ACTAUNIVERSITATIS

UPSALIENSISUPPSALA

2017

Digital Comprehensive Summaries of Uppsala Dissertationsfrom the Faculty of Science and Technology 1596

Computer Simulations of PolymerGels

Structure, Dynamics, and Deformation

NATASHA KAMERLIN

ISSN 1651-6214ISBN 978-91-513-0144-0urn:nbn:se:uu:diva-332575

Dissertation presented at Uppsala University to be publicly examined in Polhemssalen,Ångströmlaboratoriet, Lägerhyddsvägen 1, Uppsala, Tuesday, 19 December 2017 at 09:15 forthe degree of Doctor of Philosophy. The examination will be conducted in English. Facultyexaminer: Professor Jan Forsman (Lund University).

AbstractKamerlin, N. 2017. Computer Simulations of Polymer Gels. Structure, Dynamics, andDeformation. Digital Comprehensive Summaries of Uppsala Dissertations from theFaculty of Science and Technology 1596. 69 pp. Uppsala: Acta Universitatis Upsaliensis.ISBN 978-91-513-0144-0.

This thesis presents the results of computer simulation studies of the structure, dynamics,and deformation of cross-linked polymer gels. Obtaining a fundamental understanding of theinterrelation between the detailed structure and the properties of polymer gels is a challengeand a key issue towards designing materials for specific purposes. A new off-lattice methodfor constructing a closed network is presented that is free from defects, such as looping chainsand dangling ends. Using these model networks in Brownian dynamics simulations, I showresults for the structure and dynamics of bulk gels and describe a novel approach using sphericalboundary conditions as an alternative to the periodic boundary conditions commonly used insimulations. This algorithm was also applied for simulating the diffusion of tracer particleswithin a static and dynamic network, to illustrate the quantitative difference and importanceof including network mobility for large particles, as dynamic chains facilitate the escape ofparticles that become entrapped.

I further investigate two technologically relevant properties of polymer gels: their stimuli-responsive behaviour and their mechanical properties. The collapse of core-shell nanogelswas studied for a range of parameters, including the cross-linking degree and shell thickness.Two distinct regimes of gel collapse could be observed, with a rapid formation of smallclusters followed by a coarsening stage. It is shown that in some cases, a collapsing shellmay lead to an inversion of the core-shell particle which exposes the core polymer chainsto the environment. This thesis also explores the deformation of bimodal gels consisting ofboth short and long chains, subject to uniaxial elongation, with the aim to understand the roleof both network composition as well as structural heterogeneity on the mechanical responseand the reinforcement mechanism of these materials. It is shown that a bimodal molecularweight distribution alone is sufficient to strongly alter the mechanical properties of networkscompared to the corresponding unimodal networks with the same number-average chain length.Furthermore, it is shown that heterogeneities in the form of high-density short-chain clustersaffect the mechanical properties relative to a homogeneous network, primarily by providingextensibility.

Keywords: computer simulations, Brownian dynamics, polymer gel, microgel, sphericalboundary conditions, hypersphere, core-shell, deswelling, mechanical properties, uniaxialelongation

Natasha Kamerlin, Department of Chemistry - Ångström, Physical Chemistry, Box 523,Uppsala University, SE-75120 Uppsala, Sweden.

© Natasha Kamerlin 2017

ISSN 1651-6214ISBN 978-91-513-0144-0urn:nbn:se:uu:diva-332575 (http://urn.kb.se/resolve?urn=urn:nbn:se:uu:diva-332575)

A scientist in his laboratory is not a mere technician: he is also a child confrontingnatural phenomena that impress him as though they were fairy tales.

— Marie Curie

To Schubby.

List of papers

This thesis is based on the following papers, which are referred to in the textby their Roman numerals.

I Kamerlin, N., Ekholm, T., Carlsson, T., Elvingson, C. Construction of

a closed polymer network for computer simulations, J. Chem.Phys., 2014, (141), 154113

II Kamerlin, N., Elvingson, C. Tracer diffusion in a polymer gel:

Simulations of static and dynamic 3D networks using spherical

boundary conditions, J. Phys. Condens. Matter, 2016, (28), 475101

III Kamerlin, N., Elvingson, C. Collapse dynamics of core-shell

nanogels, Macromolecules, 2016, (49), 5740-5749

IV Kamerlin, N., Elvingson, C. Deformation behaviour and failure of

bimodal networks, Macromolecules, 2017, (50), 7628-7635

V Kamerlin, N., Elvingson, C. Deformation behaviour of homogeneous

and heterogeneous bimodal networks, Macromolecules, submitted

Reprints were made with permission from the publishers.

Contents

1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111.1 Description and modelling of polymers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

1.1.1 Polymer chain models and conformations . . . . . . . . . . . . . . . . . . . . . 151.1.2 Effect of solvent quality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

1.2 Polymer gels . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 191.2.1 Gel topology and inhomogeneities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 191.2.2 Mechanical properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 211.2.3 Swelling and volume phase transition . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

2 Computer Simulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 242.1 Coarse-graining and effective interaction potentials . . . . . . . . . . . . . . . . . . . . 252.2 Modelling of a closed network . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 262.3 Boundary conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

2.3.1 Periodic boundary conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 282.3.2 Spherical boundary conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 282.3.3 Free boundaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

2.4 Brownian dynamics simulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 292.4.1 The Langevin equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 302.4.2 Algorithm for Brownian dynamics in IR3

. . . . . . . . . . . . . . . . . . . . . . 302.4.3 Algorithm for Brownian dynamics on S3

. . . . . . . . . . . . . . . . . . . . . . . 31

3 Simulations of Polymer Networks on the 3-sphere . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 323.1 Construction of a closed polymer network on S3

. . . . . . . . . . . . . . . . . . . . . . . . . 323.2 Diffusion of particles through a polymer gel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

4 Investigating the Collapse Dynamics of Core-Shell Nanogels . . . . . . . . . . . . . . . 374.1 General features of a collapsing homopolymer gel . . . . . . . . . . . . . . . . . . . . . 384.2 Core collapse dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 404.3 Shell collapse dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

5 Investigating the Mechanical Properties of Bimodal Gels . . . . . . . . . . . . . . . . . . . . . . 435.1 Uniaxial deformation of homogeneous bimodal gels . . . . . . . . . . . . . . . . . . 445.2 Role of structural heterogeneities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46

6 Summary and Outlook . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50

7 Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53

8 My Contribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55

9 Summary in Swedish . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59

Nomenclature

Latin alphabet

a Polymer unit radiusb Bond lengthb Bond vectorCs Normalised cluster sized′ Scaled tracer particle diameterD Diffusion coefficientD0 Diffusion coefficient for free diffusionfs Fraction of short chainskB Boltzmann constantL Inter-wall distanceL0 Initial inter-wall distancen Number of bonds in a chainN Number of units in a chainnc(c) Number of clusters of size cP2 Second order Legendre polynomialr Position vectorR 3-sphere radiusR′

ee Scaled end-to-end distanceR′

g Scaled radius of gyrationT TemperatureV Volume

Greek alphabet

α Strainγ Core/shell mass ratioε ′ Scaled Lennard-Jones energy parameterζ Angular displacementη Viscosityθ Angle between two bond vectorsρn Cross-linking densityσ Stressσ ′ Scaled stressφ Volume fraction

1. Introduction

What is life?It is the flash of a firefly in the night.It is the breath of a buffalo in the wintertime.It is the little shadow which runs across the grass and loses itself in the sunset.— Isapo-Muxika

The question of how life emerged from the absence of life has inspired a lotof speculation. It is estimated that the first steps of life appeared on this planetover 3.8 billion years ago in the form of simple unicellular organisms fromwhich complex multicellular organisms eventually evolved. [1] Life relies onsome form of self-replicating molecule which has the capacity to store theinformation required to synthesise biological compounds and to transfer thatinformation from generation to generation. In more modern cells, these tworoles are fulfilled by the polymers DNA and RNA. Thus, in perhaps the mostsophisticated example of their importance, polymers define life itself.

Polymers are macromolecules that are ubiquitous in nature and permeateevery facet of our lives through both natural and man-made materials. Therange of applications of polymers far exceeds those of other classes of mate-rials, extending from adhesives, foams, coatings, plastic containers to textiles,electronic devices, optical devices, and many more (see Figure 1.1). Plantsand animals use natural polymers as building blocks, storage substances, andin biochemical reactions. Polymers account for a great part of the humanbody, for example in the form of proteins and carbohydrates. Cellulose isthe main structural component of plants and the most abundant organic com-pound on this planet. It is also the major constituent of this thesis. Anotherpolymer produced by plants is rubber, which can be harvested in the form oflatex from, e.g., Hevea brasiliensis (more commonly referred to as the rubbertree). These are all examples of different types of polymers that serve radi-cally different functions. Variations in the sequence and spatial arrangementof the units in polymers give rise to materials with a tremendous physical andmechanical diversity.

While the reader will encounter many different types of materials everyday, ancient cultures were limited to the native plants, soil, and stones thatwere available to them. Throughout human history, technological advanceswere made possible by mixing and modifying natural materials to obtain im-proved ones. For example, processed rubber was in use already 1600 BCE

11

Figure 1.1. Examples of some different uses of polymers. a) Agar plate used to culture bac-teria. b) Plastic (PET) bottles. c) CDs and DVDs made of polycarbonate. (Source: Multimediadiscount,

Wikimedia.) d) Paint containing polymer binders. (Source: Joyful spherical creature, Wikimedia.)

by Mesoamericans, where latex was mixed with the juice of a local vine tocreate bouncing rubber balls for ritual ballgames and for offerings in tem-ples. [2] Some millennia later, in 1839, Charles Goodyear accidentally dis-covered vulcanisation, a cross-linking process which combines natural rubberwith sulphur under heat to produce a significantly more durable material thanlatex. Vulcanised rubber is most commonly used today in automobile tires.Leo Baekeland invented one of the first synthetic plastics (called Bakelite) in1907. Intrigued by the differences between natural and synthetic polymers,Hermann Staudinger studied a variety of polymers and in 1920 publishedhis classic paper entitled "Über Polymerisation", which proposed that poly-mers are macromolecules formed by many repeating units that are covalentlybonded to form long chains. [3] At a time when the reigning theory was that allchemical compounds were low molecular weight entities that could potentially"aggregate", the theory of polymers as macromolecules was met with skepti-cism.1 Nonetheless, Staudinger’s publication heralded intense research, andpolymers gradually started to become recognized as separate from colloidalsystems. Since the 1930s, with significant steps taken by Wallace Carothersin the manufacturing of synthetic polymers through the creation of nylon andneoprene, and the subsequent inventions of polyethylene, Teflon, Kevlar, etc.,

1In fact, in response to a lecture given by Staudinger in favour of the existence of macro-molecules, he was reportedly told by a fellow scientist to "leave the concept of large moleculeswell alone; organic compounds with a molecular weight above 5000 do not exist... There canbe no such thing as a macromolecule." Met with this hostile reception, Staudinger concludedhis lecture with the words "Here I stand, I cannot do otherwise." [4]

12

the polymer industry boomed and has now become larger than the aluminium,copper, and steel industries put together. [5]

The type of polymeric structure that is of particular interest for this thesisis one in which the polymer chains are linked together by cross-links, form-ing a 3-dimensional network that is swollen in a solvent. Such a system iscalled a polymer gel. Cross-linking means that the chains lose some of theirability to move as individual entities and, unlike polymer solutions, a gel canmaintain its shape under its own weight. Gels can be fabricated into a diverserange of architectures and there is a trend towards gels with increasingly com-plex and sophisticated morphologies, such as multilayered films, [6] topologi-cal gels where the chains are linked by slide-rings, [7] and interpenetratingnetworks. [8] Among some of the properties of polymer gels that are of im-portance for their use in applications are their swelling/deswelling ability andmechanical properties. Stimuli-responsive polymers, i.e., polymers that canundergo a dramatic change in volume in response to a small change in theirenvironment (such as the temperature, pH, or salt concentration), are of greatinterest, notably in medical applications as controlled drug delivery vehicles,actuators and artificial muscles, and tissue engineering scaffolds. [9–21] Someapplications also rely on a load-bearing function, for example in tissue engi-neering as bio-scaffolds to guide the growth of new tissue. [22, 23] Conven-tional hydrogels, however, are known to be mechanically weak and unable towithstand large deformations. These poor mechanical properties may be a ma-jor limitation. The swelling and mechanical properties of gels are influencedby the topology of the network, such as the degree of cross-linking as well asthe presence of defects (such as free chain ends and loops) and entanglements.Considering the abundance of polymers in modern society, it is of great impor-tance from a fundamental and technological viewpoint to understand the basicstructure-property relation of polymer gels, for example to guide experimentalprocedures to prepare gels with tailored properties.

