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. On the other hand, there are a huge number of microscopic states available to a typical fluid sample, due essentially to the enormous number of molecules that generally make up such a sample.

3 Underpinning all basic statistical mechanical calculations of equilibrium properties are two basic postulates. The first postulate is known as the principle of equal a priori probability, which says that a collection of molecules has equal probability to be in any of its myriad of microscopic states. These microscopic states, as a collective, make up the so-called phase space of the system and the phase state variable, Γ , defines the positions, orientations and momenta (translational and angular) of all the particles in the sample. For a system of N simple atoms the number of dimensions of its phase state, Γ , is 6N. The second postulate is the so-called ergodic hypothesis, which says that all possible microscopic states are visited by the system,

consistent with the applied constraints. Any equilibrium thermodynamic property, Ae , can in principle be obtained as the average of an appropriate function of the phase space variables. For equilibrium properties, the system is assumed to have all the time it needs to explore these states (in accordance with the ergodic hypothesis), until a stable

average is obtained. That is, an average is obtained over an extremely long time te, e.g.,

1 te Ae = lim A Γ t dt 1.1 t →∞ ∫ 0 ( ( )) e te where integrating time to infinity completely negates the finite equilibration time. Thus time becomes an irrelevant variable to equilibrium averages. This makes it possible to obtain such averages using an alternative strategy, by utilizing the construct of so-called ensembles.

4 1. Ensembles

It is convenient to use an ensemble when calculating averages over phase space.

As the name implies an ensemble is a collection of an enormously large number of replicas of the system of interest. Each replica is in the same thermodynamic state, which is enforced by external constraints. Different constraints give rise to various ensembles, each one appropriate to particular thermodynamic scenarios. The thermodynamic state of the ensemble is usually defined by a small set of parameters such as the number of particles, N, the temperature, T, and the pressure, P. At least one of these parameters needs to be extensive, otherwise the system size will fluctuate uncontrollably.

1.1 The Canonical Ensemble

From an experimental point of view, the most commonly studied systems at equilibrium are at constant temperature T, hence most ensembles use constant temperature as a constraint on the allowed microstates. The Canonical Ensemble is defined as a collection of replicas (microsystems) where, apart from fixed T, each replica has a fixed number of particles N in a volume V . To accomplish this, it is assumed that in each replica particles are constrained by surfaces, which are heat penetrable. In this way, the ensemble as a whole serves as a heat bath for each of the separate replicas. The central quantity derived from the ensemble is the partition function, Q NVT

1 1 1.2 QNVT = 3N dΓ exp(−H(Γ) / kBT) N! h ∫

5 Where the factor N! accounts for identical particles and h is Plank’s constant. H (Γ) is

the Hamiltonian of the system, kB , is Boltzmann’s constant and the integral is over all phase space. From the partition function, one can essentially obtain all thermodynamic properties of the system, e.g., the Helmholtz free energy, A , is defined by,

A = −kBT lnQNVT 1.3

The probability density ρNVT (Γ) that any randomly chosen replica in the ensemble is in the phase state, Γ , is given by

exp(−H(Γ) / kBT) ρNVT (Γ) = 1.4 QNVT

1.2 The Grand Canonical Ensemble

If a system is in equilibrium with a reservoir with which it is allowed to exchange particles, another ensemble is appropriate. This is the so-called Grand

Canonical Ensemble. In this case, the number of particles is allowed to fluctuate, with the constraint that the chemical potential, µ , is held fixed along with the volume and temperature.

The grand partition function QVTµ is similarly defined as for the Canonical

Ensemble above, but here N is a fluctuating variable, which means it must be summed over, as follows

1 1 Q d exp(( H N) / k T) µVT = ∑ 3N ∫ Γ − + µ B 1.5 N N! h

The corresponding grand potential is given by

6 Ω = −PV = −kBT lnQµVT 1.6 which is a Legendre transform of the Helmholtz free energy, i.e.,

Ω = A − µN 1.7

The probability that the system is in the combined state ( Γ , N) is given by

exp((−H(Γ)+ µN) / kBT) ρµVT (Γ, N) = 1.8 QµVT

1. 3 Isothermal-Isobaric ensemble

The isothermal-isobaric ensemble is defined to study systems where the temperature and number of particles is constant, but the volume of the system is permitted to fluctuate, constrained by the pressure of its surrounds. That is, apart from

N and T, the pressure, P, is also fixed.

The partition function of the isothermal-isobaric ensemble is given by,

1 1 1.9 QNPT = 3N dV dΓ exp((−H − PV) / kBT) N! h ∫ ∫ and the corresponding thermodynamic potential is Gibbs’ free energy,

G = −kBT lnQNPT 1.10 which is the following Legendre transform of the Helmholtz free energy,

G = A + PV 1.11

The probability that the system is in the state ( Γ , V) is given by

7 exp((−H(Γ)− PV) / kBT) ρNPT (Γ,V) = 1.12 QNPT

1.4 Microcanonical ensemble

The microcanonical ensemble is an ensemble with a constant number of particles, volume and energy, E. The partition function of the microcanonical ensemble is given by,

1 1 1.13 QNVE = 3N dΓδ (H(Γ)− E) N! h ∫ where δ(x) is the Dirac delta function. The corresponding thermodynamic potential is the entropy,

S = kB lnQNVE 1.14

The probability density is given by,

δ (H(Γ)− E) ρNVE (Γ) = 1.15 QNVE

These are some of the most common ensembles used in Statistical Mechanical calculations, though there are many others which are used for more specific applications.

Examples include expanded ensembles1, used for obtaining free energy differences between a number of different thermodynamic states, and the Isotension Ensemble2, designed to consider non-uniform fluids in the presence of surfaces.

While, in principle, it is possible to calculate thermodynamic properties by evaluation of the partition function for an appropriate ensemble, the direct evaluation of the integrals in Eq.s (1.2), (1.5) or (1.9), is generally a difficult computational task when 8 dealing with typical fluids (except under simplifying assumptions). Fortunately, efficient simulation methods have been specifically designed to deal with the calculation of thermodynamic averages, usually without recourse to a direct evaluation of the partition functions.

2. Monte-Carlo simulation

In Monte Carlo (MC) simulations of fluids one usually generates a so-called

Markov chain of microscopic states of the system under study with a probability distribution consistent with a given ensemble. In this way, the thermodynamic averages are obtained as direct averages over the “trajectory” of the Markov chain. There are many ways to generate the Markov chain, but it is probably fair to say that the

Metropolis scheme is the most well-known and most commonly used. A typical

Metropolis Monte Carlo simulation of say the Canonical Ensemble will generally consider the following3.

1. The System Size

For bulk systems, the volume, which contains the particles, is generally a cubic

“box” with side length L, though other shapes have also been used.4 For the study of non-uniform systems, say in the presence of surfaces, rectangular boxes are common.

The number of particles simulated and the desired particle density determines the size of the box. Due to finite computational memory, the number of particles in a simulation is constrained. A simulation of up to 106 particles is usually possible on modern computers

9 2. Boundary Conditions

The effect of surfaces on particles in a confining box is inevitable. Particles at a surface experience a different physical environment compared to those in the bulk. For this reason, one usually employs some kind of periodic boundary conditions in MC simulations so that the effect of surfaces is ameliorated. In a periodic box, there is no physical wall at the boundary. If an arbitrary displacement was to move a particle out of the box on one side, its periodic image enters the box on the opposite side. Therefore, the number of particles in the box remains constant. In such simulations, the minimum image convention is often used. Here the interactions between particles are truncated to essentially a half box length, so that a particle cannot interact with its own image. To work out the potential energy of a particle we place it at the center of a sphere with radius equal to the potential range. The particle then interacts with all other particles that fall within the sphere. Generally, the system size is chosen so that the interaction is weak at the cut-off range. However, if the potential is exceptionally long-ranged, e.g., ionic or dipolar interactions, the minimum image convention may not be satisfactory, as the potential remains significant at the cut-off point. Other methods, such as Ewald sums5 have been used to address these kinds of systems.

3. Constructing the Markov chain: The Metropolis scheme

The Metropolis scheme (or an algorithm like it) is “at the heart” of any MC simulation program. The Metropolis scheme usually contains the following three steps

(or similar acting variations):

10 1. Randomly choose a particle from among the particles in the system and

calculate its potential energy, U(r N ), with all other particles, using the

minimum image convention (or other boundary condition, as appropriate).

2. Randomly make a trial displacement of this particle to a new position and

recalculate the potential energy in this new position U(r'N ). The distance the

particle moves in this trial move is a random fraction of an a priori chosen

maximum displacement parameter Δmax.

N N N N 3. If U(r' ) >U(r ), we accept the move if the quantity exp{−β#"U(r' )−U(r )%$}

is bigger than a generated random number within the interval [0,1] , otherwise

the particle stays where it is. If U(r'N )

4. Acceptance rate

One of the goals of computer simulations is to develop an algorithm that achieves the simulation task (i.e. calculation of averages) as efficiently as possible and with the lowest statistical error. One way of achieving this in an MC simulation is to have trial moves as large as possible without too many rejected trials. The question then will be: what is the optimal choice for the maximum displacement parameter,

Δmax ? Moves will be rejected often if Δmax is too large due to the expected large

(positive) energy difference between the new configuration and the old one. On the

other hand, if Δmax is too small, most moves will be accepted but the simulation will be inefficient, as too many configurations will be correlated (see below). The optimal

value of Δmax depends mainly upon the interaction potentials being used, but as a

11 general rule of thumb it should be chosen such that the acceptance rate (number of accepted moves over total number of trial moves) is in the range 0.2 to 0.5.

5. Convergence

The determination of when a Monte Carlo simulation has been run for long enough is a function of the particular property being sampled. In a Monte Carlo

simulation, the average of the observable A over all configurations Ntot is simply

1 Ntot A A n 1.16 r = ∑ ( ) Ntot n=1

This is the same as an ensemble average as the Metropolis algorithm guarantees states in the Markov chain are distributed according to the ensemble probability distribution.

One way to check convergence, for example, would be to plot cumulative averages of the properties of interest against the number of configurations. The simulation has converged when the change in the properties is less than a certain value. One should note that some quantities may take longer to converge; this is usually caused by long- ranged (or slow) fluctuations in the system. We would generally initiate a simulation for a period of time, where no averages are accumulated. This is the so-called equilibration period. It allows the initial configuration to “settle” down into one typical for the system parameters, before we begin to collect averages. An adequate length for the equilibration period is also vital in order to obtain proper convergence.

6. Calculating errors

As in all scientific experiments, computer simulations will contain some degree of statistical and perhaps systematic errors. Factors such as size-dependence, possible effects of random number generators and insufficient length for the equilibration period, 12 may give rise to systematic errors, which can be tracked down and minimized or removed. Statistical errors cannot be avoided, but generally become smaller as the simulation increases in length.

Consider the sampling of a property, A(n), over the Markov chain. If all the

A(n) are uncorrelated, we could calculate the variance in the mean,

N tot 2 σ 2 A = N −2 ∗ A(n)− A 1.17 ( r ) tot ∑( r ) n=1

In an actual simulation, however, there will be correlations between the A(n) within a sequence of adjacent configurations. For this reason, the error is estimated over a series

of “block-averaged” data. Suppose there are Nb blocks within which there are nb sampled configurations. The average of A over configurations in a block b is given by,

nb A n−1 A(n) 1.18 b = b ∑ n=1

We assume that the size of the blocks are sufficiently large so that the sub-averages determined in each block are uncorrelated. Hence we obtain the variance of the means as,

N b 2 σ 2 A = N −1 A − A 1.19 ( b ) b ∑( b r ) b=1

The degree of correlation between the samples in the chain can be easily determined from the central limit theorem. Fiedberg and Cameron6 defined a measure of statistical inefficiency, s , as the following ratio,

13 2 nbσ A ( b ) 1.20 s = lim 2 nb→∞ σ (A) where

N tot 2 σ 2 A = N −1 A(n)− A 1.21 ( ) tot ∑( r ) n=1

The degree of inefficiency depends upon the property being measured, as some properties relax more quickly than others

3. Molecular Dynamics simulations

While we are not going to be concerned with dynamical properties in this thesis, for completeness, we will take a small detour and briefly describe relevant simulation methods that can be used to describe both equilibrium and time-dependent macroscopic properties. The use of Monte Carlo simulations is usually not appropriate for dynamical properties, as the Markov chain of configurations are not linked by proper dynamics.

Instead, to investigate time-dependent phenomena, one uses molecular dynamics (MD) simulations. Here the sequence of system microstates are generated by a suitable equation of motion, e.g., Newton’s equations or Brownian dynamics. We will briefly describe MD simulations using Newton’s classical equations of motion below7.

3.1 Equation of motion

Consider a system with N particles. According to Lagrange’s equation of motion,

14 d ∂L (∂L / ∂q!k )− = 0 1.22 dt ∂qk

where qk are the set of (generalized) coordinates, q!k is its first derivative with respect to time and L(q, q!) is the Lagrangian which is a combination of kinetic and potential energy,

L = κ −ν 1.23 where κ and ν are the kinetic and potential energy respectively. In a Cartesian system,

Eq. (1.22) can be written as

Fi = mi!r!i 1.24

where Fi is the force on particle i , mi is its mass, and !r!i is its acceleration.

Normally, there are two options to compute the trajectory of particles. Either to solve

Eq. (1.24) or the following differential equations,

r!i = pi / m 1.25

1.26 p!i = −∇iν = Fi

where r! is the velocity of particle i, pi is its momentum with its first-order time

derivative given by p!i .

3.2 Solving the equation of motion

There are several techniques used to solve these differential equations. One of the most commonly used is the so-called Verlet algorithm. This algorithm solves Eq.

(1.24) to give the predicted position at time t +δt as,

15 2 r(t +δt) = 2r(t)− r(t −δt)+δt !r!(t) 1.27 where δt is the time interval and !r!(t) is the acceleration at time t . The velocities are obtained by Taylor expansion about r(t) ,

1 r(t +δt) = r(t)+δtv(t)+ δt 2!r!(t)" 1.28 2

1 1.29 r(t −δt) = r(t)−δtv(t)+ δt 2!r!(t)" 2 to give the velocities at time t ,

r(t +δt)− r(t −δt) v(t) = 1.30 2δt

The advantages of the Verlet algorithm are its compactness, simplicity and excellent energy conservation properties. A modified version of the Verlet scheme, known as the

“leap-frog” algorithm, has also been widely used. Here the velocities are obtained as follows,

1 r(t +δt) = r(t)+δtv(t + δt) 1.31 2

1 1 1.32 r(t + δt) = r(t − δt)+δt!r!(t) 2 2

1 1 1 1.33 v(t) = (v(t + δt)+v(t − δt)) 2 2 2

16 4. Monte Carlo Simulation of Polymers

We return now to the main thrust of the thesis: polymers. Polymers are macromolecules that consist of repeated monomer units. We find polymers almost everywhere in our daily lives, in our food, the clothes we wear and the detergents we use to clean them. This thesis will primarily concern itself with the behaviour of polymers in the presence of particles. The wide number of applications of such systems has generated many experimental and theoretical studies in an effort to understand the relevant physical mechanisms that determine their properties. From a theoretical point of view, the simulation of polymer molecules (especially long polymers) is no easy task, due to the strong correlations between monomers in a chain. For example, in an MD simulation, dealing with both the “fast” moving monomers and “slow” motion of the chains as a whole, leads to a separation in time-scales that is difficult to deal with without significant numerical expense. As we stated above though, we will be using

Monte Carlo simulations, rather than Molecular Dynamics. However, problems are faced in MC simulations, which are somewhat related to the issues seen in MD. In MC simple monomer displacement moves will be inefficient, given the general bonding constraints within the chains. One advantage of MC, however, is that fictitious moves

(that have no dynamical counterpart) are allowed, provided they satisfy the requirement of microscopic reversibility8,9. Furthermore, MC simulations can be carried out in simplified spaces, such as lattices, as we discuss below.

4.1 Lattice Simulations

One of the simplest methods used to simulate polymers is the lattice model.

There are many types of lattices that can be used, sometimes (but not always) reflecting

17 the chemical structure of the polymers under study. One of the least realistic from a chemical point of view is the cubic lattice. In a cubic lattice model, the bond length between monomers is equal to the distance between two neighbouring sites and the bond angles are either 90° or 180°, see Fig. (1.1). One of the very first simulations of polymers on a lattice was carried out by Montroll10. In that case, monomers were allowed to occupy any particular lattice site on the condition that they do not overlap with each other. Under such conditions, a given polymer configuration can be considered as the path followed in a “self-avoiding walk”. This polymer model is essentially a linear chain of bonded hard-sphere monomers confined to a lattice.

Additional longer-ranged forces, e.g., van der Waals and/or electrostatic interactions could also be assumed in particular applications. Of course, all these interactions will affect the acceptance rate of moves (via the Metropolis algorithm in the manner described earlier).

Fig. 1.1 Diagram of a polymer on a simple lattice (dotted lines). The monomers are represented as spheres and solid lines are bonds between monomers.

18 4.2 Simulations in Real Space

There have been a number of clever algorithms developed to simulate polymer fluids off-lattice, i.e., in normal 3-dimensional space in both bulk and non-uniform systems. Examples include the Rosenbluth-Rosenbluth11 method and a variation to this by Frenkel and Smit7, the so-called configurational-bias Monte Carlo method. Such techniques are particularly useful for treating polymers in a Grand Canonical Ensemble

(GCE), where particle insertions are particularly difficult, due to the high probability of overlap. Woodward and Svensson12 introduced the Isotension Ensemble to treat polymer solutions and melts in a GCE in the presence of surfaces. Also Turesson et al.13 have developed a simple grid method to study polyelectrolytes in the presence of charged surfaces.

The systems that we will consider in this thesis are particularly simple, as the polymer will generally be treated as ideal. That is, the monomers are essentially ideal gas particles, linked by rigid bonds. While this may seem an enormous simplification, there are real systems that can be approximated by this model, i.e., when a polymer solution is under so-called theta (Θ) conditions. This will be described in more detail in the next chapter. Treating the polymers as ideal means that simpler MC moves can be used, due to our lack of concern for monomer-monomer overlaps. While this is true with respect to mutual interactions between monomers, in this work, we will have other hard spherical particles mixed within the polymers. Depending on their density, these particles may substantially restrict polymer configurations. Furthermore, polymer configurations will still need to be generated with “special” moves developed specifically for bonded systems.

19 4.3 Generating polymer configurations

In this thesis, we will report MC simulations of a system of spherical particles immersed in a solution of neutral polymer under Θ conditions. Our simulations will be carried out in the Canonical Ensemble. We will compare these results with a new effective potential approach whereby the particles are assumed to interact via a potential of mean force, mediated by the polymer. As will be described in more detail later, the interesting thing about this potential of mean force is that it is a many-body potential.

We will report these comparisons in Chapter 5. In Chapter 3, we will also apply the effective approach to treat two spherical particles immersed in an ideal polymer fluid.

We will show that the A-O model works well in the cases when polymers are ideal and short, but it becomes inaccurate when polymers are large compared to the size of particles.

We used three specific types of moves in order to generate polymer configurations‡. They are listed as follows:

1. Reptation moves: The reptation move was first introduced by Wall and

Mandel14,15. As the name implies, in reptation, the polymer moves in a fashion

analogous to the way a snake or reptile does. To carry out a reptation move one

selects a polymer at random and then chooses an end at random. The monomer

at that end is deleted and another monomer is added at a random orientation to

the other end. A diagram of this move is shown in Fig. (1.2).""The usual

Metropolis algorithm is then applied to see whether this move will be accepted

or rejected.

‡ We assessed the results of our simulations to make sure (known) properties such as the radius of gyration had reached their equilibrium values. 20 "

Fig. 1.2 Schematic representation of a polymer reptation move. An end monomer was randomly chosen and deleted (shown in dashed line) and a new monomer is attached with random orientation to the other end (solid line and sphere).

2. Crankshaft move: The crankshaft move16 creates a conformation change inside

the polymer. To carry out a crankshaft move, a polymer molecule is chosen at

random. A monomer is then randomly chosen within that chain. This monomer

is then randomly rotated around an imaginary axis drawn between the two

adjacently bonded monomers, while preserving the bond lengths. End

monomers can be randomly rotated about an axis defined by the nearest and next

nearest neighbouring monomers. The crankshaft move is illustrated in Fig. (1.3).

Fig. 1.3 Schematic representation of a crankshaft move. The arrow indicates the rotation of one monomer around the imaginary axis joining the centers of adjacent monomers (shown in black solid sphere and line).

3. Translation move: Reptation and crankshaft moves will change the polymer

conformations, translation simply moves the polymer as a whole16. That is, an

21 identical displacement vector translates every monomer. The translation move

is shown in Fig. (1.4).

Fig. 1.4 Schematic representation of a translation move. One polymer is moved as a whole molecule to a new location.

5. Approximate Methods for Polymers

Despite advances in efficient algorithms, the simulation process (for polymers especially) is still quite time consuming and numerically expensive. Surprisingly, this remains the case even for ideal polymers. For this reason, more approximate statistical mechanical methods have been developed to study these systems. This includes integral equation theory, self-consistent field theories and density functional theory. We will use both self-consistent field theory and density functional theory in this thesis.

However, we will also briefly review other methods that have been applied to fluids in general, and polymers specifically.

22 5.1 Integral Equation Theories for Fluids

Integral equation theory for fluids generally concerns itself with the pair correlation functions. There has been great interest in these methods as only a small number of equations generally need to be solved and, within certain approximations, this can sometimes be done analytically. Various versions of integral equation theory have appeared over the years, e.g., the Born-Green-Yvon equation17and Kirkwood equation18,19. However, it is fair to say that the celebrated Ornstein-Zernike equation20 has generally been the main focus of most integral equation studies of fluids over the last fifty years or so.

1. The Ornstein-Zernike Equation

Ornstein and Zernike first presented the Ornstein-Zernike (OZ) equation in 1914.

It relates the pair correlation function, h(r), of the fluid to an ostensibly shorter ranged direct correlation function, c(r). For a pure, simple fluid the OZ equation has the form,

1.34 h(r12 ) = c(r12 )+ ρ ∫ c(r13 )h(r23 )dr3 with ρ = N /V , where N is the number of particles in a volume V. The direct correlation function is a type of “influence function” whose effect can be realized by repeated convolutions over correlated pairs of particles. Even in mathematical terms its physical character is evident, e.g., in terms of cluster integral diagrams c(r) is the sum of all cluster diagrams that constitute h(r), but without nodal circles8,9. The convolution on the RHS of Eq. (1.34) introduces the nodal circles in the diagrammatic sum for h(r).

While the OZ equation is formally exact, it clearly cannot be solved unless we define another relation between h(r) and c(r) in terms of known functions. This is called a

23 closure relation. It is in the closure equation that approximations are made. Some common closures are briefly described below.

2. Common Closures

The Percus-Yevick (PY) approximation21 is obtained by approximating the direct correlation function by subtracting the so-called indirect part of the pair

correlation function gindirect (r) from the total correlation function, i.e.,

" $ −βw(r) −β#w(r)−u(r)% 1.35 c(r) = g(r) − gindirect (r) = e − e = f (r) y(r)

Here w(r) is the potential of mean force and u(r) is the pair potential that acts between fluid particles. We have also used

−βu(r) f (r) = e −1 1.36

βu(r) 1.37 y(r) = e g(r)

If we expand the indirect term in Eq. (1.35) we obtain instead the hypernetted-chain

(HNC) equation,

c(r) = e−βw(r ) −1+ β [w(r)− u(r)] = f (r)y(r)+ [y(r)−1− ln y(r)] 1.38

These are typical approximations used for simple fluids. The PY closure has been found to be reasonably accurate for fluids with short-range interactions, e.g., hard spheres, whereas the HNC is more appropriate for long-ranged potentials, e.g., electrostatic interactions. However, the OZ equation has also been applied to polymer fluids.

24 3. Polymer Reference Interaction Site Model

The polymer reference interaction site model (PRISM) uses a generalization of the OZ equation to molecular fluids that was developed by Chandler and coworkers22-25.

Consider a system of molecules with number density ρ = N /V . The total interactions between molecules are treated as a sum of site-site interactions from so-called interaction sites in molecules, which normally correspond to atomic centers. We define the site-site radial distribution function gαγ (r) ,

N 2 αγ α γ ρ g (r) = ∑ δ (ri )δ (rj − r) 1.39 i≠ j=1 where α and γ are two interaction sites and the sum is carried out over separate molecules i and j. The site-site Ornstein-Zernike (SSOZ) equation is given by,

" $ 1.40 h(r) = ∫∫ dr1 dr2ω ( r - r1 )C( r1 - r2 )#ω (r2 ) + ρh(r2 )% where h(r) , C(r) and ω (r) are n × n matrices, with n the number of interacting sites in the molecules. The elements of these matrices have the form, hαγ (r) = gαγ (r)−1,

Cαγ (r) and ωαγ (r) . The last two terms are the site-site direct correlation function and the intra-molecular distribution, respectively. The site-site direct correlation function is a generalization of the direct correlation function in single site fluids, while the intra- molecular distribution accounts for the internal structure of the molecules. If the molecules are made up of fused hard spheres then the RISM theory assumes the following (Percus-Yevick) closure relations, which allows solution of the SSOZ equation,

25 gαγ (r) = 0 r < dαγ 1.41

Cαγ (r) ≅ 0 r > dαγ

Here, dαγ , is the distance of closest approach between sites α and γ on two different molecules. Schweizer and co-workers26 generalized the RISM theory to treat polymer fluids, this theory has been coined as polymer RISM, or PRISM theory. This theory is widely used to study flexible polymers solutions and melts.

