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1. Ensembles 5 Theoretical Studies of Particle/Polymer Mixtures! Haiqiang Wang A thesis in fulfilment of the requirements for the degree of Doctor of Philosophy School of Physical Environmental and Mathematical Sciences March 2015 Acknowledgements First and foremost I would like to express my sincere gratitude to my supervisor Prof. Clifford. E. Woodward for his patience, motivation, enthusiasm and advice throughout my PhD period. Without your help and encouragement I would not be able to achieve and learn so much in this short period. Your suggestions on my research and methods of logical thinking have been priceless and will be of great asset for my future career. In full gratitude I would also like to thank my supervisor Prof. Jan Forsman at Lund University for his great support and dedication to assist me with simulation and suggestions on debugging my programs. I will always remember your, “A good PhD student has to program fast as well as accurately.” My sincere thanks also go to my co-supervisor Prof. Havi Hidhu for those great suggestions on presenting my work and research. A very special thanks goes to Prof. Grant Collins for introducing me to Cliff and opening the gate to another area of chemistry. Your sense of humor as well as supports made my PhD an enjoyable and smooth period. I must also acknowledge my friend Michael Cronin, Mark Bali, Lubna Al-Rawashdeh for coffee and great discussions on everything. A special thanks goes to Ute and Daniel for their beautiful friendship, care, support and everything else. I will be forever thankful to all my other friends, both in Australia and Sweden. My research would not be accomplished without your encouragement and friendship. I would also like to thank my beloved Xiaowen Zhu for everything you have given me. iv At last, I would like to thank my parents for bringing me into this wonderful world and giving me so much joy and satisfaction in life. v $ Optional Preliminary Page – publications or presentations 1. Wang, H.; Woodward, C. E.; Forsman, J., Exact evaluation of the depletion force between nanospheres in a polydisperse polymer fluid under Θ conditions. J. Chem. Phys. 2014, 140 (19), 194903/1-194903/9. 2. Wang, H. (2014 June). Many-body interactions in nano-particle polymer mixtures. Poster session presented at: the 4th. International Colloids Conference, Madrid, Spain. vi Table$of$Content$ CHAPTER 1 1 INTRODUCTION 1 BASIC STATISTICAL MECHANICS 3 1. ENSEMBLES 5 1.1 THE CANONICAL ENSEMBLE 5 1.2 THE GRAND CANONICAL ENSEMBLE 6 1. 3 ISOTHERMAL-ISOBARIC ENSEMBLE 7 1.4 MICROCANONICAL ENSEMBLE 8 2. MONTE-CARLO SIMULATION 9 3. MOLECULAR DYNAMICS SIMULATIONS 14 3.1 EQUATION OF MOTION 14 3.2 SOLVING THE EQUATION OF MOTION 15 4. MONTE CARLO SIMULATION OF POLYMERS 17 4.1 LATTICE SIMULATIONS 17 4.2 SIMULATIONS IN REAL SPACE 19 4.3 GENERATING POLYMER CONFIGURATIONS 20 5. APPROXIMATE METHODS FOR POLYMERS 22 vii 5.1 INTEGRAL EQUATION THEORIES FOR FLUIDS 23 5.2 MEAN-FIELD THEORIES OF POLYMERS 27 5.3 CLASSICAL DENSITY FUNCTIONAL THEORY OF FLUIDS 30 5.4 POLYMER DENSITY FUNCTIONAL THEORY 33 CHAPTER 2 39 A BRIEF SUMMARY OF SOME BASIC POLYMER PHYSICS 39 1. POLYMER MELTS 39 2. POLYMER SOLUTIONS 40 3. POLYMERS AT SURFACES 46 3.1 POLYDISPERSE AND LIVING POLYMERS 48 3.2 ADSORPTION OF POLYDISPERSE POLYMERS AT SURFACES 50 3.3 AMYLOID PROTEINS ARE LIVING POLYMERS 52 4. MIXTURES OF PARTICLE AND POLYMERS 54 CHAPTER 3 60 EXACT EVALUATION OF THE DEPLETION INTERACTION BETWEEN TWO SPHERES IN THE PROTEIN REGIME 60 1. INTRODUCTION 60 1.1 POLYMER INDUCED INTERACTIONS 60 1.2 THEORETICAL STUDIES OF DEPLETION FORCES 63 viii 2. THEORY 70 2.1 POLYDISPERSE POLYMERS 71 2.2 TWO-CENTRE EXPANSION FOR 76 2.3 THE POLYMER FREE ENERGY (POMF) 80 3. RESULTS 83 3.1 ANALYTICAL RESULTS FOR IDENTICAL SPHERES IN THE PROTEIN LIMIT 83 3.2 NUMERICAL EVALUATIONS 89 4. CONCLUSION 94 CHAPTER 4 99 DENSITY FUNCTIONAL THEORY OF EQUILIBRIUM RANDOM COPOLYMERS: APPLICATION TO SURFACE ADSORPTION OF AGGREGATING PEPTIDES 99 1. INTRODUCTION 99 2. THEORY 101 2.1 DENSITY FUNCTIONAL THEORY FOR POLYDISPERSED POLYMERS 101 2.2 SELF-ASSEMBLING RANDOM COPOLYMERS 104 2.3 END-EFFECTS 110 2.4 PROTEIN AGGREGATION ON SURFACES 110 ix 2.5 FILAMENT DISTRIBUTION IN THE BULK 111 2.6 SIMPLIFIED MODEL FOR PROTEIN ADSORPTION 115 2.7 THE EXCESS FREE ENERGY 118 2.8 THE SURFACE POTENTIAL 119 3. RESULTS 120 3.1 IDEAL PEPTIDE MODEL 120 3.2 STERIC INTERACTIONS – NON-LOCAL THEORY 124 3.3 STERIC INTERACTIONS – LOCAL THEORY 129 4. CONCLUSION 132 CHAPTER 5 136 MANY-BODY INTERACTION BETWEEN CHARGED PARTICLES IN A POLYMER SOLUTION: THE PROTEIN REGIME 136 1. INTRODUCTION 136 2. THEORY 139 2.2 ELECTROSTATIC INTERACTION BETWEEN PARTICLES: THE POISSON – BOLTZMANN EQUATION 151 3. MONTE CARLO SIMULATIONS 154 3.1 MATCHING IMPLICIT AND EXPLICIT POLYMER MODELS 154 3.2 PAIR-WISE ADDITIVITY OF THE POMF 158 x 4. RESULTS AND DISCUSSION 159 4.1 COMPARISON WITH SIMULATIONS 159 4.2 COMPARISON WITH THE 2-BODY POMF 164 4.3 PHASE TRANSITIONS DRIVEN BY DEPLETION 