Modelling the Mechanical Properties of Colloidal Gels

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2022

UNIVERSIDADE DE LISBOA FACULDADE DE CIÊNCIAS DEPARTAMENTO DE FÍSICA

Modelling the Mechanical Properties of Colloidal Gels

João Lourenço Costa Grade Neves

Mestrado em Engenharia Física

Dissertação orientada por:

Dr. Cristóvão de Sousa Dias

Prof. Dr. Nuno Miguel Azevedo Machado De Araújo

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Acknowledgments

I would like to thank everyone that has accompanied me throughout the years of 2021-2022 when this work was developed. I want to thank professors Nuno Araújo and Cristóvão Dias for, firstly, four years prior to this work introducing me to the world of Numerical Simulations and Computational Physics, proving that it was (and always will be) possible to be marvelled by science again and again. And secondly, for the support and mentorship given throughout this work, helping not only in the project’s scientific development but in my development as a critical person and a researcher as well.

I would also like to thank my family for the care and support given and for the opportunity of pursuing a higher education, a chance which is, most unfortunately, reserved for a few. I would like to thank my girlfriend Marta which was always by my side, for the support, love, and specially, for believing me the most, even when I do not. Thank you. I want to thank the Ginkgo Biloba organization, family as well, for the constant presence, the reassuring smile and companionship which filled this process with joy.

And finally, I would like to thank everyone involved in my journey at the Faculty of Sciences. My friends, for the help given, the sense of union and, for in fact, the friendship. My teachers, for their patience and the opportunity to learn. I would also like to thank, specially, professor Guiomar for the genuine care and attention given to us students during every moment and professor Jo˜ao Sim˜oes for showing from a young age the beauty in science and the passion in learning.

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Abstract

The identification of the necessary conditions for the emergence of elasticity in a gel is among the most fundamental challenges in gelation. Recent confocal microscopy experiments of colloidal gels suggest that mechanical metastability results when particles with at least six neighbors percolate. At CFTC, the spatial distribution of the coordination number at the onset of elasticity was studied in a systematic way. Numerical simulations of a model system with limited valence and directional revealed that the onset of elasticity coincides with percolation of particles with three or more neighbors. This result gives support to the percolation of the local isostatic environment as the necessary condition for mechanical metastability and provides insight into the elasticity of low-valence colloidal gels. However, it is not clear how the rotational degrees of freedom affect the onset of mechanical stability. In this thesis, the previously developed model is extended to vary the bending energy up to the limit of mobile linker on the surface of each particle. We study the effect of the bending energy on the mechanical properties of the gel.

Keywords —gels, colloids, molecular dynamics.

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Resumo

As condições necessárias para a gelificação, processo pelo qual um material se torna um gel, são ainda pouco entendidas e um dos desafios da f´ısica. A origem da estabilidade mecânica e a clara definição de gel são dois assuntos sob os quais a comunidade cientifica se debruça hoje em dia.

No âmbito deste trabalho, as propriedades mecânicas de um gel de coloides foram estudadas com recurso a métodos numéricos, mais propriamente, o conceito debending energyfoi introduzido num modelo de gel com valência e direcionabilidade limitadas. O gel de coloides é particularmente interessante devido às suas caracter´ısticas. A dimensão das part´ıculas que o compõe dá azo a energias de ligação altas e dif´ıceis de quebrar, além disso trata-se de um material elástico, ainda que a baixa densidade, amplamente utilizado em áreas como cosmética, indústria farmacêutica, construção, etc. . .

O trabalho feito no âmbito desta tese tem a sua origem em dois resultados importantes. Em 2019 Tsu- rusawa et al. demonstraram que a estabilidade mecânica em gels de coloides coincide com a percolação de part´ıculas com seis vizinhos ligados a si. No entanto, resultados recentes obtidos numericamente no CFTC mostram que, embora esta percolação seja uma condição suficiente para a estabilidade mecânica, não é necessária. Esta pode ser obtida através de part´ıculas com apenas 3 dos seus vizinhos ligados a si.

Este resultado foi obtido usando um modelo depatchy particlescom valência 3 e interações direcionais, tal como neste trabalho. Aspatchy particlessão núcleos duros com áreas na sua superf´ıcie que são usa- das para criar ligações entre as part´ıculas, patches. As part´ıculas agregam-se, criando ligações através dospatches, um gel será um agregado de part´ıculas cuja dimensão se estende ao longo de todo o espaço de simulação.

O comportamento de um material pode ser visto como tendo caracter´ısticas de um sólido (elástico) ou flu´ıdo (viscoso). A viscoelasticidade é a resposta complexa no tempo de um material a um certo est´ımulo, para a estudar é aplicada ao material uma deformação de shear e é medida a sua tensão de shear. Estes valores são normalmente representados numa curva de tensão-deformação.

A capacidade que um material tem de armazenar ou dissipar energia é medida através de dois co- eficientesG^′ eG^′′, o módulo de armazenamento e de perda. O primeiro representa o comportamento elástico do material enquanto que o segundo representa o comportamento viscoso.

De modo a examinar o material, três técnicas foram utilizadas, Reologia de Oscilação,Step Sheare Microreologia. A técnica deStep Shearconsiste em aplicar uma deformação deshearprogressivamente ao material e medir a sua tensão, obtendo no final uma curva de tensão-deformação. A Reologia de Oscilação é feita aplicando pequenas deformações de shear periodicamente com uma certa frequência.

A tensão é medida e a partir da´ı são calculados os coeficientesG^′₁ eG^′′₁ que representam os primeiros termos deG^′eG^′′. A microreologia utiliza apenas o movimento browniano para calcular os coeficientes G^′eG^′′, não aplicando quaisquer perturbações externas ao material. Esta técnica pode ser feita utilizando o deslocamento quadrado médio regular ou o deslocamento quadrado médio de dois pontos.

A simulação do material foi feita com recurso a Dinâmica Molecular. Dinâmica Molecular usa métodos numéricos para calcular a velocidade e trajetória de muitas part´ıculas cujo cálculo anal´ıtico é

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imposs´ıvel ou extremamente dif´ıcil. Para os cálculos foi utilizada a biblioteca de livre usoLAMMPS, este software utiliza a integração de Verlet e as equações do movimento de Newton para calcular as trajetórias e velocidades das part´ıculas eficientemente.

De modo a simular as interações do flu´ıdo onde estão inseridas as part´ıculas foi utilizada dinâmica de Langevin. As equações de Langevin foram resolvidas nas suas componentes de translação e rotação para cada part´ıcula. Estas equações têm em conta não só as forças conservativas, mas também um termo viscoso e um estocástico. As forças conservativas são dadas por um potencial gaussiano atrativo entre patchese um potencial repulsivo do tipo Yukawa entre part´ıculas.