Polymer science is a multidisciplinary scientific field that brings togetherresearchers from a very wide range of academic areas. In the quest for ex-ploring the nature of matter, scientific methods and theories have been appliedsince early times to extract information from the world around us. With theadvent of computers in the 1940s, a new approach to studying chemical phe-nomena became accessible. No longer confined to laboratories, amid beakers,burettes, and Bunsen burners, scientists could now perform studies virtuallyby implementing the mathematical models in computer programs that mimiclaboratory settings. Tremendous advances in high performance computerssince then have made computer simulations, in combination with theory andexperiments, a powerful tool in driving innovation and for reaching a deeperunderstanding of the properties of, e.g., polymers.

In modelling the complex behaviour of polymer systems, the researchersimplifies the problem by focusing on aspects of the system that are sup-posedly relevant to the observed behaviour. Simulations have, for example,

13

been useful in improving our understanding of the dynamics of polymer melts[24, 25] and rubber elasticity. [26–28] Simulations of cross-linked polymernetworks started in the 1980s,2 notably by Eichinger et al. [30–32] Today,there are a variety of simulation methods to choose from to study the equilib-rium and dynamic properties of macromolecular systems. Unlike experiments,where the network topology may contain a number of unknown structural fea-tures in the form of network imperfections and inhomogeneities, [33–35] themolecular architecture in a simulation can be carefully chosen. This makesit possible to decouple different effects in a systematic manner and study asuccession of network models in atomistic detail under fully controlled condi-tions, in order to better understand the relationship between the structure andproperties of polymer gels. Simulations of polymer gels can, however, becomevery time-consuming due to the large system sizes and long relaxation timesinvolved, meaning that there is a need to find more efficient ways to modelthese systems.

In this thesis, we employ Brownian dynamics simulations, which is a coarse-grained approach that facilitates the simulation of much larger systems com-pared to atomistic simulations, to study the structure, dynamics, and deforma-tion behaviour of cross-linked polymer gels. Chapter 1 will introduce somesimple theoretical models and shape descriptors of polymer chains, which canalso be applied for the chains in a polymer network, as well as the topologyand properties of polymer gels. Chapter 2 will give an overview of how toperform a computer simulation, including the Brownian dynamics algorithm.Here, we also present a method to form a closed, defect-free polymer net-work as well as a novel approach for avoiding surface effects in simulationsof polymer gels, by using spherical boundary conditions (SBC). The latter isachieved by working in a higher dimensional space, namely the surface ofa ball in four dimensions. The results of simulations of tracer particles ofvarious sizes diffusing through a static and dynamic network with SBC arepresented in chapter 3. Using this method of constructing model networks,we next examine the deswelling and mechanical properties of polymer gels.Specifically, in chapter 4 we investigate the response of nanogel particles witha core-shell structure when the solvent quality is switched to induce a col-lapse. The physical coupling of the two compartments means that both thecore and the shell dynamically influence each other. This is illustrated by theeffect of shell thickness and cross-linking density on the detailed dynamics ofthe core-shell particles. In chapter 5, we examine the mechanical propertiesof bimodal gels subject to a steady uniaxial elongation. Bimodal networkshave previously been found to be mechanically strong and tough. Differenttypes of bimodal structures are considered, with the aim of understanding theunderlying reinforcement mechanism.

2A review of these early simulation works has been published by Escobedo and de Pablo. [29]

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1.1 Description and modelling of polymersA polymer is generically defined as a macromolecule composed of a sequenceof many copies of one or more repeating units, called monomers, that arejoined through covalent bonds. The average number of monomers per molecule,referred to as the degree of polymerisation, may be in the range of severaltens of thousands.3 These monomers can be arranged in many different ways,forming, e.g., linear or branched polymer chains, such as the ones shown inFigure 1.2, as well as more sophisticated architectures such as star polymers,comb polymers, and brush polymers. Of particular interest in this thesis is thecase where polymer chains are cross-linked into a large 3-dimensional networkstructure.

Figure 1.2. Examples of some different architectures of polymers, ranging from linear chainsto branched chains and cross-linked networks.

Before delving into the properties of polymer networks, we begin by dis-cussing some of the models that are used to describe the structure of singlepolymer chains.

1.1.1 Polymer chain models and conformationsDue to rotations about the bonds in the polymer backbone and the large de-gree of polymerisation that is typically involved, a polymer chain with a givensequence of monomers can assume an enormous number of possible confor-mations or ways of arranging itself in space. It is therefore suitable to discussthe conformations by applying statistical methods, where the shape and size ofa polymer are characterised by averaging over many different polymer chains.

As an example, let us consider a linear polymer chain consisting of N unitsconnected by n = (N −1) bonds, shown in Figure 1.3. The units in the chainmay be represented by their position vectors (r1, ...,rN) or by the bond vectors(b1, ...,bn) connecting consecutive units (where bi = ri+1 − ri). The anglebetween bi and b j is denoted by θi j.

There are various ways of characterising the size of the polymer chain (seeFigure 1.4). We could, for example, use the contour length, L , which can

3For example, ultra-high molecular weight polyethylene used in orthopaedic applications typ-ically has a degree of polymerisation of as much as 200,000 repeating units, with a molecularweight between 3.5−7.5 ·106 g/mol. [36]

15

Figure 1.3. A schematic of a polymer chain modelled as N spherical beads.

be pictured as the length the chain would have if it were in a fully extendedrod-like state. The contour length is given by L = ∑n

i=1 |bi|. Another measure

Figure 1.4. The contour length (L ), end-to-end distance (Ree), and radius of gyration (Rg) fora linear polymer chain.

of the chain size is given by the end-to-end distance. The end-to-end vector,Ree, is the vector connecting the first and last unit in the chain,

Ree = rN − r1 =n

∑i=1

bi. (1.1)

The maximum end-to-end distance corresponds to the contour length. A draw-back of using the end-to-end distance is that it is only well-defined for linearchains. A more general and intuitively meaningful size descriptor that is alsovalid for, e.g., branched and cyclic chains, is the radius of gyration, Rg, whichcorresponds to the root mean-square distance of all polymer units from thecentre of mass (rcm) of the polymer,

R2g =

1N

N

∑i=1

(ri − rcm)2. (1.2)

Alternatively, the radius of gyration can be computed from the gyration tensordefined by [37]

Gmn =1N

N

∑i=1

(r(m)i − r(m)

cm )(r(n)i − r(n)cm ), (1.3)

16

where r(m) is the m-th Cartesian component of the position vector. The sum ofthe corresponding eigenvalues, λk, squared gives R2

g = λ 21 +λ 2

2 +λ 23 . These

eigenvalues can also be combined to describe the chain asphericity, [38]

S =27

⟨∏3

i=1(λi − λ̄ )⟩⟨(

∑3i=1 λi

)3⟩ , (1.4)

where λ̄ is the average eigenvalue and < ... > denotes an ensemble averageover chain configurations. Being bounded by the interval −0.25 ≤ S ≤ 2,positive values correspond to prolate ellipsoid-like shapes and negative valuesto oblate shapes.

There exists a variety of models that are used to describe a polymer chain.Depending on the type of constraints that are imposed, the resulting proper-ties may vary between different models. The simplest model used to describea polymer chain is the ideal freely-jointed chain model. In this model, thepotential energy only consists of the pairwise energy term that describes theconnectivity of successive units. All bonds have a fixed length of b and thedirections of the sequential bond vectors are completely uncorrelated, mean-ing that the bond angles may assume all values with equal probability, i.e.,< cosθi j >= 0 for i �= j. The lack of correlation in the direction of consecu-tive units along the chain means that the conformation may be represented bya 3-dimensional random walk. Since the directions of the bond vectors arestatistically independent in this model, the average end-to-end vector becomes< Ree >= 0. Using the mean-square end-to-end distance instead gives

< R2ee >= b2

n

∑i=1

n

∑j=1

< cosθi j >= nb2. (1.5)

Taking the freely-jointed chain model and restricting the bond angles toa fixed value (while maintaining that all torsion angles are equally probableand independent) gives the freely-rotating chain model. This additional con-straint of a fixed bond angle is reflected in the mean-square end-to-end dis-tance, which now becomes [39]

< R2ee >= nb2

(1+ cosθ1− cosθ

− 2cosθn

(1− cosn θ)(1− cosθ)2

). (1.6)

It may be noted that the scaling law < R2ee >∝ n is a common feature of ideal

chains (for large n).Imposing bond angle restrictions introduces correlations that are transferred

along the bond vectors in the backbone. The bond orientational correlationbetween two chain segments decays exponentially, [40]

< cosθ(s)>= e−s/P, (1.7)

17

where θ(s) is the angle between the tangent vectors at two different pointsseparated by length s along the chain and the characteristic length scale of de-cay, P, is called the persistence length. On length scales that are much smallerthan the persistence length (s << P), there are strong bond angle correlationsand the chain segment essentially behaves as a stiff rod. Over larger lengthscales (s >> P), the bond angle correlations become negligible and the chainappears flexible. The persistence length may thus be used to characterise thestiffness of the chain.

1.1.2 Effect of solvent qualitySo far, the discussion of polymer chain conformations has assumed ideal con-ditions, where secondary interactions, such as those between different chainsegments and between the polymer chain and solvent molecules, were notconsidered. Treating the polymer chain as a random walk allows chain seg-ments to overlap and pass through each other. In reality, chain segments havea physical size, prohibiting multiple occupancy of the same region in space.This excluded volume interaction is treated in the self-avoiding random walk.The introduction of an excluded volume between non-bonded units reducesthe number of available conformations and causes the chain to expand rela-tive to a random walk chain. The effective size of the chain is also influ-enced by the solvent quality. A good solvent is one in which the attractionbetween polymer segments and the solvent molecules is energetically morefavourable than that between the polymer segments themselves. This meansthat the polymer swells upon mixing with solvent, adopting an expanded coilconformation. In a poor solvent, the polymer self-interactions dominate overthe polymer-solvent interactions, and there will be a net attraction between thechain segments instead. The chain collapses into a compact globular state inorder to minimise the free energy.