166 4.4 FLUID-FLUID PHASE DIAGRAM OF CHARGED PARTICLE/POLYMER MIXTURES 168 5. CONCLUSIONS 171 CHAPTER 6 175 CONCLUSION 175 xi xii Chapter 1 Introduction Particle/polymer mixtures are very important in both industry and nature. Though it has been a major goal of researchers for many decades, understanding the physical behaviour of this system from a theoretical perspective is exceptionally challenging, due to the complexity introduced by the internal configurations explored by the polymer. We will treat this problem using an effective potential approach wherein the configurations of polymer molecules are averaged and the system is described in terms of a potential of mean force between the particles only. While this has been tried before by theoreticians, this strategy has only really found application in the so-called colloidal regime, where the particles are large in comparison to the average spatial extent of the polymers. In this case, it is possible to treat the particles as if they were interacting via a potential of mean force consisting of simple 1-body and 2- body terms. This allows the full machinery of traditional fluid state theory (which focuses on pair potentials) to be used. In the opposite, so-called protein regime (small particles and large polymers), the effective potential approach has generally failed, due to the increased importance of higher order many-body interactions (3-body and so on). Thus effective potential theory for the protein regime has languished and researchers have turned to other more approximate theoretical treatments, or numerically intensive computer simulations. The main body of my thesis is arranged as follows: In Chapter 3, we develop most of the mathematical machinery used for the many-body theory, by considering the 1 case of just two particles immersed in a polymer solution. Here we also present some analytic results, useful for the cross-over regime between the protein and colloidal limits. In the course of these studies, we also considered the single particle regime, wherein a single particle surface with small curvature, which we approximate as flat. We used this geometry to develop a new polymer density functional theory for random copolymers adsorbing onto surfaces and applied it to the case of amyloidogenic peptide aggregation. In recent years there has been accumulating experimental evidence that the formation of amyloidal aggregates (responsible for many diseases, such as Alzheimer’s and Parkinson’s disease, as well as Type II diabetes) is facilitated by the presence of surfaces. We show in Chapter 4 that a plausible explanation of this phenomenon can be found in an adsorption transition that occurs when living polymers come into contact with attractive surfaces. We have revisited the effective potential approach with fresh eyes and one of the purposes of this thesis is to report a new theory for many-body interactions in the protein regime. In Chapter 5, we use it to treat mixtures of charged particles with polymers and assess its accuracy by comparing with simulations on an explicit polymer model. Finally, in Chapter 6, we will briefly describe new directions, which have been seeded by the work reported in this thesis. But before we get started on this, we need to understand some of the basic statistical mechanics that underpin the theoretical studies presented in this thesis. 2 Basic Statistical Mechanics Is there a connection between the macroscopic thermodynamic properties of fluids (such as free energy and entropy) and their microscopic properties, e.g., the specific interactions between the molecules? The answer is, of course, yes! The discipline of statistical mechanics is the bridge between thermodynamics and the underlying microscopic properties of fluids. In principle, one could calculate any macroscopic thermodynamic property of a fluid, from a complete knowledge of the molecular interactions and hence be able to predict the outcomes of experimental measurements. This is true for the case of both equilibrium and dynamic properties. There are differences between these cases though, when it comes to the theoretical foundations of statistical mechanics. For example, general extremum principles have proved useful in the establishment of mathematical relations in equilibrium statistical mechanics, but such is not the case for dynamic systems. This notwithstanding, general theoretical methods such as molecular dynamics simulations (to be described later) have played a valuable role in determining the macroscopic dynamic properties of fluids from their microscopic variables. We mention this only for completeness, as this thesis will concern itself wholly with the equilibrium thermodynamic properties of polymer solutions in the presence of large and small particles. Only a few thermodynamic variables are usually required to ascertain the macroscopic state of a fluid, and hence determine all its equilibrium thermodynamic properties.
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