Numa primeira fase aspatchy particlessão integradas enquanto um só corpo r´ıgido, cada part´ıcula tem três patches que estão sempre fixos na sua superf´ıcie. Numa segunda fase ospatchespassam a ser móveis e independentes. Para isto há que definir ângulos (entre patches) e ligações (entre part´ıcula e patch) dependentes de um potencial harmónico centrado, ora numa distância de ligação, ora num ângulo cujo vértice é a part´ıcula central. Os ângulos são deixados a variar segundo a constante harmónica K_A, as ligações têm uma distância fixa a partir de um algoritmo de SHAKE. Este algoritmo utiliza multiplicadores de Lagrange para garantir que certos constrangimentos sejam satisfeitos. As estruturas foram sujeitas a uma análise de percolação, para isto foi utilizado o algoritmo de Hoshen-Kopelman.

Os resultados experimentais deste trabalho dividem-se em dois grupos, os resultados referentes ao modelo com ligações r´ıgidas de valência limitada, onde ospatchesestão fixos na part´ıcula, e os resultados referentes ao modelo com ligações móveis de valência limitada. Primeiramente algumas amostras de gels são analisadas quanto à sua transição de percolação durante o crescimento. De seguida, a análise é feita quanto à percentagem de part´ıculas de trêspatchesnecessárias para percolação de todas as part´ıculas e de part´ıculas com três ligações.

Numa próxima fase, a análise de step shear é descrita tendo em conta três ritmos diferentes de aplicação de deformação, bem como a variação do número de ligações criadas e destru´ıdas ao longo dessa deformação. Nas curvas de tensão-deformação está expl´ıcita uma primeira zona linear, uma zona de aumento de stress não linear seguida de um decl´ınio não linear também. É dada especial atenção ao regime linear inicial (0−20%) pois é a´ı que serão realizadas as restantes medidas. De seguida, são realizadas medições de reologia de oscilação para determinar o comportamento dos coeficientes G^′₁ e G^′′₁ em relação à frequência de oscilação. O gel é predominantemente elástico a baixas frequências mas torna-se mais viscoso após uma frequência de transição. Esta técnica foi testada com diferentes energias de ligação entre patches, as medições foram todas semelhantes. O comportamento dos coeficientes de viscoelasticidade é estudado novamente desta vez com recurso a microreologia, ora com o desvio médio quadrático, ora com o desvio médio quadrático de dois pontos. O comportamento assemelha-se ao medido previamente o mas as medições são muito mais erráticas e ruidosas pelo que a reologia de oscilação foi adotada para todas as restantes medidas.

Seguidamente são expostos os resultados do modelo de ligações móveis com valência limitada. Neste caso as part´ıculas epatchesjá não são integrados como um único corpo, são independentes. Ospatches mantêm a sua distância à particula central através do algoritmo de SHAKE, os ângulos entre eles va- riam através de um potencial harmónico centrado em120^◦, a constante harmónicaK_Avaria abending energy no sistema. Da mesma maneira que anteriormente, estruturas com diferentesK_A são sujeitas a uma análise de percolação com respeito à percentagem de part´ıculas com três patches necessárias à percolação. O comportamento é semelhante entre todos osK_Ae ao caso de ligações r´ıgidas. De seguida

é feita uma análise com reologia de oscilação, o comportamento é semelhante ao caso r´ıgido, elástico a baixas frequências e viscoso após uma frequência de transição. A frequência de transição, no entanto, varia com o valor deK_A, crescente ao in´ıcio e após um pico decresce à semelhança de uma lei de potência.

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Sob esta observação, foi explorada a dependência da diferença entre a frequência de transição para vários KAe a frequência de transição r´ıgida. O comportamento de lei de potência confirma-se paraKAaltos, a frequência de transição tende para o caso r´ıgido com o aumento deK_A. Foi também feita uma análise para frequências baixas que evidencia a existência de um plateau paraG^′₁e um m´ınimo paraG^′′₁ aK_Aal- tos. Esta análise confirma também o ponto anterior de uma tendência para o caso r´ıgido à medida queKA

aumenta. Adicionalmente, foi testada a dependência do sistema ao algoritmo deSHAKE, determinou-se que a sua aplicação era equivalente a aplicar um potencial harmónico de constante harmónica KBonds

alta.

Este trabalho reforça a ideia de que a percolação de part´ıculas com três ligações é condição necessária para estabilidade mecânica, e que a introdução debending energynão afeta este fator. Conclui-se ainda que a bending energy pode ser usada como uma variável para definir a janela de elasticidade de um gel.

Palavras chave —gels, coloides, dinˆamica molecular.

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List of Figures

1 Network Snapshots of the Modelled Gel . . . 2

2 Patchy Particles Representation . . . 3

3 Patchy Particle Gel with Magnified Patchy Particle . . . 4

1.1 Representation of Strain in Shear Deformation . . . 8

1.2 Stress-Strain Curves . . . 9

1.3 Periodic Strain and Stress Representation . . . 10

1.4 Viscoelasticity Coefficients for Viscoelastic Fluids and Viscoelastic Solids . . . 10

1.5 Cubic Simulation Box . . . 11

1.6 Periodic Boundary Conditions Spatial Representation . . . 12

1.7 Period Boundary Conditions and Lees Edwards Boundary Conditions Representation . . 13

1.8 Displacement between two Particles for Two Point Microrheology . . . 15

2.1 Cell List and Verlet List Representation . . . 18

2.2 Flowchart of the Molecular Dynamics Pipeline . . . 19

2.3 Representation of Attractive and Repulsive Potentials . . . 20

2.4 Gaussian Potential Representation . . . 21

2.5 Yukawa-like Potential Representation . . . 21

2.6 Patchy Particle Representation with Angles and Bonds . . . 22

2.7 Flowchart of the AppliedSHAKEAlgorithm . . . 23

2.8 Representation of the Displacement Correction in theSHAKEAlgorithm . . . 24

2.9 Flowchart of the Adapted Hoshen-Kopelman Algorithm . . . 25

3.1 Rigid Bond Particle Representation . . . 27

3.2 Gel Structure and Patchy Particle Representation . . . 28

3.3 Gel Growth Representation Through Percolation Analysis . . . 29

3.4 Percolation Analysis for Gel Growth with Oscillatory Rheology Measurements . . . 30

3.5 Percolation Analysis for Rigid Bond Gel . . . 31

3.6 Step Shear Sequence Representation . . . 32

3.7 Step Shear Analysis for Rigid Bond Gel (100%) . . . 32

3.8 Step Shear with Broken and Created Bonds Analysis for Rigid Bond Gel (100%) . . . . 33

3.9 Step Shear Analysis for Rigid Bond Gel (20%) . . . 33

3.10 Oscillatory Rheology Analysis for Rigid Bonds Gel, Amplitudes of 1% and 10% . . . . 34

3.11 Oscillatory Rheology Analysis for Rigid Bond Gel with 3 Different Values of Energy . . 35

3.12 Mean Squared Displacement of Particles in a Rigid Bond Gel with Smoothing Applied . 36 3.13 Passive Microrheology using MSD on a Rigid Bond Gel . . . 36