By assuming that polymers are long enough so that all sites in the polymer molecule becomes essentially equivalent (at least for homopolymers), Schweizer and co-workers rewrote the SSOZ equation as,

h(r) = ∫∫ dr1 dr2ω ( r1 − r )C(r1 − r2 )[ω(r2 )+ ρmh(r2 )] 1.42

where ρm = nρ is the site density and we define the average intramolecular distribution,

n 1 αγ ω (r) = ∑ ω (r) 1.43 n α,γ=1

The difference with this theory and the RISM, as applied to small molecules is that in the latter the intra-molecular structure is generally assumed to be rigid. On the other hand, for polymers the intramolecular distribution will be a diffuse function, representing the flexibility of large polymer molecules. The PRISM theory has been used to study not only polymer fluids and solutions, but also particle/polymer mixtures.

26 5.2 Mean-field theories of Polymers

Much of our theoretical understanding of polymer solutions is thanks to the pioneering work of people such as Flory27 Edwards28 and Helfand29. These researchers were instrumental in developing so-called self-consistent-field (SCF) theories, which have provided useful approximate methods to treat both uniform and non-uniform polymer fluids. Their work has undergone a modern transformation in the SCF theory of Scheutjens and Fleer30. Even de Gennes’31 ground-breaking work, applying renormalization group methods to polymer physics, used SCF theories as a scaffold on which to incorporate his scaling ideas.

At their heart SCF theories are inherently mean-field. That is, each monomer in a given chain feels an average field due to all other monomers. This field must then be solved for self-consistently, hence the nomenclature “self-consistent-field”. From a computational point of view, the SCF approach is often algorithmically simple as it deals with 1-point functions. However, solutions may be time consuming to obtain, due to the iterative nature of the problem and the need for convergence to a predetermined tolerance. Furthermore, the application of the SCF equations may not be theoretically well-justified, especially in situations where the polymer solution is strongly fluctuating, which can occur in, e.g., semi-dilute solutions (see Chapter 2). This notwithstanding, the SCF approach has proved itself to be remarkably useful and sometimes quite accurate, particularly when the system is dominated by external potentials giving rise to significant non-uniform structure. The simplest manifestation of SCF theory is the celebrated Flory-Huggins theory, which is used to treat uniform polymer solutions.

27 1. Flory Huggins Theory

The Flory Huggins theory, developed by Flory27 and Huggins32, generalized the

Bragg-Williams lattice model for simple fluids9 to tackle the problem of polymer solutions. Unlike the Bragg-Williams model, which treats all solvent and solute molecules as equal-sized particles, the Flory-Huggins theory takes care of the significant size difference between polymer and solvent molecules by allowing the polymer to occupy several lattice sites.

Consider a polymer solution wherein all molecules are spread on a lattice with a

total number of sites equal to Mo. The numbers of solvent and polymer molecules are denoted as N1 and N2, respectively (1 = solvent, 2 = polymer). Therefore we have Mo =

N1 + MN2 where M is the number of sites each polymer occupies (the degree of

polymerization). The volume fractions of solvent and monomers are φ1 and φ2 , respectively:

N1 φ1 = N1 + N2M 1.44

N2M φ2 = N1 + N2M

Flory and Huggins introduce a (temperature dependent) mixing parameter

z χ = − (ε11 +ε22 −ε12 ) 1.45 2kT where z is the number of neighbour-sites in the lattice (e.g. z = 4 in a two-dimensional

square lattice). ε11 is the interaction energy between solvent molecules in contact, ε22 is

the interaction energy between monomers in contact and ε12 is the interaction between a

28 monomer and a solvent molecule in contact. The change in energy upon randomly mixing polymer with solvent molecules is assumed to be given by the following mean- field expression,

ΔE m = χM φ φ 1.46 kT 0 1 2

The total free energy of mixing is,

A m = N lnφ + N lnφ + χM φ φ 1.47 kT 1 1 2 2 0 1 2

The Flory-Huggins theory assumes the solution is incompressible, and hence the volume change in the process of mixing is not taken into account. In addition, being a mean-field theory, it does not consider correlations between particles33. Nevertheless, it does contain a lot of the basic physics of polymer solutions, including the relatively minor role that the polymer molecules contribute to the entropy of mixing compared to the solvent. The generalization of the Flory-Huggins theory to non-uniform polymer fluids was carried out by Scheutjens and Fleer30.

2. Edwards’ SCF Equation for Continuous Chains

While the Flory-Huggins theory is a lattice-based model, self-consistent field theory has also been used to treat polymers in continuum space. In particular, Edwards’ equation28 has been used to describe the configurations of long polymers, treated as a continuous chain in the presence of a self-consistent field, generated by its own monomers as well as an external potential source. We will use this equation later in

Chapters 3 and 5 of this thesis.

29 Consider a polymer segment of length s, the 2-point distribution function

G(r, r ';s) is the probability that this segment has ends at r and r '. Assuming a continuous chain, Edwards’ differential equation is obtained as

2 ∂G(r, r';s) σ 2 = ∇ G(r, r';s)−ψ(r)G(r, r';s) 1.48 ∂s 6 where σ is the so-called Kuhn length, which is essentially the bond length for ideal polymers and ψ(r) is the self-consistent field. It can be written as

ψ(r) = ψex(r) + ψmon(r), where the first term represents the potential due to external sources and the second term is the mean-field due to the averaged monomer-monomer interactions. In the work presented in this thesis, we will apply the Edwards’ equation to ideal polymers and thus ψmon(r) = 0. On the other hand, ψex(r) will in general be non- trivial and correspond to the field imposed by a number of fixed particles.

5.3 Classical density functional theory of fluids

Classical density functional theory (DFT) was developed as an analogue to the quantum DFT, primarily to study non-uniform fluids34,35. The theory is based on an exact extremum principle, which states that the system’s free energy is minimized by the equilibrium particle density distribution. There are various versions of DFT, but one of the first to exploit the concept of weighted densities was developed by Nordholm and co-workers36,37 in 1980. In his work with simple fluids he introduced a nonlocal treatment of the excluded volume effects, which was coined the generalized van der

Waals theory. We will give a brief description of DFT in a way that is more akin to the first exposition by Nordholm. While a formal derivation can be obtained in terms of the expansion of the free energy in terms of higher-order direct correlation functions, the

30 original ideas of Norholm followed a more field-theoretic approach, and it is interesting to present it here.

We consider first the instantaneous particle densities of the system, n(r), as a sum of Dirac delta functions,

N

n(r) = ∑δ(r − ri ) 1.49 i=1

therefore the configurational partition function, Qc , is given by,

Qc = ∑exp[−βE(n)] 1.50 n

Note that the configurational partition function, accounts for the part of the total partition function due to the positions (configurations) of particles and the interaction potential. The kinetic part (which multiplies the configuration term to get the total) is rather trivial and will add a simple temperature dependent term to the free energy. The potential energy E(n) is given by the functional,

1 1 E(n) = drdr'n(r)n(r')φ( r − r' )− Nφ(0) 1.51 2 ∫ ∫ 2 where φ( r − r' ) is the fluid-fluid interaction and the last term accounts for the self- interaction. The sum in the partition function is over all densities, which is equivalent to an integration over all particle positions. We now introduce a parameterization of the particle density, using the parameters ζ and γ. We assume the following generic form

for Qc as,

1.52 Qc = ∫ dζ ∫ dγ[exp{−βE(ζ,γ)}]ρ(ζ,γ)

31 where ρ(ζ,γ) is the density of states. The parameter ζ defines the “coarse-grained” structure of the particle densities and γ the fine-grained information not present in ζ.

We can view this in a somewhat more precise way by considering a Fourier expansion of the particle densities and splitting the contributions into that of long and short

wavelength components. We now suppose that the main contribution to Qc is

dominated within a certain region around a maximal ζm ,

1.53 Qc ≈ C ∫ dγρ(ζm,γ)exp{−βE(ζm,γ)}

With the width C fixed and small and therefore can be ignored. Furthermore, consistent

with a mean-field treatment, we assume that E(ζm,γ) only depends on ζ. Thus we obtain,

Qc ≈ W(ζm )exp{−βE(ζm )} 1.54 where the density of states gives,

1.55 W(ζm ) = ∫ dγρ(ζm,γ)

The derivative of lnQc with respect to ζ is equal to 0 when ζ = ζm,

∂[lnW(ζ )− βE(ζ )] = 0 1.56 ∂ζ

We define the free energy as,

Fc (ζ ) ≅ −kBT lnW(ζ )+ E(ζ ) 1.57

32 Where apart from the energy, we have also defined an entropy contribution (the first term in Eq. (1.57). Nordholm presented a simple van der Waals-like form for this entropy, in terms of the 1-particle density,

−1 1.58 lnW(n) = ∫ drn(r)ln[(n(r)) −υ0 ]

Here υ0 can be interpreted as the excluded volume per particle. The free energy of the system can now be written as a functional of the 1-particle density, n(r),

−1 1 F (n) = −k T drn(r)ln[(n(r)) −υ ]+ drdr'n(r)n(r')φ( r − r' ) 1.59 c B ∫ 0 2 ∫ ∫ where we have implicitly removed the self-energy from the final term.

5.4 Polymer Density Functional Theory

In 1991, Woodward38 generalized the classical DFT for simple fluids, to the case of polymer fluids. The resultant polymer DFT (PDFT) was used to describe the behaviour of polymer melts and solutions in the presence of external fields. The theory has proved itself very accurate in its description of polymer structure in the presence of external potential sources, such as hard planar surfaces and spheres.

We consider a fluid of r-mers, consisting of linear chains of identical monomers.

This is just for simplicity of algebra and is not a requirement of the theory (in fact in

Chapter 4 we will generalize the polymer DFT to study random co-polymers of arbitrary length). The configuration of a given polymer molecule is thus specified by

the collective co-ordinate, R = (r1,…, rr ), where ri is the position of monomer i . We shall assume that the bonding potential between monomers creates a freely jointed chain,

33 with a bond length equal to σ m . The bonding constraints in the chain can then be described by the (unnormalized) distribution

r−1 −βVb (R) 1.60 e ∝∏δ( ri+1 − ri −σ m ) i=1

whereVb (R) is the bond potential between monomers, δ(x) is the Dirac delta function.

This type of bonding constraint was used in a model proposed by Yethiraj and Hall39 in which polymer chains consisted of hard-sphere monomers with each sphere able to roll freely on its neighbours’ surface to form a chain, i.e., each monomer had a diameter equal to the bond length. For the moment, however, we shall assume that the monomers are ideal. In this case, the free energy functional of the (ideal) polymer fluid,

id Fr , is given by,

F id !N R # N R ln!N R # 1 d R β r " ( )$ = ∫ ( )( " ( )$− ) 1.61 ext +β ∫ N (R)(Vb (R) +V (R))d R

ext wherein the polymer density is denoted as N(R) . Furthermore, V (R) denotes the applied external potential. While we do model ideal polymers in Chapter 3 and 5, in

Chapter 4 we also consider monomers which exclude each other via steric interactions.

In this case, the ideal functional must be augmented with an excess contribution to the free energy, which accounts for these interactions. The equilibrium conditions are defined by the extremum principle, i.e., that the free energy is minimized with respect to variations in the polymer density N(R), given that numbers are constrained by a

constant chemical potential µpol. Therefore, for the ideal polymer functional, we

34 minimize Eq. (1.61) with respect to N(R) , which gives the following statement of chemical equilibrium,

∂F id r − µ = 0 1.62 ∂N(R) pol

The equilibrium polymer distribution density, Neq (R),

ext Neq (R) = Φ p exp(−βVb (R)− βV (R)) 1.63

where Φ p is the density of bulk polymer molecules corresponding to the given chemical potential. In Eq. (1.63), we have used the well-known relationship between the chemical potential and the bulk polymer density, valid for the ideal polymer case. Then

the monomer density at equilibrium, nm (r) , is given by

r

nm (r) = ∫ ∑δ(r − ri )Neq (R)dR 1.64 i=1

The PDFT has been proven to be an effective approximate approach to treat non- uniform polymers40. Since the derivation by Woodward38, there have been a number of alternative PDFT approaches, generally sharing the original formulation41-47.

35 References

(1) Lyubartsev, A. P.; Martsinovskii, A. A.; Shevkunov, S. V.; Vorontsov-Velyaminov, P. N. J. Chem. Phys. 1992, 96, 1776-1783.

(2) Parrinello, M.; Rahman, A. Phys. Rev. Lett. 1980, 45, 1196-1199.

(3) Allen, M. P.; Tildesley, D. J. Computer Simulation of Liquids; Oxford University Press: New York, 1987.

(4) Wang, S. S.; Krumhansl, J. A. J. Chem. Phys. 1972, 56, 4287-4290.

(5) Ewald, P. P. Annalen der Physik 1921, 369, 253-287.

(6) Friedberg, R.; Cameron, J. E. J. Chem. Phys. 1970, 52, 6049-6058.

(7) Frenkel, D.; Smit, B. Understanding Molecular Simulation: From Algorithms to Applications; Academic Press: San Diego, 2001.

(8) McQuarrie, D. A. Statistical Mechanics; Harper and Row: New York, 1976.

(9) Hill, T. L. An Introduction to Statistical Thermodynamics; Dover Publications: New York, 1986.

(10) Montroll, E. W. J. Chem. Phys. 1950, 18, 734-743.

(11) Rosenbluth, M. N.; Rosenbluth, A. W. J. Chem. Phys. 1955, 23, 356-359.

(12) Svensson, B.; Woodward, C. E. J. Chem. Phys. 1994, 100, 4575-4581.

(13) Turesson, M.; Aakesson, T.; Forsman, J. J. Colloid Interface Sci. 2008, 329, 67-72.

(14) Wall, F. T.; Mandel, F. J. Chem. Phys. 1975, 63, 4592-4595.

(15) Mandel, F. J. Chem. Phys. 1979, 70, 3984-3988.

(16) Binder, K. Monte Carlo and Molecular Dynamics Simulations in Polymer Science; Oxford University Press: New York, 1995.

(17) Born, M.; Green, H. S. Proc. R. Soc. London, Ser. A 1946, 188, 10-18. 36 (18) Kirkwood, J. G. J. Chem. Phys. 1946, 14, 180-201.

(19) Kirkwood, J. G. J. Chem. Phys. 1947, 15, 72-76.

(20) Hansen, J. P.; McDonald, I. R. Theory of Simple Liquids; Academic Press: San Diego, CA, 1990.

(21) Percus, J. K.; Yevick, G. J. Phys. Rev. 1958, 110, 1-13.

(22) Chandler, D. J. Chem. Phys. 1973, 59, 2742-2746.

(23) Chandler, D.; Andersen, H. C. J. Chem. Phys. 1972, 57, 1930-1937.

(24) Lowden, L. J.; Chandler, D. J. Chem. Phys. 1974, 61, 5228-5241.

(25) Lowden, L. J.; Chandler, D. J. Chem. Phys. 1973, 59, 6587-6595.

(26) Schweizer, K. S.; Curro, J. G. Adv. Polym. Sci. 1994, 116, 319-377.

(27) Flory, P. J. J. Chem. Phys. 1941, 9, 660-661.

(28) Edwards, S. F. Proc. Phys. Soc., London 1965, 85, 613-624.

(29) Helfand, E. J. Chem. Phys. 1975, 62, 999-1005.

(30) Scheutjens, J. M. H. M.; Fleer, G. J. J. Phys. Chem. 1979, 83, 1619-1635.

(31) De Gennes, P.-G. Scaling Concepts in Polymer Physics; Cornell University Press: Ithaca, N.Y., 1979.

(32) Huggins, M. L. J. Chem. Phys. 1941, 9, 440.

(33) Fried, J. R. Polymer Science and Technology; Prentice Hall PTR: Englewood Cliffs, N.J. , 1995.

(34) Likos, C. N.; Ashcroft, N. W. The Journal of Chemical Physics 1993, 99, 9090- 9102.

(35) Denton, A. R.; Ashcroft, N. W. Phys. Rev. A 1989, 39, 4701-4708.

(36) Nordholm, S.; Haymet, A. D. J. Aust. J. Chem. 1980, 33, 2013-2027.

(37) Nordholm, S.; Johnson, M.; Freasier, B. C. Aust. J. Chem. 1980, 33, 2139-2150. 37 (38) Woodward, C. E. J. Chem. Phys. 1991, 94, 3183-3191.

(39) Yethiraj, A.; Hall, C. K. J. Chem. Phys. 1989, 91, 4827-4837.

(40) Woodward, C. E.; Yethiraj, A. J. Chem. Phys. 1994, 100, 3181-3186.

(41) Lima, E. R. A.; Jiang, T.; Wu, J. Colloids and Surfaces A: Physicochemical and Engineering Aspects 2011, 384, 115-120.

(42) Hooper, J. B.; Pileggi, M. T.; McCoy, J. D.; Curro, J. G.; Weinhold, J. D. The Journal of Chemical Physics 2000, 112, 3094-3103.

(43) Yethiraj, A. The Journal of Chemical Physics 1998, 109, 3269-3275.

(44) Turesson, M.; Forsman, J.; Aakesson, T. Phys. Rev. E: Stat., Nonlinear, Soft Matter Phys. 2007, 76, 021801/1-021801/15.

(45) Yu, Y.-X.; Wu, J. J. Chem. Phys. 2002, 117, 2368-2376.

(46) Li, Z.; Cao, D.; Wu, J. J. Chem. Phys. 2005, 122, 174708/1-174708/9.

(47) Li, Z.; Wu, J. Phys. Rev. Lett. 2006, 96, 048302/1-048302/4.

38 Chapter 2

A Brief Summary of some Basic Polymer

Physics

1. Polymer Melts

As the name implies, polymer melts correspond to the neat polymer fluid (the pure polymer above its melting temperature and below its boiling point). According to

Flory’s self-consistent field theory1,2, the net force acting on every monomer in a three- dimensional polymer melt is zero. Hence, any particular monomer behaves as if it is ideal, which simplifies the description of the system enormously. As pointed out by de

Gennes3, while this may be true in three-dimensions, a chain in one dimension must obviously be stretched and in two dimensions it is somewhat swollen. More

specifically, de Gennes defines the problem of a “probe” chain of length N1 immersed in a melt of chains of length N, with the dimensionless parameter

2−d/2 ζ = N1 / N 2.1 indicating whether the probe chain is ideal or not (d is the spatial dimension). Choosing

N1 equal to N (for d = 3), we obtain that approaches zero for large N, and the polymer is ideal. For d = 2, similar considerations give a value of ≈1. Hence the

39 concentration inside the chain is comparable to the total polymer concentration and therefore the polymers in the system are segregated.

The three-dimensional polymer melt was studied by Woodward using the polymer density functional theory (PDFT)4, wherein it was possible to take advantage of the ideal behaviour of the intra-molecular polymer configurations. The use of the

PDFT for polymer solutions may be prone to error, however, due to the fact that in solutions the monomer density undergoes stronger fluctuations, which can lead to, e.g., significant swelling in good solvents. The polymer DFT is a mean-field theory and ignores such fluctuations. On the other hand, the main advantage (and strength) of the

PDFT is that one can use the theory to investigate polymer fluids in non-uniform environments. Notwithstanding the presence of these density fluctuations, it is the perturbation of the monomer density brought about by external potentials which dominates the equilibrium state, especially in the presence of strong fields such as hard surfaces. This assertion is essentially at the heart of all mean-field applications to non- uniform polymer solutions. In this thesis we will explore scenarios where the polymers are in a Θ solvent, so they can be considered to be ideal in any case. So let us now consider polymer solutions and explain what is meant by a Θ solvent.

2. Polymer Solutions

In this thesis, we will be considering polymer solutions into which are mixed particles of various sizes. As these particles will exclude monomers, they will have an influence on the polymer configurations. In order to understand those effects, we need

40 to know something about polymer solutions, so we will give a brief discussion about them here.

One way to deal with the description of polymer solutions, which ties in nicely with some of our analysis later in the thesis is to use an effective potential approach. A way to think about effective potentials is as follows. Now, we consider the case

(without particles) where polymer molecules are dispersed in a solvent (e.g. water or organic solvent). We imagine all the polymers frozen in a given configuration and allow the solvent to relax and reach equilibrium in the field imposed by the polymer molecules. We then calculate the free energy of the fully relaxed solvent in the presence of these frozen polymers. This free energy becomes an effective potential, and is a function of the configuration space of the polymers. In Chapter 3 and 5, we will show more complex systems containing particles as well as polymer and solvent molecule. In that case, however, it is the particles that will be frozen and the other two components (the polymer and the solvent) will relax to equilibrium.

Hence the polymer solution can be treated as a dilute polymer fluid wherein the monomers interact via this effective potential and the solvent is treated implicitly. If this is done rigorously, the effective potential will be a many-body interaction, consisting of 1-, 2-… n-body (temperature-dependent) terms. The simplest approach is to truncate it at the 2-body level. This idea can be illustrated with the classic Flory-

Huggins theory5,6, introduced earlier in Chapter 1.

Recall that the Flory-Huggins theory for bulk polymer solutions gives the following expression for the free energy of mixing,

41 Am = N1 lnφ1 + N2 lnφ2 + χM 0φ1φ2 2.2 kBT

where φ1 and φ2 are the volume fractions of solvent and polymer molecules. The parameter χ is the mixing parameter, which is a function of the temperature T and describes the interaction strength between monomers and solvent. Invoking the incompressibility assumption (all lattice sites are occupied with either monomer or

solvent), one obtains that the total volume fraction is unity, φtot = φ1 +φ2 =1, and Eq.

(2.2) can be re-expressed to obtain the following free energy per volume

Am φ2 = lnφ2 + (1−φ2 )ln(1−φ2 )+ χφ2 (1−φ2 ) 2.3 M 0kT M

Thus, via the incompressibility assumption, we have been able to rewrite the polymer solution’s free energy as a function of the monomer volume fraction only. That is, the system’s thermodynamics has been expressed in terms of an effective interaction between polymers with no explicit reference to the solvent density. The first term on the RHS of Eq. (2.3) is just the ideal entropy of the polymer and the last term is second

order in φ2 . On the other hand, the second term on the RHS of Eq. (2.3), the potential of mean force between monomers, is higher-order than quadratic in φ2 and would suggest that the potential of mean force (POMF) is a many-body one. However, a closer analysis reveals that this term can be considered a consequence of effective excluded volume interactions between monomers. This is very similar to the excluded volume term which appears in Nordholm’s DFT7,8 (Eq. 1.59). It is fruitful to expanded

the “excluded volume” term for small φ2 and obtain,

42 Am φ2 1 2 1 3 = lnφ2 + φ2 (1− 2χ)+ φ2 +! 2.4 M 0kBT M 2 6

The second term on the RHS of Eq. (2.4) can be considered as being due to an effective

2-body interaction between monomers at low concentrations, which includes contributions from the direct interaction (through the mixing parameter χ) as well as an excluded volume contribution. This allows us to classify solvent types for polymers on the basis of the value of χ:

1. When χ < 0.5, we have a good solvent. The association between polymer and

solvent molecules is energetically favoured and monomers appear to repel one

another. Thus polymers in good solvents will tend to expand to maximize their

contact with solvent.

2. When χ > 0.5, we have a poor solvent. The association between polymer and

solvent molecules are not energetically favoured and monomers will appear to

attract one another. Polymers in poor solvents shrink to minimize their contact

with solvent molecules.

3. When χ = 0.5, we have a Θ (theta) solvent. Ignoring terms in Eq. (2.4), which

are of higher order than quadratic in φ2 , the polymer behaves as if it is ideal.

As stated above, the quality of the solvent will affect the configuration of polymer molecules. De Gennes developed a scaling approach to illustrate the universal property of polymer statistics, depending upon the quality of the solvent. 3 To illustrate the scaling approach, we consider a linear polymer with N monomers. The linear dimension of this polymer, as it explores configuration space, can be characterized by the polymer radius of gyration,

43 1 N 2 2 Rg = ∑(rk − rm ) 2.5 N k=1

where the vector rk is the vector from the origin to monomer k and rm is the position of the center of mass of the polymer. Depending on the type of solvent, one has the following:

3/5 0.588 1. Good solvents: Rg ~ N (more refined estimates give N )

2. Poor solvents: polymer is assumed to be collapsed.

1/2 3. Θ solvents (ideal polymers): Rg ~ N .

One can use the radius of gyration to define different concentration regimes of polymer

solutions. The so-called overlap concentration, Coverlap, is defined as that where the polymer molecules are assumed to be barely overlapping each other,

N Coverlap ≅ 4π 3 2.6 R 3 g

Let Cp be the concentration of polymer: if Cp < Coverlap, the polymer solution is in the

dilute regime; if Cp > Coverlap we have the semi-dilute regime, where the polymer coils overlap but the volume fraction of monomer remains low. At even higher polymer concentration, the solution is in the concentrated regime. The semi-dilute regime generally attracts a lot of attention, as it is here that density fluctuations have a significant influence on the thermodynamic properties. At concentrations higher than the semi-dilute regime, de Gennes describes polymer solution thermodynamics generically in terms of the correlation length, ξ , which measures the length scale of

44 correlated density fluctuations within the polymer solution. For example, the osmotic pressure, Π, can approximated as

Π ≅ T /ξ 3 2.7

In Chapter 3 and Chapter 5, we will use the concept of an effective potential to describe particle/polymer mixtures in Θ solvents. In that case, it will be an effective interaction between particles as mediated by the polymer solution. The correlation length, ξ , of the polymer solution determines the range of that effective potential. The effective potential will generally be a many-body one. The smaller the particles, the more many- body terms need to be included. In Chapter 3 and 5 we will be mainly concerned with relatively small particles, and the full many-body characteristics of this effective potential become important. The 1-body free energy corresponds to the excess chemical potential for insertion of a particle into the polymer solution at infinite dilution.