The effective interactions may be expressed in terms of the dimensionlessparameter χ , introduced in the lattice-based Flory-Huggins theory and definedas [41]

χ =z

kBT(εps − (εpp + εss)/2), (1.8)

where z is the coordination number of the lattice (i.e., the number of nearestneighbours to each lattice point) and εAB is the interaction energy betweenspecies A and B (where subscript p stands for polymer and s for solvent). Theχ-parameter characterises the relative strength of attraction between differentspecies, such that χ > 1/2 corresponds to a poor solvent and χ < 1/2 to agood solvent. [40] Since the conformation adopted by the chain depends on thesolvent quality, changing the solvent composition or temperature can induce acollapse transition, referred to as the coil-to-globule transition (Figure 1.5).In terms of the scaling law <R2

ee >∝ n2v, the exponent v changes from v≈ 3/5in a good solvent to v = 1/3 in a poor solvent. At the theta temperature, the

18

Figure 1.5. Typical example of a coil-to-globule transition of a polymer chain, induced byvarying the temperature.

polymer self-interaction balances the polymer-solvent interaction and v= 1/2,which can be recognized as the ideal chain relation, described by Eq. (1.5).

1.2 Polymer gelsA polymer gel is on a basic level an assembly of polymer chains that are linkedtogether into a 3-dimensional network that is swollen in a solvent. Dependingon the cross-linking mechanism, gels are commonly classified as "physical" or"chemical".4 In physical gels, the chains are held together by relatively weakand temporary associations between chains, which may include hydrophobicinteractions or hydrogen bonds. Due to the reversible nature of these cross-links, chain segments are able to easily break and reform. Chemical gels, onthe other hand, are formed by covalent bonds that are difficult to break onceformed. We will focus solely on chemical gels in this thesis.

In this section, we will explore the topology and properties of polymer gels.

1.2.1 Gel topology and inhomogeneitiesThe structure of a polymer gel may be envisioned as a sponge penetrated bynano- to millimeter-sized pores, as illustrated in Figure 1.6. By a perfect net-work structure, we mean one in which all chains are connected at both endsto two different junction sites and there are no entanglements between thechains. In reality, most chemically cross-linked polymer gels, in particularthose obtained by conventional radical copolymerisation of monomers andcross-linkers, are structurally complex and display deviations from a perfectstructure. There may be different types of defects and inhomogeneities that

4Not all gels easily fit into these two categories. For example, the slide-ring gels developed byOkumura and Ito [7] are formed by using figure-eight shaped cross-links that are free to slidealong the chain. Such topological gels which are held together by physical constraints could beconsidered a different class of gels.

19

Figure 1.6. Environmental scanning electron microscopy (ESEM) images showing variousporous structures of synthetic hydrogels. Reprinted with permission from Institute of Macro-molecular Chemistry AS CR.

span multiple length scales. [33–35] A schematic of some of these is shown inFigure 1.7.

Figure 1.7. Illustration of different types of network imperfections and defects. Danglingchains are connected to the network at only one chain end, while loops are connected at bothends to the same cross-linker. Trapped entanglements can only be released by breaking bonds.Clusters consist of regions of highly cross-linked chains.

Gels typically contain an unknown number of dangling chains that are at-tached to the network only by one end, loops that have both ends of a chainconnected to the same cross-linker, as well as trapped entanglements betweenchains. Aside from chain length polydispersity, there may also be a heteroge-neous spatial distribution of cross-links, with highly cross-linked domains orclusters. Such imperfections at the molecular level may impact the materialproperties. For example, dangling chains and loops typically do not contributeto the network elasticity and may result in a mechanically softer gel, while

20

chain entanglements may act as supplementary cross-links and improve thestiffness of the gel. An analysis of structure-property relations is complicatedby the presence of unknown or uncontrollable features in the gel structure.

Experimental approaches have been developed that allow for greater con-trol of the structure, in order to produce more homogeneous gels with fewerdefects,5 including end-linking pre-existing polymer chains, [46] although thesynthesis of highly regular and reproducible topologies with a desired distribu-tion of elastically active strands remains an important challenge for syntheticchemistry.

1.2.2 Mechanical propertiesThe chemical cross-linking of polymer chains means that the gel will behaverubber-like when it is extended, producing a restoring elastic force whichbrings back the material to its original shape once the external force is re-moved. For sufficiently large deformations, beyond the elastic limit, covalentbonds are severed and the gel is ultimately fractured. The mechanical proper-ties of materials may be obtained by performing, e.g., uniaxial tensile testing.In an experimental setup, a specimen of length L0 and cross-sectional area Ais mounted between two grips and a tensile force is applied in one directionwhich elongates the sample. The stress, σ , is defined as the applied force perunit area. The associated deformation per unit length is called the strain, α ,and is defined as the ratio of the change in length in the stretching directionrelative to the initial length, i.e., α = ΔL/L0. By running the tensile testinguntil fracture and monitoring the stress-strain behaviour, it is possible to ob-tain information about several mechanical properties, such as the stiffness orelastic modulus6 through the initial slope of the stress-strain curve. The ten-sile strength corresponds to the maximum stress before the material breaks,and the energy required to reach fracture, referred to as the toughness of thematerial, is given by the area under the curve.

The mechanical properties of polymers are important in many practical ap-plications that rely on their load-bearing function, for example in tissue en-gineering as bio-scaffolds. [22, 23] Scaffolds used for bone tissue growth, forexample, require good mechanical properties to be able to provide tempo-rary mechanical integrity upon implantation. [47] The biocompatibility of hy-drogels (i.e., water-swollen networks) makes them appealing candidates inbiomedical applications. Depending on the application, however, it mightbe necessary to improve the mechanical characteristics to make the materialmore suitable. Much research has been devoted to finding ways of improvingthe mechanical properties. For example, Sakai et al. used end coupling of

5See, for example, [42–45].6For a tensile stress, the elastic modulus is called the Young’s modulus. Under shear forces (i.e.,forces parallel to the surface), one obtains the shear modulus.

21

tetra-arm polyethylene glycol (PEG) chains to form tetra-PEG gels that have arelatively homogeneous network structure and high mechanical strength. [44]Gong et al. combined two networks with very different cross-linking densitiesto form a double network that exhibited a greater toughness, which are promis-ing candidates in, e.g., bioengineering. [48–53] Mark et al. also employed abimodal molecular weight distribution by end-linking a mixture of very shortand long chains to form bimodal elastomers (i.e., rubber-like solids) that havein some cases shown substantial improvements in mechanical strength andtoughness. [54–60]

1.2.3 Swelling and volume phase transitionWhen a polymer network is immersed in a good solvent, the chain segmentsand solvent molecules tend to mix, whereby solvent molecules permeate andswell the network. The strength of this tendency to mix can be represented bythe osmotic pressure, Π. Using the Flory-Huggins lattice model of polymersolutions, the osmotic pressure may be written as [41]

Π(φ) =kBTvc

(− ln(1−φ)−φ −χφ 2) (1.9)

where vc is the volume of a lattice site. At the same time, network dilationentails deformation of the chains, which produces an elastic reaction that actsagainst the swelling. The gel reaches its equilibrium swelling level when themixing contribution and the elastic contribution are balanced, which may bewritten as [41]

Π(φ) = G0

(φφ0

)1/3

, (1.10)

where G0 is the shear modulus of the network before swelling, and φ0 and φare the polymer volume fractions before and after swelling, respectively. Ifφ << 1 then Eq. (1.9) may be simplified and Eq. 1.10 can be solved for φ togive

φ = φ0

[G0vc

kBT φ 20

1(1/2−χ)

]3/5

. (1.11)

It can be seen that if (1/2− χ) is large then φ is small and the gel is swollen.A rapid shrinking takes place (i.e., φ increases) as χ approaches 1/2.

The effective interactions between the polymer and solvent may be highlysensitive to external conditions (such as the temperature, pH, or salt concen-tration). Gels that undergo a dramatic change in volume in response to a smallchange in their environment are called stimuli-responsive or smart gels. Thisreversible swelling/deswelling ability makes these materials of great interestin medical applications. For example, stimuli-responsive polymers have beenused as self-adjusting valves. [16, 17, 61] If the liquid that flows through the

22

valve changes properties, the gel can respond by either opening or closingthe valve. Charged gels that react to changes in an electric field are usedin the development of artificial muscles, for example in the construction ofrobots. [62] Stimuli-responsive gels are also used as drug carriers to deliverdrugs to a specific site in the body, by exploiting variations in physiologicalconditions. [12, 18, 20, 21] The design of gels with a core-shell architecture,which are composed of a different type of stimuli-responsive polymer gel inthe core and the shell compartment, have been particularly interesting as multi-responsive materials (i.e., materials that are responsive to two or more differ-ent stimuli). [12, 63–68]

23

2. Computer Simulations

The good news about computers is that they do what you tell them to do. Thebad news is that they do what you tell them to do.— Ted Nelson

Owing to the rapid development of computers, simulations have become anindispensable tool in a number of disciplines, for elucidating phenomena ob-served in experiments or for making predictions of systems that have not yetbeen studied. Various simulation methods exist, such as Monte Carlo, Mole-cular dynamics, Brownian dynamics, and dissipative particle dynamics, whichhave been extensively utilised for studying the properties of polymer materi-als. [69] The choice of method depends on the purpose of the study. Sincethe concept of time is lacking in traditional Monte Carlo simulations, this istypically used to study equilibrium properties by generating random configu-rations.1 If the aim is to follow time dependent dynamic properties, thenMolecular dynamics simulations could be used. A typical Molecular dynamicssimulation uses Newton’s equations of motions to compute the trajectories ofall particles in the system (usually at an atomistic level). It is also possible todescribe the dynamics of only a selected group of particles (e.g., the polymerunits), with the remainder of particles (e.g., the solvent molecules) acting as acontinuum bath. The implicit-solvent approach is the premise of the Langevinequation, used in Brownian dynamics simulations, and has the advantage ofbeing computationally less expensive, thereby allowing the study of largersystems during longer time scales. As we are interested in the long-time dy-namics of large polymeric systems, Brownian dynamics simulations are usedin this thesis and will be discussed more extensively in section 2.4.

One of the first tasks in performing a computer simulation is to decide onthe level of detail necessary to capture the essential features of interest. Inorder to keep calculations tractable, it is common to use coarse-graining whenmodelling polymer chains. Next, the influence of essential intermolecular andintramolecular interactions are introduced by choosing a functional form of theeffective interactions, which may include bonded and non-bonded terms. Aninitial configuration must then be generated for the system that is ideally closeto the state of interest. This initial configuration might need to be relaxed,

1Monte Carlo schemes have been developed which could be used to study time dependent phe-nomena. [70–72]

24

during which the structural properties of the system may be monitored untilequilibrium is obtained. The equilibrated system is then used as a startingstructure to produce trajectories which can be subsequently analysed for theproperties of interest.

In this section, we will look at some of the steps and procedures of perform-ing a simulation study of a polymer system.

