The shear-thinning

behavior of the system is found to be magnified when the fraction of small particles is

increased, thus promoting flow, and at the same time, the viscosity increases for the

systems with more small particles. As a result of this phenomenon, the onset of shear-

thickening is shifted to higher shear stresses. Shear-thickening of the suspension is found

to be decreased when the volume fraction of the small particles increases. However, increasing the volume percentage of small particles to more than 50% does not change the rheological behavior significantly, since the behavior is controlled mainly by small particles.

One can explain the rheological response of suspensions by the concept of maximum packing fraction. As it was mentioned before, the maximum packing fraction for a suspension is a function of particle size distribution. For instance this value for random packing of mono-sized hard-spheres is 64%, while it increases to more than %73 for the

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bimodal suspensions. One can calculate the effective radius of a poly-dispersed suspension

33 using the volume average particle size RReff= ∑φ i i , where Reff is the effective radius of i

colloidal particles and φi is the composition of colloidal particles with the radius of Ri.

Figure 5.24. Reduced viscosity vs. reduced stress of bimodal systems with small particle- large particle compositions of: a)10-90, b)25-75, c)50-50 and d)75-25. Experimental result from N. J. Wagner et al.

5.10.2. Effect of size ratio

The effect of the size ratio in bimodal systems was studied by simulation of

different size ratios (from 2:1 to 6:1) for ϕeff = 64vol % suspensions with two different

compositions of 10-90 and 25-75 for the small-large volume fraction in the system and the results are given in the figure 5.25.

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Figure 5.25. Reduced viscosity versus shear rate for the different size ratios in bimodal systems of with the combination of 10-90 (a) and 25-75 (b) for the small-large colloidal particles. The results in figure 5.25 suggest that while the shear-thinning effect does not change significantly, reducing the size ratio between the two particle sizes promotes the shear-thickening behavior. Thus combination of both size ratio, and the composition together was studied for a system of high size ratio (6:1) and high fraction of small particles

(50-50) and the results were compared again with experimental data of D’Haene and

Mewis [145, 149] for bimodal dispersions of poly(12-)hydroxysteric acid grafted PMMA samples (Figure 5.26).

D’Haene and Mewis used particles of 129 nm and 823 nm core size with 10 nm of grafted polymer to make bimodal suspensions of 6:1 ratio between the large and small particles. Since the system is a stabilized suspension with a hard-sphere core coated by a soft layer of grafted polymer, the interactions are the same as our system of interest and method. Figure 5.26 shows that while the addition of 10% of the small particles (with size ratio of 6:1) results in the shear-thickening behavior, increasing this fraction to 50% thins the suspension with no evidence of thickening through the whole range of shear rate.

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Figure 5.26. Reduced viscosity (Experimental data from D’Haene and Mewis) versus reduced stress for bimodal systems of with ratio of 6:1 and combination of 10-90 (a) and 50-50 (b) for the small-large colloidal particles. 5.11. Proposed mechanism

Now that we have validated our modified DPD model’s ability to reproduce a

complete rheological measurement on the mono-sized and bimodal suspensions with

different characteristics, we will present different analysis and studies in order to explain

the underlying mechanisms that give rise to such macroscopic behavior.

5.11.1. Force Analysis

Validation of the model allows the contribution of different forces to the total force

acting on the system to be studied, in order to determine the nature of fluid’s response in

the different flow regions. As explained in the introduction part, these governing forces in

general are classified in three general categories given in equation 19.

dv mF=++Brownian F Inter− particle F Hydrodynamic , (eq. 19) dt

In core-modified DPD the Hydrodynamic interactions are captured via the lubrication forces. The random and dissipative forces together form the canonical

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ensemble, as explained before and thus are the source for Brownian contributions. The

inter-particle interactions are described by core-potential or the contact forces in addition

to the conservative forces in the modified DPD model (eq. 20-22).

FFHydrodynamic= H , (eq. 20)

FBrownian= FF D + R , (eq. 21)

FInter− particle= FF Contact + C , (eq. 22)

The contribution of each of these forces was studied and the results are given in figure 5.27. The graph shows dominancy of Brownian forces at very low Peclet numbers.

Increasing the shear rate gives rise to the inter-particle forces in the shear-thinning regime, and especially in the viscosity minimum region. Hydrodynamic forces remain approximately constant and negligible until the onset of shear-thickening, but increase abruptly afterwards and become dominant in the shear-thickening regime. This leads to the formation of hydroclusters in the system, which explains the shear-thickening behavior observed at high Pe. On the other hand, the predominance of Brownian forces in the

Newtonian plateau region leads to a well-dispersed structure.