It is mostly ignored as trivial, but depending upon how the many-body forces are decomposed, it may play an important role in phase equilibria of the particle/polymer mixtures. It is also true that for very large particles, the 1-body free energy may dominate the thermodynamics. This is certainly the case when particles are so large that they essentially present a planar surface to the polymer solution. In these circumstances the particles can drive surface phase transitions of the polymer solution, quite separate to those that occur in the bulk. This scenario is the subject of Chapter 4 of this thesis.

45 3. Polymers at Surfaces

The physical behaviour of polymers in the bulk is known to be different from polymer at surfaces. The interfacial properties of polymers is of great importance when large particles, e.g., colloids, are mixed into a polymer solution. In the case of excluded polymers and non-adsorbing surfaces, at a high polymer concentration, the surface density of polymers can be higher than that in the bulk, due to hard sphere packing.

However, if the polymer is ideal, the density of polymers adsorbed on the surface is lower than it in the bulk. The reason for this is that polymers at a surface explore fewer configurations than polymers in the bulk. The corresponding loss in configurational entropy close to a surface causes the concentration of polymers there to be relatively low. This effect is known as depletion. On the other hand, for adsorbing surfaces, the opposite is true. Here we find that, due to cooperativity, polymer molecules tend to adsorb very strongly to the surface. This cooperativity is caused by adsorption of a given monomer in the polymer chain, which then aids the adsorption of other bonded monomers. The correlation length within a polymer chain in the dilute regime is of

order Rg. In the limit of infinite polymers in a Θ solvent, cooperative adsorption leads to the well-known (second-order) adsorption transition3. This manifests itself as a

“bound-state” solution to the Edwards SCF equation9, which dominates the polymer configurations as the polymer length becomes infinite. This terminology exploits the analogy between Edwards’ equation and the time-dependent Schrödinger equation. For finite length polymers, the bound-state solution still asserts itself as the dominant contribution (for long polymers), though there is strictly speaking no phase transition in this case.

46 The picture is more complex in the case of finite length polymers under good solvent conditions. In a model developed by Jenkel and Rumbach10, the structure of polymers adsorbed at flat surfaces usually contains three components namely: trains, loops and tails, as shown in Fig. (2.1). The overall adsorbed polymer concentration plays a role in determining these configuration types. When the bulk polymer concentration is low so is adsorption and the polymers on the surfaces tend to have a flat structure (trains). However, if the bulk polymer concentration is high, there are generally many polymers present at the surfaces and they will tend to stick out to form a brush-like structure (loops and tails). This results from polymers in a good solvent competing for space, due to the effective excluded volume interactions. Those two characteristic types of surface adsorption structures are shown in Fig. (2.2).

Fig. 2.1 A schematic diagram showing the train-loop-tail structure for an adsorbed polymer.

47

Fig. 2.2 A schematic diagram displaying two kinds of structure of adsorbed polymer on the surfaces. The figure on the left shows the flat structure, while the brush-like structure is shown by the figure on the right.

Woodward and Forsman11-13 have also used the PDFT to study polymers at interfaces. The PDFT proved to be an accurate method to study polymers in a good solvent at adsorbing surfaces11. It has since been generalized to treat infinitely long flexible and semi-flexible polymers at surfaces12 and to include polydispersity in polymers, which is often encountered in many experiments13.

3.1 Polydisperse and Living Polymers

Polymers, whether they are synthesized or natural, all contain a certain degree of polydispersity. In polydisperse polymer solutions, there is a range of molecular weights characterized by an average value and distribution width. From a theoretical point of view it is thus desirable to include some degree of polydispersity in the modelling. One of the most common ways of describing the molecular weight distribution in a polydisperse polymer sample is via the so-called Schultz-Flory14 (S-F) chain length distribution,

48 n+1 n (n) κ s F (κ, s) = exp(−κs / s ) 2.8 Γ(n +1) s n+1

Here s is the polymer length, Γ(n +1) is the gamma function and κ and n are variables that determine the shape of the distribution. As κ and n increase, the distribution becomes more monodisperse. Furthermore, if κ = n +1 , the variable s represents the average length of the polymers in the sample. In some circumstances polydisperse polymers are equivalent to so-called living polymers.

The term living polymer was first used by Szwarc15 in 1956. Somewhat colourfully, he compared the termination of the polymerization process with the end of life and thus referred to a polymer as “living” if its polymerization process was not yet terminated. If the polymerization process is reversible, this system remains in dynamic equilibrium. That is, the local molecular weight distribution in any part of the polymer solution is sensitive to the immediate environment. This means that the polymer molecular weight is able to adjust to influences, such as the presence of attractive or repulsive surfaces. If one is interested in the equilibrium rather than the dynamic response of living polymer solutions, it is possible to treat the living polymer solution as a non-living polydisperse polymer with an exponential molecular weight distribution

(equivalent to the lowest-order, n = 0, S-F distribution). That is, the equilibrium response of living polymers to an external potential is the same as that of a polydisperse bulk polymer solution in contact with the external potential. This can be obtained as the minimization of an appropriate free energy. Woodward and Forsman 12 have shown this in a general formulation of polymer density functional theory. Indeed, the theory by

Woodward and Forsman is actually applicable to polymer solutions having a general

Schulz-Flory distribution, not just for living polymers. A similar approach, but specific

49 only to living polymers, was developed by van der Gucht and coworkers3 within a modified form of the Scheutjens-Fleer self-consistent field lattice theory16.

3.2 Adsorption of Polydisperse Polymers at Surfaces

From a theoretician’s perspective it would seem to be easier to consider the case of surface adsorption of monodisperse polymers, though real-life scenarios would seem to make the generalization of theory to polydisperse systems an imperative. As stated above, for monodisperse polymers of finite length in a Θ solvent, there is no adsorption transition. But the same is not necessarily true for a polydisperse polymer solution.

Interestingly, it was found that if the molecular weight distribution possessed an exponential tail, then an adsorption transition can be found at some temperature, even though the average polymer length is finite17. Sear17 applied scaling ideas3 to present a simple argument which balances the lowering in energy, due to a short-ranged adsorption potential, and the loss in configurational entropy due to surface exclusion of monomers. If one assumes an exponential molecular weight distribution, obtained from the S-F distribution with κ =1, we obtain the following expression for the density of polymer chains with at least one end close to the surface,

ρ ∞ ρ ~ exp[s(ε / T −ε / T )− s / s ]ds a s ∫ 0 a 2.9 (ρ / s ) ~ −1 2 Te < T < Ta s − (ε / T −ε / Ta )

Here, ρa , is the adsorbed polymer density at the surface, ρ is the bulk polymer density,

s is average polymer length and ε is the attraction strength of the surfaces. Ta is the adsorption transition temperature (for the infinitely long polymer). At temperature T =

50 Ta the monomer density profile at the surface is assumed to have zero slope which implies that there is no excess adsorption at the surface ρ . We see that for the case of an exponential length profile, an adsorption transition will occur at a lower temperature

Te , given by,

T a Te = −1/2 2.10 1+ N (Ta /ε)

Using Edwards’ SCF theory9 and PDFT4, Woodward and Forsman also studied the phase behaviour of polymers displaying S-F polydispersity at both non-adsorbing and adsorbing surfaces18. That work generalized the analysis by Sear to account for the presence of two surfaces. In Chapter 4, we will use PDFT to treat living random copolymers at surfaces. Copolymers contain different species of monomers, so in a living, random copolymer those monomers are reversibly bonded in a random way (see

Fig. 2.3). We use this new PDFT to study an example of significant biological importance.

51

Fig. 2.3 Schematic diagram showing adsorption of a living copolymer on an attractive surface. Orange and blue spheres represent different monomer types.

3.3 Amyloid proteins are living polymers

Living polymers (and analogues to them) are often seen in biological systems.

One example are amyloidal proteins, which are known to aggregate into thread-like crystals called fibrils, as well as a number of intermediate aggregate structures (see Fig.

2.4)‡.

‡ A diagram to show the schematic formalism of the living polymer chain used to simulate Amyloid proteins is shown in Chapter 4. 52

Fig. 2.4 Sketch of the process of forming fibrils where amyloid intermediates have to be formed before the formation of filaments. Filaments aggregates to form fibrils in the end.

A particularly topical area of research is the study of how surfaces, e.g., cell- membranes or nanoparticles, are able to facilitate the aggregation of these proteins19.

These types of studies are particularly important as amyloidal diseases, such as

Alzheimer’s and Type II diabetes, present serious health concerns in modern society.

Furthermore, the interplay of human biology and modern technology (as epitomised by the current nanoparticle revolution) is an emerging and challenging area of research of vital importance. These types of studies require the expertise of researchers across many disciplines from biochemistry to theoretical physics. With this in mind, we have initiated a model study of amyloidal proteins and the influence of surfaces on their aggregation. We modeled the protein aggregates as a living polymer chain, consisting of two different types of monomers. Each monomer represents a different

53 conformational state of the protein, one being a globular structure and the other an

“amyloidogenic” form. The latter corresponds to configurational forms of the protein, which allows for the formation of so-called cross-β sheets, in long stretches of the protein chain.

4. Mixtures of Particle and Polymers

Particle/polymer mixtures are especially important in many industries and in the natural world. Therefore, a great number of both experimental and theoretical studies have set out to explore the fundamentals of these systems. One of the very first studies of this type was carried out by Traube who observed particle flocculation when mixing water-soluble polymers with natural rubbers20. It is now well known that this phase separation is due to the so-called depletion forces induced by polymers. We will consider these in more detail in Chapter 3, but for now we can briefly describe them as due to the exclusion of polymer from the region between particles. This is caused by the depletion effect discussed earlier and lowers the osmotic pressure between particles compared to the bulk. Thus particles are pushed together by the external polymer concentration. The range of the depletion force depends on the correlation length in the polymer solution, which is equal to the radius of gyration in a Θ solvent. The strength of the depletion force depends on the bulk polymer concentration or, more precisely, the osmotic pressure.

A particle/polymer mixture will adopt different kinds phases depending on the conditions (see Fig. 2.5). For example, one may observe a fluid-fluid phase separation wherein one phase (the “gas”) is low in the particle concentration and high in polymer 54 and the other (the “liquid”) is high in particle concentration and low in polymer. A crystal phase, consisting of ordered particles is also possible (see Fig. 2.5). All of these are equilibrium phases, in the sense that they are the stable phases predicted from equilibrium thermodynamics. The fluid-fluid coexistence generally becomes metastable when the range of the depletion interaction is small compared to the size of the particles.

This is known as the colloid limit, and will be discussed in more detail in Chapter 3 and

Chapter 5. In the colloid limit, the fluid-solid transition is stable compared to the fluid- fluid transition. The opposite limit is the protein limit, where the range of the depletion attraction is relatively longer than the radius of the particles. Under these conditions the fluid-fluid transition becomes more stable.

In the colloidal regime, one is more likely to observe so-called non-equilibrium phases as well. These include gels (strong attraction at low particle volume fraction) and glasses (weak attraction and high particle volume fraction), also shown in Fig. 2.5.

These non-equilibrium phases are defined kinetically rather than thermodynamically and are experimentally identified by the slowing down of dynamic processes in the fluid.

The number of experimental and theoretical studies of these non-equilibrium phases has grown enormously over the last decade or so, which is indicative of their importance in many technologies21-24. One particularly promising area of study has been the incorporation of electrostatic interactions between the particles in the mixture25-28. The addition of a long-ranged (electrostatic) repulsion in conjunction with short-ranged

(depletion) attraction, has led to many interesting effects. This includes the appearance of a cluster phase at intermediate particle volume fractions, due mainly to depletion causing aggregation and the electrostatics limiting cluster sizes25,26,29,30.

55

Fig. 2.5 Schematic phase-diagram of particle/polymer mixtures. The polymer concentration (y-axis) is plotted as a function of particle volume fraction (x-axis). G + L denotes the gas-liquid mixture and C for crystal. The non-equilibrium phases of gels and glasses are also indicated.

The transition from the colloidal to protein limits presents many problems to theory, primarily because polymer configurations play a larger role in determining much of the thermodynamic behaviour in the protein limit. Depletion forces in the colloidal limit are relatively simply described as 1-body and 2-body potentials. This allows us to employ the full machinery of the statistical mechanical theory, which has been developed over the last hundred years, mainly on fluids with pair-wise additive interactions. On the other hand, in the protein limit, n-body interactions (n > 2) become important and the theory becomes difficult. This makes it hard, for example, to study the transition regime in the phase diagram, where non-equilibrium (gel and glass) phases start to become suppressed as one proceeds from the colloidal to the protein

56 limits. The role played by many-body forces in this regime is not well understood. In

Chapter 5 we will describe a new theoretical approach that accounts for many-body forces in a particle/polymer mixture in the protein limit, wherein the particles also carry an electrostatic charge. We shall use the theory to predict fluid-fluid phase diagrams and the effect of charge on these.

57 References

(1) Flory, P. J. J. Chem. Phys. 1949, 17, 303-310.

(2) Flory, P. J. J. Macromol. Sci., Phys. 1976, B12, 1-11.

(3) De Gennes, P.-G. Scaling Concepts in Polymer Physics; Cornell University Press: Ithaca, N.Y., 1979.

(4) Woodward, C. E. J. Chem. Phys. 1991, 94, 3183-3191.

(5) Flory, P. J. J. Chem. Phys. 1941, 9, 660-661.

(6) Huggins, M. L. J. Chem. Phys. 1941, 9, 440.

(7) Nordholm, S.; Haymet, A. D. J. Aust. J. Chem. 1980, 33, 2013-2027.

(8) Nordholm, S.; Johnson, M.; Freasier, B. C. Aust. J. Chem. 1980, 33, 2139-2150.

(9) Edwards, S. F. Proc. Phys. Soc., London 1965, 85, 613-624.

(10) Jenckel, E.; Rumbach, B. Z. Elektrochem. Angew. Phys. Chem. 1951, 55, 612-618.

(11) Forsman, J.; Woodward, C. E.; Freasier, B. C. J. Chem. Phys. 2003, 118, 7672- 7681.

(12) Woodward, C. E.; Forsman, J. Phys. Rev. E: Stat., Nonlinear, Soft Matter Phys. 2006, 74, 010801/1-010801/4.

(13) Woodward, C. E.; Forsman, J. Macromolecules 2009, 42, 7563-7570.

(14) Schulz, G. V. Z. Physik. Chem. 1939, B43, 25-46.

(15) Szwarc, M. Nature (London, U. K.) 1956, 178, 1168-1169.

(16) Scheutjens, J. M. H. M.; Fleer, G. J. J. Phys. Chem. 1979, 83, 1619-1635.

(17) Sear, R. P. J. Chem. Phys. 1999, 111, 2255-2258.

(18) Woodward, C. E.; Forsman, J. Phys. Chem. Chem. Phys. 2011, 13, 5764-5770.

(19) Vacha, R.; Linse, S.; Lund, M. J. Am. Chem. Soc. 2014, 136, 11776-11782.

58 (20) Traube, J. Gummi-Ztg. 1925, 39, 434-435.

(21) Stradner, A.; Sedgwick, H.; Cardinaux, F.; Poon, W. C. K.; Egelhaaf, S. U.; Schurtenberger, P. Nature 2004, 432, 492-495.

(22) Saunders, B. R.; Laajam, N.; Daly, E.; Teow, S.; Hu, X.; Stepto, R. Adv. Colloid Interface Sci. 2009, 147-148, 251-262.

(23) Mezzenga, R.; Schurtenberger, P.; Burbidge, A.; Michel, M. Nat. Mater. 2005, 4, 729-740.

(24) Stradner, A.; Romer, S.; Urban, C.; Schurtenberger, P. Prog. Colloid Polym. Sci. 2001, 118, 136-140.

(25) Stradner, A.; Thurston, G. M.; Schurtenberger, P. J. Phys.: Condens. Matter 2005, 17, S2805-S2816.

(26) Cardinaux, F.; Stradner, A.; Schurtenberger, P.; Sciortino, F.; Zaccarelli, E. EPL 2007, 77, 48004/1-48004/5.

(27) Atmuri, A. K.; Bhatia, S. R. Langmuir 2013, 29, 3179-3187.

(28) Denton, A. R.; Schmidt, M. J. Chem. Phys. 2005, 122, 244911/1-244911/7.

(29) Klix, C. L.; Murata, K.-i.; Tanaka, H.; Williams, S. R.; Malins, A.; Royall, C. P. Sci Rep 2013, 3, 2072.

(30) Campbell, A. I.; Anderson, V. J.; van Duijneveldt, J. S.; Bartlett, P. Phys. Rev. Lett. 2005, 94, 208301/1-208301/4.

59 Chapter 3

Exact Evaluation of the Depletion

Interaction between Two Spheres in the

Protein Regime

1. Introduction

A great number of theoretical1-6 and experimental7-11 studies involving particle/polymer suspensions have been carried out over many decades. These mixtures are attractive systems, not only due to their theoretical interest, but also their practical applications in, e.g., food, cosmetics, paints and the pharmaceutical industry 12,13. When mixing polymers with particles considerable interfacial phenomena will also be observed and the physical properties of the whole system, such as its phase behaviour, will also be fundamentally changed14-16.

1.1 Polymer induced Interactions

When polymer molecules are added to particle dispersions a range of (polymer mediated) forces will be active. In a classic experiment in 1925, Traube added a water- soluble polymer to natural tubber (latex spheres) and observed aggregation17. The

60 water-soluble polymer did not adsorb onto the hydrophobic latex spheres and it is now widely accepted that phase separation (flocculation) can be achieved upon addition of even small amounts of non-adsorbing polymer to particle dispersions. This is brought about via so-called depletion attractions. When non-adsorbing polymers are added to particles, each particle creates a depletion zone extending beyond the particle’s surface with a range determined by the correlation length of the polymer solution. In a Θ

solvent that correlation length is equal to the radius of gyration, Rg . When particles approach each other their respective depletion zones overlap and the difference in polymer concentration between the particles and in the surrounding gives rise to a depletion force which causes the formation of two-particle clusters. This phenomenon is an entropy-driven phenomenon as polymer chains have fewer configurations close to excluding surfaces16,18 (see Fig. 3.1).

61

Fig. 3.1 Schematic diagram dipicting the depletion between two spheres. The curled polymer (solid black line) is depleted by two approaching spheres and moves to a new location (black dashed-line).

When polymer molecules are instead attracted to the particles they are able to bridge across the spaces between particle surfaces. In this way, they are also able to cause particle attraction, by ‘dragging’ them together18. This is called bridging (see Fig.

3.2). In this chapter, we will focus mainly on depletion forces.

62

Fig. 3.2 Schematic view of the bridging effect among mixture of long polymer/particles.

1.2 Theoretical studies of Depletion forces

A considerable number of theoretical studies on depletion interactions have been carried out on idealised models of particle/polymer mixtures. The theoretical modelling of these systems falls into two general classes:

(i) Effective potential models: It is quite natural to discuss the system in terms of

effective interactions between particles only, as mediated by the other

components (polymer and solvent) in the solution. In this approach we treat the

particles as an effective 1-component fluid. The interactions between these

particles are modified by the presence of the polymers, but the polymer

molecules are treated implicitly. The polymer-mediated interactions are

analogous to dispersion forces between molecules. They are calculated by

holding the particle fixed in a given configuration and then allowing the

polymer solution to relax in the field of the fixed particles. The free energy of

the polymer solution then plays the role of the effective interaction between the

particles. This effective interaction will generally be temperature dependent (as

it is a free energy) and will formally consist of 1-body, 2-body … n-body 63 interactions. Such potentials are generally known as a potential of mean force

(POMF). The practical implementation of such an approach must generally

truncate this series at the 2-body level. This is because the treatment of many-

body interactions ( n ≥ 3 ) is too demanding numerically. The truncation of the

POMF at 2-body interactions is a reasonable approximation in the colloid limit

(large particles, small polymers). But it becomes a poor approximation in the

protein limit (small particles, large polymers).

(ii) Mixture models. The full mixture problem amounts to treating the polymer

explicitly. The solvent can be usefully treated implicitly, as described in

Chapter 2. This approach is the usual one for dealing with the protein limit.

Unfortunately, treating the polymer explicitly, including all its degrees of

freedom, also creates a very difficult problem and further simplifications are

generally needed. As described in Chapter 1, one such simplification is to

restrict polymer configurations so that they become defined by a space-filling

lattice3. However, simulation work with lattice polymer models are still

extremely time consuming, and very few studies have appeared in the

literature19-21. Another approach is to use even simpler models for the polymer,

as in the Asakura-Oosawa A-O model22,23.

In the A-O model, particles are described as hard spheres. The polymers are also

assumed to be spheres with diameter 2Rg (shown in Fig. 3.3) which cannot penetrate into the particles, but are themselves mutually penetrable. Thus, it is understood that the traditional A-O model can only be used under Θ conditions. The range of the depletion interaction in the A-O model is determined by the radius of gyration and its

64 strength by the osmotic pressure of the polymers. For example, the POMF between just two particles immersed in the polymers can be expressed as:

% ∞ r ≤ 2RS ' V(r) = & −Π pVo (r) 2RS < r ≤ 2RS + 2Rg 3.1 ' 0 r > 2R + 2R (' S g

where Π p is the osmotic pressure induced by the polymers and Vo (r) is the volume of the overlapping region between two particles at a distance separated by r. The A-O model contains all the many-body effects alluded to above, as it is a 2-component mixture model. This approach, and variations of it, has been widely adopted by workers in the field as one of the ‘cornerstone’ approaches in this area, so much so, that the effective potential approach has been somewhat neglected in the literature to date.

Fig. 3.3 Schematic view of the A-O model illustrating the colloidal aggregation caused by depletion forces. The region between the dotted lines and the colloidal surface is called the “depletion zone” which excludes all polymers.

65 The polymer reference interaction site model (PRISM)24-27, described in Chapter

1, has been applied to particle/polymer mixtures. Unlike the A-O model‡, no simulations are necessary, as all thermodynamic quantities can, in principle, be directly calculated within the theory. PRISM has also been used to also calculate the (2-body)

POMF between just a pair of particles. This is obtained by considering the particle- particle pair correlation function in the limit of infinite dilution in the particles. The full many-body problem can be solved by allowing the particle density to be finite of course.

However, the extraction of a many-body POMF is not possible, as this is convoluted within the correlation functions.

In recent work, Woodward and Forsman28 revisited the effective potential approach for a model of hard sphere particles and fully flexible ideal chains. From the point of view of the mathematical analysis, it was an advantage to treat a system of polydisperse polymers with a molecular weight dispersion described by the Schulz-

Flory distribution. Serendipitously, a model that incorporates polydispersity in the polymer molecular weight is also more relevant to experimental conditions. Using the model, Woodward and Forsman were able to obtain a formal expression for the POMF between a pair of particles as an infinite multipole expansion, wherein the coefficients are obtained analytically. This POMF is essentially the 2-body component of the many- body interaction between particles described above. But as we stated earlier, the 2-body term alone is not sufficient for the description of particle dispersions when the ratio

q = Rg / RS becomes large. In order to overcome this problem, a so-called "spherical approximation" has been proposed by Woodward and Forsman29,30. The consequence of this approximation is that only monopole terms are required to describe many-body

‡ A-O model cannot be solved analytically. Therefore, a proper treatment of the A-O model requires simulations. 66 forces, leading to an enormous simplification of the resulting expression for the free energy. This will be described in more detail in Chapter 5. However, we can give a general physical feel for this approximation as follows.

In the calculation of the POMF, we suppose that particles are fixed in a typical fluid configuration within the polymer solution. The polymer molecules then relax in the field of these particles. It is then reasonable to assume that the average density of monomers, close to the surface of any chosen particle, is approximately spherically symmetric about that particle's centre, due to the averaging effect of the surrounding particles. To put it in more mathematical terms, we can express the local equilibrium monomer density in the vicinity of a chosen particle in a basis of multipoles (centred on the particle); these multipoles can be thought of as being due to the “polarization” of the monomer density caused by the external potential. The polymer solution is analogous to a “sea of electrons” being polarized by the disordered crystal of fixed particles. The free energy is then a complex function of these particle-centred multipoles. The aforementioned spherical approximation amounts to assuming that the monopole terms will dominate the interaction free energy, as the higher-order induced moments about any particle will tend to cancel in a typical fluid configuration. This argument is plausible at high or moderate particle densities, but seems less so at low density. That is, at low particle densities typical particle configurations will be dominated by 2-body

“collisions”. In this regime, one expects that dipolar contributions to the free energy will also be important and the spherical approximation may fail. It is evident that if we go to infinitely dilute solutions, the dominant free energy term is the 1-body POMF and the spherical approximation is exact. These arguments are summarized in Fig. (3.4).

67

Fig. 3.4 Illustration of assumptions made for the POMF with different particle concentrations. At high particle concentration (top), a particle (black) is in a sea of other particles where the monopole term dominates the interaction free energy. When the particle concentration is low (middle), the 2-body terms are important, whereas the 1-body term dominates at infinitely dilute concentration (bottom). The dotted line illustrates the spherically symmetric environment about the black particle.

68 In order to ascertain the applicability of the spherical approximation, it would be useful to consider the low density regime (middle diagram in Fig. 3.4) as it is in this case that we expect to see major deviations from the spherical approximation. Hence, in this chapter we will calculate the 2-body POMF, which is the effective potential that acts between just two particles immersed in the polymer solution. Apart from testing the validity of the spherical approximation, the work presented here is useful for another reason. In their previous work28, Woodward and Forsman only reported the asymptotic behaviour of the 2-body POMF. Thus, our results will provide an accurate calculation of the full 2-body interaction, which will be useful for say describing colloid/polymer mixtures (where the 2-body truncation of the many-body potential is valid).

Furthermore, in the course of our analysis we will also present an analytical expression for the 2-body POMF in cases where the spherical approximation needs to be augmented by the inclusion of a dipolar correction. In addition to this, we also note that the calculations presented here generalize those in the work of Woodward and

Forsman28, by considering the case where the particles are attractive to the monomers.