2.1 Coarse-graining and effective interaction potentialsA physical system can often be described on a wide range of time and lengthscales. When modelling a system in a computer simulation, we would ideallytry to describe it as accurately as possible.2 If the aim is to study the quali-tative long-time dynamics of polymer chains as a whole, then the atomic scaleis generally less important to capture the relevant physics of interest. Thesimulation efficiency may be improved by merging several atoms into oneunit.

The most common representation of a polymer chain, which is also adoptedin this thesis, is the generic bead-spring model, where a chain is described asa string of beads of radius a. The beads are connected through a harmonicstretching potential,

Us =ks

2 ∑i(bi −2a)2, (2.1)

where ks is the stretching force constant and 2a is the equilibrium bond length.Although the harmonic potential does not take into account bond dissociation,a simple way of including finite extensibility is to introduce a cut-off for thepotential at a desired bond breaking distance.

Chain stiffness is accounted for through an angular potential,

Ub =kb

2 ∑i(cosθi − cosθ0)

2, (2.2)

where θ0 is the angle of minimum bending potential energy, θi is the anglebetween bond vectors bi and bi+1, and the bending force constant, kb, is deter-mined from the persistence length, [73]

P =< b >1−< cosθ >n

1−< cosθ >(2.3)

where < b > and < cosθ > denote the chain-averaged bond length and bondangle cosine, respectively.

2This would entail working in the quantum mechanical realm. The number of units in a simu-lation of polymers, however, is often in the order of tens of thousands. A detailed descriptionof the electronic structure of all molecules would not be feasible for simulating the long-timedynamics of such large systems on present-day computers. When atomic detail is not necessary,larger systems may instead be studied using classical mechanics.

25

In a real polymer chain, two or more segments cannot occupy the samespace. This excluded volume effect may be accounted for by introducing asoft repulsive potential between non-bonded particles, such that the force isgiven by [74]

Frepi =

εrepkBT4a2 ∑

j

(2ari j

−1)

ri jn̂i j, (2.4)

where εrep is a dimensionless force constant and n̂i j is the unit vector fromparticle j to i.

To model the effect of solvent quality, a Lennard-Jones (LJ) potential iscommonly used, which is composed of a short-range repulsion term and along-range attraction term. The potential is usually truncated in order to speedup the computing time. To avoid a discontinuity in the forces at the cut-offdistance, we use the following form of the LJ potential, [75]

ULJ = 4εLJ

[σ12

LJr12 − σ6

LJr6 +Cr6 +D

], r < rc (2.5)

where r is the distance between the interacting units, εLJ is a measure of theinteraction energy, σLJ is effectively the diameter of one particle, and rc is thecut-off distance for the potential. The constants C and D correspond to

C =−σ6LJr−12

c +2σ12LJ r−18

c , D = 2σ 6LJr−6

c −3σ12LJ r−12

c . (2.6)

The quality of the solvent can be modified by varying the attraction via the LJinteraction parameter. As εLJ increases, the strength of the attraction betweenthe chain units increases, representing a decrease in solvent quality.

Although not considered in this thesis, there may also be electrostatic forcespresent in the system. The most common way to handle electrostatics in a sys-tem with periodic boundaries (see section 2.3) is by using the Ewald summa-tion technique. [76–78] Alternatively, it is possible to use spherical boundaryconditions by embedding the 3-dimensional system onto the surface of a ballin four dimensions, as discussed in section 2.3.2.

2.2 Modelling of a closed networkComputer simulations can provide valuable information of cross-linked poly-mer gels by enabling us to work with well-defined network structures, wherethe topology is known a priori or can readily be determined. Different mod-els of polymer networks have been used, ranging from simple idealised latticestructures, [28,79–84] typically a diamond or cubic lattice, with a highly regu-lar topology to more realistic models that include more sophisticated dynamicpolymerisation and cross-linking procedures that naturally introduce structuraldefects during network formation. [85–93] In this thesis, a new algorithm has

26

been developed for generating an off-lattice, closed network (papers I and II)that may either be used to form discrete gel particles, or, in combination withsuitable boundary conditions, to study bulk properties. Structural defects orinhomogeneities may be introduced in a controlled manner, meaning that theireffect on the material properties can be examined systematically.

Figure 2.1. Schematic of the various steps involved in forming a closed network. See the textfor a description of each step.

A schematic of the network formation procedure is shown in Figure 2.1.The tetravalent cross-linking nodes are randomly distributed within a speci-fied volume, under the restriction that no two particles are generated within acertain minimum distance from one another (step (i)). Each node is then con-nected to its four nearest neighbours that are unsaturated, i.e., that have fewerthan four connections (step (ii)). As nodes within proximity start to becomefully occupied, this procedure will eventually result in some of the connectedpair of nodes being far apart. These long connections are reduced by a processof reconnection (step (iii)), described in paper II. Next, the polymer chains aregenerated by placing the constituting beads along the path connecting eachpair of nodes, forming a closed network (step (iv)). The number of beadsper chain may either be fixed, or it may be determined iteratively for eachnode-node connection by using the equations for the persistence length andthe end-to-end distance (see paper I), forming a network with a distribution ofchain lengths and pore sizes. Finally, the network structures are equilibrated(step (v)).

27

2.3 Boundary conditionsIn a simulation, it is also important to specify the boundary conditions, e.g.,free or no boundaries, rigid boundaries, periodic boundaries, or a mixture ofboundary conditions. The choice of boundary condition will depend on thetype of system being investigated.

A simulation is usually performed on a comparatively small number of par-ticles, due to limitations in memory and/or CPU time. For small system sizes,a large fraction of particles will reside on the surface of the sample.3 If theinterest is in bulk properties, then surface effects are commonly overcomeby implementing periodic boundary conditions. An alternative route, usingspherical boundary conditions, is presented in this thesis for the simulation ofpolymer networks, which should also be advantageous for charged systems.

2.3.1 Periodic boundary conditionsPeriodic boundary conditions (PBC) are a common choice for mimicking abulk phase. Here, all units in the simulation cell (typically a cubic box) arereplicated throughout space to form an infinite lattice. A displacement of aparticle in the main cell leads to a displacement of all its mirror images in thereplicas. The original particles can interact with other particles in the cell aswell as with the infinite number of mirror images. If the interactions are shortranged, it is common to use the minimum image convention, which meansthat for each particle, only the interaction with the nearest neighbour of eachof the remaining particles is taken into account. For systems with long-rangeinteractions, several neighbouring particles might need to be included. Forelectrostatic forces in particular, an infinite number of replicas should in prin-ciple be included, which results in a slowly converging sum over all charges.There are different methods for dealing with electrostatics, of which the mostpopular one has been the Ewald summation method, which entails splittingthe sum into two rapidly converging sums. [76, 77] This method is, however,still computationally demanding.

2.3.2 Spherical boundary conditionsAlthough periodic boundary conditions are the most common choice to avoidpossible surface effects, we could approach the problem by thinking outsidethe box. In a simulation with spherical boundary conditions (SBC), the 3-dimensional system is embedded on the surface of a 4-dimensional ball (i.e.,a hypersphere), denoted S3, which is a closed space without boundaries. [94]Living on a spherical surface, distances must be computed along geodesics in

3For example, for 1000 particles in a 10x10x10 cube, about 49% of the particles will appear onthe surfaces. For 106 particles, the percentage at the surfaces decreases to 6%.

28

order for the particle trajectories to be confined to the surface, meaning thatthe algorithm used to displace the particles will need to be slightly modifiedcompared to the case in IR3. When simulating charged systems, SBC offerthe advantage that calculations are considerably simplified. Rather than need-ing to calculate a sum over periodic images, the electrostatic forces on S3 aregiven by a closed expression, making the calculations more computationallyefficient.4 This property was developed in a series of studies by Caillol andLevesque. [95–98]

Despite their potential merit, relatively few simulation studies have usedSBC. [94, 99–105] For polymer systems, the diffusion of free chains has beeninvestigated on S3 and in IR3 and found to give identical results with respectto both structure and dynamics. [105] In this thesis, the use of SBC has beenextended to simulate polymer networks embedded on S3. Various ways offorming a defect-free, closed network on a hypersphere are presented in papersI and II, as well as the diffusion of tracer particles within such a gel (paper II).

2.3.3 Free boundariesWhile surface effects are limited using PBC or SBC in the study of bulkproperties, in some cases it may be more suitable or even necessary to workwith free boundaries, for example to study the structure of finite-sized gels[106] or to simulate the diffusion of molecules from nanoparticles. [107]

In studies of the mechanical properties of polymer networks under uniaxi-al tension, it is common to impose an affine deformation, [40] which meansmodifying the periodic simulation cell size and rescaling particle coordinatesto the new box size at each time step, followed by relaxation. [108–118] Thisis contrary to an experimental setup, where the strain is transmitted to thesample through the imposed motion of a boundary in the form of grips. Somestudies have cleared periodicity and used a boundary driven deformation in-stead. [119–121] A boundary driven deformation is achieved by replacingPBC with free boundaries and introducing molecular grips perpendicular tothe strain axis, which can move the sample ends apart while allowing for allpossible chain movements. Free boundaries are used in papers III-V.

2.4 Brownian dynamics simulationsIn 1827, during a microscopic study of pollen grains submerged in water, aScottish botanist named Robert Brown noted that the tiny particles discharged

4For systems consisting of a few hundred molecules, the computation time may be reduced bya factor of 2-3 compared to simulations using the Ewald method with a cubic box and periodicboundary conditions. [95]

29

by the pollens undergo an erratic random motion.5 Although it took nearlya century before this perpetual irregular motion was explained, [122–124] wenow know that Brownian motion is due to incessant collisions with the solventmolecules.

2.4.1 The Langevin equationAt the core of a Brownian dynamics simulation is the Langevin equation,which is a stochastic differential equation used to describe the motion of aparticle in a viscous continuum. The Langevin equation is given by [124,125]

mdui

dt=− f ui +Bi +Fi, (2.7)

where m is the particle mass, ui is the velocity of particle i, f = 6πaη is thefriction coefficient, η is the viscosity, Bi is a random force, and Fi =−∇U(r)is the total direct force arising from, e.g., interactions between particle i andother particles.

The basic premise of the Langevin equation is to replace the explicit solventmolecules and describe their collective effect on the dynamic behaviour ofthe solute by adding a stochastic force (Bi) and by imposing a frictional dragforce on the motion of the particle through the solvent which is proportionalto the velocity of the particle. By treating the solvent molecules implicitly andonly considering their average properties, this simulation method becomes lesscomputationally intensive than, for example, fully atomistic molecular dyna-mics simulations which require the input of all forces present. The reductionin computation time is not only due to the smaller number of explicit particlespresent, but also due to that larger time steps may be used, enabling the studyof phenomena that occur over relatively long time periods.6

2.4.2 Algorithm for Brownian dynamics in IR3

In order to calculate the displacements of particles over successive time inter-vals, it is necessary to integrate the Langevin equation. For the simulationsin this thesis, we use the algorithm developed by Ermak. [127, 128] In thisscheme, the time evolution of the position of unit i is given by

ri(t +Δt) = ri(t)+D0ΔtkBT

Fi +Ai, (2.8)

5This phenomenon was initially thought to be unique to living organisms, with the movementsbeing forms of life. However, Brown soon recognized that the random motion persists evenwith inorganic particles.6For comparison, Widmalm and Pastor performed Molecular dynamics (MD) and stochasticdynamics simulations of an ethylene glycol molecule together with 259 water molecules. [126]The MD simulation required 1500 hours whereas the stochastic dynamics simulation of solutealone required just 24 minutes.