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Figure 5.27. Contribution of different forces (%) versus Peclet number for monodisperse colloidal system of with radius of 1.5 (a) and 3.0 (b) for the colloidal particles. In order to confirm the validity of this analysis we have performed the same type of study on bimodal suspensions with corresponding rheology curves presented in figure

5.26. The results of these force contributions are presented in figure 5.28. The typical thinning-thickening transition in the 90:10 suspension as opposed to the pure shear- thinning behavior observed for 50:50 suspension can be explained by the graph of contribution of the different forces (Figure 5.28) in which the hydrodynamic force never dominates for the system with no shear-thickening behavior.

Figure 5.28. Contribution of different forces (%) into total force acting in the system versus Peclet number, for bimodal systems of with ratio of 6:1 and combination of 10-90 (a) and 50-50 (b) for the small-large particles.

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5.11.2. Hydro-cluster formation

In order to gain an insight to the microstructural evolution of the suspension, snapshots of different flow regimes are displayed in Figure 5.29 (clusters are shown as red agglomerates). The formation of hydro clusters in the system was studied using the local density of particles and the criteria of N > 6 for the number of colloidal particles in a

cluster. Hydro-clusters were identified and tracked down over 50,000 time steps. This

definition is based on the proposed criterion by Cheng and coworkers [102].

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Figure 5.29. Snapshots of the simulation system for different Peclet numbers. The particles contributing to clusters are shown in red. The average number of clusters in the system was used as characteristic parameter.

Figures 5.30 show the number of clusters in the system versus Peclet number. At very low shear stresses there is no evidence of clustering, but increasing the Peclet number gives rise to the formation of clusters that continue to grow until the onset of shear-thickening where they increase abruptly.

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Figure 5.30. The number of hydro-clusters versus the Peclet number for monodisperse colloidal system of with radius of 1.5 (a) and 3.0 (b) for the colloidal particles. While the number of hydro-clusters formed in the suspension of the small particles is more than the one with the larger particles and this suggests more shear-thickening effect in this system, the flow curve shows opposite. Thus the percentage of particles contributing in formation of the clusters shall be used instead of the number of hydro-clusters. Figure

5.31 shows the percentage of particles in the clusters versus Peclet number for two systems.

Figure 5.31 shows that for the system with the larger particles at high Peclet numbers majority of colloids contribute in formation of the hydro-clusters, which can explain the enhanced shear-thickening behavior in that suspension.

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Figure 5.31. Percentage of particles contributing into clusters for monodisperse colloidal system of with radius of 1.5 (a) and 3.0 (b) for the colloidal particles. Figure 5.32 shows that for the suspensions (previously discussed in regards to viscosity and the force contributions) higher size ratios (6:1) with large fraction of small particles, hydro-clusters are not formed effectively and thus the suspensions do not exhibit shear-thickening behavior. Looking at the contribution of particles into cluster formation in the system reveals that while for the system with low fraction of small particles at high

Peclet number 25% of particles are in present in the form of hydro-clusters, for the suspension with high fraction of small particles this value never reaches to 1%.

Figure 5.32. Contribution of particles (%) in formation of hydro-clusters versus Peclet number for bimodal systems of with ratio of 6:1 and combination of 10-90 (a) and 50-50 (b) for the small-large colloidal particles.

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5.11.3. Contact network formation

The concept and the criterion for identifying the hydroclusters in a suspension under shear is somewhat arbitrary. In other words, since the structures under high shear rate flows are subject to very fast changes and the criterion of 6 or more particles in a close vicinity does not include the string like clusters in the flow direction, one might question the reliability of the method. Thus here in this section we present the snapshots of our simulations where the colloidal particles are replaced by imaginary contact bonds between the colloidal particles. In these snapshots, a bond is drawn only if the distance between the colloidal particles is smaller the surface roughness of colloidal particles presented in section 5.7. In other words, this is the distance where semi-hard contact potentials are activated and also hydrodynamic stresses are very high. The representative snapshots of the simulations of 58% suspensions at different shear rates is presented in figure 5.33.