The results that we derive here for attractive particles will be crucial to the success of our study of the full many-body POMF, carried out in Chapter 5.

The theoretical description that we will employ for the polymer solution will use

Edwards SCF theory, outlined in Chapter 1. This theory gives a good description of flexible polymer chains, especially under Θ conditions, wherein the self-consistent field

(due to monomer-monomer interactions) is zero.

69 2. Theory

We consider two spherical particles α (α = A or B) immersed in a polymer

solution under Θ conditions. The radius of sphere α is Rα and their centres are fixed at a separation R . We assume that the polymer can be treated as ideal flexible chains which will sample their full configuration space, as allowed by the excluding spheres.

Those configurations are described by the end-end segment distribution function,

G(r, r';s), which denotes the joint probability that a chain segment of length s has ends at r and r' . This distribution satisfies the following Edwards’ equation, which is analogous to the “time-dependent” Schrödinger equation,

2 ∂G(r, r';s) σ 2 = ∇ G(r, r';s)−ψ(r)G(r, r';s) 3.2 ∂s 6 with the initial boundary condition

G(r, r';0) = δ(r - r') 3.3

Here, σ is the Kuhn length and ψ(r) is the external field acting on monomers. As we are dealing with a Θ solvent, we ignore the interactions between monomers and express

ψ(r) as a sum of spherically symmetric potentials centred on each particle. We assume that these are short-ranged on the length-scale of the polymers. The affect of the spheres can then be described by applying the following homogenous boundary condition at their surfaces,

1 ∇ G(r, r';s)⋅ nˆ = −ε ∀r ∈ S 3.4 G r A A

70 where SA is the surface of the sphere A . A similar equation holds for the surface SB .

These boundary conditions assume that the spheres impose a short-ranged adsorption interaction on the free monomers in the polymer/particle mixture, in addition to steric exclusion. The outcome of this can be expressed as a gradient term at the surfaces. The

quantity εA is a measure of the adsorption energy between polymers and sphere A and

nˆ is a unit vector pointing outward and acting normal to the surfaces of the spheres. If

εA is negative we have a repelling surface, and if it is positive we have an attractive surface.

2.1 Polydisperse Polymers

Solving Eq.s (3.2) – (3.4) is facilitated by considering the more general problem of a polydisperse polymer fluid where the molecular weight distribution can be described by the class of Schulz-Flory (S-F) polydispersities28 (see Fig. 3.5). As described in Chapter 2, these correspond to the following normalized probability distributions,

n+1 n (n) κ s F (κ, s) = exp(−κs / s) 3.5 Γ(n +1) s n+1

Here s is the degree of polymerization and, Γ(x) , is the gamma function. In Eq. (3.5), we have also introduced the independent variable, κ . When κ = n +1, then s is the average polymer length and κ (and n) determine the shape of the distribution.

71

Fig. 3.5 Plot of the Schultz-Flory molecular weight distribution with respect to the polymer length s with different n values.

For the polydisperse fluid, G(r, r';s) must be averaged over the S-F molecular weight distribution to give Gˆ(r, r';κ) . We begin our analysis by considering the n = 0 case, which corresponds to an exponential distribution. Later in the analysis we show how this can be generalized to higher-order S-F distributions. Thus for n = 0, we obtain

∞ κ Gˆ(r, r';κ) = ds exp(−κs / s)G(r, r';s) 3.6 ∫ 0 s

Averaging both sides of Eq. (3.2) and using the boundary condition, Eq. (3.3), we get,

R2 g ∇2Gˆ(r, r')−Gˆ(r, r') = −δ(r - r') 3.7 κ r

2 2 where Rg = sσ / 6 is the average square radius of gyration. Integrating one polymer end over the space external to the spheres (denoted as V ' ) gives the following end- distribution function,

72 gˆ(r) = dr'G(r, r') 3.8 ∫V ' where gˆ(r) is normalized to unity in the bulk (far from the spheres). The corresponding differential equation for gˆ(r) , ( r ∈ V ' ) is given by,

∇2gˆ(r)− λ 2gˆ(r)+ λ 2 = 0 3.9

2 2 where λ = κ / Rg . The boundary conditions at the surfaces of the spheres are easily obtained from Eq. (3.4),

1 ∇gˆ(r)⋅ nˆ = -ε ∀r ∈ S 3.10 gˆ α α

As an aside, we note that perfectly depleting spheres correspond to ε → −∞ and is α characterized by Dirichlet boundary conditions,

gˆ(r) = 0 ∀r ∈ S 3.11 where S is the union of the two spherical surfaces (this is the case solved by Woodward and Forsman28). In addition to the surface boundary conditions, we also have the bulk condition that gˆ(r) →1 far from the spheres.

To proceed, we use the Green’s function for the associated Helmholtz equation in free space31,

2 2 ∇r G0 ( r − r' )− λ G0 ( r − r' ) = −δ(r - r') 3.12

Given the bulk boundary conditions for gˆ(r) , we seek that G (r) → 0 as . The 0 r → ∞ required solution is given by,

73 exp(−λr) G (r) = 3.13 0 4πr

The solution to Eq. (3.9) can then be formally expressed using the Greens function as,

2 gˆ(r) = λ dr'G ( r − r' )+ dsG ( r − r' )Λ(s) 3.14 ∫V 0 !∫ S 0

The quantities Λ(s) provides multipole source terms at each of the surfaces. They are determined by the boundary condition Eq. (3.10).

The first integral in Eq. (3.14) is over all space and is equal to unity. The second integral is over the surfaces of the two spheres. Thus, we obtain

gˆ(rA ) =1+ dsˆG0 ( rA − s )Λ A (sˆ)+ dsˆG0 ( rA − s )ΛB (sˆj ) 3. 15 !∫ SA !∫ SB where the origin is chosen to be at the centre of sphere A. The exact solution to this problem is unknown, but it is useful to expand gˆ(r) in basis of spherical harmonics,

L l ˆ ˆl l ˆ 3.16 g(rA ) ≈ ∑∑ gm (rA )Ym (rA ) l=0 m=−l where rˆ denotes the angular coordinates (θ,φ) of the vector r with respect to the frame at sphere A. For m ≥ 0 , we use the following definition for the spherical harmonic functions.32

l m 2l +1 l − m l im Y (θ,φ) = (−1) ( ) ( )P (cosθ)e φ 3.17 m 4π l + m m

l where Pm (cosθ) is an associated Legendre polynomial, and i = −1 . For m < 0 , we use the complex conjugate form,

74 l m l * Ym (θ,φ) = (−1) Ym (θ,φ) 3.18 where we have used the notation m = −m .

The expansion, Eq. (3.16), becomes exact in the limit that L approaches ∞ but in any practical calculation, L will be finite. As the problem is cylindrically symmetric it makes sense to place the z-axis along the line joining the sphere centres, which allows us to set m = 0 in Eq. (3.16). A similar expansion can be made about the sphere B.

The surface multipoles can also be expanded in a similar basis,

L l l ˆ Λ A (s) ≈ ∑Λ0 (A)Y0 (s) l=0 3.19

L' l l ˆ ΛB (s) ≈ ∑Λ0 (B)Y0 (s) l=0 where s is a vector from the centre to the surface of the sphere (A or B) and we note that in general the upper limit in the sums may be different for sphere A and B. We note

l l l here that for identical spheres we would have reflection symmetry, Λ0 (B) = (−1) Λ0 (A) and, in this case, we could choose L = L' ; we shall use this result later when we perform explicit calculations.

The first integral on the RHS of Eq. (3.15) is a surface convolution of multipoles centered on sphere A (which is the site of the origin). Thus it can be usefully re- expressed by using the following, and well-known, 1-centre expansion of the Green’s

function, G0.

∞ l l ˆ l ˆ * 3.20 G0 ( r − r' ) = λ∑kl (λr> )il (λr< )∑ Ym (r> )Ym (r< ) l=0 m=−l

75 Here r< and r> are the lesser and greater of r and r' respectively. The modified spherical Bessel functions of the first and second kind are defined as33,

π i (x) = I (x) l 2x l+1/2 3.21 π k (x) = K (x) l 2x l+1/2

where Iv (x) and Kv (x) are the modified Bessel functions of the first and second kind respectively. The second integral on the RHS of Eq. (3.15) is a surface convolution of multipoles on sphere B, expressed in the co-ordinate system centered on sphere A.

Hence, we need to use a 2-centre expansion for G0 about both spheres. This is not as well-known as the 1-center expansion and was only recently derived by Woodward and

Forsman28.

2.2 Two-centre expansion for G0 ( r − r' )

The 2-centre expansion for the free space Green’s function G0 ( r − r' ) , around the centres of the two spheres is obtained using the following decomposition,

r − r' = r − r''− R 3.22 where R is the vector from sphere A to sphere B and r'' , is the vector between the centre of sphere B and the vector r' (see Fig. 3.6).

76

Fig. 3.6 Definition of the coordinate variables for the two-center expansion.

Assuming R ≥ r − r'' , which will be true in our application, one obtains

G ( r − r''− R ) = λ k (λR)i (λr)i (λr'') 0 ∑ ∑∑ l l1 l2 l1,m1 l2,m2 l,m 3.23 Q C(l l l;000)C(l l l;m m m) Y l (Rˆ)Y l1 (rˆ)*Y l2 (rˆ'')* × l1l2l 1 2 1 2 1 2 × m m1 m2

where C(l1l2l;m1m2m) is a Clebsch-Gordan coefficient and

1 1 2 l "(2l +1)(2l +1)% Q ( 1) 2 (4 )2 1 2 3.24 l1l2l = − π $ ' # (2l +1) &

We note that

C(l l l;000)C(l l l;m m m)×Y l (Rˆ)Y l1 (rˆ)*Y l2 (rˆ'')* ∑ 1 2 1 2 1 2 m m1 m2 3.25 m1m2m

77 are rotational invariants. The appearance of these combinations of spherical harmonics

is a consequence of rotational symmetry of G0 (r) . Choosing the z - axis to coincide with the Rˆ direction we obtain the following simplification,

G ( r − r''− R ) = λ k (λR)i (λr)i (λr'') 0 ∑∑ l l1 l2 l1,l2,l m 3.26 ( 1)l2 A C(l l l;000)C(l l l;mm0)Y l (Rˆ)Y l1 (rˆ)Y l2 (rˆ'') × − l1l2 1 2 1 2 m m m

1 where, A (2l 1)(2l 1) 2 and we have taken the complex conjugate of the RHS l1l2 = [ 1 + 2 + ] of Eq. (3.26).

Substituting these expansions for G0 ( r − r' ) into the RHS of Eq. (3.15) and using the spherical harmonic expansions for the surface multipoles give the following expansion coefficients for the function gˆ(r) about sphere A.

ˆl l * g0 (r) = 4πδ0 + Γl (A)kl (λr)+il (λr)∑Gl,l' (λR)Γl' (B) 3.27 l' where we have defined renormalized multipole field strengths,

l Γl (X) = λΛ0 (X)il (λRs ) 3.28

* l and used the nomenclature Γl (X) = (−1) Γl (X) . Finally,

2 G (λR) = A k (λR)C 3.29 l1l2 l1l2 ∑l l l1l2l;000

Using Eq. (3.27) and the boundary conditions at each surface, we are able to solve for

the surface multipoles, Γl (X) . For example, we have at the surface of sphere A,

78 * l Γl (A)Κl (λRA )+ Ιl (λRA )∑Gl,l' (λR)Γl' (B) = −εA 4πδ0 3.30 l'

Where we have defined the following combinations of modified spherical Bessel functions (note the prime nomenclature for the derivatives),

' Κl (x) = εα kl (x)+ λkl (x) 3.31 and

' Ιl (x) = εαil (x)+ λil (x) 3.32

The general equation is most conveniently written in matrix form as,

! $ Γ0 (A) ! ε / Ι (λR ) $ # & # A 0 A & # ! & # 0 & ! $ # (A) & JA G ΓL # ! & # &×# & = − 4π # & 3.33 T # * & # G JB & Γ0 (B) # εB / Ι0 (λRB ) & " % # & ! # 0 & # & # & # * (B) & ! " ΓL' % " %

where the (L +1)×(L +1) submatrix J A is diagonal, with elements

# Κ (λR ) & % 0 A 0 ! 0 ( % Ι0 (λRA ) ( % Κ (λR ) ( % 0 1 A ! 0 ( JA = % Ι1(λRA ) ( 3.34 % ( % " " # " ( Κ (λR ) % 0 0 ! L A ( % ( R ) ( $ ΙL λ A '

with a similar expression for the (L'+1)×(L'+1) submatrix JB . Furthermore, the expression for the (L +1)×(L'+1) submatrix G is given by,

79 ! G ! G $ # 00 0L' & G = # ! " ! & 3.35 # G ! G & "# L0 LL' %& where the elements are given by Eq. (3.29). We shall denote the matrix on the LHS of

Eq. (3.33) as M = J +G , where J is the diagonal matrix made up of the submatrices

J A and JB , and G is the residual component consisting of the G submatrices. It is useful to note that the elements of G approach zero, as the distance between spheres becomes large.

2.3 The Polymer Free Energy (POMF)

The excess free energy of the polymer fluid in the presence of the spheres

(beyond the bulk value), is given by,

βΔω = −Φ dr gˆ(r)−1 3.36 tot p ∫V { }

where β =1/ kBT . This simple expression for the free energy follows from the fact that the polymers can be treated as ideal, and therefore no excess monomer-monomer interactions are present. If they were, the free energy would be a more complex function of the monomer density.

Using Edwards’ differential equation for gˆ(r) , Eq. (3.9), we can rewrite the free energy as,

Φ p 2 3 βΔω = − dr∇ gˆ(r)+ Φ 4π R / 3 3.37 tot 2 ∫V ' p ∑ λ i=A,B

80 Gauss’s Theorem allows us to re-express the volume integrals in Eq. (3.37) in terms of surface integrals of ∇gˆ(r) at each sphere,

dr∇2gˆ(r) = − R2 dσ ⋅ ∇gˆ ∫V ' ∑ i ∫ S 3.38 i i=A,B

where Si represents the surface of sphere i and σ is the unit surface vector, which points outward from the spherical centre. From the boundary condition, Eq. (3.10), we then obtain,

2 0 dσ ⋅ ∇gˆ(rˆ) = − 4π R ε gˆ (R ) 3.39 ∫ S i i 0 i i where the spherically symmetric component of gˆ(r) has been symmetry selected, by the integration over the spherical surface. Using this result, we can rewrite the free energy as,

Ns 4π 2 l 3 R gˆ (R ) 4 R / 3 βΔωtot = −Φ p 2 ∑ i εi 0 i + Φ p ∑ π 3.40 λ i=A,B i=1

Using Eq. (3.27) we obtain for the component of the gradient perpendicular to the surface of sphere A ,

l ∂gˆ (r) ' ' * 0 = Γl (A)λkl (λr)+ λil (λr)∑Gl,l' (λR)Γl' (B) 3.41 ∂r l'

Combining Eq.s (3.41) and (3.27) using the boundary condition Eq. (3.10), we obtain,

* l Κl (λRA )Γl (A)+ Ιl (λRA )∑Gl,l' (λR)Γl' (B) = −εA 4πδ0 3.42 l'

Using the following relationship, which is easily derived from the properties of the modified spherical Bessel functions,

81 λ i (x)Κ (x) = k (x)Ι (x)− 3.43 l l l l x2

we multiply Eq. (3.42) by il (λRA ) to obtain

* Γ (A) l ( R )k ( R ) (A) ( R ) G ( R) (B) l i ( R ) 4 Ιl λ A l λ A Γl + Ιl λ A ∑ l,l' λ Γl' − 2 = − l λ A εA πδ0 3.44 l' λRA which we re-express as,

l l Γ (A) l Ι (λR ) g (R )− 4πδ − l = −i (λR )ε 4πδ 3.45 l A { 0 A 0 } 2 l A A 0 λRA and finally rearrange to obtain,

' l l λi (λR ) Γ (A) l A l g0 (RA ) = 4πδ0 + 2 3.46 Ιl (λRA ) Ιl (λRA )λRA

Substituting back into Eq. (3.40), we obtain the following general expression for the free energy,

2 3 ) βΔω 3/2 &ε σ i (σ ) ε Γ (i) σ + tot − i i 1 i i 0 i * = −κ ∑ ' + − * 3.47 4πΦ p i=A,B ( Ι0 (σ i ) 4π Ι0 (σ i ) 3 ,+

' * 3 where we have used the fact that i0 (x) = ii (x) and defined σ i = λRi and Φ p = Φ pRg .

From the matrix Eq. (3.33) we obtain the following expressions for the monopoles in terms of elements of the matrix, M = J +G ,

4πεA −1 Γ0 (A) = − M00 3.48 Ι0 (σ A )

82 4πεB −1 Γ0 (B) = − ML+1L+1 Ι0 (σ B )

3. Results

3.1 Analytical results for identical spheres in the protein limit

From here on, we shall assume identical ( A = B ) spheres and set εi = ε and

Ri = RS . This notwithstanding, the analysis presented does not rely on this

* simplification. The problem simplifies, by noting that, Γl (A) = Γl (B) = Γl and hence we are able to write,

# Γ & # 1/ Ι (σ ) & % 0 ( % 0 ( [M ]×% ! ( = − 4πε % 0 ( 3.49 % Γ ( % ! ( $% L '( $% '(

where the reduced matrix M(= J + G), has dimensions (L +1)×(L +1). σ = λRS and

# Κ (σ ) & % 0 0 ! 0 ( % Ι0 (σ ) ( % Κ (σ ) ( % 0 1 ! 0 ( J = % Ι1(σ ) ( 3.50 % ( % " " # " ( Κ (σ ) % 0 0 ! L ( % ( ) ( $ ΙL σ '

Hence the exact solution to Eq. (3.49) is given by

4πε −1 Γ0 = − M00 3.51 Ι0 (σ )

83 −1 Closed analytic expressions can be obtained for M00 with finite L, although these rapidly become more complex as L increases. However, in the protein limit

( RS / Rg → 0 ) the solution is rather simple. Considering Eq. (3.30) and letting σ i → 0 we see that only the monopole (l = 0) term survives this process. Mathematically, this

follows from the fact that il (0) = 0 for l > 0, and i0 (0) = 1. Physically, this is because the influence due to sphere B on the value of gˆ(r) on the surface of sphere A is essentially constant. This is due to the relatively small size of the spheres compared

with Rg and therefore, only monopole source terms are required to ensure surface boundary conditions. Hence we obtain,

− 4πε Γ0 = 3.52 Κ 0 (σ )+ Ι0 (σ )G00 (λR)

where G00 is,

e−x G (x) = 3.53 00 x

Substituting this result into Eq. (3.47) and subtracting the 1-body contributions to the free energy, we get the following interaction free energy between the spheres,

(0) 2 βΔω 3/2 ε G (R) 2 − 00 * = −2κ 3.54 4πΦ p Κ 0 (σ )(Κ 0 (σ )+ Ι0 (σ )G00 (R))

Explicit expression for the modified spherical Bessel functions are: i0 (x) = sinh(x) / x ,

' 2 ' 2 k0 (x) = exp(−x) / x , i0 (x) = {x cosh(x)−sinh(x)} / x and k0 (x) = −exp(−x)(x +1) / x

2 Small σ expansion gives the following, Ι0 (σ ) ~ ε + λσ / 3, and Κ 0 (σ ) ~ ε /σ − λ /σ which leads to,

84 (0) 2 4 βΔω 3/2 ε σ G (R) 2 − 00 * ~ −2κ 2 3.55 4πΦ p (εσ − λσ − λ) which, in explicit terms, is

*2 (0) * 2 ε exp(− κ R / Rg ) ~ 8 (R / R ) 3.56 βΔω2 − πΦ p S g * 2 (ε −1− κ RS / Rg ) κR / Rg

* where we have defined the dimensionless adsorption strength ε = εRS . For attractive spheres, ε > 0 , we note that it is possible that the denominator term in the prefactor of

Eq. (3.56) can become zero and the free energy diverges. This corresponds to a surface adsorption transition, which has been previously shown to occur on planar surfaces in the presence of a polymer solution with a Schulz-Flory distribution of molecular weights34. The physical picture that emerges here is that the infinitely long polymer molecules in the bulk are recruited to participate in an adsorption transition onto and across the spherical surfaces of the spheres (see Fig. 3.2). This phenomenon was described for planar surfaces in Chapter 2, but it occurs here in the presence of a pair of spheres, leading to an infinite bridging attraction. This transition will be suppressed in good solvents, where monomers will exclude each other. The more monodisperse the polymer solution becomes (κ → ∞ ), the greater the adsorption strength required to achieve this transition. For monodispersed solutions of finite polymer length, an infinite adsorption strength is necessary. Interestingly, as the spheres become smaller in size, a larger adsorption is also required to achieve a similar result.

It is possible to obtain the corresponding free energies for higher-order Schulz-

Flory distributions by using the following relation28,

85 n m m (n) (−κ) ∂ (0) (n) (0) Lˆ 3.57 Δω2 = ∑ m Δω2 = κ Δω2 m=0 m! ∂κ

(n) where Δω2 (κ) with κ = n +1, is the interaction between two spheres in the presence of a polymer fluid with a Schulz-Flory distribution given by F (n) (κ, s) in Eq. (3.5).

This relation follows from the fact that the polymer free energy is a linear function of the end point density, and that the S-F distribution itself satisfies Eq. (3.57). As, n → ∞, the polymer solution becomes more monodisperse. Alternatively, from Eq. (3.6) it is straightforward to see that the interaction in the case of monodispersed polymers is

(0) given by the inverse Laplace transform of Δω2 /κ . That is, substituting the interaction free energy into Eq. (3.6), we have,

∞ κ −1Δω (0) = dte−κtΔω (∞) (st) 3.58 2 ∫ 0 2

(∞) where Δω2 (s) is the interaction between two spheres in a monodisperse polymer solution with polymer length s . Hence,

(∞) −1 −1 (0) 3.59 Δω2 (s) = L {κ Δω2 }(t =1) where L−1 denotes the inverse Laplace transform. For the case of perfectly depleting spheres, ε* → −∞, we get,

(0) * 2 exp(− κ R / Rg ) βΔω2 ~ 8πΦ p (RS / Rg ) 3.60 κR / Rg which upon substitution into Eq. (3.59) gives the known expression for the depletion interaction between small spheres (protein limit) in an ideal monodispersed polymer, first derived by Eisenriegler and coworkers35,

86 (∞) * 2 32π 2 R βΔω2 ~ −Φ p (RS / Rg ) i erfc( ) 3.61 R / Rg 2Rg where i2erfc(x) is the twice-iterated complementary error function33. On the other hand,

* for small spheres with finite ε , such that ε , κ RS / Rg <<1, we obtain

(0) * 2 *2 exp(− κ R / Rg ) βΔω2 ~ −8πΦ p (RS / Rg ) ε 3.62 κR / Rg

When we consider the transition to more monodispersed polymer solutions, it is

assumed that the spheres are small enough so that the condition, κ RS / Rg <<1, applies as κ increases36. Thus, for finite ε , progression to the monodisperse case gives the following sequential limit (in the protein limit),

(∞) * *2 2 32π 2 R βΔω2 ~ −Φ pε (RS / Rg ) i erfc( ) 3.63 R / Rg 2Rg

This result shows that the pair interaction for small spheres is symmetric about the non- adsorbing case (ε = 0 ). As far as we are aware, this is the first time this result has been reported.

(0) The expression for the POMF, Δω2 , given in Eq. (3.54), is the result of applying the spherical approximation, described in the Introduction. This approximation strictly only applies in the protein limit, i.e., when the spheres are small

(0) compared to the polymer radius of gyration. Corrections to Δω2 , for larger spheres can be obtained by analytically inverting the matrix M , obtained with a sufficient number, L, of spherical harmonic basis functions. The expressions are complex for large L, but is straightforward for low L . For example, for L =1 , we obtain,

87 # & G00 (R)+ Κ 0 (σ ) / Ι0 (σ ) G01(R) M = % ( 3.64 G (R) G (R) ( ) / ( ) $% 10 11 + Κ1 σ Ι1 σ '(

$ ' −1 1 G11(R)+ Κ1(σ ) / Ι1(σ ) −G01(R) M = & ) 3.65 det(M) G (R) G (R) ( ) / ( ) %& − 10 00 + Κ 0 σ Ι0 σ () giving,

G11(R)+ Κ1(σ ) / Ι1(σ ) Γ0 = − 4πε 3.66 Ι0 (σ )det(M)

Where det(M) is the determinant of M . Also we have,

e−λR (λR +1) G (R) = 3 3.67 01 (λR)2

and G10 (R) = G01(R) .

(λR)2 e−λR + 2e−λR (λR2 + 3λR + 3) G (R) = 3.68 11 (λR)3

The expression for Γ0 can be substituted back into Eq. (3.47) to obtain a higher order, analytic expression for the potential for the free energy,

(0) βΔω 3/2 2 G (R)+ ΔG(R) 2 − 00 * = −2κ ε 3.69 4πΦ p Κ 0 (σ ){Κ 0 (σ )+ Ι0 (σ )(G00 (R)+ ΔG(R))} where

G (R)2 ΔG(R) = − 01 3.70 G11(R)+ Κ1(σ )Ι1(σ )

88 Comparing with Eq. (3.54), we see that Eq. (3.69) introduces a dipolar correction to the spherical approximation. While it is possible to obtain analytic expressions for even larger L, they become cumbersome, so we chose to solve cases for L > 1 numerically.