30

where Δt is the time step and D0 = kBT/(6πηa) is the diffusion coefficient.The stochastic displacement, Ai, is given by a Gaussian vector with the prop-erties

< Ai >= 0, < Ai ·A j >= 6D0Δtδi j. (2.9)

In a typical simulation in this thesis, the input parameters, such as the timestep, temperature, and viscosity, are first specified. An initial configurationis generated according to the method of constructing a network (as describedin section 2.2). A random number is generated for each unit which describesthe Brownian motion, and the total force on each particle is calculated. Thepolymer units in the system are displaced according to the integrated Langevinequation of motion (Eq. (2.8)). These last two steps are repeated, resulting ina trajectory that specifies how the particle positions vary with time, which maythen be analysed.

2.4.3 Algorithm for Brownian dynamics on S3

The Brownian dynamics algorithm for generating a trajectory using sphericalboundary conditions looks slightly different compared to the one in IR3, al-though the principles are the same. The new position of particle i originally atposition pi is given by [104]

pi,new = cos(|Δri|/R)pi +Rsin(|Δri|/R)Δri/|Δri|, (2.10)

where R is the radius of the 3-sphere and

Δri =D0ΔtkBT

Fi +Ai. (2.11)

The functional form of the forces will be different since distances are now cal-culated along geodesics instead of straight lines. While the random numbers,Ai, in IR3 are sampled from the solution of the corresponding diffusion equa-tion (i.e., Gaussian random numbers), on S3 these are now obtained from thesolution of the diffusion equation on the 3-sphere, which is given by

P(ζ ,Δt) ∝1

sinζ

∞

∑k=−∞

(ζ +2πk)exp(−R2(ζ +2πk)2/4D0Δt), (2.12)

where ζ denotes the angular displacement. For simulations of polymer sys-tems, the solution can be simplified with very good accuracy by keeping onlythe k = 0 term in the sum. The details and derivation of this algorithm can befound in [104].

31

3. Simulations of Polymer Networks on the3-sphere

It is sometimes important for science to know how to forget the things she issurest of.— Jean Rostand

Computer simulations of bulk systems commonly use periodic boundary con-ditions to reduce surface effects. As discussed in section 2.3, spherical bound-ary conditions are an appealing alternative to periodic boundary conditions. ABrownian dynamics algorithm for simulating diffusion processes of particleson S3 has previously been developed, [104, 105] where it was also shown thatusing spherical boundary conditions results in the same structural propertiesand dynamics for polymer chains as in the ordinary Euclidean 3-dimensionalspace. With this algorithm available, the use of spherical boundary conditionshas been extended in this thesis to also include bulk gels, by developing a novelmethod for constructing a closed polymer network. This model would, for ex-ample, be a suitable option in further applications exploring the properties ofpolyelectrolyte gels.

3.1 Construction of a closed polymer network on S3

Papers I and II outline alternative ways for constructing a closed network thatis embedded on the 3-sphere. The task of forming a closed network is onethat requires bookkeeping in order to keep track of the connectivity. In a firstapproach, the network is formed in a multi-step method, starting in IR3, byplacing the cross-linking nodes on a cubic lattice structure (as described inpaper I). Next, specific nodes on the boundaries of two such identical cubesare connected together and the coordinates of all nodes are mapped to a diskand finally to the northern and southern hemispheres of S3. The structure ofthe resulting networks is analysed in paper I. A possible disadvantage of thisapproach, however, is that the underlying regular starting structure introducesa bias in the chain length distribution. Also, the symmetry of the constructionand the way in which the two cubes are connected mean that the number ofcross-linking nodes cannot be chosen freely. A more direct way to form aclosed network is to randomly distribute the cross-linking nodes on S3 and

32

connect each node to its four nearest neighbouring nodes that have vacancy (asoutlined in paper II). This method offers the advantage of avoiding any bias inthe initial node distribution. A straightforward implementation, however, leadsto the presence of a number of long chains, as nodes within proximity startto become fully occupied. This may be corrected by performing a process ofreconnection to reduce long distances of the node-node connections. The basicprinciples of this approach are the same as described in section 2.2, except thatdistances here are measured along geodesics.

0

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

0.18

0 5 10 15 20 25 30 35 40

Re

lative

fre

qu

en

cy

End-to-end distance, R’ee

φ=0.02φ=0.02φ=0.03φ=0.05

(a)

0

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

0 5 10 15 20

Re

lative

fre

qu

en

cy

Pore size

φ=0.02φ=0.02φ=0.03φ=0.05

(b)

Figure 3.1. Equilibrium probability distribution for (a) the end-to-end distance and (b) the poresizes, both scaled with the diameter of a network unit. Unfilled symbols represent a smaller sys-tem size and φ = 0.02. The error bars correspond to the standard deviation for five independentinitial network structures.

33

For the purpose of studying the diffusion of particles in a bulk gel, polymernetworks with three different volume fractions (φ ) are generated directly onS3 and subsequently equilibrated. The relaxed structures are analysed withrespect to the end-to-end distribution and the pore size distribution, wherethe latter is determined by using an expanding bubble scheme. [129] In thisscheme, a small bubble is inserted at a non-overlapping random position withina fixed network configuration and allowed to diffuse and expand at each timestep. Excluded volume interactions between the bubble and the network unitsmean that the bubble will eventually become trapped and can no longer ex-pand. The final bubble diameter is used to define the size of the pore in whichit has become lodged. Repeating this process for a sufficiently large num-ber of bubbles inserted at random positions results in a distribution of poresizes. The method of inserting the bubbles, however, is biased towards largerpores, which may be corrected for by only accepting unique bubbles that donot overlap with each other.

The pore size distribution is correlated with the chain end-to-end distribu-tion, as shown in Figure 3.1 for different volume fractions. A lower volumefraction corresponds to a wider distribution of end-to-end distances, whichtranslates to a broader distribution of pore sizes.

3.2 Diffusion of particles through a polymer gelIt is rather common in computer simulations of diffusion processes to treat acrowded environment as an assembly of immobile obstacles. [129–133] Thismay be a reasonable approximation for stiff network structures, but may notcapture the correct properties for more flexible networks. It is therefore rele-vant to also explore the role of both static and dynamic obstacles on the diffu-sive motion of a tracer particle.

A single tracer particle is introduced at a random, non-overlapping positionin the network, where the tracer particle diameter, d′, is chosen from the poresize distribution (shown in Figure 3.1b). A range of particle sizes is selectedto probe different length scales of the porous network structure. The displace-ment of the tracer particle as it diffuses in the gel is monitored as a function oftime for 100 different starting positions.

For a particle diffusing freely in IR3, the mean square displacement is givenby the simple relation < r2 >= 6D0t. This proportionality is, however, onlyvalid in the limit of short times for particles diffusing freely on S3, when theyhave not yet started to experience the curvature. Since the particles are unableto escape to infinity, the long-time limit of the mean square displacement onS3 is a constant given by [104]

< r2 >= R2(

π2

3− 1

2

), (3.1)

34

where R is the radius of the 3-sphere.The diffusion coefficient may be obtained through the relation [134]

< cosζ >= exp(−3Dt/R2), (3.2)

where ζ denotes the angular displacement of the diffusing particle. The dis-placement of the tracer particles, expressed as − ln< cosζ > as a function ofthe reduced time t/τ (where τ = R2/3D0), as they diffuse through both a staticand a dynamic network is shown in Figure 3.2. The smallest tracer particles

0

0.05

0.1

0.15

0.2

0 0.05 0.1 0.15 0.2

-ln<

cos

ζ>

t/τ

d’

d’=1.0d’=2.5d’=3.5d’=4.5d’=7.5d’=9.5

d’=11.0d’=12.5d’=14.0d’=15.0

Figure 3.2. The logarithm of < cosζ > for tracer particles with different sizes diffusingthrough a network with φ = 0.02. For each label, upper (black) curves represent a dynamicnetwork whereas the corresponding lower (red) curves represent a static network.

experience the environment as a connected porous medium with little obstruc-tion, diffusing freely with a constant slope close to unity. On this length scale,the dynamics of the network has no effect on the mean square displacement.For tracer sizes more comparable to and greater than the average size of thepores (i.e., for larger tracer sizes or polymer volume fractions), the networkobstructions become more significant and the reduced accessible volume forthe particles leads to lower diffusion rates and a non-linear time dependenceof the mean square displacement. While the qualitative trends are the samefor diffusion in the static and dynamic networks, the quantitative differenceincreases as the environment becomes more crowded. Within the static net-works, the largest particles appear to have become permanently trapped. Thesetrapped particles are forced to escape by their own Brownian motion. For thedynamic networks, however, there will be transient openings through whichthe particle can escape. By looking at the individual particle displacements,it can be seen that although each individual particle may not have necessarily

35

traveled farther in the dynamic networks, the particles that do manage to es-cape through the fluctuating pores contribute to a larger displacement, leadingto overall greater diffusion rates.

0.001

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φ=0.02φ=0.03φ=0.05

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Figure 3.3. Reduced diffusion coefficients for static (open symbols) and dynamic (filled sym-bols) networks of different volume fractions as a function of tracer size, shown on a logarithmicscale. The inset shows the same plot on a linear scale.

The reduced diffusion coefficient, D/D0, is shown in Figure 3.3 on a loga-rithmic scale as a function of tracer size. The diffusion coefficient decreasesfor larger tracer sizes, being larger where mobility of the polymer network isincluded. The plot quantitatively shows the importance of the dynamics of thegel matrix in facilitating the diffusional transport for larger particle sizes thatwould otherwise have become permanently trapped. One should also note thatdiffusion is still possible even for particle sizes well above the mode (i.e., thepeak) of the pore size distribution.

36

4. Investigating the Collapse Dynamics ofCore-Shell Nanogels

The most exciting phrase to hear in science, the one that heralds new discove-ries, is not ’Eureka!’ (I found it!) but ’That’s funny ...’— Isaac Asimov

A central issue in many technological fields is the development of new poly-meric materials, where the properties have been tailored for a specific end-use. Multi-stimuli responsive composites, which incorporate a combinationof two or more stimuli-responsive polymers, play an important role in thedevelopment of novel smart materials. One type of multi-responsive designconsists of spatially separating the different kinds of polymers into a core anda shell, forming so-called core-shell particles. These particles are, for exam-ple, of interest as controlled drug delivery vehicles.1 Various factors influencethe degree of swelling/deswelling of the two compartments, such as the shellthickness and the cross-linking density. [135–138] Tailoring core-shell partic-les for specific applications thus requires knowledge of the detailed struc-tural properties under different swelling conditions. The local swelling andcollapse of polymer gels have been investigated in simulations using perio-dic boundary conditions. [139–145] Some simulations have considered dis-crete gel particles [146–149] and fewer have looked at core-shell nanoparticlegels. [107, 150]

In this chapter, we will explore the dynamics of core-shell particles whenthe solvent quality is changed (paper III). The gel particles are created witha homogeneous distribution of cross-linkers and equilibrated with a soft re-pulsive potential between non-bonded units (Eq. (2.4)). To cause a collapse,the inter-bead potential is switched to include an attractive term through a LJpotential (Eq. (2.5)). The quality of the solvent may be changed by varyingthe LJ interaction parameter, such that a decrease in solvent quality corres-ponds to increasing εLJ . The correlated changes in the structure of the coreand shell that arise from the mechanical coupling of the two compartmentsare analysed for different cross-linking densities (ρn), core/shell mass ratios(γ), and εLJ . The results below are expressed using the dimensionless para-meter ε ′ = εLJ/(kBT ) and time τ = D0t/4a2, and all lengths are scaled withthe diameter of a polymer unit (2a).