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Figure 5.33. Contact networks formed at different shear rates in 58% suspensions and modulus of 25000. The snapshots of contact networks in different shear rates in 5.33, reveals that first of all the contact networks are stable over the course of simulation times and do not change as the strain units are accumulated. This is in contradiction with the contact networks observed for discontinuous shear-thickenings reported by Mari et al. [105]. Secondly,

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considering the fact the onset of shear-thickening for this suspension is at Pe=16, only the first row suspensions are in the shear-thinning regime. Interestingly, it is evident that at this low shear rate, contact bonds are scattered and do not form a network or cluster under shear. This is changed at higher shear rates where a network of contacts in the flow direction, with some extents of alignment in the compressional axis is formed. This network can be also considered as the stress network, as at these distances both the hydrodynamics stresses and the normal repulsive forces are very large. Thus one can conclude that these networks at high shear rates are capable of bearing high stresses exerted on colloidal particles which consequently gives rise to shear-thickening behavior.

Similar to the approach taken in section 5.7.3, we monitored formation of the contact networks in presence and lack of lubrication forces for the same suspensions

(highest modulus of 25000, and 58% and the Peclet number of 320). The results which are shown in figure 5.33 prove that in case of un-lubricated suspensions, the contact network is not effectively formed, which subsequently leads to transient shear-thickening behavior followed by strong thinning region for hard-sphere particles. This can be explained as follows: at close proximities, the colloidal particles start to defy one another through very large contact forces; however the lubrication potential keeps the particles in these small separation distances and provides an energy barrier that the particles cannot escape. In lack of this energy barrier (without the lubrication force), the colloidal particles simply repel each other and thus only exhibit thinning behavior.

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Figure 5.34. Snapshots of contact networks formed with (left) and without (right) lubrication potentials. 5.12. Potential energy analysis

Inspired by the idea introduced into scientific community for the first time for super-cooled liquids [150, 151], we looked at the potential energy of the interacting components in our system to gain insight into microstructural evolutions of colloidal suspensions under shear. Lacks [152] suggested that the potential energy of a non-

Newtonian fluid increases as the fluid undergoes a shear-thinning regime. In this section, we present the potential energy of each component and the nature of pair potentials in order to better understand and correlate the microstructural changes of a suspension to its macroscopic rheological features. To do this, one can calculate ensemble average potential energies using the force between each pair of particles with regards to the distance between two interaction species. It should be mentioned that since dissipative and lubrication depend on the relative velocities of interacting particles and thus include kinetic element in their formalism, cannot be included in the potential energy calculations. As a result, only forces which solely depend on the position of particles, namely conservative and contact

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forces are used in calculation of potential energy. Furthermore, by doing this, one can look at the contribution of each interaction type and correlate the macroscopic measures into specific type of interactions in the system.

The first step is to study the potential energy of suspensions at different volume

fractions at equilibrium conditions. The results of solvent and colloidal potential energies

are presented in figure 5.35. The potential energy of solvent and colloids, tend to increase

with increasing the colloidal content linearly. This is expected as the inter-particle distance between the colloidal particles become limited by increasing the concentration, and solvent particles as well as colloidal particles become more fixated in space.

Figure 5.35. Dimensionless potential energies of solvent and colloidal particles as a function of solid concentration. One can calculate the potential energy of suspensions under flow as well. The

results of the same suspensions being subject to shear flows are presented in figure 5.36. It

171 can be concluded that as the potential energy of the suspensions remain rather unchanged in the shear-thinning region (with slight increases), it begins to exponentially grow at the onset of shear-thickening. This confirms formation of microstructures where colloidal particles are strictly fixed in their locations.

Figure 5.36. Potential energy of suspensions with different volume fractions under flow. Similar to the approach discussed in section 5.7.2 one can study the effect of contact modulus on the potential energy of the suspensions, and attempt to correlate the rheological response of the suspensions to their potential energy state. This was done by performing the same analysis on the 58% suspensions with different contact moduli and the results are presented in figure 5.37. These data suggest the not only the potential energy state of a suspensions is directly dependent on the rigidity of colloidal particles, but also that the

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potential energy can be correlated to the rheological response of a suspension, with or

without the lubrication potentials.

Figure 5.37. Potential energy of 58% suspensions with (solid line and symbols) and without (dashed lines and empty symbols) lubrication potential, for various contact moduli. Since in the DPD model colloidal and solvent particles are modeled separately, it

is possible to look at the evolution of potential energies of these species individually under

flow. Figure 5.38 shows the average potential energy per particle for the colloids (bottom row) and the solvent (top row) as a function of shear rate for different volume fractions/contacts.

The average colloidal energy as a function of Peclet suggests that: (i) there is a distinct change of behavior in presence/absence of lubrication, (ii) potential energy of a colloid begins to increase exponentially as the suspension thickens at high shear rates, and

(iii) the potential energy decreases as the suspension undergoes a second thinning regime.