3.14 Viscoelasticity Coefficients Measured using Two-Point Microrheology . . . 37 xiii

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4.1 Mobile Bond Patchy Particle Representation . . . 40

4.2 Percolation Analysis for four Gel Samples with DifferentKA . . . 41

4.3 Mobile Bond Oscillatory Rheology . . . 42

4.4 Transition Frequencyω_tfor DifferentK_A(Linear and Logarithmic Scale) . . . 43

4.5 Transition Frequencyωtfor Different1/KA(Linear and Logarithmic Scale) . . . 43

4.6 Difference between the Transition Frequency ω_t and the Rigid Bonds Transition Fre- quencyω0 . . . 44

4.7 Difference between the Transition Frequency ω_tand the Ideal Rigid Bonds Transition Frequencyω₀According to a Power Law Fit . . . 44

4.8 Mobile Bond Oscillatory Rheology for Low Frequency . . . 45

4.9 G^′₁ andG^′′₁ for Low Frequencies at DifferentK_ACompared to Rigid Bonds Values . . . 46

4.10 Ratio ofG^′₁/G^′′₁at Different Low Frequencies with Dependence onK_A . . . 47

4.11 Transition Frequencyωtfor DifferentKAand1/KA(SHAKEand Harmonic Potential) . 47 4.12 G^′ and G^′′ for Low Frequencies at Different K_A Compared to Rigid Bonds Values (SHAKEand Harmonic Potential) . . . 48

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List of Abbreviations

PBC Periodic Boundary Conditions TPM Two-point Microrheology MSD Mean Squared Displacement

TP-MSD Two Point- Mean Squared Displacement MD Molecular Dynamics

CFTC Centro de F´ısica Te´orica e Computacional (Theoretical and Computational Physics Centre) GSER Generalized Stokes-Einstein Relation

LAMMPS Large-Scale Atomic/ Molecular Massively Parallel Simulator MPI Message-passing Interface

LJ Lennard-Jones

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Introduction

The necessary conditions for mechanical stability in gels is still an open question [1]. In this work the mechanical properties of a colloidal gel are studied using numerical methods. In particular, bending energy is introduced in a limited valence and directionality model of a colloidal gel.

Colloidal gels have several advantages comparing to other materials, specially soft matter materials.

This is one of the reasons why there has been a big focus in studying their properties. The large relaxation timescale besides giving colloidal gels very particular features, allows for single-particle resolution in a few experimental techniques such as confocal microscopy [2] [3]. The possibility of tuning and tweaking parameters in order to construct a gel which has specific characteristics is also one of the reasons of interest. Manipulating the composition and ansiotropy of a gel can lead to the design of a specific structure whose properties satisfy the needed requirements [4]. One possibility is by functionalizing colloidal particles as patchy-like particles with selective binding [5].

Although some work has been done at high density, in the case of low density gels the interactions at play are not that well understood and research still needs to be done in order to fully unravel the relevant mechanisms. The main problem is that there is a wide variety of gels to study and a wide variety of results, some even contradicting each other. There is also the challenge that some systems are too complicated and conclusions regarding that system are system specific [1].

Colloidal gels are used in a wide range of areas which go from foods to cosmetics and pharmaceu- ticals. The most obvious examples of known colloids and colloidal gels are foods such as milk and jelly, therefore, food science deals a lot with colloidal gels and the research of their properties. Recently, research on this topic is focused on the properties of colloidal gels formed by macromolecules such as proteins [6] [7] and on the nutrient delivery properties of them [8]. These delivery properties have been used by the pharmaceutical industry as well in drug delivery applications through the encapsulation of drugs using colloidal gels [9] and continues to be applied further on. Applications of this material can also be found in construction (cements, ink) and biomedicine [4]. This diversity of applications is one of the reasons of interest in the material and one of the motivations of this work.

The work done on this thesis takes its fundamental basis on two important results, one from Tsuru- sawa et al. [2] and recent results in our group. The onset of elasticity for a colloidal gel has been the main focus of experiments performed by others [10]. The appearance of an elastic-like material which is capable of bearing weight at a low density is a useful and promising situation that motivates its study.

The properties of this elastic state and its necessary conditions are the main aim of this work. In addition to what has already been done, bending energy is introduced as the adjustable property on which the system will also depend.

Tsurusawa et al. have shown in 2019 that mechanical stability in colloidal gels is due to the isotropic percolation of isostaticity [2]. The criterion for isostaticity in a particle is satisfied if it has six neighbours, that is six connections to other particles, limiting all six degrees of freedom, complying with a local Maxwell criterion [11]. This is similar to considerations previously done regarding granular mate-

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rials [12]. If a structure has a system-spanning connected network of isostatic particles then this would mean mechanical stability is guaranteed. The experimental work shows that isotropic percolation [13]

of isostatic particles happens simultaneously with the transition of elasticity when G^′ > G^′′ in dilute systems, although it should not be exclusive to dilute systems. Tsurusawa et al. performed experimental measurements on two systems, a dense and a dilute one. They showed that the direct percolation of all particles and the isotropic percolation of isostatic particles happen simultaneously on the dilute system, this is also the timeτgelat which the system becomes mechanically stable. On the dense system however the two percolations are far apart, and it is clear thatτ_gel is coincident with the time of isotropic percolation of isostatic particles. Therefore, the claim is that isostatic percolation is a necessary condition for mechanical stability.

Numerical simulations on this matter have shown that although the isostatic percolation might be a sufficient condition for mechanical stability it is not a necessary one. Recent results from CFTC (and also shown in this work) suggest that mechanical stability needs but only 3 neighbours per particle instead of 6 as was theorized before. Using patchy particles models with low valence (3 maximum bonds possible) and directional interactions, percolation of particles with at least 3 neighbours is enough to guarantee the elasticity of a gel. On Fig. 1 a modelled gel that was analysed in this work is shown together with a representation of the very network that the particles in the gel form, moreover the connections which come fromthree bonds particlesare highlighted in yellow in the bottom image.

Figure 1: Three snapshots of the same modelled gel, on the top left the particles that make the gel are shown forming a percolative network that is the gel. On the top right the simulation box shows this very connected network with particles appearing in purple and their connections in blue. On the bottom the network is again shown with the particles which have three bonds highlighted in yellow.

The probing of the mechanical characteristics of a gel model using numerical simulation was done in 2014 by Del Gado [14], an athermal quasistatic shear protocol was used to recover information on the rheology of such model. This allowed to acquire the stress-strain curve as well as other analysis.