1For example, a poorly water-soluble drug could be localised in the core compartment, with theshell acting as a protective barrier from the hydrophilic external environment.

37

4.1 General features of a collapsing homopolymer gelWe first examine the collapse of homopolymer gel particles with differentcross-linking densities. An example of such a collapse is shown in Figure4.1 at two different stages, illustrating the initial local formation of clustersand the final compact globule.

Figure 4.1. Simulation snapshots using VMD [151] of a homogeneous network taken shortlyafter starting the collapse (left image), illustrating locally formed clusters, and at the end of thesimulation (right image), when the network has collapsed into a globule. Here, ρn = 0.59 ·10−2

and ε ′ = 2.0.

The time dependence of the radius of gyration of the nanoparticle is shownin Figure 4.2. A faster collapse rate is achieved by increasing the cross-linkingdensity or the LJ parameter, where increasing the latter also leads to a morecompact final state. While the radius of gyration captures the macroscopic

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45

50

0 2000 4000 6000 8000 10000 12000

Rg’

τ

ε’=0.5ε’=0.75

ε’=1.0ε’=2.0

Figure 4.2. Time dependence of the radius of gyration for homopolymer networks col-lapsing with different ε ′. The filled symbols correspond to a higher degree of crosslinking,ρn = 2.0 ·10−2, and the hollow symbols to ρn = 0.59 ·10−2. The total number of network unitsis approximately the same.

38

volume change, this parameter alone is not sufficient to distinguish betweenthe different stages of the network collapse.

The coil-to-globule transition of single polymer chains is often describedby a two-stage model proposed by de Gennes, [152] where connected blobsof locally collapsed regions are first formed along the chain backbone, fol-lowed by a slower process of chain contraction and thickening to form acompact globule.2 Computer simulations have demonstrated the formation of

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<C

s>

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(b)

Figure 4.3. (a) Number of clusters as a function of time (where the inset shows an extendedtime scale), and (b) normalised average cluster size over time for homopolymer networks.

2Since de Gennes, other variations have been proposed to describe the collapse. See, for exam-ple, refs [153–155].

39

pearl-necklace conformations during the collapse of polymer chains in poorsolvents. [156–162] Such conformations have also been observed in simula-tions of polymer gels. [141] To be able to identify the different stages of acollapse, we define a cluster as the smallest set of non-bonded units that arewithin a critical inter-particle distance. The growth and merging of the clus-ters formed during the collapse is followed by computing the average clustersize, normalised by the total number of polymer units (M) in the network, i.e.,< Cs >= 1

M∑cnc(c)∑nc(c)

, where c is the cluster size, and nc(c) is the number ofclusters of size c.

The time evolution of the number of clusters, as well as the average clustersize, are shown in Figure 4.3. The results suggest that the network collapsealso takes place in two stages, similar to the kinetics of a single chain un-der poor-solvent conditions. In the first stage, there is a rapid formation oflocally compact small-sized clusters, followed by a cross-over to a coarsen-ing stage, where the clusters merge and the network forms a more compactglobule. Both through visualisation and by analysis, it can be seen that theclusters are initially formed near cross-linking nodes, due to the higher den-sity of units in these regions. For larger values of the LJ parameter, < Cs >reaches unity, while for lower values of ε ′, the attractive forces are too weak tocause a complete compaction and instead there are fluctuations, with clustersmerging and separating.

4.2 Core collapse dynamicsHaving looked at the collapse of homopolymer networks, we now turn to thedynamics of nanoparticles with a core-shell morphology. By switching thesoft repulsion to a LJ potential between a limited set of chains, namely thosewithin a predefined distance from the centre of mass of the nanogel particle, itis possible to bring about a collapse of only the core, while the shell remainsswollen.

Similar to the homopolymer collapse, there is first a rapid stage where smallclusters are initially formed near the cross-linking nodes in the core, followedby the merging of clusters and ultimately the collapse of the core. Since thecore and shell are coupled through covalent bonds, the collapse of the core willnecessarily exert a retracting force on the shell. This can be seen by looking atthe radius of gyration of the entire particle as a function of the size of the core,shown in Figure 4.4. (The inset shows the decay of the radius of gyrationas a function of time.) The relaxation time for the shell units is, however,significantly longer than the core collapse transition time; the shell continuesto relax well after the core has formed a globule. For larger values of γ , i.e., athinner shell, the shell units are more strongly affected by the collapsing core.

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Rg’

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Figure 4.4. The radius of gyration of the entire particle shown as a function of the size of thecore (for ρn = 2.0 ·10−2 and γ = 0.2). The inset shows the decay of the radius of gyration forthe entire particle (filled symbols, upper set) and the core (hollow symbols, lower set).

4.3 Shell collapse dynamicsFor the reversed case, the LJ potential is only introduced between units in theshell, thereby inducing a shell collapse around a swollen core. From the radiusof gyration, shown in Figure 4.5, it can be seen that the final size of the gelparticle with a collapsed shell is smaller than that of the original core. Thishas also been observed in experiments, suggesting that the shell restricts thecore to a more compact structure. [136] In the first stage, the initial forma-tion of clusters in the shell causes the core to slightly expand, shown as a firstrecoil (in Figure 4.5). During the coarsening stage, as the clusters begin tomerge, the core starts to become compressed. An interesting feature, however,is that there is a second recoil for the larger LJ parameters. Visualisation of thetrajectories reveals a new type of structure, viz. a core-shell inversion, wherethe shell has collapsed into the core compartment and squeezed out the core(see Figure 4.6). This inversion becomes more pronounced with decreasingdegree of cross-linking and increasing LJ parameter. Increasing the core/shellmass ratio prevents the shell units from permeating the core to the same ex-tent, instead resulting in large clusters nested inside the core compartment butsegregated by the bulky core. Such inversions and patchy surface structureshave recently also been observed in simulations by Ghavami et al., where itwas also shown that the structural changes are reversible. [150] By carefullychoosing, e.g., the core/shell mass ratio, this inversion mechanism of exposingthe core polymers to the environment could possibly be used to design novelsmart materials.

41

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ε’=0.5ε’=0.75

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(b)

Figure 4.5. The radius of gyration of the entire particle shown as a function of the size of thecore for networks for ρn = 2.0 ·10−2, (a) γ = 0.4 and (b) γ = 0.1.

Figure 4.6. Simulation snapshots using VMD [151] taken at the end of the trajectory for acollapsing shell (with ρn = 2.0 · 10−2 and γ = 0.2), from left to right: ε ′ = 0.5,0.75,1.0, and2.0. Shell units are shown in pink and core units in blue.

42

5. Investigating the Mechanical Properties ofBimodal Gels

Research is to see what everybody else has seen, and to think what nobodyelse has thought.— Albert Szent-Györgi

Many applications of polymer materials require particular mechanical proper-ties, such as high stretchability and high toughness. In tissue engineering, forexample, materials for the replacement of damaged cartilage in human jointsor of spinal discs need to be able to deform under compression and tensionwithout breaking. Most conventional hydrogels, however, are fragile, exhibit-ing a low mechanical strength and elastic modulus. Various methods havebeen used over the past decades to reinforce polymers and this is still an on-going research field.1

One way of strengthening polymer gels is by forming cross-linked networkswith a bimodal molecular weight distribution, which may be considered a typeof double network. In some cases, these bimodal networks have been foundto be unusually tough compared to their unimodal short-chain or long-chaincounterparts, depending on, e.g., the fraction of short chains and the relativemolecular weights of the short and long chains. [54–60] While the studies onbimodal networks have primarily focused on elastomers, there has not beenany systematic study on the deformation behaviour of bimodal gels.

In this chapter, we isolate the effect of bimodality and investigate the de-formation behaviour and mechanical properties of bimodal gels by applyinga continuous boundary driven uniaxial tension (along the z-axis) until failure.The macroscopic deformation may be analysed by following the orientationand rupture of chains, as well as the stress-strain relation. The chain segmen-tal orientation along the strain axis is described by the second order Legendrepolynomial, [167]

P2 =< 3cos2 ϕ −1 > /2, (5.1)where ϕ is the angle between the direction of extension and the local chainaxis of the polymer segment. In simulations, the mechanical stress tensor onthe atomistic scale may be evaluated according to [41]

σ =− 1V ∑

i∑j>i

fi j(ri − r j). (5.2)

1See, for example, [48, 163–166].

43

Results below are presented using the reduced stress σ ′ = (4πa3/(3kBT ))σ .The strain is given by α = (Lz −L0)/L0, where Lz is the current distance bet-ween the grips and L0 is the initial distance.

5.1 Uniaxial deformation of homogeneous bimodal gelsThe deformation of defect-free bimodal networks with various fractions ofshort chains ( fs) is first investigated and compared with equivalent unimodalnetworks with a similar cross-linking density (see paper IV). The networks aregenerated as cylindrical specimens, with a homogeneous distribution of shortand long chains of fixed lengths. Bonds that extend beyond a distance of 3aduring an elongation are severed.

Simulation snapshots taken at various stages during a deformation are shownin Figure 5.1.

Figure 5.1. Simulation snapshots using VMD [151] taken during uniaxial elongation of asystem with 5000 chains and fs = 0.90, showing the short (blue) and long (purple) chains oflength Ns = 6 and Nl = 30, respectively.

Applying tension in one direction leads to a greater average orientation ofchain segments in the direction of the strain axis, as shown in Figure 5.2,and eventually to bond breaking and network failure. The extent to whichthe networks can be stretched before failing depends on the fraction of shortchains, with lower values of fs leading to much larger ultimate strains.

Comparing the two different types of chains (in Figure 5.2), the short chainsare more strongly oriented than the long chains for all network compositions.The orientation of the short chains during deformation, however, is almostunaffected by changes in fs. This is a consequence of the homogeneous distri-bution of chains, which means that the short chains form percolating structuresbetween the mechanical grips (for the fractions of short chains considered

44

here). By looking at the initial orientation of the chains that have ruptured at

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fs=1.00fs=0.97fs=0.95fs=0.90fs=0.80fs=0.65fs=0.50

Figure 5.2. The average segmental chain orientation for networks with fs = 0.50− 1.00, forthe short (solid) and long (dashed) chains, shown separately.

failure, it can be shown that the first chains to rupture are the short chains thatare initially aligned along the strain axis.

As the applied strain eventually leads to stretching of the chain backbone,the stress, shown in Figure 5.3, increases dramatically. Networks containinga large fraction of short chains behave similar to a pure short-chain network,with a rapid increase of stress and failure after a relatively small additionalstrain, at a high ultimate stress. Decreasing fs decreases the ultimate stress.For all compositions considered, however, the stress is not uniformly sharedamong the short and long chains (see Figure 5.3b). Analogous to the orienta-tion curves, the stress on the short chains is much larger for all network com-positions, whereas the long chains are in comparison only marginally affectedat smaller strains.