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The exponential increase of the energy in the ST regime suggests that colloidal particles are in microstructures with closer separations, as this abrupt increase is pronounced and shifted to lower shear rates at higher volume fractions. In presence of lubrication force and at small contact forces, the potential energy becomes stable at elevated shear rates (second shear-thinning regime) as the particles start to deform elasto-hydrodyanmically and thus the separation distance between the particles remain rather unchanged. Increasing the modulus of particle on the other hand hinders the elastic deformation and results in higher lubrication forces. As a result the rigid clusters continue to grow and hence the potential energy. This is however not observed when lubrication potential is absent in simulations.

At intermediate shear rates and without the lubrication, the microstructure of colloidal particles start to form; nonetheless since the energy barrier for these structures to grow and resist to the flow is absent, increasing the shear rate (and thus increasing the repulsive contacts) destructs them and the potential energy decreases abruptly. The potential energy of the solvent slightly increases (<15% at most) at high shear rates. The origin of this solvent energy increase at high shear rates can also be attributed to the formation of hydroclusters at high shear rates. The physical consequence of the colloidal structure formation in the solvent particles is: (i) expelling the solvent particles from the clusters and thus formation of “solvent-free” regions, or (ii) trapping the solvent particles inside the colloidal structures; both of which increase the density of the solvent-rich regions, and consequently higher potential energy levels.

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Figure 5.38. The potential energy of the colloids (bottom) and solvent (top) as a function of Pé for different volume fractions (left) and contact forces (right), normalized by its value at the equilibrium condition with (solid) and without (open) lubrication. A more detailed information can be extracted by taking a closer look into the nature

of pair potentials and their energy product. Thus figure 5.39 shows the normalized potential energy originated from each type of interaction. The solvent-solvent/colloid energies (left and middle columns in fig. 5.39) agree with our previous discussions, suggesting formation of solvent-rich regions both outside and also trapped inside the colloidal structures formed at high shear rates. One should note that in both of these interaction types, the increase of the potential energy is very minimal. In contrary, one can clearly observe a significant change in the energy calculated from the colloidal interactions, showing similar trends as

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the one presented in figure 5.38. This proves that the changes in the potential energy of a

suspension is dominated by the colloid-colloid interactions.

Figure 5.39.Potential energy measured from different types of interactions: (left) solvent-solvent, (middle) colloid- solvent and, (right) colloid-colloid, with (solid) and without (open) lubrication potential, for a range of volume fractions (top row) and contact potentials (bottom row), normalized by its value at equilibrium. The results in figure 5.38 and 5.39 clearly show evidences of structure formation at the

shear-thickened state; however, in order to better explain the structure formation and

correlation of the contact and lubrication potentials, we have monitored the number of

colloidal contacts at each simulation. Figure 5.40 depicts the number of particles in contact

normalized by the total number of colloidal particles. The data presented in figure 5.40

confirms our previous explanation of the potential energy changes. Regardless of the

volume fraction of the colloidal particles, the number of contacts in a suspension is

minimum at equilibrium, which remains fairly unchanged over the shear-thinning regime,

followed by an abrupt increase at high Pé. The increase of the contacting particles tracks down similar trend as the potential energy and the number of hydroclusters in a shear-

176 thickened suspension [18]. The number of contacting colloids for different contact forces suggests that higher contact forces form smaller/fewer clusters in ST regime, as opposed to weaker contact forces where each colloidal particle is in contact with more than one other particle (on average). One should note however, that these fewer/smaller colloidal structures are significantly stronger and thus hold larger stresses. On the contrary, without the lubrication force barrier the number of contacts at high Pé decreases very quickly, meaning that local colloidal structures within close contact are being destructed.

Figure 5.40. The fraction of particles in contact normalized by the total number of colloids with (solid) and without (open) lubrication interactions for different volume fractions (left) and contact potentials (right). 5.13. Colloidal Gels

5.13.1. Introduction

During the past two decades a novel class of materials, called “colloidal gels”, have attracted a lot of attention from the scientific community. One can argue that in general arrested dynamics of species in the field of soft matter in general has emerged as the one of the most interesting and fast-growing areas of active research. Among different classes of soft matter, colloidal suspensions are associated with large time and length scales that

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make them suitable for experimental studies in order to characterize and quantify these

arrested dynamics. Thus the field of colloidal gels have been progressively growing over

the past decade in respect to synthetic chemistry, to physics and applications [153].