3.2 Numerical Evaluations

Using ©Matlab 2007, we solved the case of two perfectly depleting spheres

(ε → −∞ ) with varying ratios q = Rg / RS . The upper angular momentum was varied

* from L = 2,…9 . The POMF, βΔω / (4πΦ p ) , was then obtained as a function of the separation between the spheres. We investigated how many spherical harmonics were required to obtain accurate solutions, for various values of q . The results are given in the figures below.

* In Fig. (3.7), we compared the value of βΔω / (4πΦ p ) for L = 0,1, 9 . The results for L = 0 and 1 are solved analytically while those with L = 9 are obtained by inverting the resulting matrix. It is clear that for q =10 (protein regime) there is no significant difference between curves for the ranges L = 0 to L = 9 . From q = 5 to

q =1, we see the spherical approximation becomes progressively less accurate.

However, we note that the analytical dipolar term, Eq. (3.70), provides an excellent correction to the spherical expression, as the spheres approach.

89

90

* Fig. 3.7 The quantity βΔω / (4πΦ p ) as a function of the distance between spheres for various L values. The curves for L = 0,1 are solved analytically (see text) while the cases with L = 9 are obtained numerically. (a) RS = 0.1Rg (b) RS = 0.2Rg (c) RS = 0.7Rg (d) RS = Rg.

91 In order to explore the colloidal regime, we pushed q to even smaller values,

q =1/ 3,1/ 7 . These results are shown in Fig. (3.8). Here it is clear that even the dipolar correction is not able to provide a significant improvement to the spherical approximation, and a larger number of spherical harmonics are required to obtain an accurate solution.

As expected, in the protein regime, the surface multipoles can be approximated with just a small number of spherical harmonics. In the colloidal regime, a large number needs to be used, in order to get a reasonable value for the POMF. The reason for this is that the polymer in the colloidal regime has a greater spatial asymmetry than is in the case of the protein regime. In the latter, the distribution function gˆ(r) is much less affected by the asymmetry of the 2-sphere system, and the spherical harmonic expansion of gˆ(r) can be truncated at relatively small L .

92

* Fig. 3.8 The quantity βΔω / (4πΦ p ) as a function of the distance between spheres for various L values. The curves for L = 0, 1 are solved analytically while the cases with L = 2, 3, 9 are obtained numerically.

(a) RS = 3.0Rg (b) RS = 7.0Rg.

93 The POMF obtained for L = 9 in the cases investigated in this study essentially correspond to an exact solution of the continuous chain model. In other words, the corrections due to the inclusion of higher L contributions would be negligible. A comparison with exact solutions of a discrete chain model was made in the earlier work28. The numerical solution of the discrete model showed excellent agreement with the analytical solution of the continuous chain model in the asymptotic regime. It is worth noting that the asymptotic expression corresponds to the L = 0 solution presented here. At close surface separations, the asymptotic expression predicts too high an attraction, which is qualitatively similar to the behaviour seen in this study.

4. Conclusion

The analysis carried out in earlier work on the interaction between spherical particles in an ideal polymer solution with a Schulz-Flory polydispersity has been generalized to the case of arbitrary adsorption strength. Analytical results have been obtained in the case of small particles (protein limit), which are consistent with earlier analytical work by Eisenriegler and coworkers35. Furthermore, we have carried out a numerical evaluation of POMF, which takes into account a more extensive multipole expansion. The interesting result that emerges is that a low order truncation of the multipole expansion becomes more accurate, as the particles are made smaller, relative to the average polymer radius of gyration. Indeed, for perfectly depleting spheres and

Rg / Rs ≥10 it appears that the monopole term is largely sufficient to provide an accurate representation of the POMF, even when the spheres are at contact.

Furthermore, analytic corrections due to the addition of dipolar terms are easily derived

94 in the framework of our analysis, and provide an adequate correction for cases in the

range 10 > Rg / Rs ≥1.

The two-particle system we describe herein provides a stringent test of the spherical approximation that will be invoked by us in Chapter 5 to treat the many-body

POMF in particle dispersions. We have shown that the l = 0 approximation embodied in that assumption is justified in the protein limit, which is precisely when many-body forces (beyond POMF) are expected to the most important. Thus far the protein limit has been considered the most difficult one to treat theoretically. On the other hand, the colloidal limit can be easily dealt with using the pair (2-particle) approximation. Our results show that the effective potential approach for depletion forces can be significantly simplified in the protein regime.

One interesting result we obtained is embodied in Eq. (3.56). Here we noted a polymer adsorption transition that had a profound effect on the POMF between the two spheres. While this was treated as a bit of a minor curiosity in the present chapter, it is sufficiently interesting to warrant further pursuit. Thus we digress from a discussion of effective potentials for the moment and consider the infinitely dilute particle/polymer mixture. More specifically, we shall consider a single large sphere in contact with a polymer solution. We shall assume the sphere is so large it essentially presents a planar surface to the polymer solution and explore the consequences of the surface transition just described in an interesting biological context. This will be done in the next chapter.

95 References

(1) Dijkstra, M.; van Roij, R. Phys. Rev. Lett. 2002, 89, 208303/1-208303/4.

(2) Jimenez, J.; Rajagopalan, R. Eur. Phys. J. B 1998, 5, 237-243.

(3) Meijer, E. J.; Frenkel, D. Phys. Rev. Lett. 1991, 67, 1110-1113.

(4) Stoll, S.; Buffle, J. J. Colloid Interface Sci. 1998, 205, 290-304.

(5) Syrbe, A.; Bauer, W. J.; Klostermeyer, H. Int. Dairy J. 1998, 8, 179-193.

(6) Zherenkova, L. V.; Mologin, D. A.; Khalatur, P. G.; Khokhlov, A. R. Colloid Polym. Sci. 1998, 276, 753-768.

(7) Cowell, C.; Li-In-On, R.; Vincent, B. J. Chem. Soc., Faraday Trans. 1 1978, 74, 337-347.

(8) De Hek, H.; Vrij, A. J. Colloid Interface Sci. 1981, 84, 409-422.

(9) Poon, W. C. K.; Selfe, J. S.; Robertson, M. B.; Ilett, S. M.; Pirie, A. D.; Pusey, P. N. J. Phys. II 1993, 3, 1075-1086.

(10) Ilett, S. M.; Orrock, A.; Poon, W. C. K.; Pusey, P. N. Phys. Rev. E: Stat. Phys., Plasmas, Fluids, Relat. Interdiscip. Top. 1995, 51, 1344-1352.

(11) Calderon, F. L.; Bibette, J.; Biais, J. Europhys. Lett. 1993, 23, 653-659.

(12) Dickinson, E. Colloids Surf. 1989, 42, 191-204.

(13) Dickinson, E.; McClements, D. J. Advances in Food Colloids; Blackie Academic & Professional: New York, 1996.

(14) Fleer, G. J.; Tuinier, R. Phys. Rev. E: Stat., Nonlinear, Soft Matter Phys. 2007, 76, 041802/1-041802/4.

(15) Fuchs, M.; Schweizer, K. S. J. Phys.: Condens. Matter 2002, 14, R239-R269.

(16) Tuinier, R.; Rieger, J.; de Kruif, C. G. Adv. Colloid Interface Sci. 2003, 103, 1-31.

(17) Traube, J. Gummi Zeitung 1925, 434.

96 (18) Striolo, A.; Colina, C. M.; Gubbins, K. E.; Elvassore, N.; Lue, L. Mol. Simul. 2004, 30, 437-449.

(19) Meijer, E. J.; Frenkel, D. J. Chem. Phys. 1994, 100, 6873-87.

(20) Bolhuis, P. G.; Louis, A. A.; Hansen, J. P. Phys. Rev. Lett. 2002, 89, 128302/1- 128302/4.

(21) Chou, C. Y.; Vo, T. T. M.; Panagiotopoulos, A. Z.; Robert, M. Phys. A (Amsterdam, Neth.) 2006, 369, 275-290.

(22) Asakura, S.; Oosawa, F. J. Polym. Sci. 1958, 33, 183-192.

(23) Vrij, A. Pure Appl. Chem. 1976, 48, 471-483.

(24) Chandler, D. J. Chem. Phys. 1973, 59, 2742-2746.

(25) Chandler, D.; Andersen, H. C. J. Chem. Phys. 1972, 57, 1930-1937.

(26) Lowden, L. J.; Chandler, D. J. Chem. Phys. 1974, 61, 5228-5241.

(27) Lowden, L. J.; Chandler, D. J. Chem. Phys. 1973, 59, 6587-6595.

(28) Woodward, C. E.; Forsman, J. J. Chem. Phys. 2010, 133, 154902/1-154902/9.

(29) Forsman, J.; Woodward, C. E. Soft Matter 2012, 8, 2121-2130.

(30) Woodward, C. E.; Forsman, J. J. Chem. Phys. 2012, 136, 084903/1-084903/5.

(31) Arfken, G. B.; Weber, H.-J. Mathematical Methods for Physicists; Elsevier: Boston, 2005.

(32) Gray, C. G.; Gubbins, K. E.; Joslin, C. G. Theory of Molecular Fluids 2: Applications. International Series of Monographs on Chemistry; Oxford University Press: New York, 2011.

(33) Abramowitz, M.; Stegun, I. A. Handbook of Mathematical Functions; Dover: New York, 1972.

(34) Forsman, J.; Woodward, C. E. Langmuir 2012, 28, 4223-4232.

(35) Eisenriegler, E.; Bringer, A.; Maassen, R. J. Chem. Phys. 2003, 118, 8093-8105.

97 (36) Verwey, E. J. W.; Overbeek, J. T. G. Theory of the Stability of Lyophobic Colloids; Dover Publications: Mineola, N.Y., 1999.

98 Chapter 4

Density Functional Theory of

Equilibrium Random Copolymers:

Application to Surface Adsorption of

Aggregating Peptides

1. Introduction

Polymers are profoundly affected by the presence of surfaces. This behaviour is relevant to many industrial and biological phenomena, such as colloidal stability and protein crystallization1,2. Polymer density functional theory, PDFT, has proven to be a powerful theoretical tool for the treatment of these kinds of systems3-7.

In the past, PDFT has been focused on monodisperse chains3. This is despite the fact that laboratory samples of polymers rarely posses chains of a fixed molecular weight. Indeed polydispersity can be viewed as a potentially useful control parameter in experimental systems, which has made it desirable to find theoretical treatments for polydispersed systems. Tuinier and Pethukhov have used a product function approximation to study polydispersity effects on depletion by ideal chains8 and the case of equilibrium or living polymers has been studied by van der Gucht and co-workers9-11,

99 using Scheutjens-Fleer theory12 as well as Edwards’ self-consistent-field (SCF) theory13,14. In other work, the effect of polydispersity has been treated by numerically averaging monodispersed solutions over the molecular weight distribution function using quadrature methods15-17. Yang et al18 have also studied the effect of polydispersity on the depletion interaction between non-adsorbing surfaces.

Recently, Woodward and Forsman showed how the PDFT could be generalized to include polydispersity when the molecular weight distribution is of the Schulz-Flory

(S-F) form19. Surprisingly, it was found that the algebraic structure of the PDFT was simpler than that for the monodispersed fluid and this manifested itself in a numerically more efficient solution algorithm. The S-F distribution encompasses the exponential molecular weight distribution (MWD), characteristic of living polymers. Indeed, at thermodynamic equilibrium, the cases of living polymers (with reversible bonding of monomers) and an exponential MWD with permanent bonds, are indistinguishable.

There are many types of macromolecular systems that can reversibly aggregate into linear structures under appropriate conditions. A particularly interesting example is the aggregation of amyloidal proteins into linear filaments. These structures are believed to be part of the process that form fibrils, which are implicated in diseases such as Alzheimer’s and Parkinson’s Disease and Type II diabetes. There have been a number of theoretical attempts to model the kinetic and equilibrium behaviour of these protein aggregates. For example, a simple model has been developed by van Gestel and de Leeuw20, which describes the linear aggregation of a two-state protein model. Their work is a generalization of the Zimm-Bragg treatment of helix-coil transitions in finite length polymers, to account for reversible aggregation21. In their treatment they included the possibility of the linear filaments aggregating laterally into fully-formed

100 fibrils. The work we present here will not consider fibril formation, instead focussing on the simpler linear aggregates. In particular, we will consider the effects of surfaces on the process of filament formation. Recent experimental investigations have shown that the presence of surfaces, such as cell membranes, or even man-made nanoparticles can significantly modify the aggregation of proteins compared to the bulk22. The PDFT, is particularly suited to the description of reversibly aggregating monomers next to surfaces. Here we consider an extension of the theory to include the possibility of different (countable) states of constituent aggregating monomers. In the case of aggregating proteins, this generalization would allow us to account for different conformational states of the protein molecules. The resulting theory essentially corresponds to a density functional treatment of living random copolymers.

In the next section we will develop the formalism of our new PDFT for living copolymers. As an application of the theory we generalize the model of van Gestel et al20, for associating proteins to take into account the presence of surfaces. Subsequently, results are presented for a range of model parameters and the implications of our results for the formation of protein filaments near surfaces are discussed.

2. Theory

2.1 Density Functional Theory for Polydispersed Polymers

Our derivation is based on earlier work by Woodward and Forsman on semi- flexible polydisperse polymers3,23,24. We begin with the exact canonical free energy density functional for an ideal, monodisperse polymer fluid of semi-flexible r-mers,

101 id βFr = ∫ d RNr (R)(ln[Nr (R)]−1)+ d RN (R)Φ(b) (R)+ drn (r)ψ 0 (r)+ ∫ r ∫ r 4.1 r−2 ∫ d RNr (R)∑EB (ri, ri+1, ri+2 ) i=1

Here, β , is the inverse thermal energy. The r-point density, Nr (R) , is a function of the

(b) monomer positions, R = (r1,…, rr ) where ri is the coordinate of monomer i . Φ (R) describes the intra-molecular connectivity modelled as nearest-neighbour, non- directional bonding, i.e.,

r−1 (b) (b) 4.2 Φ (R) = ∑φ ( ri − ri+1 ) i=1

The bending potential, EB , imparts stiffness to the chain. Keeping this stiffness term complicates the algebra significantly and in the exposition given in this chapter we shall assume it to be zero. Thus, we will focus on flexible random copolymers. Finally,

ψ 0 (r) , is the external potential acting on the monomers. While Eq. (4.1) is appropriate for polymers under Θ conditions, when additional non-bonding interactions are present, the ideal functional must be supplemented with a suitable excess term, F ex . This excess term accounts for short-ranged steric interactions as well as possible long-ranged attractions, depending upon the quality of the solvent.

Consider now the case of polydisperse polymers with up to nm different

monomer types (labeled by 1,…,nm ). We let the vector, c = (r, s), denote both the degree of polymerization r , and the primary sequence of monomer types,

s = (s1, s2,…, sr ) , where sk ∈ {1,…, nm } . A bulk fluid of such polymers can be

characterized by the average density of all polymer species, φ p , and the fractional

102 distribution of polymer types, F(c) . This distribution gives the fraction of species with structure given by c and is normalized according to

∑F(c) =1 4.3 c

The chemical potential of the various polymer species is given by,

βµc = ln[φ pF(c)]+ βΔµc (bulk) 4.4

where Δµc (bulk) is the excess chemical potential (which is zero for Θ conditions). In

f addition to bound species, there are free monomers (of type i ), with density ni (r) and

f chemical potential µi . When the fluid is subject to a spatially varying external potential, non-uniform density profiles result. These densities minimize the grand potential free energy functional,

id ex " α $ f f Ω = F + F #ni (r)%−∑µc ∫ d RNc (R)−∑µi ∫ drni (r) 4.5 c i

The first term on the RHS of Eq. (4.5) is the ideal contribution, and is the generalization of Eq. (4.1) to a mixture,

F id d RN (R) ln N (R) 1 drn f (r) ln"n f (r)$ 1 β = ∑ ∫ c ( [ c ] − ) +∑ ∫ i ( # i %− ) + c i dRN (R)Φ(b) (R)+ drntot (r)ψ 0 (r)+ ∑ c c ∑ ∫ i i 4.6 c i r−2 ∑ ∫ d RNc (R)∑EB (ri, ri+1, ri+2 ) c i

The fourth term on the RHS of Eq. (4.6) accounts for the fact that the monomer types

α will generally have different interactions with the external potential. We let, ni (r) , denote the density of type i monomers with connectivity denoted by α . The different

103 types of monomer connectivities are: monomers with two bonded neighbours; monomers with a single bonded neighbour, and monomers with no bonds (free). The

tot α total monomer density is ni (r) = ni (r) . As with most DFT approaches, we shall ∑α assume here that the excess free energy ex ! α #, is a functional only of these F "{ni (r)}$ monomer densities. If we assume that only steric interactions act between monomers, the system is denoted as athermal, which corresponds to a particular type of good solvent. In principle, this excess free energy term should then be sensitive to the monomer connectivity, due to cooperativity in excluded volume overlaps. In short, a free monomer excludes more volume to other particles than a bound monomer.

Furthermore, bound monomers at chain ends exclude more volume than those in the interior part of the chain. For simplicity, in the immediate analysis, we shall only

f b differentiate free and bound monomer densities, denoted by ni (r) and ni (r) , respectively. That is, we assume bound monomers consist of all those with either one or two bonded neighbours. Later, when we apply the theory to adsorbing peptides, we shall remove this distinction as well, so as not to let technicalities obscure the main results of the model.

2.2 Self-Assembling Random Copolymers

Bulk System

We now consider polymer molecules, which result from the reversible self- assembly of monomers into linear chains. Hence, the total chemical potential of all monomer species is set by the free monomers in the bulk, with densities

f {ni (bulk);i =1, nm }. For the case of only one monomer type, the bulk distribution,

104 F(c) , will depend only upon the degree of polymerization, r . For ideal polymers it will be given exactly by an exponential distribution,

F(r) = Ke−κr 4.7 with K a normalization factor and −1 r is the average polymer length in the bulk. κ = b

Even for non-ideal polymers, the exponential distribution is a reasonably good approximation for long enough polymers due to the fact that, to a good approximation, the excess chemical potential scales as the degree of polymerization, r . When different monomer types are present, the bulk distribution is determined by considering all possible monomer combinations. This is easily done by mapping the problem onto a 1- dimensional Ising chain and using the standard transfer matrix method, an approach used by van Gestel et al to study helical transitions in equilibrium polymers25. In this case, the distribution of aggregate lengths, f (r) = F(c) , will be given by ∑s

f (r) = Q(r) / ∑Q(r) 4.8 r where the Grand partition function for the r -mer (r >1) is given by, the expression,

Q(r) = u⋅ Tr−1 ⋅ f ∗ 4.9

The transfer matrix T is given by the expression

! f t f t ! f t $ # 1 11 1 12 1 1nm &

# f2t21 f2t22 ! f2t2n & T = # m & 4.10 # ! ! " ! & # f t f t ! f t & " nm nm1 nsp nm 2 nm nmnm %

where fi is the bulk fugacity of the free monomer of type i

105 f f βΔµi (bulk) 4.11 fi = ni (bulk)e

f here Δµi (bulk) is the excess chemical potential (beyond the ideal fluid) of the free

−βgij monomer of type i . The quantity tij = e , where gij is the bonding free energy of monomers of type i and j . The terminal vectors are given by

" f % $ 1 ' f ∗ = $ ! ' 4.12 $ f ' # nm & and

u = (1,1…1) 4.13

Variation of these terminal vectors allows the investigation of end-effects on the molecular weight distribution. For example, a model used to study associating proteins found that the lengths of filaments that formed can be significantly affected by the conformational state of the terminal proteins20. The Mass Action Law gives the

following expression for the bulk copolymer densities, Nc (bulk) , with primary structure c,

r−1 N (bulk) = f T 4.14 c sr ∏ si,si+1 i=1

We note that Nc = φ pF(c) and Tij are the elements of the transfer matrix, Eq. (4.10).

Non-uniform System

0 Suppose the polymer fluid is subject to an external potential, ψk (r) which acts

on the different monomer types k =1…nm . We shall assume that the external potential

106 is zero in the bulk fluid. The equilibrium polymer density becomes spatially dependent and is obtained by minimizing the free energy functional, Ω. This gives the following self-consistent expression for the density of aggregates of dimers and greater (r >1),

r r−1 b (bulk) βψ (r ) N (R) = N (bulk)eβΔµc e si i Θ ( r − r ) 4.15 c c ∏ ∏ sisi+1 i i+1 i=1 i=1

Here we have defined,

ex b δF 0

ψi (r) = b +ψi (r) 4.16 δni (r)

b ex where ni (r) is the density of the bound monomers and F is the excess free energy of the non-uniform fluid, from Eq. (4.5). Also,

(b ) ( r r ' ) −φkl − 4.17 Θkl ( r − r ' ) = e

The excess (beyond the ideal) bulk chemical potential of the chains with primary structure c , is given by

r r−1 βΔµ (bulk) = βψ b (bulk)− ln[ dr'Θ ( r − r ' )] 4.18 c ∑ si ∑ ∫ sisi+1 i=1 i=1

The first term on the RHS of Eq. (4.18) describes the effect of the medium on the chemical potential of the bound monomers. It is equal to the RHS of Eq. (4.16) in the bulk, i.e.,

b b βψi (bulk) = ψi (r) for all r ∈ bulk 4.19

The second term on the RHS of Eq. (4.18) is the free energy of the ideal chain with one end pinned. Substituting this expression for the chemical potential into Eq. (4.15) gives,

107 r r−1 b Θ ( r − r ' ) −βΔψsi (ri ) sisi+1 Nc (R) = Nc (bulk) e 4.20 ∏ ∏ dr'Θ ( r − r ' ) i=1 i=1 ∫ sisi+1 where b (r) b (r) b (bulk) . Finally, using Eq. (4.14), we obtain Δψi = ψi − βψsi

r r−1 −βΔψ b (r ) N (R) = f e si i T ( r − r ) 4.21 c ∏ si ∏ sisi+1 i i+1 i=1 i=1 where

Θlk ( r − r ' ) Tlk ( r − r ' ) = tlk 4.22 ∫ dr 'Θ( r − r ' )

Eq. (4.21) is a non-uniform generalization of Eq. (4.14). The above equations represent a closed system, which must be solved self-consistently in order to generate the thermodynamic properties of the system.

In order to calculate the required monomer densities, it is useful to introduce the following modified distribution functions for chain segments of length n (≥ 2)

n−1 b b −βΔψs (ri ) b −βΔψs (r )/2 c (n;r ) = dr … dr … f e i T ( r − r ) f e n 4.23 sn n ∫ 1 ∫ n−1∑ ∑∏ si sisi+1 i i+1 sn s1 sn−1 i=1 and, for n =1,

b −βΔψk (r )/2 4.24 ck (1;r) = fk e

The average total density of bound monomers of type k is given by,

∞ ∞ b 2 nk (r) = ∑∑ck (n;r)ck (m;r)− ck (1;r) 4.25 n=1 m=1

108 where the second term on the RHS of Eq. (4.25) corrects for the spurious non-bonded term (n, m =1) . This expression can be rewritten more succinctly as,

(b) 2 nk (r) = ck (r)ck (r)− ck (1;r) 4.26 where

∞ k 4.27 c (r) = ∑ck (n;r) n=1

The density of free monomers is given by

f f −βΔψk (r ) nk (r) = fke 4.28 where,

ex f δF 0

Δψk (r) = f +ψk (r) 4.29 δnk (r)

Given the definitions Eq.s (4.23) and (4.27), one can obtain the following simple recursion formula,

nm b −βΔψk (r )/2 k l 4.30 c (r) = ∑ ∫ dr'c (r')Mlk (r, r ')+ fk e l=1 where

b b −βΔψk (r')/2 −βΔψk (r )/2 Mlk (r', r) = fl e tlkTlk ( r'- r ) fk e 4.31 which can be written in matrix form as,

c = cM + v 4.32

109 where M is the symmetric matrix with elements, Mlk (r', r) with the vector

b b (r )/2 −βΔψ1 (r )/2 b −βΔψnm c = (c1(r)…cnm (r)) and v = ( f e f e ) . The generalized matrix 1 … nm vector product is given by

nm l 4.33 ∑ ∫ dr'c (r')Mlk (r', r) l=1

2.3 End-Effects

The expressions above make the assumption that all monomers in the aggregate chains are in equilibrium with the bulk. It may be desirable to consider situations where the distribution of terminal monomer types in the bulk are fixed a priori. We use α to label an end of a linear aggregate which is modified from its equilibrium distribution via the weights, P(α ) (p(α ), , p(α ) ). These weights affect the end vector as follows, = 1 … nm

b −βΔψ b (r ) v(α ) (p(α ) f e−βΔψ1 (r )/2 p(α ) f e nm ) and the recursion formula, Eq. (4.30), = 1 1 … nm nm becomes,

(α ) (α ) (α ) c = c M + v 4.34

These new end-point distributions can be used to generate the (bound) monomer densities in an equilibrium random copolymer, with different types of ends, α and β .

These are given by,

(α ) (β ) b b (α ) −βΔψk (r ) (β ) 4.35 nk (r) = ck (r)ck (r)− pk fke pk

2.4 Protein Aggregation on Surfaces

The theory presented above is quite general and can be applied to a wide range of molecular models over a diverse range of length-scales. Here, we will consider the 110 problem of protein aggregation on surfaces. In this case, the monomer length is of the order of nanometers. We shall make use of a protein aggregation model for the Aβ peptide (linked to Alzheimer’s Disease) proposed by van Gestel and de Lueew20. This model describes the formation of protein filaments and their association into fibrils in bulk solution. Using the general PDFT above, we will generalize this model so as to describe a non-uniform version of this theory. In particular, we will look at the interaction between Aβ peptide aggregates and a nearby surface.