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This protocol is crucial for the shear analysis and oscillatory rheology done further in this work, together with two point microrheology [15] these are the main tools used on the characterization of a model’s mechanical properties.

The studied colloidal gel is a gel structure whose solute particles are colloids suspended in a fluid solvent. A colloidal particle’s size is about 1 nmto 1 µm [16], and it is made of several billions of atoms, the size of these particles is one of their main characteristics and the source of their interactions.

Because of their magnitude, the range of interaction is much smaller than its size and the energies are very close to those ofk_BT[17].

The potential of particle-particle interaction is a combination of a short range repulsion and medium- range attraction. The interaction between colloidal particles can be thought of as a combination of Van der Walls and electrostatic interactions [18].

Colloidal solutions have the same principle, small solid particles (or fluid droplets) dispersed in a fluid in which the solvent particles have a much smaller size than the solute particles dispersed in it [16]. Whether these solutions will flow or not depends on the density of solute, and in what way the colloid particles are connected. In a colloidal gel a structure of cross-linked particles spanning the whole dimension of the material is needed in order for stability. The conditions for gelation though, the process of becoming a gel are not specified with regard to their minimum conditions [19].

The lifetime of the bonds between colloids, however, is something that will define the characteristics of the gel, these bonds are physical and can be reversibly broken. A gel is usually characterized by long living bonds and therefore the structures relax in a timescale much longer than the observed time window [1]. Increasing the lifetime of the bonds can be done by increasing the attraction potential, but this comes at the cost of possible phase separation. One way to prevent it is to include long-range repulsion between the colloids, as in a charged colloid-polymer mixture, another one is limiting the valence and directionality of the colloids, this is done by considering patchy particles at low temperatures.

In an attempt to model the particular structure of a colloidal gel, a patchy particle approach was used [20]. As said before, a gel can be thought as an aggregate of particles suspended in a solvent with connections between the colloidal particles spanning the whole system. In this model we use patchy particles to simulate the colloidal particles in the gel.

A patchy particle is, in our case, a hard sphere with a specific mass, density and size. The spheres will interact between each other and the fluid creating a dynamic structure. Each one having, equally spaced on their surface, a few areas which will be called the patches. On Fig. 2 a representation of the two patchy particles models used is shown. These bodies are composed of the central particle in grey and the surrounding patches in blue. Both models will be explained further on in detail but the main difference between them is the way in which the patches behave relatively to the central particle.

Figure 2: Representation of the two patchy particles used throughout this work. Both cases will be explained in detail in Chapters 3 and 4. On both of them the spherical particle (on grey) is surrounded by three patches (in blue), the difference between the two models is in how the patches move, or not, relatively to the sphere.

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The patches are spots in the central particle’s surface with the dimension of a point, they have no mass or density and are responsible for the connections between the spheres, the bonds. Each patch belongs to a certain particle (they cannot interchange between particles) and is subject to an attractive potential towards other patches such that, when in close distance, these patches stick to each other and form a bond. As to maintain coherence with the physical description of colloid gels, the attractive potential’s energy should be much higher thank_BT so that inter-particle connections remain long-lived and stable.

The resulting structure will then be of a wide network of connected particles spanning across the whole system as seen in Fig. 3. On this figure the gel structure is shown as an interconnected aggregate of patchy particles, one of these particles is then magnified.

Figure 3: Scheme of a patchy particle gel, on the left we see a magnified patchy particle with 3 patches. This is the constituent of the structure, on the right we see the grown structure formed by a network of interconnected patchy particles.

This patchy particle model allows for limited bonding valence depending not only on the number of patches per particle but also on their geometrical distribution. The distribution will ensure anisotropic interactions among particles and, together with a short-range interaction (typically 10 to 20% of the particle’s diameter), a tendency to not form multiple patch bonds [5].

Colloidal gels are an out of equilibrium structure, this means that the growth of a gel is a process which will have a different outcome every time. Two gels grown in the same conditions will have certainly different configurations, however statistically they will have the same properties. This is one of the reasons why this matter should be looked through the scope of physics.

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Chapter 1

Mechanical Properties

In order to study the mechanical properties of the obtained structure, several measurements were performed. Some of them analyse only data from the Brownian movement [21] of the gel at a certain temperature (passive microrheology [22]) while others impose a small perturbation to particles of the gel. Oscillatory rheology [14], for example, imposes small amplitude oscillatory shearing while the stress is measured. To know whether the amplitude is actually small the stress suffered by the material by the application of step shear is also measured. The stress-strain measurements, in the case of oscillatory rheology, give information on the viscoelasticity of the material. After a brief explanation of the concepts at hand an explanation of how the mechanical measurements were done will follow.

1.1 Viscoelasticity

The behaviour of a material can be classified according to its capacity to either store or dissipate energy.

A perfectly elastic material will store all mechanical energy applied to it, whereas a Newtonian viscous fluid will not be able to store any, all energy is dissipated. Soft matter however can behave in ways which are comparable to both, the viscoelasticity [23] of a material is its complex time response to a stimulus such as a shear force, which combines a viscous and an elastic response. In order to measure the viscoelasticity of a material one must be familiar with the concept of shear stress and shear strain.

1.1.1 Shear Strain

Strain γ is a measure of deformation of a material, in the case of shear deformation a displacement is applied to all particles following the transformation of a shear matrix. We can define strain as the relative displacement of the structure, that is, the ratio between the magnitude of the displacement and the initial magnitude of the material [18].

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x y

z

L

x y

z

L

δL

Figure 1.1: Representation of the deformation measurement strainγin the case of shear deformation.

For the example in Fig. 1.1 we define,

γ = δL

L . (1.1)

1.1.2 Shear Stress

Shear stress is defined as the forceF per unit area acting on a surfaceAwhich is parallel to it [18], σ= F

A. (1.2)

It can also be defined in terms of the stress tensor, the stress tensor completely defines the forces acting in a body [24]. The forceFper unit area acting on a surface perpendicular to the normal vector⃗ncan be defined asF =nσ,σhere being the stress tensor

σ=







σxx σxy σxz

σ_yx σ_yy σ_yz σzx σzy σzz





 , (1.3)

whereσ_ij is thejcomponent of the force acting on a surface perpendicular to the direction ofi. There- fore, if we are interested in the shear stress acting on a surface perpendicular toxin the direction ofy, componentσ_xy of the stress tensor will give us that information.

1.1.3 Stress-Strain Curve

The plotting of the two aforementioned quantities together is a usual way of getting information from a certain material. Experimentally, what is done is to apply a force to the material (a shear force) and measure its degree of deformation (strain), several mechanical apparatus are used to provide this type of data [25]. A curve like this can tell a lot about a material, an elastic material will be characterized by its Young modulus which can be measured by inspecting the linear part in a stress-strain curve. The Young modulus determines the stiffness of a material and is calculated by the ratio between stress and strain ^σ_ϵ, a higher Young modulus means the material can sustain higher amounts of stress until deformation.