While bimodal networks have been found to show improved mechanicalproperties compared to their constituent pure short-chain and long-chain net-works, it is also of interest to compare their behaviour with equivalent uni-modal networks having the same number-average molecular weight of chains,to see if this reinforcement is indeed a result of the bimodal distribution ofchain lengths. At small fractions of short chains, the ultimate properties ofthe bimodal networks are poorly predicted by the equivalent unimodal net-works. In the absence of defects and trapped entanglements, a bimodal molec-ular weight distribution alone is sufficient to alter the mechanical propertiesat lower fractions of short chains. The bimodal networks can be stretchedmuch further than the equivalent unimodal networks before reaching failure,giving rise to a greater toughness (i.e., area under the stress-strain curve). Thisdifference in mechanical properties is, however, reduced when fs is increased.

45

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(b)

Figure 5.3. (a) The scaled stress-strain curves for networks with fs = 0.50−1.00. The dashedlines correspond to unimodal networks with similar cross-linking densities (N = 8 (yellow), 10(green), 16 (purple)). (Note that the unimodal network with N = 16 has a cross-linking densitylying between fs = 0.50 and fs = 0.65). (b) The corresponding stress-strain curves, showingthe stress from the short and long chains. The solid lines represent the short chains and thedashed lines long chains.

5.2 Role of structural heterogeneitiesIn the previous section, we considered bimodal networks where the shortand long chains were homogeneously distributed. Experiments have, how-ever, found that bimodal networks often are spatially heterogeneous, as theshort chains tend to segregate into clusters during synthesis. [168–173] Whilemacroscopically phase-separated bimodal networks have generally been foundto be weak, [174, 175] some studies have found that bimodal networks withnanoscale clusters are reinforced, suggesting that there may also be a structuralcomponent to the reinforcement mechanism. [60,176] Structural effects on the

46

Figure 5.4. Schematic of a homogeneous and heterogeneous bimodal network, where the filledcircles represent the cross-linking nodes and the bold and thin lines correspond to short and longchains, respectively.

deformation behaviour and mechanical properties are considered by generat-ing bimodal networks containing dense clusters of cross-linked short chainsand comparing these with the homogeneous networks of the correspondingcross-linking densities (see paper V). A schematic of a homogeneous and het-erogeneous bimodal network is shown in Figure 5.4.

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’ ee)

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Homogeneous,shortHomogeneous,long

Heterogeneous,shortHeterogeneous,long

Figure 5.5. Probability distribution for the chain end-to-end distances (scaled with the particlediameter) for the short and long chains in a homogeneous and heterogeneous bimodal networkwith fs = 0.65. The solid lines show the undeformed equilibrium distribution and the dashedlines show the distribution for the deformed networks at failure.

Looking at the probability distribution of the end-to-end distances, shownin Figure 5.5, it can be seen that the long chains on average assume larger end-to-end distances at equilibrium in the unperturbed heterogeneous network, asclustering removes some of the constraints at the junction points. During elon-gation, both short and long chains are pulled apart in the homogeneous net-work, due to the intimate coupling between the chains, and it can be seen thatboth maxima have shifted at network failure. In contrast, the short chains in

47

the clusters are substantially shielded from stretching. The end-to-end dis-tribution of the short chains is almost unchanged after elongation, with onlya minor shoulder appearing as chains at the periphery of the clusters are de-formed.

During deformation, the shielding of short chains within the clusters meansthat these chains are significantly less oriented than their counterparts in thehomogeneous bimodal network, as shown in Figure 5.6. The short-chainorientation of the heterogeneous networks, however, increases with increas-ing fraction of short chains, due to the presence of larger agglomerates ofclusters that eventually percolate between the top and bottom grips. This trendis also observed in simulations of dynamically end-linked bimodal networks,which tend to form clusters during the end-linking. [177] Similar to the ho-

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Figure 5.6. Segmental chain orientation with increasing strain for the short chains in a homo-geneous (dashed lines) and heterogeneous (solid lines) bimodal network with fs = 0.65, 0.80and 0.90.

mogeneous bimodal networks, the first chains to rupture in the heterogeneousnetworks are the ones that are already aligned along the strain axis in theirunperturbed state.

Looking at the stress-strain curves, shown in Figure 5.7, it can be seenthat the ultimate stress is much lower for the heterogeneous networks thanthe corresponding homogeneous networks, even at large fractions of shortchains. Furthermore, the stress on the short chains is also significantly lowerin the heterogeneous networks (Figure 5.7b). These results are sensitive tohow firmly the clusters are embedded within the gel matrix, which is reflectedby the average number of connections between a cluster and nodes that donot belong to this cluster. When the number of connections is decreased, theshort chains in the clusters become even more isolated and the number ofelastically active short-chains is reduced. Consequently, the ultimate stressis decreased even further. While clustering does not affect the maximum ex-tensibility at small fractions of short chains, it does delay fracture at large fs,

48

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(b)

Figure 5.7. (a) Stress-strain curves for bimodal networks with fs = 0.65− 1.0. The solidlines correspond to the heterogeneous networks, the dashed lines to the homogeneous networks,and the dotted lines to the unimodal networks of the corresponding cross-linking densities.(b) The corresponding stress-strain curves, showing the stress from the short and long chains,for homogeneous and heterogeneous networks with fs = 0.65 and fs = 0.90. The solid linescorrespond to the short chains and the dashed lines to the long chains.

thereby slightly improving the toughness, also compared to equivalent uni-modal networks with the corresponding number-average chain length. Thiswould suggest that clustering might play a role in the reinforcement mecha-nism at large fractions of short chains.

49

6. Summary and Outlook

Polymers are widely used materials that can be found almost everywhere inour daily life. Polymer gels, in particular, are involved in many importantapplications. From both a fundamental and technological viewpoint, a deeperunderstanding of the structure-property relation for polymer gels could leadto the design of materials with properties tailored to specific applications. Forthis purpose, computer simulations can shed light on the underlying mecha-nisms of important phenomena in polymer systems, through the possibilityto vary different parameters independently of each other and to correlate thedetailed structural features with the physical properties.

In this thesis, Brownian dynamics simulations have been used to investigatethe structure, deswelling and deformation of chemically cross-linked polymergels with controlled structures, with the aim of contributing to the understand-ing of various structural parameters on the dynamics of polymer gels. In thefirst part, a new method was developed for constructing a closed, cross-linkedpolymer network. Simulations of polymers entail a large amount of computa-tional effort, due to the large system size and long simulation times involved.Methods for accelerating the simulation speed are thus invaluable for facilitat-ing more extensive studies. A novel approach for simulation polymer gels hasbeen proposed using spherical boundary conditions (SBC) to avoid possiblesurface effects, which could also be a suitable option to more efficiently simu-late charged systems. This is achieved by embedding the system on the surfaceof a ball in four dimensions, i.e., a hypersphere. Using SBC, the diffusion oftracer particles of various sizes through both a static and dynamic network ofdifferent volume fractions was simulated. A slowdown with increasing tracerparticle size was observed, eventually leading to trapped diffusion for particlesizes well above the peak in the gel pore size distribution. The quantitativeeffect of the network dynamics was illustrated through an increase in the dif-fusion coefficient of the tracer particles compared to a static gel structure.Mobile polymer chains facilitate the escape of trapped tracer particles, andit was shown that this escape mechanism becomes particularly important forlarge particles.

With the method of constructing a closed network at hand, two techno-logically relevant properties of polymer gels were investigated, namely theirstimuli-responsive behaviour and mechanical properties. The collapse dyna-mics of core-shell nanogel particles of various degrees of cross-linking andmass ratios between the core and shell was studied. Core-shell particles areparticularly interesting as multi-responsive materials that can respond to varia-tions in their environment, e.g, for use as drug carriers. In the simulations, a

50

collapse was induced by switching the pair interactions from an effective ex-cluded volume to a Lennard-Jones (LJ) potential, for different strengths of theLJ interaction parameter to show the influence of the solvent quality. Twodistinct stages could be observed after changing the quality of the solvent: aninitial rapid formation of small-sized clusters near the cross-linking nodes anda subsequent cross-over to a slower coarsening stage. A faster collapse ratewas observed by increasing the degree of cross-linking or the LJ interactionparameter, where the latter also resulted in a more compact final state. Incontrast, for the smallest value of the LJ parameter, the attractive forces weretoo weak to cause a complete collapse, resulting instead in the formation oftransient clusters. In the case of a shell collapsing around a swollen core, theparticles formed structures with a patchy surface through core units protrud-ing from the shell. One particularly interesting feature was observed in somecases, namely an inversion of the core-shell particle. This mechanism of ex-posing the core polymer chains to the environment could provide a novel routefor designing functional nanomaterials.

The final parts of the thesis concerned the deformation of bimodal gels (i.e.,networks consisting of both short and long chains) with various fractions ofshort chains, subject to uniaxial elongation. Understanding the tunability ofthe mechanical properties of materials is important to fulfil practical require-ments and to expand the applicability of the material. Bimodal networks havepreviously shown substantial improvements in strength and toughness com-pared to their corresponding short-chain or long-chain unimodal networks.However, while previous studies on bimodal networks have focused on elas-tomers, a systematic study of bimodal hydrogels has been missing. Two typesof network structures were considered here: in the first type, the short andlong chains were distributed homogeneously within the network to isolate theeffect of bimodality. In the second type, the short chains were arranged indense clusters to also investigate the structural component, since experimentshave shown that bimodal networks are usually spatially heterogeneous, withshort chains aggregating into clusters during the synthesis. In both types ofgels, the first chains to rupture were short chains initially aligned along thestrain axis. For the homogeneous bimodal networks, the bimodal molecularweight distribution alone was sufficient to strongly alter the mechanical prop-erties of the network at lower fractions of short chains, compared to unimodalnetworks of the corresponding number-average chain length, resulting in botha greater maximum extensibility and toughness. This difference became lesspronounced for larger fractions of short chains. In contrast, the heterogeneousbimodal networks containing clusters were generally weaker in comparison,due to the substantial shielding of the short chains within the clusters. Net-works with small scale clustering did, however, show an enhancement in theultimate strain and also a minor improvement in the toughness for large frac-tions of short chains. These results suggest that clustering can be anticipated toplay a role in the reinforcement mechanism at large fractions of short chains.

51

In real gels, however, there may be collateral effects from a short-chain lengthpolydispersity, network defects, and trapped entanglements, which have so farbeen excluded from the model. Further studies are needed to investigate theseeffects on the mechanical properties of polymer gels.

52

7. Acknowledgements

Many people have contributed to this thesis in one way or another over theyears. Although this list might not be comprehensive, those not mentioned arecertainly not forgotten. I would like to specifically thank the following people:

First and foremost, my supervisor, Christer Elvingson, for always keeping hisdoor open to discuss questions, ideas, and problems. For his time, guidance,and attention to details, from the moment I first stepped into his office lookingfor a project and into the world of polymers.

My co-supervisor, Tobias Ekholm, for many insightful discussions and forsharing his knowledge of objects in higher dimensions.

Per Hansson, for discussions on experimental aspects of polymer gels and forsharing ideas.