In the theory of van Gestel and de Lueew, a filament consists of a random linear aggregate of two types of “monomer”. One is the peptide in a β -strand conformation, which forms so-called cross − β sheet hydrogen bonds with neighbouring peptides having a similar conformation. The other monomer type is the disordered conformer, which associates more weakly, but is the preferred structure at the terminals of the filaments. We shall label these as the ( β ) and (d ) conformers of the Aβ peptide. The theory of van Gestel and de Lueew also describes fibrils, which are formed by the lateral aggregation of several filaments via steric zipper interactions. While fibril formation is often associated with amyloidal diseases, it is now widely thought that smaller, oligomeric aggregates are more likely responsible for cell death, presumably through adsorption and eventual disruption of cell membranes. It is likely that these oligomers are not mature fibrils, but have a structure better represented by the simpler filaments. Thus, in this study, we shall only consider the interaction between linear peptide filaments and surfaces.

2.5 Filament distribution in the bulk

From Eq. (4.15), we see that the non-uniform theory requires the distribution of aggregates in the bulk fluid as input. In the theory of van Gestel and de Lueew20, the 111 water solvent is treated implicitly and it is assumed that the long-ranged (dispersion) interaction between the peptides is relatively weak. The primary structure of a filament is determined by the nearest neighbour bonding free energy and the availability of

monomers, determined by their bulk fugacity, fk . Monomer types at the terminals of the filaments can be predetermined without introducing significant additional complexity into the modelling and these were assumed to be in the d conformation.

This choice was based on the idea that β -strands are stabilized by a sufficient number of adjacent peptides especially when they are also in a β conformation.

The thermodynamic properties of the bulk can be obtained by mapping onto a 1- dimensional Ising chain in a magnetic field, which is then solved using standard transfer matrix methods. The details are provided in the original reference20 so we will only report the results of relevance to the current study, remaining consistent with the nomenclature used there. For example, the number density of oligomers of length, r

(≥ 2) , is given by,

ρ(r) = zrq(r) 4.36 where z is the fugacity of the free monomers of the disordered peptide and q(r) is the essentially the partition function of an oligomer containing r monomers (of either type).

An explicit expression for q(r) can be obtained in terms of the eigenvalues of the 2 × 2 transfer matrix,

r−2 r−2 r−1 q(r) = (xλ1 + yλ2 )k 4.37

with x = (λ1 − s) / (λ1 − λ2 ) and y = (s − λ2 ) / (λ1 − λ2 ) . The eigenvalues are given by

112 1 s ((1− s)2 + 4sσ )1/2 λ = + ± 4.38 1,2 2 2 2

These parameters reflect the various interaction energies between the bound monomers:

k = exp(−M ) ; s = exp(−P) , and σ 1/2 = exp(−R) . Here M is the binding free energy between d conformers. P is the additional free energy upon binding two peptides in the

β conformation. Finally, R , represents the cost of creating an interface between a bound peptide pair in β and d conformations respectively (see Fig. 4.1).

Fig. 4.1 Schematic diagram depicting the aggregated filament and free energy parameters used in the theory. The β conformers are shown as parallel arrows, whereas the disordered peptides are depicted as ovals.

We can define the normalized distribution for r-mers, r (≥ 2) ,

113 zrq(r) f (r) = ∞ 4.39 zrq(r) ∑r=1

Here, we make the implicit substitution, q(1) = 1, in the denominator sum. Upon substitution of Eq. (4.37), we obtain,

r r f (r) = K1(λ1kz) + K2 (λ2kz) 4.40 where

x 2 λ1 K1 = 4.41 x λ1kz y λ2kz 2 + 2 λ1 1− λ1kz λ2 1− λ2kz and

x 2 λ2 K2 = 4.42 x λ1kz y λ2kz 2 + 2 λ1 1− λ1kz λ2 1− λ2kz

In this study, we set the parameters s = 70 and σ = 0.1, as in the work by van Gestel and de Leeuw20. These parameters contain the free energy terms associated with β - β bonding and β -d interfaces in excess of an underlying chain of d conformers. The free energy of this underlying chain is determined by zk which is the excess fugacity associated with removing a peptide from the bulk and creating a (shared) d-d bond. In order to obtain values for the parameters z and k, we first specify a value for the average filament length in the bulk r and solve for the product from the following b zk expression (obtained as an appropriate average over the distribution f(r)),

114 zk +∑ zrkq(r)r r = r=2 4.43 b zk + zrkq(r) ∑r=2

Having chosen the average fibril length, we then choose the bulk protein density

ntot (bulk) and hence obtain both z and k values, using the following expression for the number density of r-mers,

tot tot n (bulk) # zkx zky & Φ = = z + z% + ( 4.44 p r 1 zk 1 zk b $ − λ1 − λ2 '

tot where Φ p , is the overall oligomer density of the bulk.

2.6 Simplified Model for Protein Adsorption

The oligomer distribution Eq. (4.40) provides a starting point for a simplified model for protein surface aggregation. In this model, we shall assume the presence of a flat surface, which adsorbs both β and d conformers of the peptide equally well. This treatment ignores scenarios where conformational changes in the peptide affect their surface interactions in a significant way, e.g., functionalized surfaces may interact with specific peptide sites, which become accessible upon conformational change. Instead, we consider situations where the conformational changes associated with the d → β transition have greater implications on the peptide-peptide interaction, rather than on the peptide-surface interactions. This is likely to be the case for the Aβ peptide, which forms strong cross−β sheets. It is worth nothing, however, that the generalization of the theory to account for conformation dependent surface interactions is relatively straightforward. We shall also assume that conformational changes will not cause significant changes in the size (excluded volume) of the peptides. With these

115 assumptions, the local excess chemical potential for bound monomers, Eq. (4.16), becomes independent of monomer type. Thus it is useful to define,

ex b δF 0 ψ (r) = +ψ (r) 4.45 δnb (r) and

Δψ b (r) = ψ b (r)−ψ b (bulk) 4.46 where nb (r) is the total bound monomer density, and ψ 0 (r) is the generic surface- protein interaction. Additionally, the bonding factor has the following simpler form,

Tlk ( r'− r ) = tlkt( r'− r ) 4.47 with t( r'− r ) describing monomer-monomer bonding. In this application we shall assume the bonding has the simple form,

δ( r'− r −σ ) t( r'− r ) = 4.48 4πσ 2 with δ(r) the Dirac delta function and σ is the nearest-neighbor bonding length-scale of the aggregating peptides. Using Eq. (4.25), we sum over the monomer type k to get the total bound monomer density,

∞ ∞ 2 b n (r) = ∑ ∑∑ck (n;r)ck (m;r)− ∑ ck (1;r) 4.49 k=β,d n=1 m=1 k=β,d

From the definition of ck (n;r) in Eq. (4.23), we can rewrite Eq. (4.49) as,

∞ ∞ b 2 n (r) = ∑∑Φ p f (n + m −1)c(n;r)c(m;r)− Φ p f (1)c(1;r) 4.50 n=1 m=1

116 where we define the simpler end-point distribution,

n−1 b (r ) b (r )/2 c(n;r ) = dr … dr e−βΔψ i t( r − r )e−βΔψ 4.51 n ∫ 1 ∫ n−1∏ i i+1 i=1 which satisfies the following recursion formula

b b c(n +1, r) = ∫ dr'e−βΔψ (r )/2t( r − r' )e−βΔψ (r')/2c(n, r') 4.52 with initial condition,

c(1, r) = exp(−βΔψ b (r) / 2) 4.53

Using Eq. (4.40), we obtain

∞ ∞ b K1 n m n (r) = Φ p ∑∑ (λ1kz) c(n, r)(λ1kz) c(m, r) n=1 m=1 λ1kz 4.54 ∞ ∞ K2 n m 2 +Φ p ∑∑ (λ2kz) c(n, r)(λ2kz) c(m, r)− Φ p f (1)c(1;r) n=1 m=1 λ2kz

Defining,

∞ n ci (r) = ∑(λikz) c(n, r) 4.55 n=1 where i =1,2 we obtain

b −1 K1 K2 2 n (r)Φ p = c1(r)c1(r)+ c2 (r)c2 (r)− f (1)c(1;r) 4.56 λ1kz λ2kz

From Eq. (4.53) and the definition Eq. (4.55), the following recursion formula applies,

−βΔψ b (r )/2 −βΔψ b (r')/2 i i 4.57 c (r)− λikc(1, r) = λikz ∫ dr 'e t( r - r' )e c (r')

117 Eq.s (4.45), (4.56) and (4.57) provide a self-consistent set of equations which can be used to solve for the density of peptides which are adsorbed onto surfaces as filaments.

2.7 The excess free energy

Different versions of PDFT arise from the choice of the excess functional

F ex 3-7,26-32. We used three different models for the excess free energy. In the first case we assumed that the water was a Θ solvent for the peptides, so that the excess was zero.

In the other two cases, only steric interactions acted between the peptides, which we assumed could be modeled as effective hard spheres with a radius independent of the peptide conformation ( d or β ). For hard sphere interactions we have found good accuracy with the so-called Generalized Flory-Dimer (GFD) functional for the excess free energy, F ex 4,28. This functional will be used in the present problem.

It is convenient to express the excess free energy in the following form,

ex ex βF = ∫ drntot (r)a (η)dr 4.58

ex where ntot (r) is the total peptide density. Here a (η) is the local free energy per particle and η is a functional of the total peptide density. In the GFD approximation aex (η) is given by the general expression,4,28

2 ex c +1 (2c − 2a − 4)η + (3− b + a − 3c )η a (η) = − i ln(1−η)− i i i i i 4.59 2 (1−η)2 where i = 1, 2 and we have,

118 a1 =1 , a2 = 2.45696

b1 =1 , b2 = 4.10386 4.60

c1 = 0 , c2 = −3.75503

We denote the two versions of the steric model as local and non-local. The local model is similar in sprit to the incompressibility assumption, used in the Flory-Huggins theory.

If one assumes incompressibility at every point in the solution, we obtain a simplified excluded volume term between monomers. If we use the GFD functional to treat this, the functional η in Eq. (4.58) is given by a local volume fraction, i.e.,

3 η = [(πσ ntot (r)) / 6]. The other model employs a non-local version of this functional with a weighted density that accounts for the possibility of short-ranged structuring.

3 The functional has the form, η = [(πσ ntot (r)) / 6] where,

3 n (r) = dr 'n (r ') 4.61 tot 4πσ 3 ∫ r-r '<σ tot

2.8 The surface potential

The interaction between the surface and peptide conformers is modeled as a truncated and shifted Lennard-Jones potential, integrated over the half-space of the surface, i.e.,

" $ ωLJ (z)−ωLJ (zc ), for z < zc ψ 0 (z) = # 4.62 $ 0 otherwise % where

2 σ 9 a σ 3 βω (z) =10π[ ( ) − ω ( ) ] 4.63 LJ 45 z 3 z

119 Here β = kBT with kB the Boltzmann’s constant and the temperature T, z is the

perpendicular distance from the surface and we have set zc = 4σ . The strength of the

surface attraction is determined by the parameter, aω . Due to the form of this external potential, this system has a planar geometry and the densities which minimize the free energy will be z dependent.

3. Results

3.1 Ideal peptide model

Fig. 4.2 shows the density of peptide bound in filaments as a function of distance from the surface for various values of the attractive part of the surface potential,

as parameterized by the quantity aω . The average filament length in the bulk was set at

r = 2. In the ideal model, the steric effect of the peptides is not considered. At low b surface attraction, it is clear that there is a depletion of monomers close to the surface due to the decrease in configurational entropy. However, as the attraction on the

surface increases, the peptide concentration starts to build. Indeed, for aω slightly greater than 0.07 we find that the monomer density adjacent to the surface goes to infinity (for obvious reasons this cannot be displayed in Fig. 4.2). When the surface attraction reaches a certain critical value, the density of the adsorbed peptide diverges, displaying the signature of a second order surface phase transition. This is the same adsorption transition observed by de Gennes for infinite polymers in a Θ solvent14,33.

However, it occurs here in a living polymer formed by aggregating peptide, displaying a dual exponential molecular weight distribution, Eq. (4.40). Hence this is essentially equivalent to the circumstances of the surface phase transition predicted by Sear34 (as 120 described in Chapter 2) and which has also been reported by van der Gucht et al11 as well as Forsman and Woodward35. This transition is accompanied by an enormous growth in the average length of filaments close to the surface.

According to Eq. (2.9) (Chapter 2), the surface phase transition occurs when the

2 attraction is larger than a a (eq) 1/ r where a (eq) is the value which gives w ≈ w + b w essentially zero excess adsorption, θ ex = 0. The excess adsorption is defined as,

∞ θ ex = (nb (z)− nb (bulk))dz 4.64 ∫ 0

b where n (z) is the bound peptide density. The value of aw must be somewhat greater

than aw (eq) for the surface transition to be observed. The larger the average bulk

filament length, the closer to aw (eq) at which this transition will occur. In Fig. 4.3, we show the bound monomer density as a function of the distance from the surfaces for a number of average filament lengths. The strength of the surface adsorption is set at

ex aw = 0.05 . It is clear from the profiles that θ < 0, i.e., aw < aw (eq). Increasing the average length of the peptides in the bulk only serves to decrease the peptide density at the surfaces, due to the depletion effect. Indeed, one finds that the average length of peptide filaments close to the surfaces actually decreases below that of the bulk value

(not shown). In Fig. 4.4, we show the bound peptide density profiles at varying average filament lengths with a = 0.065 b 3 ex w and n (bulk)σ = 0.001. In this case θ > 0, for all the

r investigated. The longer filaments have a higher density near the surface than the b shorter ones, due to cooperative binding, and increasing the value of r eventually b leads to the adsorption transition. We note here that for ideal polymers, the bulk density serves only as a multiplicative factor for the density profiles. This will not be the case

121 for the peptide models which include steric interactions. Though diagram as presented show only a section of the density profile for the sake of clarity, we have confirmed numerically that the density plateaus to the bulk value at large distance.

Fig. 4.2 Plot of nb (z)σ 3 as a function of z / σ for ideal polymer for increasing surface b 3 attraction and n (bulk)σ = 0.001 . The arrow indicates the direction of increasing aw .

122

b 3 Fig. 4.3 Plot of n (z)σ as a function of z / σ for different average lengths, aw = 0.05 and b 3 r n (bulk)σ = 0.001 . The arrow indicates the direction of increasing b .

123

b 3 Fig. 4.4 Plot of n (z)σ as a function of z / σ for different average lengths with aw = 0.065 b 3 r and n (bulk)σ = 0.001 . The arrow indicates the direction of increasing b .

The infinite surface density that is implied by the adsorption transition is clearly unphysical. It occurs in the ideal model because of the lack of steric interactions. We expect that the latter should act to supress the transition. This can be verified by considering the models which include steric interactions, as will be done in the next section.

3.2 Steric interactions – non-local theory

In this section we consider the peptide model which includes steric interactions.

The bound peptide density profiles for the case r = 2 are plotted in Figs 4.5 and 4.6 b

for different values of attraction parameter aw . In Fig 4.5, we see a similar behaviour to what is seen in the ideal model. However, when the surface attraction increases to quite

124 large values (as shown in Fig. 4.6), the density remains finite due to peptide excluding one another at the surface. The peptides already adsorbed on the surface stop more from being absorbed. We note that the density adjacent to the surface can go to a large number provided that the integral of the density within a certain distance of the order of

σ from the surface is finite. In Fig. 4.6, there is clear evidence of peptide layering at the surface. This is due to the non-local functional, Eq. (4.61), which is able to account for hard sphere structuring.

Fig 4.5 Plot of nb (z)σ 3 as a function of z / σ for polymer with hard sphere terms for increasing surface attraction and nb (bulk)σ 3 = 0.001 . The arrow indicates the direction of increasing aw .

125

Fig. 4.6 Plot of nb (z)σ 3 as a function of z / σ for polymer with hard sphere terms and larger attraction from the surface. The bound peptide density in the bulk nb (bulk)σ 3 = 0.001 .

In Fig. 4.7 we plot the excess adsorption, θ ex , of bound peptide at the surface as a

function of the attraction strength, aw, for various average filament lengths. Obviously, at very weak adsorption potential strength, the excess adsorption is low and clearly

approaches negative values. We note that aw (eq) is rather independent of the average filament length, as can be ascertained by the quite sudden transition between positive to negative θ ex . As the peptides are able to sterically exclude one another, the second- order adsorption transition does not occur in this system. However, we do expect to see

a sudden increase in the excess adsorption at a particular value of aw, which echoes the adsorption transition of the ideal system. This should appear as a point of inflection in the adsorption plot. Interestingly, we do not see strong evidence for this in Fig. 4.7, which would imply that steric effects suppress this behaviour. 126 In Figs. 4.8 and Fig. 4.9, we plot similar adsorption profiles, but at much smaller bulk peptide densities. This should have the effect of diminishing the steric effect.

While showing similar behaviours to the adsorption profiles at higher bulk density, we do note the presence of an inflection point in these profiles suggesting a sudden (orders

of magnitude) increase in the adsorbed peptide concentration at a particular value of aw.

We reiterate that this is not a true phase transition, but reflects the adsorption transition that occurs in the ideal system − a “soft transition”. Indeed, we see in Fig. 4.9 that the soft transition in the surface adsorption for r = 2, occurs at a value of a , b w which is very close to the (estimated) adsorption transition point in the ideal system.

As expected the soft transition occurs at smaller adsorption strengths, the larger is r . b

ex Fig. 4.7 Plot of the excess adsorption on the surface θ / (Φ p < r >b σ ) as a function of attraction strength on the surface, aw , for different average lengths. The bound peptide b 3 r density in the bulk n (bulk)σ = 0.001 and the arrow indicates the direction of decreasing b .

127

ex Fig. 4.8 Plot of the excess adsorption on the surface θ / (Φ p < r >b σ ) as a function of attraction strength on the surface, aw , for different average lengths. The bound peptide density in the bulk nb (bulk)σ 3 =10−6 and the arrow indicates the direction of decreasing r . b

128

ex Fig. 4.9 Plot of the excess adsorption on the surface θ / (Φ p < r >b σ ) as a function of attraction strength on the surface, aw , for different average lengths. The bound peptide b 3 −9 density in the bulk n (bulk)σ =10 and the back short arrow indicates the direction of r decreasing b . The red arrow shows the point where the adsorption of ideal polymer on the r surface goes to infinity for the case b = 2.

3.3 Steric interactions – local theory

The non-local theory described above is able to describe structuring in the density profiles of the adsorbed peptide filaments. However, at very high adsorption strengths, the iterative solutions of the PDFT become rather laborious. On the other hand, the local model may be a viable alternative approach, especially at lower adsorption strengths where the peptide density profiles are less structured.

In Figs. 4.10 and 4.11 we plot the bound peptide density profiles for the local model using the same set of parameters as those in Figs. 4.5 and 4.6. The local and

non-local theories show good agreement at low values of aw (compare Figs 4.10 and 129 4.5). On the other hand, for large adsorption strengths (Figs. 4.6 and 4.11) there is qualitative disagreement, as expected. The local theory shows less structure and certainly no indication of the expected layering behaviour.

b 3 Fig. 4.10 Plot of n (z)σ as a function of z / σ for polymer with hard sphere terms for increasing attraction from the surface. The bound peptide density in the bulk b 3 −3 n (bulk)σ =10 and the arrow indicates the direction of increasing aw .

130

Fig. 4.11 Plot of nb (z)σ 3 as a function of z / σ for polymer with hard sphere terms and larger b 3 −3 attraction from the surface. The bound peptide density in the bulk n (bulk)σ =10

However, the excess adsorption density profiles from the local model are similar to the results of the non-local model. That is, while the density profiles are qualitatively different, the integrated densities of both approaches give similar values.

131

ex Fig. 4.12 Plot of the excess adsorption on the surface θ / (Φ p < r >b σ ) as a function of attraction strength on the surface, aw , for different bulk polymer lengths. The bound peptide density in the bulk b 3 −6 and the arrow indicates the direction of decreasing r . n (bulk)σ =10 b

4. Conclusion

In this work we have generalised the PDFT to treat living random copolymer systems. This theory was then applied to the problem of amyloidal peptide adsorption onto surfaces at various degrees of adsorption attraction. We used the model of van

Gestel and de Lueew20 for the Aβ peptide, which is known to aggregate into peptide filaments, wherein the monomers consist of peptide conformers in either a β or disordered state. The former is able to form cross- β sheets with neighbouring β conformers and build up long peptide filaments.

132 Our results show that the presence of an attractive surface is able to facilitate the formation of these filaments to an exceptional degree. The reason for this is an underlying second-order adsorption transition. This transition is seen in infinitely long polymer systems under Θ conditions, as exemplified in the work by de Gennes14,33, and later shown for polydispersed polymers11,34,35. We have shown that this transition is also exhibited in the generalized amyloid model of van Gestel and de Lueew, in the case of ideal peptides. When we allowed for steric interactions between the peptides, this adsorption transition was suppressed, due to the exclusion effect of peptides already strongly adsorbed onto the surface. However, the peptide adsorption clearly showed the echo of the adsorption transition. There was an order of magnitude increase in the adsorption at a critical adsorption strength, which decreased for larger average filament length. We emphasise, the increase in the adsorbed peptide density corresponds to a significant increase in filament lengths adjacent to the surfaces. This work gives a mechanistic explanation to the experimental observations that surfaces are able to enhance the formation of amyloidal fibrils22. Furthermore, the PDFT theory developed here can be further applied to many other systems, which display reversible bonding of co-monomers.

133 References

(1) Tuinier, R.; Rieger, J.; de Kruif, C. G. Adv. Colloid Interface Sci. 2003, 103, 1-31.

(2) Poon, W. C. K. J. Phys.: Condens. Matter 2002, 14, R859-R880.

(3) Woodward, C. E. J. Chem. Phys. 1991, 94, 3183-3191.

(4) Woodward, C. E.; Yethiraj, A. J. Chem. Phys. 1994, 100, 3181-3186.

(5) Yethiraj, A. J. Chem. Phys. 1998, 109, 3269-3275.

(6) Forsman, J.; Woodward, C. E.; Freasier, B. C. J. Chem. Phys. 2002, 117, 1915-1926.

(7) Yu, Y.-X.; Wu, J. J. Chem. Phys. 2002, 117, 2368-2376.

(8) Tuinier, R.; Petukhov, A. V. Macromol. Theory Simul. 2002, 11, 975-984.

(9) van der Gucht, J.; Besseling, N. A. M. Phys. Rev. E: Stat., Nonlinear, Soft Matter Phys. 2002, 65, 051801/1-051801/10.

(10) Van der Gucht, J.; Besseling, N. A. M.; Fleer, G. J. J. Chem. Phys. 2003, 119, 8175-8188.

(11) van der Gucht, J.; Besseling, N. A. M.; Fleer, G. J. Macromolecules 2004, 37, 3026-3036.

(12) Scheutjens, J. M. H. M.; Fleer, G. J. J. Phys. Chem. 1979, 83, 1619-1635.

(13) Edwards, S. F. Proc. Phys. Soc., London 1965, 85, 613-624.

(14) De Gennes, P.-G. Rep. Prog. Phys. 1969, 32, 187-205.

(15) Sides, S. W.; Fredrickson, G. H. J. Chem. Phys. 2004, 121, 4974-4986.

(16) Matsen, M. W. Eur. Phys. J. E 2007, 21, 199-207.

(17) Cooke, D. M.; Shi, A.-C. Macromolecules 2006, 39, 6661-6671.

(18) Yang, S.; Tan, H.; Yan, D.; Nies, E.; Shi, A.-C. Phys. Rev. E: Stat., Nonlinear, Soft Matter Phys. 2007, 75, 061803/1-061803/7.

(19) Woodward, C. E.; Forsman, J. Phys. Rev. Lett. 2008, 100, 098301/1-098301/4. 134 (20) van Gestel, J.; de Leeuw, S. W. Biophys. J. 2006, 90, 3134-3145.

(21) Zimm, B. H.; Bragg, J. K. J. Chem. Phys. 1959, 31, 526-35.

(22) Vacha, R.; Linse, S.; Lund, M. J. Am. Chem. Soc. 2014, 136, 11776-11782.

(23) Forsman, J.; Woodward, C. E.; Freasier, B. C. J. Chem. Phys. 2003, 118, 7672- 7681.

(24) Forsman, J.; Woodward, C. E. Macromolecules 2006, 39, 1261-1268.

(25) van Gestel, J.; van der Schoot, P.; Michels, M. A. J. J. Phys. Chem. B 2001, 105, 10691-10699.

(26) Wertheim, M. S. J. Chem. Phys. 1987, 87, 7323-7331.

(27) Chiew, Y. C. Mol. Phys. 1990, 70, 129-143.

(28) Honnell, K. G.; Hall, C. K. J. Chem. Phys. 1991, 95, 4481-4501.

(29) Song, Y.; Lambert, S. M.; Prausnitz, J. M. Macromolecules 1994, 27, 441-448.

(30) Hooper, J. B.; Pileggi, M. T.; McCoy, J. D.; Curro, J. G.; Weinhold, J. D. J. Chem. Phys. 2000, 112, 3094-3103.

(31) Forsman, J.; Woodward, C. E. J. Chem. Phys. 2004, 120, 506-510.

(32) Turesson, M.; Forsman, J.; Aakesson, T. Phys. Rev. E: Stat., Nonlinear, Soft Matter Phys. 2007, 76, 021801/1-021801/15.

(33) De Gennes, P.-G. Scaling Concepts in Polymer Physics; Cornell University Press: Ithaca, N.Y., 1979.

(34) Sear, R. P. J. Chem. Phys. 1999, 111, 2255-2258.

(35) Woodward, C. E.; Forsman, J. Phys. Chem. Chem. Phys. 2011, 13, 5764-5770.