A metal for example will have a linear strain-stress curve, both quantities will be proportional and, the curve will be equal in loading and unloading as in Fig. 1.2 (left). A viscoelastic material like rubber on the other hand can be elastic, meaning it will not suffer a permanent deformation, but there will be a hysteresis effect on the unloading as seen in Fig. 1.2 (right). This hysteresis has to do with the energy lost due to viscous effects as the material is viscoelastic.

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In Fig. 1.2 the loading curve is plotted on a solid line whereas the unloading curve is plotted on a dashed line, loading corresponds to applying the force and unloading to de-apply it. The two curves being coincident means the material is ideally elastic, and no energy is lost due to viscous effects, if they do not overlap though, the material does lose energy due to viscous effects, this energy is proportional to the area between the two curves.

Figure 1.2: On both images the graph shows the loading curve of the material on a solid line, and the unloading curve on a dashed line. On the left image the material is a metal, ideally elastic, and both the loading and unloading curve are equal which means there is no energy lost in the process. On the other hand, the image on the right corresponds to a rubber, the loading and unloading curve do not match, the area between the two curves represents the magnitude of the energy lost due to viscous effects.

In this work viscoelastic materials will be studied so the stress-strain curve will not be linear and, if loading and unloading curves were presented (they will not), the unloading would presumably show hysteresis. In the case of a colloidal gel the stress-strain curve would have the same behaviour as seen in Del Gado et al. [14]. In fact, there are several examples that show the stress-strain curve of a colloidal gel, one of them is computationally simulated, Del Gado et al. [14] and the other, made experimentally with samples at different concentrations and strain rates, by Laurati et al. [26].

Both these works show that the behaviour is similar, at low strain the stress increases with it, after which it reaches a maximum and decreases non-linearly. On Del Gado et al., together with load curve of the material the amount of broken and newly formed bonds is also shown, an analysis which will also be done in this work [14]. On the work of Lauratti et al., four different samples are used to plot the stress-strain curve at three different strain rates [26]. It is shown that higher shear rates will result in higher amounts of stress.

1.1.4 Viscoelasticity Coefficients

The relation between the two quantities defined before is what characterizes the viscoelasticity of a material, they are usually plotted together on a stress-strain curve and the behaviour of this curve is, as seen previously, particular to several materials.

Viscoelastic materials have properties both elastic and viscous [23]. In order to probe the viscoelastic properties of a material, a periodic strain is applied, and the stress is measured. In the case of a perfectly elastic material both quantities are in phase,ϕ= 0, whereas in the perfectly viscous caseϕ= 90^◦[25].

Viscoelastic materials on the other hand have a value ofϕbetween these two values as seen in Fig. 1.3.

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t Strain, Stress

Strain Stress ϕ

Figure 1.3: Representation of the periodic strain applied to a material when measuring its viscoelasticity. The difference in phaseϕbetween Stress and Strain is representative of the material’s viscoelasticity. Ifϕ = 0the material is elastic, if it is viscous thenϕ= 90^◦. For viscoelastic materialsϕis between these two.

In order to obtain the viscoelasticity coefficients of a material one must subject them to an oscillatory strain of small amplitude (linear regime), this is called oscillatory rheology [14]. Applying a strain of the sinusoidal formγ(t) =γ₀sin(ωt)will result in an also sinusoidal stress dependent on two different coefficientsG^′(ω)andG^′′(ω), these coefficients define a complex module

G(ω) =G^′(ω) +iG^′′(ω), (1.4) where G^′(ω) is the storage module and G^′′(ω) the loss module [24]. The first one, storage module, is the real part of the complex modulus and represents the elastic behaviour of a material whereas the second one,loss module, represents the viscous behaviour of the material, it is the imaginary part of the complex module. These coefficients vary with the frequency of imposed oscillations, the storage module is in phase with the applied strain and is related to the amount of elastic energy the material is able to store whereas the loss module is in phase with the strain rate and is related to the dissipation of energy through viscosity. The magnitude of the ratio between these two quantities tells on whether the material has a predominantly elastic or viscous behaviour (solid-like or liquid-like) by inspecting which of these is of greater magnitude.

For a typical viscoelastic fluid and solid the behaviour of the viscoelasticity coefficients is as follows on Fig. 1.4. At low frequencies it is clear that on the right the storage module is much greater than the loss module, this means we’re in the presence of a solid, in this case a gel, whereas on the left the behaviour is opposite therefore the material is considered a fluid. It is also clear that viscoelastic materials can have different behaviour throughout the spectra of frequency [18].

ω G^′′

G^′

ω G^′ G^′′

Figure 1.4: Representation of the viscoelasticity coefficients with dependence on the frequency. On the left the complex module of a viscoelastic fluid is represented, at low frequencies the loss module is greater than the storage one, although their magnitude can interchange throughout the spectra of oscillation. On the right the complex module of a viscoelastic solid, a gel, is represented, at low frequencies the storage module is greater than the loss one.

A viscoelastic solid maintains its form when unperturbed, at low frequency the material does not flow and its behaviour is elastic storing energy when deformed, just like a rubber or gel. At high frequencies however, it is able to flow. The other contemplated case is a viscoelastic fluid which can be a polymer melt for example, this type of materials flow but will store energy when subject to high frequency perturbations.

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1.2 Mechanical Measurements

The characteristics mentioned above were probed using the aforementioned methods of Step Shear, Os- cillatory Rheology and Microrheology.

1.2.1 Step Shear

The stress-strain curve of the material can be obtained by application of a shear stress, the stress applied makes the material follow a shearing flow which in this case resembles the one obtained by the plane Couette geometry [24].

The plane Couette geometry creates a shearing flow by containing the fluid between two parallel plates, then one of the plates slides in a certain direction and generates a flow created by the shear stress that is being exerted onto the material. Computationally what is done is deforming the simulation box step by step at a certain rate [14].

Initially the simulation box is defined by three vectorsA⃗ = (α,0,0),B⃗ = (0, β,0)andC⃗ = (0,0, ζ) (Fig. 1.5 left), in this specific case the magnitude ofα,β andζis the same which makes the simulation box a cube, in order to allow for shear to happen the simulation box is shifted from orthogonal to triclinic and the vectors that define the new box will be given by⃗a = (α,0,0)⃗b = (δ, β,0)and⃗c = (0,0, ζ) (Fig. 1.5 right). The tilt factorδis increased at everyN steps, as this value changes one of thexzplanes of the simulation box will slide towards the positive xaxis leaving the otherxzplane unchanged as it would happen in the two sliding plates case.

x y

z

A⃗ B⃗

C⃗

x y

z

⃗a

⃗b

⃗c

Figure 1.5: Cubic simulation box with vectorsA, ⃗⃗ B, ⃗C (at the left), and triclinic simulation box with vectors⃗a,⃗b, ⃗c(at the right).