By definition, gel is generally referred to a state of fluid that exhibits solid-like behavior. This gel can be quite elastic or contrarily can be notably soli; nevertheless, the viscoelastic response of the material is different from the one usually observed in liquids.

One of the most important properties that a gel is classified based on, is the yield behavior.

This is due to the fact that at absence of external forces, the structure/network formed within the gel is capable of keeping the mechanical integrity of the material intact and thus a gel does not flow similar to a viscous liquid. It should be noted that this definition of gel based on the mechanical state of the matter fundamentally differs from the one usually referred to cross-linked networks in polymer chemistry. Although, the state of a cross- linked polymer network mechanically qualifies it as a gel, the term gelation in chemistry is commonly referred to the point where the cross-linking between the reacting polymer chains leads to formation of a stable network.

In contrary to the chemical definition of gels, a physical gel arises from the physical interactions between the components of a system. Since these bonds are of physical nature and are not chemically linking the two interacting species, they can be broken and formed many times which makes the dynamics of these systems more attractive to physicists. This behavior is best exemplified and studied in colloidal and soft particles as well as associative polymers.

Colloidal gels can be achieved by introducing short-range attraction potentials between the colloidal particles [129, 153-157]. Brownian Dynamics simulation studies as

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well as numerous experimental studies have shown that by tuning the short-range attractive forces between the colloidal particles, one can tune the morphology of colloidal gels into different desired structures [154, 158-161]. Nonetheless, another route for gel formation in colloidal systems is through the depletion forces generally observed in polymer-colloidal mixtures. This is particularly important in polymer mixtures, because by changing the size of polymer chains and their functionalities one can tune the structure of the gel [153, 162-

166].

Although during the past decade the number of reports and publications focusing on colloidal gels has been continuously increasing, most of these studies have focused on the quasi-equilibrium behavior of colloidal gels. This can be attributed to two different facts: firstly, the field of colloidal gels is very young and there are many unexplored areas even at no-flow conditions compared to very well-studied neutral suspensions, and secondly, at high shear rates and when the flow forces dominate the inter-particle potentials specific to gels, a colloidal gel behaves similar to neutral suspensions. Time-dependent behavior of colloidal gels have been reported experimentally where the average number of colloidal bonds, and the distribution of these bonds under flow were studied and correlated to the elasticity of the gel structure [164, 167-170].

The detailed study and characterization of colloidal gels, including the type of interactions, and longtime dynamics of gels at equilibrium conditions has been studied in our group and is to appear in a Doctoral Dissertation by Arman Boromand. However, here we present some key characteristics of colloidal gels under flow conditions.

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5.13.2. Equilibrium properties

Capturing the colloidal gels in our simulation can be achieved by simply removing the conservative interactions between the colloidal particles. This can be explained as follows: the solvent species always interact via the conservative (repulsive) force of 25.0 with one another giving them the correct compressibility characteristics, also in order to avoid overlap between the colloidal particles and the solvent ones, they interact through repulsive parameters of 50.0 with a cubic decay function. The cubic decay function is chosen in order to give the solid-liquid interaction identity to these type of potentials. On the other hand, if the colloidal particles do not interact with one another through conservative interactions, since relative to solvent-solvent and solvent-colloidal interactions it generates attraction between two colloidal neighbors, an osmotic pressure from the solvent pushes the colloidal particles together. Consequently, solvent-rich and solvent-free zones begin to form which eventually results in formation of a gel network.

The snapshots of colloidal gels being formed at different times are presented in figure 5.41.

In these snapshots, the colloidal particles are color coded based on their number of bonds for visual purposes and to show the dense structure of colloidal gels. In this coding, the warmer the color of a colloidal particle, the more surrounding interacting neighbors it has.

In other word, a blue particle means that individual colloidal particles are not interacting with any other colloid, while sharp orange colors indicate many colloidal neighbors. One should note that, we define the colloidal bond based on short-range interactions which means that colloidal particles are within the 1st minima of radial distribution function

curves. Based on the definition by Solomon and coworkers [167], the cut-off distance of

0.3R was chosen as the bond length for colloidal particles.

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Figure 5.41. Snapshots of gel formation in 15% colloidal systems at different times.