135 Chapter 5

Many-body Interaction between Charged

Particles in a Polymer Solution: the

Protein Regime

1. Introduction

Particle/polymer mixtures are commonplace in our daily lives. Understanding the properties of these systems is of great importance for many modern applications and technologies. This includes the food industry1-3, cosmetics4,5 and pharmaceuticals6-8.

The addition of polymers to particle dispersions can be used in conjunction with other components in the dispersing medium to influence the thermodynamic state of the mixture in a way that achieves specific behaviours. As described in Chapter 2, the incorporation of electrostatic repulsions between the particles in the mixture has been useful in allowing researchers to fine-tune the system via long-ranged (electrostatic) repulsion in conjunction with short-ranged (depletion) attraction9-12. The work in this chapter describes a new theory for charged particles in the presence of added polymer, using the ideas developed in Chapter 3.

136 The theoretical treatment of particles interacting via electrostatic forces in the presence of electrolyte has generally made use of the so-called DLVO theory developed by Derjaguin and Laudau13 as well as Verwey and Overbeek14. This theory, traditionally, married attractive van der Waals interaction with screened electrostatic repulsions between particles. The van der Waals interaction is due to correlated fluctuations within and between the degrees of freedom contained in the atoms and molecules that make up the particles, which are moderated by similar fluctuations in the solvent. The combination of these fluctuations generally gives rise to attractions between the particles15. These fluctuations span a range of frequency domains. At the lower frequency end they correspond to atomic and molecular displacements, which can be treated classically, giving rise to the so-called “zero frequency” van der Waals interactions. High frequency correlations due to electronic and nuclear motions give rise to attractive mechanisms that are quantum in nature. Work by Ninham and others16-

19 have shown that the zero frequency van der Waals interaction is intrinsically linked with the (classical) statistical mechanical theory used to treat the electrostatic part of the problem, while the higher frequency (quantum) contributions, remains a separate and additive component. This is because the slow moving classical modes occur on a different time-scale to that of electronic degrees of freedom.

In this work, we shall treat the non-polymer part of the particle interactions, using the classic DLVO theory. This approach is somewhat inconsistent from the point of view of the discussion above, as image interactions due to the difference in dielectric constant between particles and the water solvent are usually ignored. Generally, we expect that (non-metallic) particles will have an interior dielectric constant, which is much lower than that of water. In the model used here, the repulsive electrostatic

137 interaction between the particles will be screened by the electrolyte solution in which they are immersed. The repulsion is usually approximated as a screened Coulomb or

Yukawa interaction14. On the other hand, image effects would tend to push ions away from the particle surfaces, thus increasing the repulsion between the particles beyond the Yukawa interaction. This repulsive image effect counters the zero frequency van der Waals contribution, so that the latter becomes effectively screened by the electrolyte.

Indeed the zero frequency van der Waals should have an exponential decay under these conditions, rather than the 1/r6 tail typical of usual dispersion interactions19.

Notwithstanding this, we shall leave these concerns aside in the present study; indeed we shall ignore the dispersion term completely and assume that the usual DLVO approach is appropriate to deal with the electrostatic component alone. As the polymer molecules in our study are assumed to be neutral, they do not confound the electrostatic treatment. Instead, the interactions contributed by the polymers can be simply added to the electrostatic part. As in Chapter 3 we will consider largely non-adsorbing polymer molecules, which induce an attractive force between the particles due to depletion. We will also consider the polymer solution to be under Θ conditions, so that the range of

the depletion interaction is of the order of the polymer radius of gyration Rg.

When non-adsorbing polymer is introduced into a particle dispersion, a

depletion attraction is established between them. It is useful to define the ratio, q = Rg /

RS where RS is the radius of the particles. In the colloid limit, the particle radius, RS, is much larger than the polymer radius of gyration Rg, so q is small. Here, the close approach of particles, and expulsion of polymers between them is generally accomplished by pair collisions between particles. Thus, the polymer-mediated potential of mean force (POMF), described in Chapter 3, can be truncated at the 2-body

138 level. On the other hand, in the protein limit, Rg is large compared to RS (large q) and multi-body collisions occur between the particles and polymers. Under these conditions, the 2-body truncation is insufficient and one needs to use a full many-body POMF instead. Using a similar mathematical development to that presented in Chapter 3 (for two particles) it is possible to derive a many-body POMF, using a few mild and plausible approximations20,21. This derivation is carried out in the following section.

2. Theory

2.1 Polymer mediated depletion interactions between colloids: The many-body Hamiltonian

We consider a number, N, of spherical particles in a volume V, which thus defines the particle density in this dispersion, N/V. The particles have the same radius,

RS, and are at fixed positions, {Ri ; i = 1, N}. They are also charged, but that will not concern us for the moment as we will only consider the depletion forces mediated by uncharged polymers here. In other words, the depletion forces are unaffected by the electrostatics. The dispersion is in chemical equilibrium with a bulk reservoir that contains electrolyte and a polydisperse polymer fluid with bulk polymer concentration

Φp. Everything is immersed in a water solvent, which is treated implicitly (see Fig. 5.1).

139

Fig. 5.1 Schematic diagram depicting the semi-grand model. There are exchanges of salt and polymers with the bulk but not the particles. The particle-particle interactions are the sum of depletion and screened Coulomb interactions.

The solution is under Θ conditions and we assume the polymer can be modelled as a flexible continuous chain. We note here that a discrete version of this chain, consisting of ideal monomers, linked by freely rotating bonds of fixed length, σ, will be used in Monte Carlo simulations (as described below). As was also done in Chapter 3 we shall assume that the polymer is polydisperse with a chain length distribution described by the Shultz-Flory (S-F) distribution,

n+1 n (n) κ s F (κ, s) = exp(−κs / s ) 5.1 Γ(n +1) s n+1

We recall that s is the polymer length, Γ(x) is the gamma function and κ and n are variables that determine the shape of the distribution. As κ and n increase, the

140 distribution becomes more monodisperse. When κ = n +1 then s is the average length of the polymers, but generally these shape variables can be treated as independent.

Edwards’ diffusion equation formulation is used to describe the configurations of polymers (treated as continuous chains) in the presence of the particles. The configurations of a polymer segment with length s are determined by the end-end distribution function G(r, r ', s) , which gives the statistical weight of the segment with ends at r and r' . This distribution satisfies the following equation,

2 ∂G(r, r ', s) σ 2 = ∇ G(r, r ', s)−ψ(r)G(r, r ', s) 5.2 ∂s 6 with the initial boundary condition,

G(r, r ', 0) = δ(r − r ') 5.3

Here σ is the Kuhn length and ψ(r) is the potential which acts on the monomers. This potential is a sum of two terms. One comes from any external sources (here due to the fixed colloidal spheres) and the other comes from monomer-monomer interactions. As we assume Θ conditions, the second contribution is identically zero. Thus the polymer is only subject to the excluding forces that prevent them from penetrating the fixed particles. This potential is effectively a sum of spherically symmetric terms centered on each particle.

We firstly consider the case of equilibrium polymers (n = 0 in the S-F distribution). The end-end distribution of a segment of the polymer chain of length s, averaged over the S-F distribution, is denoted as Gˆ and is given by the following

Laplace transform of G(r, r ', s),

141 ∞ κ Gˆ(r, r ') = ds exp(−κs / s )G(r, r ', s) 5.4 ∫ 0 s

Integrating one polymer end over the space external to the spheres (denoted as V ' ) gives the following average end distribution,

gˆ(r) = dr 'Gˆ(r, r ') 5.5 ∫V ' where gˆ(r) is normalized so that it is equal to unity in the bulk and satisfies the following equation,

∇2gˆ(r)− λ 2gˆ(r)+ λ 2 = 0 5.6

2 2 with λ = κ / Rg . The free energy change associated with the insertion of the N colloid particles within the volume V is given by,

βΔω = −Φ {gˆ(r)−1} 5.7 tot p ∫V

where Φp is the average polymer density in the reservoir and β =1/ kBT is the reverse thermal energy. The effect of the excluding spheres, can be mimicked as a boundary condition at their surfaces. Though we assume that the spheres are non-adsorbing, it is useful for the moment to consider adsorbing spheres. We let the quantity ε define the strength of the adsorption potential. Thus, the boundary conditions at the surfaces of the fixed particles is given by,

1 ∇gˆ(r)⋅ nˆ = −ε ∀r ∈ S 5.8 gˆ where S denotes all the surfaces of the fixed colloids. For the case of non-adsorbing particles, we have ε → −∞ which follows from the assumption that monomers are

142 completely repelled from depleting surfaces and that this can be described by the

Dirichlet condition,

gˆ(r) = 0 ∀r ∈ S 5.9

It is worth spending a little time discussing these boundary conditions. The contact condition, Eq. (5.9), for continuous chains at the particle surfaces does not give equivalent results with those of the discrete non-adsorbing chains that we will use in our simulations. The discrete model shows a finite contact density of monomers at the particle surfaces, while the continuous chain boundary condition assumes this to be zero.

Ultimately, we will make comparisons between explicit simulations on discrete chains, with the analytic results of the continuous chain. So in order to ameliorate this discrepancy in the contact boundary conditions we will use the more general result of

“adsorbing” chains in the continuous chain model and fit the value of ε to give the best agreement with the discrete model. This procedure will be described in more detail below.

In addition, to the contact condition, we have the usual bulk boundary value,

gˆ(r) →1 far from the spheres. Using the Green’s function for the associated Helmholtz equation in free space22,

2 2 ∇rG0 ( r − r ' )− λ G0 ( r − r ' ) = −δ( r − r ' ) 5.10

exp(−λr) G (r) = 5.11 0 4πr the solution to Eq. (5.6) can be formally expressed as,

143 N

gˆ(r) =1+ dsˆG0 ( r − s )Λi (sˆ) 5.12 ∑!∫ Si i which is the appropriate generalization of Eq. (3.16). Here the integrations are over the

surfaces (Si) of the particles and we sum over all of them. The function Λi (sˆ) represents the multipole source terms at the surface of particle i and is determined by the boundary condition in Eq. (5.8). As in Chapter 3, we expand gˆ(r) in spherical harmonics about an arbitrarily chosen spherical particle,

lmax l ˆ ˆl l ˆ g(r) ≈ ∑∑ gm (r)Ym (r) 5.13 l=0 m=−l where rˆ denotes the angular coordinates (θ,φ) of the vector r with respect to the frame at the sphere center. A similar expression can be used for the surface multipoles,

lmax l l ˆ Λi (s) ≈ ∑ΛmYm (s) 5.14 l=0 where sˆ is a vector from the center to the surface of the chosen sphere. Unlike what was done in chapter 3, we cannot use cylindrical symmetry to simplify the expressions here. Using the 1- and 2- center expansions of the Green’s function, we obtain the following expression for the expansion coefficients for the function gˆ(r) , about the chosen sphere (labeled i),

l' l l gˆ (r ) 4 (i)k ( r ) i ( r ) G ' ' ( R ) ' ' ( j) m i = πδ0 + Γlm l λ i + l λ i ∑∑ ∑ lm,l m λ ij Γl m 5.15 j≠i l' m'=−l'

where the sum j is over all other particles and Rij is the distance between spheres i and j.

The renormalized multipole field strengths are defined as,

144 l Γlm (X) = λΛm (X)il (λRS ) 5.16 and we have

G (λR) = k (λR)i Q C(l l l;000)C(l l l;m m m) l1m1,l2m2 ∑ l l1 l1l2l 1 2 1 2 1 2 5.17 l,m

Note that the expressions have become more complex, compared to those in Chapter 3, due to the apparent lack of symmetry in the many sphere problem. Using Eq. (5.15) and logarithmic boundary conditions at each surface,

εgˆ + ∇gˆ(r)⋅ nˆ = 0 ∀r ∈ S 5.18

Now we are able to solve for the surface multipoles Γlm (X) using,

l' l (i) ( R ) ( R ) G ' ' ( R ) ' ' ( j) 4 5.19 Γlm K l λ S + J l λ S ∑∑ ∑ lm,l m λ ij Γl m = −ε πδ0 j≠i l' m'=−l' where,

' K l (x) = εkl (x)+ λkl (x) 5.20 and

' I l (x) = εil (x)+ λil (x) 5.21

Eq. (5.19) allows us, in principle, to solve for the induced surface multipoles on all particles. Such a solution would appear to be unfeasible, however, due to the summations over the basis set of spherical harmonics and the number of particles,

2 giving rise to a matrix of order Nlmax .

145 Using the second order differential equation for gˆ(r) , Eq. (5.6), we can rewrite the free energy in Eq. (5.7) as,

N Φ p 2 3 βω = − dr∇ gˆ(r)+ Φ 4π R / 3 5.22 tot 2 ∫V ' p ∑ S λ i=1

Gauss’s Theorem allows us to re-express the volume integrals in the above equation in terms of surface integrals of ∇gˆ(r) at each sphere,

N 2 2 dr∇ gˆ(r) = − R dσ ⋅ ∇gˆ 5.23 ∫V ' ∑ i ∫ S i i=1

where Si represents the surface of sphere i and σ is the unit surface vector, which points outward from the spherical center. From the boundary condition, Eq. (5.18) we finally obtain,

2 0 dσ ⋅ ∇gˆ(r) = − 4π R εgˆ (R ) 5.24 ∫ S i 0 i i where the spherically symmetric component of gˆ(r) has been selected, by the integration over the spherical surface. The free energy can thus be expressed as

N N 4πε 2 0 3 R gˆ (R ) 4 R / 3 5.25 βΔωtot = −Φ p 2 ∑ S 0 S + Φ p ∑ π S λ i=1 i=1

Using Eq. (5.15) we obtain for the component of the gradient perpendicular to the surface of the chosen sphere,

l l' ∂gˆ0 (ri ) ' ' (i) k ( r ) i ( r ) G ' ' ( R ) ' ' ( j) 5.26 = Γl λ l λ i + λ l λ i ∑∑ ∑ l0,l m λ ij Γl m ∂r j≠i l' m'=−l'

146 where we use the following notation, Γl (i) = Γl0 (i) . Combining the above equation and

Eq. (5.15) using the boundary condition Eq. (5.18), we obtain,

l' l ( R ) (i) ( R ) G ' ' ( R ) ' ' ( j) 4 K l λ S Γl + J l λ S ∑∑∑ l0,l m λ ij Γl m = −ε πδ0 5.27 j≠i l' m'=l'

Using the following relationship, derived from the properties of the modified spherical

Bessel functions,

λ i (x)K (x) = k (x)I (x)− 5.28 l l l l x2

we multiply Eq. (5.27) by il (λRS ) to obtain

l Γl (i) −il (λRS )ε 4πδ0 = J l (λRS )kl (λRS )Γl (i)− 2 λRS 5.29 l'

( R ) G ' ' ( R ) ' ' ( j) +J l λ S ∑∑∑ l0,l m λ ij Γl m j≠i l' m'=l' which we re-express as,

l l Γ (i) l (λR ) g (R )− 4πδ − l = −i (λR )ε 4πδ J l S { 0 i 0 } 2 l S 0 5.30 λRS and finally rearrange to obtain,

' l l λi (λR ) Γ (i) l S l g0 (Ri ) = 4πδ0 + 2 5.31 J l (λRS ) J l (λRS )(λRS )

where Ri = RS . Substituting back into Eq. (5.25), we obtain the following general expression for the free energy,

147 N * 2 * 3 ( βΔω 3/2 %ε σ i (σ ) ε Γ (i) σ tot − i 1 0 * = −κ ∑& + − ) 5.32 4πΦ p i=1 ' I 0 (σ ) 4πI 0 (σ i ) 3 *

' * 3 where we have used the fact that i0 (x) = i1(x) and defined σ = λRS , Φ p = Φ pRg and

* ε = εRS .

This expression for the POMF is the appropriate generalization of Eq. (3.48) for the many-particle problem and provides us with some physical insights. Firstly we note that, while only a single sum over the particles appears in the expression, its many-body

nature is implicit in the monopole terms Γ0 (i) . That is, these terms are dependent on the positions of all particles. For a regular arrangement of particles (e.g., a crystal) the surface multipoles will all be either equivalent, or else classifiable according to a finite

(and generally small) number of equivalent sites, i.e., those that have the same point symmetry. Thus, solving Eq. (5.27) would be significantly simplified, with the matrix

2 problem of order lmax .

However, we are not dealing with crystals, and furthermore, in the protein limit, where the polymer becomes long in comparison to the particle radius another simplification ensues. That is, the influence of surrounding particles, as measured over the surface of an arbitrarily chosen particle, becomes more spherically symmetric, due to the diminishing importance of gradient terms. We shall call this the spherical approximation (the same approximation described in Chapter 3). If we assume that we need only consider spherical terms (l = 0) in Eq. (5.27), the spherical approximation allows us to evaluate the monopole terms at each sphere i, via the following linear equation,

148

Γ0 (i)K 0 (σ )+I 0 (σ )∑k0 (λRij )Γ0 ( j) = −ε 4π 5.33 j≠i

−x where k0 (x) = e / x is the zeroth order modified spherical Bessel function of the second kind. The above linear and self-consistent equation shows that the polymer- mediated potential of mean force can be described by monopoles induced on the particles. This could be feasibly done in a simulation, by solving the matrix equation

“on the fly” in a manner akin to treating polarizable force-fields. However, this problem is much easier, as we will only be dealing with induced monopoles, which have spherically symmetric interactions.

So it seems that the spherical approximation may apply reasonably well in the protein regime, the system that has traditionally been difficult to treat by the effective potential approach. Indeed, in Chapter 3, we investigated the validity of this spherical approximation for the POMF for 2-particles and found it to be a very good approximation for q > 10. But can we do better than this? We suggest that we can.

Typically, liquid-like particle configurations give rise to approximately spherically symmetric local environments irrespective of particle size. This means that induced surface multipoles, higher than monopoles are likely to be small (on average). Thus we anticipate that it will remain a good approximation in more dense fluids.

While the spherical approximation affords an incredible simplification of the effective potential problem, we still have the added numerical expense of dealing with calculating configuration dependent induced monopole terms on the particles. Can this procedure also be simplified? We suggest that it can if the particle density satisfies

3 N /VRS >1/ q , which is likely in the protein limit. In this case, the fluid will be dominated by configurations wherein the local environment around each sphere varies 149 slowly. On the other hand, if the particles are highly diluted, the solution is dominated by isolated particles or small dimers and the induced monopoles will be more or less similar on the spheres. Thus in both concentrated and dilute regimes, one can make the

so-called local approximation and set Γ0 ( j) ≈ Γ0 (i) in Eq. (5.33). This allows us to obtain a closed expression for the multipoles, and the following compact expression for the polymer mediated many-body POMF ensues,

*2 N ε ∑ k0 (λRij ) (0) * −3/2 j≠i βΔω (κ) = −4πΦ pκ ∑ 5.34 i 1 K 0 (σ )+K 0 (σ )I 0 (σ ) k0 (λRij ) = ∑ j≠i

Note that here we have subtracted the (uninteresting) 1-particle contribution. This is the major theoretical result of this chapter.

One of the most important features of Eq. (5.34) is that it is in some sense “pair- wise additive”. That is one only requires summations over particle pairs in order to evaluate the free energy for any particular particle configuration (we show this explicitly below). Thus, implementing this POMF in a simulation is as straightforward

(and efficient) as for pair-additive potentials.

We also note that the function Δω (0) (κ) is a generating function for the potentials of mean force for higher-order S-F distributions (κ >1), which can be obtained from,

n m (n) (−κ) ∂ (0) ( ) ( ) 5.35 Δω κ = ∑ m Δω κ m=0 m! ∂κ

Thus, using Eq. (5.35) we are able to generate the POMF for progressively more monodisperse polymer, though the monodisperse limit is clearly unobtainable via this method. This will generally not be a problem when describing experimental systems, which will usually involve dealing with polydisperse polymer. However, as stated 150 earlier, we will make comparisons with simulations using an explicit discrete polymer model, which is monodisperse. So, for this reason, obtaining a many-body POMF for monodisperse polymers would be desirable. Fortunately, previous work21 suggests, that for polymers in Θ solvents and in a depleting environment, surface forces do not appear to be particularly sensitive to the degree of polydispersity, beyond the first-order S-F distribution (κ = 2 ) especially if the average radius of gyration is large, which is consistent with the protein regime. Hence, in this work, we will use the POMF derived for the caseκ = 2 , which can be obtained via the following expression,

(0) (1) (0) ∂Δω Δω (2) = Δω (2)− 2 (2) 5.36 ∂κ

We note that higher-order expressions could also be generated in a similar fashion, but these become rapidly more complex in form. Remarkably, however, the “pair-wise” additivity of the resulting many-body POMF remains.

2.2 Electrostatic interaction between particles: the Poisson –

Boltzmann equation

The particles in our mixture also carry a surface charge. This charge may originate from attached ionic groups or else adsorbed ions provided from the underlying solution in which they are dispersed. Apart from polymer, the solution consists of a water solvent as well as dissolved electrolyte species. Although the structure of neutral polymers may in reality be affected by electrolyte through dispersion and induced interactions, for this preliminary model, we simply assume no interaction between polymer and electrolyte. On the other hand the electrolyte accumulates at the particle surfaces to form the so-called double layer of charge. Two adjacent particles will have overlapping double layers, which leads to an unbalanced repulsive pressure between 151 them. The classic approach to treating the double layer is the Poisson-Boltzmann (PB) theory23.

The PB is theory is mean field in the sense that every ion is assumed to respond to the average potential determined by the particles and the other ions. One begins with the Poisson equation,

2 ρ ∇ ψ = − e 5.37 εrε0

where ψ is the mean electrostatic potential and ρe is the average local charge density, εr

is the dielectric constant of solvent and ε0 is the permittivity of the vacuum. The main approximation in the PB theory is summarized by the Boltzmann equation,

0 −z jeψ cj = cj ⋅ exp( ) 5.38 kBT

0 where cj is the concentration of ion j in the reservoir and z j is its valence. For the

case of a 1:1 salt, e.g., sodium chloride, with equal reservoir concentration c0, of anions and cations, the average charge density is given by,

5.39 ρe = c0e⋅[exp(−eψ / kBT)− exp(eψ / kBT)]

Substituting the above equation to Eq. (5.37) gives,

2 c0e⋅[exp(−eψ / kBT)− exp(eψ / kBT)] ∇ ψ = − 5.40 εrε0

Thus the Poisson–Boltzmann equation is a non-linear partial differential equation which must be solved in general for every configuration of the fixed particles. In principle, this would lead to a many-body electrostatic POMF, analogous to the polymer-mediated

152 interactions. However, if one makes the assumption that the electrolyte concentration is high enough, the electrostatic screening length is small, then a kind of colloid limit can be used on the electrostatic interactions to justify truncation at the pair level.

A full solution to the PB equation can only be obtained numerically. However, it is common to assume that the PB equation can be linearized, i.e., the potential ψ is

everywhere sufficiently small to give, exp(−eψ / kBT) ≈ 1− eψ / kBT , which leads to the so-called Debye–Hückel approximation (DHA). Even if this is not the case close to the particle surfaces, it will be true asymptotically, which leads to a linearized treatment but with effective particle charges. Using the DHA and assuming that the asymptotic result applies for the most relevant part of the particles’ configuration space, one obtains the following 2-body electrostatic POMF,

2 lBZ eff exp(−κ D (r − 2RS )) U (r,c ) = r > 2R 5.41 r s 2 r S (1+κ D RS )

14 2 This is the well-known Yukawa potential where lB = e / 4πεrε0kBT is the Bjerrum

length in water, with e the elementary charge, ε0 and εr the vacuum permittivity and

relative permittivity of water. The quantity κ D = 8πcs N AlB is the inverse Debye

length with cs the salt concentration and NA Avogadro’s number. Zeff is the effective charge on the particle surfaces and r is the center-to-center distance between particles. It can be seen that an increase in the salt concentration will decrease the Debye length.

153 3. Monte Carlo Simulations

In order to assess the accuracy of our polymer POMF, we will compare its predictions with those of a model that contains explicit polymer molecules. While we carried out Monte Carlo simulations using the many-body POMF for the polymer- mediated interactions, parallel simulations of the explicit discrete polymers in the presence of hard sphere Yukawa particles were carried out by Professor Jan Forsman at

Lund University. The explicit model chains are monodisperse and (as described earlier) are ideal monomers linked by freely rotating bonds of fixed length.

3.1 Matching implicit and explicit polymer models

The many-body POMF described above is formally developed in a semi-Grand

Ensemble (see Fig. 5.1). Here the polymer chemical potential is fixed and the mixture containing particles exchanges polymers with a bulk reservoir (also containing electrolyte). However, the comparisons will be made via Monte Carlo (MC) simulations. In both cases, the simulations will be carried out in the canonical ensemble.

That is, in the explicit polymer simulations, we will choose a priori the number of particles and polymer molecules to be used in the simulation. This leads to a problem of matching of the appropriate polymer chemical potential, or equivalently, reservoir

polymer concentration, Φ p , for use in the POMF. One solution would be to simulate the explicit model in the semi-grand ensemble, but this would be rather time consuming.

Instead, we chose to estimate the reservoir polymer density by using the polymer

154 density functional theory (PDFT), which is exact in its treatment of discrete polymer chains under Θ conditions. i) Determining the polymer reservoir concentration

We used two different methods for determining the reservoir concentration, depending upon the overall particle concentration in the mixture. In the case of a dilute particle dispersion, the distances between particles are large therefore the reservoir concentration could be estimated from a simple model, wherein a particle is placed at the centre of a spherical cell with radius appropriate to the particle concentration. The

PDFT was then solved in this spherical geometry. The bulk polymer density was then adjusted in this calculation, until the polymer density in the cell was equal to that in the explicit polymer simulations.