The structure suffers a transformation which can be defined as the following shear matrixTin which the new coordinatesr₁of the particles will be given after the first transformation by

r1 =T r0, (1.5)

in whichT can be given by

T =







1 δ_T 0 0 1 0 0 0 1





, (1.6)

this transformation is applied multiple times. The time between transformations is given by N, the number of timesteps for which the system relaxes. Increasing or decreasing this number will change the

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rate at which the shear transformation is applied, a higher N will mean a slower rate. If the integration timestep isτi, the shear rateγ˙ = ^∂γ_∂t can be calculated as

˙ γ = ∂γ

∂t = ∂γ N τi

. (1.7)

These simulations are done at zero temperature (athermal) in order to neglect effects from the random term of the Langevin dynamics, as this affects the kinetic energy.

The computation of a stress tensor is then done at each deformation by inspecting the interactions of every particle, this tensor has six different terms regarding the surface and direction at which stress is being exerted. It is calculated in the following way as described by Thompson et al. [27],

σ_ij = P_N

k r_kif_kj

V , (1.8)

the sum is done over allNparticles,i, jcan bex, y, z;r_kiis particlek’s position regarding componenti andf_kjis particlek’s force over componentj. In this case the kinetic part of the stress tensor is neglected as the velocities of the particles are low.

When performing these measurements one should also be careful with the boundary conditions, initially Periodic Boundary Conditions [28] are used. This means that particles leaving the box will be added back on the opposite plane where they left. To every coordinate the PBC function described on Eq. 1.9 is applied wherelis the coordinate andLthe length of the box,

PBC(l) =







l+L ifl <0

l−L ifl > L . (1.9)

This is equivalent to having 26 simulations boxes around the main simulation box, so that particles in its limits are also interacting with neighbouring particles that might be in the opposite part of the box.

In Fig. 1.6 a representation of this idea is made in two dimensions, particles near the edge of the box interact with particles on the opposite edge and, if they cross the limit of the box they will re-appear on the opposite edge. This periodicity in space is what characterizes and defines this type of boundary conditions, particles always interact with each other over the closest distance possible.

Figure 1.6: Representation of what happens in Periodic Boundary Conditions in two dimensions, particles always interact with particles close to them even if they’re far from each other in the simulation box. Their real distance is measured as if there were other simulation boxes around them.

When shear stress is applied, the simulation box becomes triclinic, periodic boundary conditions can 12

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no longer be considered. This is due to the particles that might leave the box in one of the planes that is suffering shear stress, or in the opposite plane, these particles are added to the box, as they normally would, with an offset equivalent to the tilt factorδ. These are Lees-Edwards boundary conditions [29], a graphical example of both aforementioned examples is shown in Fig. 1.7.

Figure 1.7: In this image the two types of boundary conditions are represented. On the right, Periodic Boundary Conditions are shown, the particle leaving the box in the inferior limit, rejoins the box in the upper limit following Eq. 1.9. On the left, Lees-Edwards conditions are shown, the particles are added back in the box like in the previous case but if they leave in a surface that is suffering shear (or the opposite plane), they are added back with a shift.

1.2.2 Oscillatory Rheology

A method to study the characteristics of the material at different timescales is also defined by imposing a periodic strain of small amplitude. This method will not tell us the lengths at which the system is able to deform and its internal stress but how this internal stress varies at each frequency of oscillation. From this definition the computation of the viscoelasticity coefficientsG^′andG^′′can be done, these values, as said before, measure the elastic and viscous response of the material.

In practice the shear deformation is done repetitively at a low strain amplitude in a periodic manner, the valueδwill vary with time with the following equation:

δ=γ0 sin(ωt), (1.10)

in which Ais the amplitude of oscillation and ω the frequency of oscillation. The amplitude must be chosen with close attention to the range within which the behaviour of the material is linear. We can see this by inspecting the stress-strain curve obtained by the step shear analysis.

As before we measure the stress tensor and use the computed values to obtain the viscoelasticity coefficients in the following way [14].

G^′₁(ω, γ0) = ω γ0π

Z t0+2π/ω t0

σxy(t) sin(ωt)dt , (1.11)

G^′′₁(ω, γ0) = ω γ0π

Z t0+2π/ω t0

σxy(t) cos(ωt)dt , (1.12) The values of the stress tensor are integrated starting from an initial random timet0 until the end of the oscillation att₀+ 2π/ω, contemplating the whole period. We obtainG^′₁andG^′′₁which are the first order viscoelastic coefficients, these represent only the response of the material in the linear regime which is the region that is being studied here. This definition comes from the Fourrier series that represents stress σ,

σ(t;ω, γ0) =γ0

X

n odd

G^′_n(ω, γ0) sin(nωt) +G^′′_n(ω, γ0) cos(nωt) , (1.13)

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linear regime contributions will be given by the first harmonicG^′₁andG^′′₁

1.2.3 Microrheology

Microrheology covers another range of methods for analysing the material’s rheology as the ones mentioned before. In these methods however there’s not a physical perturbation of the material as the application of a shear force, simple observation is done, and unperturbed particles are studied only subject to their own Brownian motion [22]. This is also called passive Microrheology as opposed to the active one previously defined. Passive microrheology usually provides more detailed measurements which are not affected by external factors. Additionally, it provides a broad range of oscillation frequencies, much wider than oscillatory rheology does.

Mean-Squared Displacement

Firstly, the Mean-Squared Displacement (MSD) is used to characterize the material’s viscoelasticity [30]

this is done by using the Generalized Stokes-Einstein Relation (GSER) [31]:

G^∗(ω) = k_BT

iω⟨∆r(ω)²⟩πa, (1.14)

whereG^∗(ω) is the viscoelasticity coefficient (a complex number),⟨∆r(ω)²⟩the Fourier transform of the MSD andathe radius of the particle we’re considering.

The MSD is calculated at each timestep as an ensemble over all particles as stated in equation 1.15,

⟨∆r(τ)²⟩=⟨(r(t+τ)−r(t))²⟩. (1.15) This MSD is not useful in the time-space but rather in frequency, however to preform a Fourier transform data from 0 to∞ would be needed. As this is not possible different strategies need to be applied to achieve⟨∆r(ω)²⟩

The method used by Mason [31] is here applied, firstly the logarithmic derivativeαis found at every timestep,

α(τ) = d ln⟨∆r²(τ)⟩

d lnτ , (1.16)

in practice this is done by checking the slope of every point in the logarithmic space.