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Similar to the approach explained in section 5.5, one can measure the zero-shear viscosity of the colloidal gels. It should be noted that since colloidal gels exhibit a yield behavior and do not flow at equilibrium condition, zero-shear viscosity cannot be defined for these systems; however, in our measurements using the Green-Kubo [58] expression we can define the zero shear modulus of gels. The viscosity (modulus) values for colloidal systems of different volume fractions are presented in figure 5.42 and are compared to experimental data [171].

Figure 5.42. Viscosity/Modulus at no-flow conditions versus the volume fraction for colloidal gels. The volume fraction dependency of the zero-shear modulus measurements at no flow conditions presented in figure 5.42 reveals two important observations: i) regardless of the length of the soft layer thickness in our model, in the absence of conservative force, gel-like behavior is recovered as opposed to neutral feely-moving hard-spheres, and ii)

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while at very low volume fractions and close to maximum packing fractions, the values of viscosity for gel systems resemble the ones in Brownian neutral suspensions, at intermediate volume fractions they significantly deviate from theoretical/empirical models of Krieger-Dougherty type. The latter can be explained by the fact that in the intermediate

volume fractions formation of a colloidal network effectively changes the dynamics of the

system and results in much higher modulus values compared to liquid-like suspensions.

The change of dynamics in such systems can be better understood by monitoring the mean

squared displacement graphs, similar to the ones presented in 5.5 for neutral hard-spheres.

The MSD curves for the colloidal gels are presented in 5.43.

The MSD curves presented in 5.43 clearly show that regardless of the fraction of

colloidal particles, the displacement of colloidal particles become significantly restricted

after an initial diffusive regime (corresponding to unit slope in the logarithmic scale). The

sub diffusive dynamics of colloidal particles at long time scales suggest that in this region

particles are in stable and bound structures. This type of behavior is most commonly

referred to as the arrested movement of colloidal particles. Another important observation

to be made from the MSD curves in 5.43 is the fact that the slope of MSD curve in linear

scale which corresponds to diffusion coefficient of the suspension, decreases by increasing

the volume fraction of solid particles. Also, by comparing the MSD curves in 5.7 and 5.43

one can conclude that the displacement of a neutral hard-sphere suspensions on average is

more than two orders of magnitude higher than the attractive or gel-forming colloidal

particles. This is expected as the motion of colloidal particles is substantially hindered as

soon as each colloid becomes in attractive contact with another one and a bond is formed.

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Figure 5.43. Mean Squared Displacement graphs of colloidal gels with different volume fraction of solid particles in linear (top) and logarithmic (bottom) scales versus time.

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Evolution of the colloidal network and formation of the gel can be also studied

based on the number of bonds for each colloidal particles in the system. Figure 5.44

presents the average number of bonds for each colloidal particle as a function of time for

different volume fractions. As it is evident from figure 5.44, the average number of bond

for a colloidal particle in a gel structure increases with time and eventually becomes

independent of time, when the stable network is formed.

Figure 5.44. Average number of colloidal bonds as a function of time for different volume fraction of colloidal particles. Also, one can argue that at very dense fractions, the dependency on the calculation

time is absent. This is expected as even in neutral hard-sphere particles in this regime, very quickly crystallization of colloidal particles into glass structures occur and particles become strictly fixed in their well-define locations. Also, it should be noted that since in our simulations, the short-range attraction is not directly applied on the colloidal particles

185 and is the result of osmotic pressure from the solvent, by increasing the volume fraction of colloidal fraction the extent of this pressure continuously decreases and eventually the system becomes comparable to the neutral Brownian hard-spheres in dense regime. Thus for the rest of this section, we focus on the low volume fraction regime, which corresponds to strongest gel in our simulations. Although the average number of bonds in a gel system identifies the average state of colloidal particles, at any given time a distribution of bond numbers exists. Figure 5.45 depicts the distribution of bond numbers at the steady state equilibrium conditions (after the microstructure becomes unchanged by time), for 15% gel system. The values on the y-axis of the curve in 5.45, correspond to probability of finding a colloidal particle with specific number of bonds, for example P(5) corresponds to fraction of colloidal particles having 5 bonds.

Figure 5.45. Distribution of number of bonds in a 15% colloidal gel system at steady state conditions.

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The pair correlation graphs of colloidal gels with different volume fractions of solid particles is presented in figure 5.46. It can clearly be observed that colloidal structures are formed even at very low volume fractions. However, suspensions in the dense regime show similar microstructures as the ones observed in neutral suspensions (presented in figure

5.8).