Dóra Bárdfalvy, Junhao Dong, and Alex Thomson, for their enthusiasm anddedication.

The support team at UPPMAX and NSC for decrypting the various error mes-sages over the years.

Pavlin Mitev, for helping solve system errors and showing me how to useTriolith. It was very much appreciated.

Martin Agback, Peter Lundström and Patrik Lindahl, for resolving localIT issues.

The administrators, including Jessica Stålberg, Eva Larsson, Eva Ohlsson,and Susanne Söderberg.

Ted Karlsson, for always managing to have a poster ready on time, even onshort notice.

My past and present colleagues of the physical chemistry group, includingEmad, Hanna, Jens, Kari, Mohammad, Robin, Sonja, Viktor, and Tobias,with most of whom I’ve had the pleasure of sharing lab teaching duties.

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The theoretical chemistry group (past and present), including Roland, Mar-

cus, Erik, Meiyuan, Morgane, Konrad, Lasse, Michael, Rahul, Orlando,and Piotr, for making me feel at home at the department. Specifically, Igna-

cio, for showing me how to use VMD, and Nessima, for offering help when itwas needed.

For making life outside of work enjoyable, I would like to thank:

The EE group, including Marisol, Tobias, Sussi, Sara, Adam, and Nils.

The badminton group, including Tabea, Christian, Ravi, Scarlet, Miryam,Juri, Tobias, Hitesh, and more.

Charlotta, for our wonderful discussions about cats, life, and research.

More distant connections that have not been forgotten, including Dimitri andMikey.

My parents, for teaching me the value of hard work.

Schubby and Xena, for unconditional love and affection.

Finally, thank you, Mickaël, for always being there to listen and to providefeedback, for keeping a sense of humour all the times I lost mine, and forsharing your life with me. I love you.

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8. My Contribution

The following is a brief summary of my specific contribution to the paperspresented in this thesis:

Paper I: Implemented the multi-step method for constructing a closed net-work on a hypersphere. Designed and implemented the method of formingthe network directly on the hypersphere. Performed all the simulations andanalyses. Took part in writing the manuscript.

Paper II: Designed the method for reconnecting the network. Implementedthe algorithm. Formulated the scientific questions together with co-author.Performed all the simulations and analyses. Main responsible for writing themanuscript.

Papers III-V: Initiated and formulated the scientific questions together withco-author. Developed and implemented the algorithms. Performed all thesimulations and analyses. Main responsible for writing the manuscript.

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9. Summary in Swedish

Det skulle vara svårt att föreställa sig ett modernt samhälle utan den typ avmolekyler vi kallar polymerer. Produkter som innehåller polymerer finns över-allt runt omkring oss, från textilier till plast, målarfärg, epoxylim och mångamånga fler. Polymerer finns också i alla levande organismer i form av t.ex.DNA, proteiner och kolhydrater.

Polymerer är stora molekyler (makromolekyler) som är uppbyggda av ettstort antal ihopkopplade mindre molekyler (s.k. monomerer). Variationer i ty-pen av monomerer och i vilken ordning monomererna är ihopkopplade gerupphov till material med en enorm mångfald av egenskaper. Många polymererbestår av en linjär kedja men de kan också bilda t.ex. grenade eller stjärnforma-de strukturer. I denna avhandling är vi intresserade av det fall där kedjorna ärtvärbundna i något lösningsmedel och bildar ett stort tredimensionellt nätverk.Denna typ av system kallas en gel. Exempel på geler är gelatin och agar. Någraav egenskaperna hos polymergeler som är viktiga för deras användning är de-ras förmåga att absorbera stora mängder lösningsmedel samt deras mekaniskaegenskaper. Polymerer som kan genomgå en kraftig strukturförändring vid enliten förändring av någon yttre variabel (såsom temperatur, pH, eller salthalt),är av stort intresse som “intelligenta material” för medicinska tillämpningar,t.ex. för att kontrollera frisättningen av läkemedel, som självjusterande ven-tiler eller som artificiella muskler. De mekaniska egenskaperna hos geler ärockså viktiga i många praktiska tillämpningar som bygger på att de kan fun-gera som stödjande material. För många tillämpningar kan det emellertid varanödvändigt att förbättra de mekaniska egenskaperna. Hos konventionella hyd-rogeler (d.v.s. geler där lösningsmedlet är vatten) är dessa ofta dåliga och ären begränsande faktor.

För att förbättra vår förståelse av material där polymerer ingår och ävenför att tolka resultaten från experiment är datorsimuleringar ett ovärderligtverktyg, genom möjligheten att kunna variera olika parametrar oberoende avvarandra samt att kunna relatera materialegenskaperna till den atomära struk-turen. Den detaljerade strukturen hos en gel är svår att kontrollera vid fram-ställningen, vare sig det är i ett laboratorium eller industriellt. Det kan finnaslösa kedjor såväl som intrasslingar av polymerkedjorna. Det kan också finnasområden som innehåller en hög koncentration av tvärbundna polymerkedjoroch mer utspädda domäner, vilket medför en heterogen struktur. De fysikalis-ka egenskaperna hos geler påverkas av hur kedjorna i nätverket är sammanlän-kade, men dessa strukturella defekter och inhomogeniteter är svåra att mäta iexperiment och komplicerar en analys av sambandet mellan den mikroskopis-ka strukturen och de efterfrågade materialegenskaperna. Strukturen hos en gel

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kan dock väljas noggrant i en simulering, vilket gör det möjligt att på ett syste-matiskt sätt studera olika effekter separat för en serie nätverksmodeller underfullt kontrollerade förhållanden. Från både ett grundläggande och ett teknisktperspektiv kan en djupare förståelse av gelers struktur och egenskaper leda tillutvecklingen av material som är skräddarsydda för specifika tillämpningar.

En utmaning är att simuleringar av polymerer vanligtvis kräver en stor be-räkningsinsats på grund av det stora antalet partiklar och de långa tidsskalornasom är inblandade. Utvecklingen av nya algoritmer och metoder för att öka si-muleringshastigheten är således ovärderliga för att underlätta mer omfattandestudier och för att kunna lära oss mer om dynamiska processer hos polymerer.

I denna avhandling används en simuleringsmetod som kallas Brownsk dy-namik, vilket är en snabbare metod för simuleringar av stora system jämförtmed atomistiska simuleringar. I den första delen har en ny metod utvecklatsför att skapa ett sammanknutet, tvärbundet nätverk utan lösa ändar, där samt-liga kedjor är elastiskt aktiva. I avhandlingen beskrivs ett nytt sätt att simulerapolymergeler genom att använda sfäriska randvillkor för att undvika yteffekter.Detta uppnås genom att bädda in systemet på ytan av en boll i fyra dimensio-ner, d.v.s. en hypersfär, och låta kedjorna röra sig på ytan av sfären istället föri en tredimensionell “låda”. Genom att använda sfäriska randvillkor har diffu-sionen simulerats för partiklar av olika storlekar genom både ett statiskt och ettdynamiskt nätverk för olika koncentrationer av polymerkedjor. Den kvantita-tiva effekten av nätverksdynamiken kan observeras i ökningen av partiklarnasdiffusionskonstant. Dynamiska polymerkedjor underlättar alltså för de infång-ade partiklarna att röra sig och i avhandlingen visas att denna mekanism blirsärskilt viktig för stora partiklar. Speciellt kan man observera en diffusion ävenför partikelstorlekar långt över maximat i porstorleksfördelningen.

Med denna metod till hands för att konstruera ett slutet nätverk undersöktestvå tekniskt relevanta egenskaper hos polymergeler: strukturändringen (“kol-lapsen”) då lösningsmedlet ändras från bra till dåligt samt de mekaniska egen-skaperna vid en töjningsdeformation. Dynamiken hos kärnan/skalet för nano-gelpartiklar studerades då dessa kollapsar i ett dåligt lösningsmedel (där detsenare modellerades med en Lennard-Jones potential mellan polymerkedjor-na). Inverkan av tvärbindningsgraden och massförhållandet mellan kärnan ochskalet vid kollapsen undersöktes. Man kunde se att dynamiken för gelkollap-sen liknar den för en enkel kedja: ett första stadium där kedjeenheterna näratvärbindningsnoderna samlas i små kluster vilket följs av en långsammare pro-cess där antingen kärnan eller skalet (beroende på vilken del som kollapsar)bildar en mer kompakt struktur. En snabbare kollapshastighet observerades ge-nom att öka graden av tvärbindning eller genom att försämra lösningsmedlet.I fallet där skalet kollapsar kring kärnan observerades en speciellt intressantegenskap, där kollapsen resulterade i en inversion av partikeln, varvid kärnanvändes ut mot omgivningen. Denna mekanism för att exponera kedjorna i kär-nan för den omgivande lösningen skulle kunna innebära en ny möjlighet förutformningen av funktionella nanomaterial.

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I denna avhandling undersöktes också deformationen av nätverk med enbimodal molekylviktsfördelning (d.v.s. med korta och långa kedjor samman-kopplade i ett nätverk) under mekanisk töjning. En förståelse av de mekaniskaegenskaperna är av stor betydelse för att kunna uppfylla praktiska krav och föratt utvidga materialets användbarhet. Bimodala nätverk har tidigare visat vä-sentliga förbättringar i styrka och seghet jämfört med ett motsvarande nätverkbestående av endast korta eller långa kedjor. Tidigare studier har emellertid fo-kuserat på elastomerer, medan en systematisk studie av bimodala hydrogelerhar saknats. Den sista delen av denna avhandling fokuserar på deformationenoch förstärkningsmekanismen för bimodala geler. Två typer av nätverkskon-struktioner studerades: i den första typen fördelades de korta och långa kedjor-na homogent i nätverket för att isolera effekten av bimodalitet utan att behövata hänsyn till möjliga effekter av exempelvis kedjeaggregation. I den andratypen bildade de korta kedjorna kluster, vilket möjliggjorde en undersökningäven av strukturkomponenten. Experiment har nämligen funnit att bimodalanätverk är heterogena, med en tendens för de korta kedjorna att aggregera un-der syntesen. I båda typerna av geler var de första kedjorna som bröts undertöjningsprocessen korta kedjor som redan från början var orienterade längstöjningsaxeln. För de homogena nätverken var den bimodala molekylviktsför-delningen tillräcklig för att kraftigt påverka de mekaniska egenskaperna vidlägre andel av korta kedjor, jämfört med unimodala nätverk av motsvaran-de genomsnittlig kedjelängd. En bimodal molekylviktsfördelning ledde till attmaterialet kunde dras ut ytterligare samt till en ökad seghet. Denna skillnadblev mindre uttalad för en större andel av korta kedjor. Däremot visade närva-ron av kluster i allmänhet inte en förbättring i de mekaniska egenskaperna, pågrund av den avsevärda avskärmningen av de korta kedjorna. Resultaten försmå kluster vid en hög andel korta kedjor visade en förbättring i den maxima-la töjningen och även i segheten. Dessa resultat tyder på att det skulle kunnafinnas en strukturkomponent i förstärkningsmekanismen. I praktiken kan detdock finnas andra bidrag från t.ex. defekter i strukturen samt intrassling av po-lymerkedjor som hittills har uteslutits från modellen. Fler studier behövs föratt undersöka effekten av dessa på de mekaniska egenskaperna.

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