In the case of more concentrated particles, the spherical cell model was replaced with a slightly more complex version, which accounts for the asymmetric environment of the particle dispersion. In this case we attempt to mimic the polymer distribution in a cubic array of particles at the appropriate particle density (see Fig. 5.2). In order to do this we:

1. Place two spherical particles in a bulk polymer solution under Θ conditions, at a

separation l, such that l3 = V / N .

2. A square pyramid is formed with its base (dimensions l ×l ) placed half-way

between the two spheres, such that the line joining their centres is perpendicular

to the base and passes through its centre. The vertex of the square pyramid

coincides with one of the sphere centres. This pyramid serves as our new cell,

within which we obtain an average polymer density, see Fig. (5.2)

155 3. The PDFT was solved in the region surrounding the two particles in cylindrical

coordinates. We note that it is only the spheres that exclude polymer, not the

pyramid surface. The bulk polymer concentration was varied until the average

polymer density in the pyramidal cell matched that of the explicit simulations.

Note that changing the polymer density in the explicit simulations only requires

a trivial rescaling with respect to Φp, whereas changing the particle density in

the simulations requires a new cell calculation.

Fig. 5.2 Schematic diagram depicting the construction of the pyramid cell model. The polymer concentration in the shaded pyramid is calculated using PDFT and matched with the explicit model simulations.

156 We also need to determine the adsorption strength ε*. As we described above, this parameter is used to match surface boundary conditions for our continuous chain model

(used to determine the many-body POMF) and the explicit chain model. ε* measures the depletion region of the polymer proximal to the surfaces at a scale where discreteness of the chain will play a role. This region will, in turn, affect the polymer- mediated interactions between spheres. ii) Determining the value of ε*

To determine ε*, we first calculate the exact interaction between two particles in a fluid of explicit chains with a given bulk concentration. This is obtained by solving the PDFT in cylindrical coordinates (see the cell model described in the last section).

The PDFT calculation is fairly rapidly performed and is exact for ideal chains. If we subtract the 1-body terms, this interaction should in principle be equivalent to the 2- body component of the many-body POMF, derived above for the continuous chain model. That is, this would be the POMF used in, say, the colloidal limit. In the continuous model, the 2-body term is of course analytic and depends upon the adsorption strength, ε*. In order to account for differences in the surface boundary conditions between discrete and continuous models, we vary ε* and use it as an effective fitting parameter to obtain the best fit between the two forms (discrete and continuous) of the 2-body potentials. In general, we found almost perfect agreement is achievable

(see Fig 5.3 as an example), validating the functional form of our analytic potential of mean force (at least at the two particle level). The value of the bulk polymer concentration is not shown in Fig 5.3 since the pair interaction scales linearly with the

* * reservoir density, i.e., ε is independent of Φp. Thus, once ε is determined, it remains fixed for all particle and polymer concentrations investigated for a given model.

157

Fig. 5.3 Fitted ε* to obtain the radial distribution function (rdf) between 2 spheres in the explicit discrete polymer model and continuous chain (implicit) polymer model. The 2-body POMF is given by –ln (rdf).

3.2 Pair-wise Additivity of the POMF

As stated above, using the many-body POMF in the simulations was essentially as efficient as using a pair-wise additive potential. This can be illustrated as follows.

From Eq. (5.34), it is easy to see that the many-body potential has the generic form,

N (0) Δω = ∑Φ[G(i)] 5.42 i=1 with the function G(i) given by the summation over pairs of particles of the form,

G(i) k R R = ∑ 0 ( i − j ) 5.43 j≠i

' If particle k moves from from Rk to Rk , the new value for the energy is obtained as

158 (0)' ' ' Δω = Φ(G (k)) +∑Φ(G (i)) 5.44 i≠k where

G' (k) k R' R = ∑ 0 ( k − j ) 5.45 j≠k and

G' (i) k R R k R R' = ∑ 0 ( i − j ) + 0 ( i − k ) 5.46 j≠i≠k

Hence, during the simulations, one only needs to update the generic functions G(i), which are just sums of pair terms. Using the generating formula Eq. (5.36) to obtain

Δω (1) (κ) will introduce an additional function with the same generic form as G(i) (this arises from the derivative with respect to κ ) hence the pair-wise additivity will be retained. The same will be true for higher-order POMF obtained via the generating formula.

4. Results and Discussion

4.1 Comparison with simulations

All simulations were carried out in the Canonical Ensemble with the side length

of the cubic box equal to 100σ. We used Rg = RS = 5σ (q = 1), which is in the intermediate regime between the protein and colloidal limits. The number of particles

* was chosen to be N = 20 and the polymer concentration in the reservoir was Φp = 0.226, which is in the semi-dilute region. According to the matching criteria described above,

159 this meant that 3360 150-mers were required in the explicit simulations. The matching of the adsorption energy gave ε* = -9. The effective charge on the spheres was chosen

as Zeff = 10 and the salt concentration was varied from 10 mM to 640 mM.

The particle-particle radial distribution functions (rdf) for various salt concentrations are shown in Figs. 5.4 - 5.7. The rdf is a good indicator of the accuracy of our implicit model, since it is sensitive to both the strength and the range of POMF.

When comparing the effort required to obtain these results between the explicit and the

POMF simulations, there was a qualitative difference. For the explicit simulations we had several hundred thousand additional monomers to account for, together with the rather slowly converging moves required for polymer simulations. Furthermore the high osmotic compressibility of the Θ solvent meant that there were enormous polymer density fluctuations in the system. This means that the explicit simulations generally took weeks to converge using a cluster with several hundred cores. On the other hand, the implicit model simulations, using the many-body POMF, required one or two minutes on a standard laptop. Despite the enormous difference in numerical effort required, it can be seen that the many-body model POMF gives quantitative agreement with the explicit polymer model at all the salt concentrations we investigated. The amount of structure in the rdf increases with the salt concentration, which reflects the reduced screening length of the electrostatic potential and the increased importance of the depletion interaction. At the lowest salt concentration, the electrostatic repulsion dominates, and the rdf at contact is lower than unity. At higher salt concentrations, the depletion interaction increases and the rdf displays a sharper peak at contact.

Interestingly, there is a slight depression at contact in the explicit model compared to the POMF results that occurs at intermediate salt. There is good evidence to suggest

160 that this is due to the slow convergence of the explicit model, rather than inaccuracy in the POMF. However, further simulations will need to be carried out in order to confirm this.

Fig. 5.4 Comparison between the radial distribution functions generated using many-body POMF and the explicit polymer model. The simulated system is a particle/polymer mixture immersed in an * * electrolyte solution with salt concentration cs = 10 mM, L = 100σ, Rg = RS = 5σ, ε = -9, Zeff = 10, Φp = 0.226 and N = 20.

161

Fig. 5.5 Comparison between the radial distribution functions generated using many-body POMF and the explicit polymer model. The simulated system is a particle/polymer mixture immersed in an * * electrolyte solution with salt concentration cs = 40 mM, L = 100σ, Rg = RS = 5σ, ε = -9, Zeff = 10, Φp = 0.226 and N = 20.

162

Fig. 5.6 Comparison between the radial distribution functions generated using many-body POMF and the explicit polymer model. The simulated system is a particle/polymer mixture immersed in an * * electrolyte solution with salt concentration cs = 160 mM, L = 100σ, Rg = RS = 5σ, ε = -9, Zeff = 10, Φp = 0.226 and N = 20.

163

Fig. 5.7 Comparison between the radial distribution functions generated using many-body POMF and the explicit polymer model. The simulated system is a particle/polymer mixture immersed in an * * electrolyte solution with salt concentration cs = 640 mM, L = 100σ, Rg = RS = 5σ, ε = -9, Zeff = 10, Φp = 0.226 and N = 20.

4.2 Comparison with the 2-body POMF

Despite the quantitative accuracy of the many-body POMF, we can still question the importance of the many-body effects. The value q = 1, puts us in that cross-over regime between colloid and protein limits, so perhaps one could do reasonably well by truncating the POMF at the 2-body level. Recall that the 2-body contribution was used to fit the adsorption strength parameter, ε*, and essentially perfect agreement was found with the PDFT solution, which is exact. Thus we can be certain that the 2-body term in the POMF is a very accurate representation of the POMF between two particles in the discrete polymer fluid.

164 A comparison between the rdf obtained using the 2-body and many-body interactions is shown in Fig. (5.8) for the system described above at a salt concentration of 160 mM. At this intermediate salt concentration, the depletion interaction is important, but the electrostatics also have a role to play in determining the structure. It is clear from these results that truncation at the 2-body interaction considerably overestimates the depletion potential. The higher order many-body contributions provide a repulsive component to the depletion attraction. This is because when the polymers are long compared with the particles, the particles actually work cooperatively in depleting polymers in any region where they may cluster. Thus the net depletion potential between two particles will be reduced, when a third particle in the vicinity is also acting to deplete the same polymers. The same cannot be said in the colloidal limit.

Here particles are so large that when two of them approach each other, and deplete polymers in the space between them, they sterically preclude a third particle from entering the region. Clearly, in this system there is an interplay between the long- ranged repulsion of the Yukawa potential and this depletion interaction. For example, we expect that the depletion effect plays a smaller role when the repulsion is high, however, an accurate representation of the many-body effects is crucial to determine the relative importance of these potentials, especially when we are near or well within the protein regime. One area where this interplay would be felt is in the fluid-fluid phase diagram for these particle polymer mixtures.

165

Fig. 5.8 Comparison between the radial distribution functions generated using 2-body POMF and the many-body POMF. The simulated system is a particle/polymer mixture immersed in an electrolyte * * solution with salt concentration cs = 160 mM, L = 100σ, Rg = RS = 5σ, ε = -9, Zeff = 10, Φp = 0.226 and N = 20.

4.3 Phase transitions driven by depletion

As we know, adding non-adsorbing polymer to particle suspensions can lead to the destabilization and flocculation of particles. However, the presence of charges on the surfaces of particles also has a significant influence on the stability of the dispersion.

Fortini et al.24 theoretically studied a mixture of charged colloids with non-adsorbing ideal polymers using free volume theory and found that the electrostatic repulsion causes a reduction in the depletion effect and shifts the fluid-fluid and fluid-solid coexistence curves. Gögelein and Tuinier25 investigated a mixture of charged colloidal spheres with non-adsorbing interacting polymers in good and Θ-solvents and found the

166 range of repulsion has a great influence on the phase behaviour of the charged- colloid/polymer mixture.

One of the most common simulation techniques to study phase transition is the

Gibbs ensemble method, which was proposed by Panagiotopoulos26. However, this method is more suitable to gas-liquid transitions and proves to be inefficient for dense systems. The semigrand-canonical ensemble method of Kofke27 and the so-called

Gibbs-Duhem integration28 have proved to be more suitable to dense phase coexistence.

Notwithstanding this, explicit polymer simulations for the study of fluid-fluid phase coexistence presents a daunting task. This was evidenced by the enormous effort that was required to obtain converged results at a single state point! Thus POMF simulations would be ideal for the determination of phase coexistence in these systems.

We note here that the POMF was derived in a semi-grand ensemble, which means that the implicit components (polymer, solvent, salt) are all kept at a constant chemical potential. Maintenance of the overall pressure and the equivalence of the free energies at constant chemical potentials, means that we can treat the particles as a one- component system, where the phase equilibrium is determined by equivalence in the particle chemical potential and the osmotic pressure in the two phases. Thus, we have in effect a kind of gas-liquid phase coexistence which could, in principle, be studied using the Gibbs’ ensemble method of Panagiotopoulos26.

On the other hand, the Gibbs’ ensemble method still requires a fairly intensive numerical effort that would have tested the computational resources available to us.

Therefore, we chose a much simpler approach to estimate the phase boundaries in these systems, that made use of the fact that long-ranged density fluctuations are expected to occur in systems on the verge of phase transition. This is especially the case in finite-

167 sized systems, where the particles are attempting to separate into dense and dilute phases, but are frustrated by the resultant surface energy. Note that in the Gibbs’ ensemble method, the surface energy is eliminated by simulating the two phases separately. These density fluctuations become evident in the tail of the radial distribution function whereby a long-ranged decay is observed, rather than a quick settling to unity. Thus, a reasonable way of determining the phase envelope for these systems, would be to monitor the slope of the rdf at long-range. If that slope is above a certain critical value, we deem that the system is attempting to phase separate. Note that this approach is similar in spirit to what is done in experimental determinations of phase boundaries by radiation scattering. In that case, one monitors the first ( k → 0 ) peak in the scattering function, which corresponds to the long-ranged part of the rdf.

When that peak becomes sizeable, the system is assumed to be at the phase boundary.

Of course, one may encounter an ergodic problem in the simulations, whereby the system may remain in a metastable dilute or condensed phase without showing signs of separation, even though one may expect that it should. This lends a certain amount of uncertainty to the determined phase envelope. However, we believe that this should be minimal, if the spinodal line is close to the phase boundary, which is expected to be the case in particle/polymer mixtures15.

4.4 Fluid-fluid phase diagram of charged particle-polymer mixtures

Fig. (5.9) shows the method we used to determine the fluid-fluid coexistence curve. In the simulation, the number of particles is fixed while we increase the concentration of the polymers in the bulk. When the rdf shows a slowly decaying tail, we assume there is a phase transition occurring in the system. This is due to the formation of large clusters in the simulation and the overall particle density and polymer 168 concentration push the system between the phase boundary and spinodal region in the phase diagram. For example, Fig. (5.10) is a snapshot taken from a MC simulation within the 2-phase region of the phase diagram. This phenomenon has been observed in experiments as well29. We found that, in general, once the phase boundary was crossed, the absolute value of the slope in the rdf increased rapidly. Hence, the signature for phase transition was clear. This was true when we crossed either from the gas or the liquid regions. The slope was calculated using linear regression over the final 10σ of the rdf. The criterion used for a gas to liquid transition was an absolute slope of 10-3 and 2 ×

10-4 was used for the liquid to gas transition. This is because, in the gas phase, the rdf is more sensitive to the changes in the particle number. The error bar in Fig. (5.11) were determined by the simulated state points, which bounded the transition region (as determined by our criterion).

Fig. 5.9 Schematic diagram displaying the determination of the fluid-fluid coexistence curve. The slope of g(r) at large r was measured. Phase transitions from gas (G) to liquid (L) and liquid to gas are determined for a finite slope < −10−3 (for the upper case) or < −2 ×10−4 (for the lower case).

169

Fig. 5.10 A snapshot from the MC showing the clusters of particles (spheres) formed in the simulation using the first-order many-body POMF. Parameters used in the simulation are L = 100σ, Rg = RS = 5σ, * * ε = -9, N = 55, ηc = 0.0288 , Φ p =1.6 , Zeff = 0.

In Fig. (5.11), we compare the phase diagrams for charged particle/polymer mixtures at a salt concentration of 640 mM, for several values of charge on the particles.

The effect of charges is very clear and obvious, that is, increasing the charge requires a higher polymer density in order to bring about phase coexistence. At this fairly high salt concentration the charge effect is relatively minor, and we expect to see a bigger influence at lower salt concentration.

170 While more simulation work is required for a complete study of the interplay between charge, salt and polymer on the phase behaviour of this system, it seems that this method provides a viable and numerically cheap approach to determining phase coexistence. Married with the many-body POMF, we are able to quickly determine phase coexistence in these systems, as opposed to the anticipated laborious calculations that would be required for explicit polymer simulations.

Fig. 5.11 Phase diagram of the particle-polymer mixtures in a canonical ensemble with L = 100σ, Rg = * RS = 5σ, ε = -9. The reduced polymer concentration is plotted as a function of particle volume fraction ηc. Error bars are also shown. In systems with salt solutions, the salt concentration cs = 640 mM.

5. Conclusions

We have studied charged particle/polymer mixtures by developing a many-body

POMF and compared results with an explicit discrete polymer model. Our model has

171 proven to be both accurate and computationally efficient for these systems, as evidenced by the excellent agreement with radial distribution functions for a range of potential parameters. The importance of many-body interactions, beyond the 2-body interaction was also demonstrated by explicit simulations. We then conducted a number of Monte

Carlo simulations using our many-body POMF to study the phase coexistence in a

Canonical ensemble, using a new method which just monitors density fluctuations.

While more work needs to be done to assess the accuracy of this method, the results are certainly plausible and show the appropriate behaviour with respect to the effect of particle charge.

172 References

(1) Guzey, D.; McClements, D. J. Adv. Colloid Interface Sci. 2006, 128-130, 227-248.

(2) Rhim, J.-W.; Park, H.-M.; Ha, C.-S. Prog. Polym. Sci. 2013, 38, 1629-1652.

(3) Youssef, A. M. Polym.-Plast. Technol. Eng. 2013, 52, 635-660.

(4) Raj, S.; Jose, S.; Sumod, U. S.; Sabitha, M. J. Pharm. BioAllied Sci. 2012, 4, 186- 193.

(5) Mihranyan, A.; Ferraz, N.; Stromme, M. Prog. Mater. Sci. 2012, 57, 875-910.

(6) Dallas, P.; Sharma, V. K.; Zboril, R. Adv. Colloid Interface Sci. 2011, 166, 119-135.

(7) Parveen, S.; Misra, R.; Sahoo, S. K. Nanomed. 2012, 8, 147-166.

(8) Couvreur, P. Adv. Drug Deliv. Rev. 2013, 65, 21-23.

(9) Stradner, A.; Thurston, G. M.; Schurtenberger, P. J. Phys.: Condens. Matter 2005, 17, S2805-S2816.

(10) Cardinaux, F.; Stradner, A.; Schurtenberger, P.; Sciortino, F.; Zaccarelli, E. EPL 2007, 77, 48004/1-48004/5.

(11) Atmuri, A. K.; Bhatia, S. R. Langmuir 2013, 29, 3179-3187.

(12) Denton, A. R.; Schmidt, M. J. Chem. Phys. 2005, 122, 244911/1-244911/7.

(13) Derjaguin, B. V.; Landau, L. Acta Physicochim. USSR 1941, 14, 633-662.

(14) Verwey, E. J. W.; Overbeek, J. T. G. Theory of the Stability of Lyophobic Colloids; Dover Publications: Mineola, N.Y., 1999.

(15) Israelachvili, J. N.; Academic Press: Burlington, MA, 2011.

(16) Parsegian, V. A.; Ninham, B. W. Nature 1969, 224, 1197-1198.

(17) Parsegian, V. A.; Ninham, B. W. Biophys. J. 1970, 10, 664-674.

(18) Mitchell, D. J.; Ninham, B. W. J. Chem. Phys. 1972, 56, 1117-1126.

173 (19) Mahanty, J.; Ninham, B. W. Dispersion Forces; IMA, 1976; Vol. 5.

(20) Forsman, J.; Woodward, C. E. Soft Matter 2012, 8, 2121-2130.

(21) Woodward, C. E.; Forsman, J. J. Chem. Phys. 2012, 136, 084903/1-084903/5.

(22) Arfken, G. B.; Weber, H.-J. Mathematical Methods for Physicists; Elsevier: Boston, 2005.

(23) Butt, H.-J.; Graf, K.; Kappl, M. Physics and Chemistry of Interfaces; Wiley-VCH: Weinheim, 2006.

(24) Fortini, A.; Dijkstra, M.; Tuinier, R. J. Phys.: Condens. Matter 2005, 17, 7783- 7803.

(25) Goegelein, C.; Tuinier, R. Eur. Phys. J. E 2008, 27, 171-184.

(26) Panagiotopoulos, A. Z. Mol. Phys. 1987, 61, 813-826.

(27) Kofke, D. A.; Glandt, E. D. Mol. Phys. 1988, 64, 1105-1131.

(28) Kofke, D. A. Mol. Phys. 1993, 78, 1331-1336.

(29) Bailey, A. E.; Poon, W. C. K.; Christianson, R. J.; Schofield, A. B.; Gasser, U.; Prasad, V.; Manley, S.; Segre, P. N.; Cipelletti, L.; Meyer, W. V.; Doherty, M. P.; Sankaran, S.; Jankovsky, A. L.; Shiley, W. L.; Bowen, J. P.; Eggers, J. C.; Kurta, C.; Lorik, T., Jr.; Pusey, P. N.; Weitz, D. A. Phys. Rev. Lett. 2007, 99, 205701/1- 205701/4.

174 Chapter 6

Conclusion

The addition of polymer into a dispersion of particles profoundly changes their behaviour. We have explored this phenomenon using an effective potential approach, wherein the particles respond to each other via an interaction mediated by the polymer.

This interaction is essentially the free energy of the polymer in the presence of the field of (fixed) particles; a so-called potential of mean force (POMF). The qualitative nature of this interaction is affected by whether the particles are repulsive or attractive to the polymer. We have focussed in Chapter 3 and Chapter 5 on the former case and shown the effect of so-called depletion interactions between particles. In Chapter 3 we developed the theory for depletion interactions between a pair of particles, and explored the effect of particle size. We found that if particles are relatively smaller than the polymer radius of gyration a spherical approximation becomes valid, which simplifies the mathematics enormously. In Chapter 5, we argued that the spherical approximation can be justified in more concentrated dispersions, merely because the field due to other particles give on average a spherical environment about any one particular particle.

This, together with a local approximation (which essentially states that the environment in the dispersion is similar for all particles) allows us to derive a compact expression for the total POMF. This interaction is a many-body one in the protein limit, where it is shown to be exceptionally accurate, via comparisons with simulations on particles with explicit polymer. In the course of these studies, we also investigated the case of a single surface, presenting an attractive adsorption energy to self-associating copolymers. We developed an extension to the polymer density functional theory (PDFT) and applied it

175 to a model for amyloidogenic peptides. We showed that experimental observations of enhanced fibril formation at surfaces could be explained in terms of a surface phase transition for living polymers. While this is a proper transition in Θ solvents, its echo remains in the case of good solvents, where monomers exclude one another. Indeed many of the results presented here have been derived with Θ solvents in mind. The reasons for this are two-fold. Firstly, the Θ solvent allows us to simplify the physics significantly, but we make no apologies for this. Theories for particle/polymer mixtures have, in the past, also relied upon the Θ solvent case (e.g. the Asakura-Oosawa model) but have not been able to solve the problem of the protein limit. Our work has gone a long way toward doing this. Secondly, Θ solutions are experimentally realizable systems, and thus there is a host of experimental results that await interpretation with our new theory.

This notwithstanding, one of the first areas for future work that suggests itself is the generalization of our theory to the good solvent case. Work has already begun in this area, but the analysis is much more complex. In the meantime, some qualitative predictions are possible to make. For example, a simple generalization of the many- body POMF (developed in Chapter 5) to a good solvent sees the replacement of the ideal osmotic pressure (of the reservoir), with the osmotic pressure of the polymer (in a

good solvent). Also, the depletion length scale, Rg, should be replaced by the polymer correlation length, ξ. defined by de Gennes (see Chapter 2). For a good solvent it is

well known that this correlation length becomes shorter than Rg beyond the overlap concentration1. This means that many-body effects will be less important at higher polymer concentration, particularly in dilute particle dispersions. On the other hand, depletion would play a role in enhancing many-body effects by reducing the polymer concentration in a concentrated dispersion (in the semi-grand ensemble). Other areas of

176 generalization of our many-body POMF include the consideration of different sized particles and the effect of surfaces on the depletion interaction.

i) Particles of different size. When we consider the depletion interaction in

particles of different size there are some qualitative differences. One point

emphasized in the thesis was that large particles (colloids) can be considered

as interacting via 2-body depletion forces, whereas progressing to the protein

limit means that repulsive many-body contributions also come into play.

This effect was shown in Figure (5.8). Thus colloidal sized particles

experience much greater depletion attractions than smaller nano-sized ones.

This means for a dispersion with a range of particles, we will, under some

conditions, see a fluid-fluid (or even a crystal- fluid) phase transition,

wherein the condensed phase will consist of larger sized particles. The

many-body potential (suitably generalized for differently sized particles) will

provide a means whereby we can investigate this phenomenon.

Experimentally, this could be used as an effective method for particle size

selection. We have already started preliminary work on generalizing the

many-body POMF to consider this problem.

ii) Effect of Surfaces. If the particle/polymer mixture is in contact with a

surface, the depletion force will be affected. If that surface is attractive to

the polymer, then the depletion forces between spheres will be enhanced,

due to the locally higher polymer concentration. The opposite will be true

for depleting surfaces. This introduces a position dependent many-body

POMF between colloids. In a Θ solvent, this POMF can be obtained by

solving the Edwards equation in the presence of boundary conditions

introduced by the surface as well. The general scenario leads to the

177 possibility of interesting phenomena. For example, the enhancement of

particle-particle attractions due to the presence of a surface can give rise to

anomalous surface wetting phenomena, associated with an extraordinary

surface transition2. Work has already begun on the generalization of the

many-body POMF to the presence of surfaces. Work has already started on

this problem.

Finally, we might also mention the work begun in Chapter 4. There is still much that can be investigated in this modelling, for example, the introduction of discriminating surface interactions with different peptide conformers. It is anticipated that a hydrogen bonding surface with an affinity for β conformers could well cause an exceptional increase in filament lengths for adsorbed aggregates, due to the strength of the cross-β sheet bonding. This will serve to further enhance filament formation in such cases. We should also point out that the theory developed in Chapter 4 assumed that the aggregates were flexible. While this did not affect the qualitative outcomes, we expect that real peptide aggregates are likely to possess a degree of stiffness. The PDFT would need to include the contribution due to next nearest neighbour interactions. It is also of interest to see how stiffness will influence the surface adsorption transition, and indeed filament aggregation in general.

We believe that the work presented in this thesis has thus, not only addressed some crucial problems of particle/polymer mixtures, but also seeded many new avenues for further research.

178 References

(1) De Gennes, P.-G. Scaling Concepts in Polymer Physics; Cornell University Press: Ithaca, N.Y., 1979.

(2) Forsman, J.; Woodward, C. E. Phys. Rev. Lett. 2005, 94, 118301/1-118301/4.

179