Figure 5.46. Pair Correlation Function graphs of colloidal gels with different concentrations. The microstructure formation at different calculation times, for the 15% fraction suspensions, is presented in figure 5.47. Formation of the colloidal structures at very short times is evident in PCF graphs of 5.47. Although, the general structure remains constant in the range of calculation times presented in 5.47 (the gel network is formed at these times), one can observe a coarsening of structure as the calculation time is increased.

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Figure 5.47. PCF graphs of 15% gel at different calculation times.

5.13.3. Rate-dependent properties

Similar to neutral freely moving hard-sphere suspensions, colloidal gels exhibit a rich rate-dependent rheological response under different flow conditions. Figure 5.48

depicts the microstructure of colloidal gels at different shear rates in different planes.

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Figure 5.48. Pair Correlation Function graphs of 15% colloidal gels at various shear rates in different planes. The PCF graphs of the gel at low shear rate in 5.48 suggest that: firstly, the structure

of the suspension is rather isotropic and, secondly, the agglomerated structure of colloidal

particles remains intact at low shear rates. By increasing the shear rate, in the flow direction

similar anisotropies as observed in neutral suspensions, start to emerge which suggests that

in the flow direction the shear forces dominate the inter-particle potentials between the

colloidal particles; however, in the vorticity directions at the intermediate shear rates, one

can clearly observe an alignment which suggests that long-range ordering occurs in the vorticity direction. This alignment disappears when the shear rate is increased to shear- thickening regime. This is expected as in this high shear rate regime, the microstructure is governed by the short-range hydrodynamics rather than the inter-particle forces (this was thoroughly discussed in previous sections). Figure 5.49 shows the snapshots of these gels at different shear rates.

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Figure 5.49. Snapshots of colloidal gel at different shear rates shown for different planes.

5.13.4. Time-dependent properties

As it was explained in the previous section, colloidal gels exhibit a rich rate- dependent rheological behavior at different flow regimes. Nevertheless, providing a flow

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curve for these systems is rather unrealistic due to two main reasons: firstly, as the colloidal

gels show yield stress at equilibrium and require a minimum stress to start flowing, very

low shear rate behavior of these suspensions cannot be explored, secondly, the rheological

response of a colloidal gel varies at different times of measurements and thus a clear steady

state for these systems cannot be defined. The time-dependent rheology and structure of colloidal gels have been experimentally observed and studied via high-speed confocal

microscopy techniques [164, 167, 170].

One can define the number of colloidal bonds for each particle based on two

different cut-off distances: the short-range attraction bond which corresponds to the

distance where the colloidal particles are within the first minima of the radial distribution

function, and the long range bond at which the colloidal particles start to see and interact

to one another and is the DPD cut-off distance in our simulations. Figure 5.50 depicts the

average number of bonds based on both definitions for 15% gel at different strain units.

The number of bonds in a gel abruptly decrease at the startup of the shear flow,

suggesting that the applied shear forces at very short times effectively break up the weaker

bonds in a system into smaller aggregates. Nonetheless, by increasing the strain units, these

smaller particle clusters once again become within the bond formation distance of one

another and form larger aggregates, which in turn increases the average number of bonds

in the system. Finally, at large strain units this change in the bond number shows a plateau

suggesting a steady state structure at long times.

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Figure 5.50. Average number of bonds versus strain units, for 15% colloidal gel system under dimensionless shear rate of 0.5. Similar to the distributions presented in figure 5.45, one can plot the distribution of bond numbers in a system under the flow conditions. We have plotted these distributions in figure 5.51; however, in order to pronounce the effect of flow rather than the absolute values of probabilities, the probability of finding each bond number at the equilibrium is subtracted from these data. In other words, the results in 5.51 show the relative probability of finding a particle with specific bond number under the flow conditions, compared to its value at rest. The results in 5.51 based on the short-range bonds suggest that by increasing the strain units the population of colloidal particles with high number of bonds decreases systematically, and in turn the number of particles with fewer bonds increases.

Interestingly, the number of colloidal particles with 7 bonds remains constant at all strain units. On the other hand the long-range bonds exhibit a completely different behavior under

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flow. Increasing the number of colloidal particles within the cut-off distance of one another

by increasing the strain units suggests at the same time as decreasing this number for short-

range interactions, suggest that although dense and closely packed gel structures are

effectively destructed into smaller particle clusters, these small colloidal assemblies are in

long range correlation which gives rise to vorticity-aligned structures under flow.

Figure 5.51. Distribution of bond numbers at different strain units for 15% colloidal gel under shear rate of 0.5, based on: left) short-range and, right) long-range definition for bond formation. The results in figure 5.51 are in agreement with experimental measurements of

Solomon and coworkers [167], however it should be noted that these graphs vary from one

concentration to another as well as under different shear rates and interaction potentials.