The module of the viscoelasticity coefficient is given then by the following expression,

|G^∗(ω)|= k_BT

πa⟨∆r²(τ = 1/ω)⟩Γ [1 +α(τ = 1/ω)], (1.17) whereΓis the gamma function, this is an approximation of the GSER.

The real and imaginary part of this viscoelasticity coefficient is then retrieved in the following way:

µ(ω) = π 2

d ln|G^∗(ω)|

d lnω , G^′(ω) =|G^∗(ω)|cos (µ(ω)), G^′′(ω) =|G^∗(ω)|sin (µ(ω)), (1.18) whereG^′(ω)represents the elastic response of the material andG^′′(ω)the viscous response.

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Two-Point Microrheology

This method is fairly similar to the previous one but in this case the motion of pairs of particles is used to compute not an MSD but a two point MSD which will be the cross correlated motion of that pair of particles. The advantages of this method are its robustness to uncertainties and its ability to probe the bulk rheology of a material in a more exact manner, it also takes away effects of the whole gel’s movement [22]. If∆r₁(τ)is the displacement of particle 1 regarding its initial position and∆r₂(τ)the one of particle 2 their correlation can be represented as

Drr(R, τ) =⟨∆r₁(τ) ∆r2(τ)⟩, (1.19) in which R is the module of vectorR⃗ that connects the two particles, the displacements∆r₁(τ) and

∆r2(τ)are calculated in the line that connects the centres of the two particles 1 and 2, so∆r1(τ)and

∆r2(τ)will be the actual displacements of the particles (∆r^real₁ and∆r^real₂ ) projected on vectorR⃗ (Fig.

1.8).

P1 R⃗ P2

∆r^real₁ (τ) ∆r^real₂ (τ)

∆r1(τ) ∆r2(τ)

Figure 1.8: Representation of the displacement for two point Microrheology. The displacement is measured in the line that connects both centres of the particles to reduce noise. The real displacement of the particles is represented, as well as the projection of this displacement on the line that connects both centres.

The cross correlated motions are aggregated depending on the distance between the two particles at an initial step. In practice what is done is the binning of the particles depending on their initial distance so that we have a certain number of points inR. Drrwill be then a two-dimensional array in space and time.

This array is transformed in the two point MSD by getting rid of the spatial dependence and re-scaling it with distance as:

⟨∆r²(τ)⟩₂= 2R

a Drr. (1.20)

Then the same procedure as before is used to transform this two-point MSD into the viscoelasticity coefficient.

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Chapter 2

Computational Methods

On this chapter an overview of the computational methods used is given. Molecular Dynamics [28] was mainly used to perform the simulation of the gel and in addition to this a percolation analysis algorithm, Hoshen-Kopelman [32], was also developed. In the next sections the parameters and details of the Molecular Dynamics simulation are explained as well as the tools used to do it.

2.1 Molecular Dynamics

Molecular Dynamics uses numerical methods to compute the movement of bodies in a many body problem whose analytical solutions are impossible [28]. The trajectories and velocities of these bodies are numerically calculated from the equations of movement. The main idea is that given the equations of motion and the respective conditions of the problem at hand the solution of the system’s dynamics for each one of its constituents can be reached [28]. This is only possible with the powerful help of comput- ers and as these and their processing power continues to evolve the method and its applications evolve as well.

We used LAMMPS (Large-Scale Atomic/Molecular Massively Parallel Simulator) [33] an open- source tool whose efficiency helps to solve MD calculations by applying parallelization. LAMMPS uses velocity Verlet integration to solve the dynamical evolution of the system through Newton’s equations.

Verlet integration [34] takes the following two steps to integrate position and velocity,

⃗

r(t+ ∆t) =⃗r(t) +⃗v(t) ∆t+1

2⃗a(t) ∆t², (2.1)

⃗v(t+ ∆t) =⃗v(t) +⃗a(t) +⃗a(t+ ∆t)

2 ∆t . (2.2)

Additionally, for efficiency and memory purposes, LAMMPS calculates velocity twice at each step and takes the value of force as the one calculated between those two half steps, efficiency comes from only calculating forces once. This is called the velocity Verlet integration, and it is done in the following way.

First the velocity at the half step is computed,

⃗

v(t+ ∆t/2) =⃗v(t) +⃗a(t)∆t

2 , (2.3)

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then the positions at the full step,

⃗r(t+ ∆t) =⃗r(t) +⃗v(t+ ∆t/2) ∆t , (2.4) with these positions the new updated forces can be calculated as well and, consequently the velocities at the full step,

⃗v(t+ ∆t) =⃗v(t+ ∆t/2) +1

2⃗a(t+ ∆t) ∆t . (2.5) Interactions between pairs of particles will be selected according to their distance to each other. This means that force calculations will be done over all pairs of particles whose distance is below a chosen threshold, rcut (Fig. 2.1), defined by the potential to be used. In addition to rcut, another value of distance,skin, is used to account forghost atomslost to parallelization.

Instead of checking all possible pairs of particles, LAMMPS employs some MD strategies that lead to faster computations. Besides parallelization, the Verlet list [35] and cell list [28] methods are used.

The simulation box is spatially divided in cell lists as shown in Fig. 2.1 (right). Then, the neighbour list (Verlet list) is updated and the distance between particles is only calculated for particles contemplated by the respective cell list. These are two processes that are employed by LAMMPS for efficiency. In Fig. 2.1 the two methods are represented, the cell list, on the right, spacially divides particles into lists according to their coordinates, then the Verlet list, on the left, is updated only considering particles which are close according to the cell lists.

r_cut +skin

Figure 2.1: Particles which have a distance of less thanrcuthave their force computed, they are added in the neighbour list (left). In order to not check all particles, a cell list division is firstly made to know which particles should have their distance computed (right).

LAMMPS provides options to run in parallel, this means that the computational work can be divided through each processor without any loss in information. This process ensures faster results are achieved as long as the number of processors is adequate to the length of the simulation, if not much time will be loss in the communication between cores. Communication which is done using the MPI (Message- passing Interface) protocol [36]. In practice the simulation domain is spatially divided into smaller subdomains in which the simulations will be performed using only one processor. Atoms in a region close to the border of a subdomain, ghost atoms, are accounted for in more than one processor which will communicate among each other. [33] The area of the subdomains whereghost atomscan be found is determined by theskinparameter.

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Velocity at half step

Positions at full step

Communication between processes

Forces at full step

Velocities at full step

Figure 2.2: Flowchart of Molecular Dynamics pipeline, including the step where communication between processes is done if parallel computation is used.

2.2 Langevin Dynamics

In the context of this work MD will be performed in a fluid which means that the fluid’s coarse-grained interactions with the particles should also be taken into account. This is done by using Langevin Dy- namics which computes the force of the liquid acting on each particle through the Langevin equation.

The resulting net force will be a sum of the conservative term given by other defined potentials, a viscous force proportional to the particle’s velocity and a noise term which represents the fluctuations in the particle collisions with fluid particles. The Langevin equations of motion were solved for the translation and rotation degrees of freedom [37] [38],

m⃗v˙ =−∇_⃗_rV(⃗r)−m τt

⃗v(t) +

r2mk_BT τt

⃗ζ(t) , (2.6)

I⃗ω˙ =−∇_⃗_θV ⃗θ

− I τr

⃗ ω(t) +

r2IkBT τr

ζ⃗(t) . (2.7)

In these equations⃗vand⃗ωrepresent the translation and angular velocities,I the inertia of the particle,V is the potential acting on the particle,mthe mass,k_Bthe Boltzmann constant,Tthe temperature,τ_tand τrthe damping times and⃗ζ(t)a stochastic term with zero mean.

The viscous term depends on thefriction constantζ_Stoke, as stated in Stoke’s equationF⃗ =−ζ_Stoke⃗v.

This constant can also be written in terms of its damping timeτ (dampin LAMMPS documentation), ζStoke =m/τ which leads to both translation and rotational viscous terms ^−m_τ

t ⃗v(t)and ^−I_τ

r⃗ω(t)[18].

This term however is not sufficient to represent the whole motion of a particle in a fluid, anoiseterm should be added, this is a fluctuating force exerted by the small liquid particles. It follows a Gaussian distribution with a mean value of zero and its magnitude is proportional to the square-root of the temperature and the Boltzmann constant. Using the Fluctuation-dissipation theorem [18] the noise terms are

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then for the translational case

q2mkBT τt

⃗ζ(t)and for the rotational one,

q2IkBT τr

⃗ζ(t). The⃗ζ(t)function is the stochastic term taken from a random distribution with zero mean.

The conservative term depends on potentialV, this potential term is a sum of the potential forces that are being exerted on the particles. Particles suffer both an attractive and repulsive potential, the nature of these will be treated in the following paragraphs.

The trajectories of the particles are then integrated using equations 2.6 and 2.7. LAMMPS is used to do this due to its efficiency.

2.2.1 Interaction Potentials

In order to account for interactions between the particles, potentials are used. In this case we define two potentials to be used between particles and patches, an attractive and a repulsive one. A pair of particle/particle or patch/patch will only interact if these are at a distance of less thanrc. The outcome of these interactions results in a force which is accounted for as potentialV in equations 2.6 and 2.7, it is the conservative term in the net force of each particle and patch.

Figure 2.3: Representation of attractive and repulsive potentials. On the left an attraction between patches is represented, this corresponds to the Gaussian Interaction. On the right, the repulsion between cores is represented, this interaction comes from the Yukawa-like potential acting on these particles.

Attractive Gaussian Interaction Between Patches

For connections between patches the Gauss potential was used [39]. This potential mimics the attraction between two patches which can form a covalent bond, it has the form,

V_G=−ϵe^−Br², r < r_c, (2.8) in whichrcis the cut radius therefore only patches that are at a distance less thanrcfrom each other will be regarded for computation of this potential. ϵhas energy units andB will represent the width of the Gaussian potential. In this case a value ofB = 1was always used so that if a patch is in a distance which is less than 1σ it willfallinto the Gaussian well and a bond will be formed. The value ofrwill be in this case the distance between patches.

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r E

ϵ

Figure 2.4: Gaussian Potential Representation

In the context of this work two particles will be considered bonded if there is a pair of patches, each from a different particle, whose distance between each other is equal or less than one. The criterion is explained by the amount of energy in the interaction between patches, at this distance the energy needed to separate the two patches will be high, given the attractive potential acting on them.

Yukawa-like Repulsive Interaction Between Colloids

The potential acting between the particles is repulsive and is given by the Yukawa/ colloid [40] functional which mimics the interaction of two colloid particles in an electrolyte. This is an extension of the Yukawa potential for colloid particles, and it has the form,

V_Y = A_Y

κ e^−κ(r−(rⁱ^+r^j⁾⁾, r < r_c, (2.9)

in whichκis the inverse screening length (units of distance⁻¹) andAY has units of energy over distance, as before only particles in the range of the cut radiusrcare considered. The distanceris in this case the distance between colloids whereasr_iandr_j are the radii of the two particles.

r E

r_i+r_j

Figure 2.5: Yukawa-like Potential Representation

Rigid Body Integration

Firstly the patchy particles will be regarded as a unique body, this means that patches are fixed in their positions regarding the central core. The patchy particle body is whole with its own mass and inertia and, it will suffer both the attraction between patches and the repulsion between colloids as in [37] [38].

The resulting interaction will be similar to the one of a colloidal solution where the particle will suffer a deep attraction when patches are close to each other, but will feel the repulsion at a higher distance.

This means that when unperturbed it will not be easy for bonds to break and also unlikely for them to be formed.

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The rigid body integration means also that the central particle and its patches will follow the translational motion of their central core and the rotational motion as well, this means that when the central particle rotates the patches cannot change their configuration but will follow the rotational likewise. Sim- ilarly, patches cannot be interchanged between particles nor can they be lost or gained, in this model the patchy particles will remain whole the entire time.

The following paragraphs will introduce concepts used when the model changes and the two bodies core and patches become independent.

2.2.2 Angles and Bonds

In some simulations angles and bonds have to be created, these will follow a harmonic potential defined around a distance or an angle as the equilibrium point.

Figure 2.6: Representation of the patchy particle together with its defined angles (left) and its defined bonds (right).

Harmonic Angle

The angles, in this case, are given between the patches with the particle being the vertex as shown in Fig.

2.6. The structure is then well-defined within a certain degree of deformation which will be represented in a form of a bending energy. This is done with the definition of harmonic angles, these angles will have a magnitude ofθ₀ = 120^o. However, the actual value can vary within a small range of values (as a spring) with the harmonic potential:

V_angle=KA(θ−θ0)² , (2.10)

in whichK_A is actually ¹₂k_Aand has units of energy/radians². By changing the value ofK_Aone can vary the energy needed to displace the angle between the patches and deform the patch configuration in a particle. This will be the method used in order to vary thebending energy.

Harmonic Bond

In the same manner, harmonic bonds can also be defined. As before, these will also be present in the connections between a particle and its own patches to maintain the patchy particle’s configuration.

Initially patches are fixed in the surface of the particle but by defining harmonic bonds they will be able to move (extend or contract) their distance to the centre of the particle, the easiness with which they do this can be, again, defined by the prefactorKBondspresent in the harmonic bond expression,

V_Bonds=K_Bonds(r−r0)² . (2.11)

A graphical representation can be found again on Fig. 2.6.

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Modelling the Mechanical Properties of Colloidal Gels

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