The detailed study of this time and rate dependent behavior in colloidal gels with different characteristics is the subject of an ongoing work within the same research group, and will be presented in a doctoral dissertation by Arman Boromand.

The snapshots of a gel system under shear rate of 0.5 at different strain units in different directions are presented in figure 5.52. These snapshots clearly show the time- dependent vorticity-aligned structure formation of these gels at intermediate shear rates.

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Figure 5.52. Snapshots of colloidal gel under 0.5 shear rate flow, in different planes at different strain units.

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5.14. Conclusions

A complete rheological analysis is reported for a range of soft to rigid suspensions,

including measurements of the shear stress, pressure, and normal stress differences. The

corresponding rheological parameters are then correlated to the microstructural changes in

each suspension by means of pair correlation function graphs. Our simulation model

predicts a continuous and strong shear-thickening behavior at high shear rates for the near

hard-sphere suspensions, while suggesting a second shear-thinning regime for soft

suspensions. This is in agreement with experimental measurements of Kalman [124] for

PMMA and Silica particles. The results of our normal stress measurements suggest that the

N1 remains negative and linearly increasing function of the shear rate at high shear rates,

with slight dependency on the softness/hardness of colloidal particles. Nevertheless, N2

shows a strong dependency on the contact modulus of the solid particles: in the shear-

thickening regime N2 is negative and linearly increases (in magnitude) by shear rate; however, at the onset of second thinning regime for soft particles N2 begins to decrease in

magnitude and eventually reverses its sign at elevated shear rates. This is not observed for

near hard-sphere particle which only exhibit large and negative N2 values. Both normal

stress differences show strong dependencies on the fraction of colloidal particles.

Furthermore our results show that while at low volume fractions NN12> for rigid

particles, N2 grows faster by increasing the colloidal fraction and eventually in the dense

regime it becomes larger than the N1. Our study suggests that at high shear rates and for

the near hard-sphere suspensions, P>>σ NN21 > . Using the stress of the suspending

fluid at different shear rates, first and second normal stress coefficients, ψ1 and ψ2 , and

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relative viscosity of the suspension, ηr , were defined and for the rigid particles, all three quantities, as well as the pressure of the suspensions found to diverge near the maximum packing fraction. By fitting into empirical models of Eilers and Maron-Pierce we calculated the maximum packing fraction of colloidal particles with different moduli, and correlated it to the rheological response of the suspensions. We showed that the maximum packing fraction decreases at high shear rates as hydroclusters are effectively formed. The pair correlation functions of the soft suspensions show clear evidence of particle deformation in the flow direction at high shear rates and in the second shear-thinning regime; also one can correlate the positive N2 to pancake-like structure formation for soft colloids at high

shear rates, where the deformation of particles dominates the anisotropy raised from the

hydrodynamic interactions. In general, our results lead us to conclude that:

- Presence of the lubrication potential is a critical factor in capturing the rheological response of a colloidal suspension at high shear rates, shear-thickening behavior, while it

does not affect the ability to predict equilibrium and low shear rate regimes,

- Increasing the contact force between colloidal particles enhances the ST behavior, as suggested by several recent reports [105-107, 113, 114]; however, the proper dynamics of a suspension can only be recovered in presence of lubrication forces,

- The potential energy as an independent microstructural characteristic of a system, tracks the macroscopic changes of a suspension and hence provides detailed information about the underlying physics and the nature of each different flow regimes,

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- Our result show that lubrication force is a prerequisite for formation of

hydroclusters and providing the close separation distances required for the contact and

frictional forces to give rise to shear-thickening of any kind in colloidal systems.

We also showed that by using a simple modification in our DPD model (removing

the conservative force between the colloidal particles), a completely different class of

colloidal suspensions usually referred to as “colloidal gels” can be simulated. We have

performed zero shear viscosity measurements and correlated the gel network formation to

the macroscopic behavior of these materials, both at no flow and under shear flows. It was

showed that colloidal gels exhibit a rich time and rate dependent rheological behavior under

flow conditions, which gives rise to specific and unexpected structure formations in

vorticity direction. Since the origin of gel-formation in colloidal system is short-range attractive potential or the depletion forces in polymer-colloidal mixtures, it can be concluded that by introducing the proper form of these potential and substituting the traditional conservative force, one can reproduce colloidal gels of different characteristics.

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