Eindhoven University of Technology MASTER. Emergent plasticity in glassy and polymer-like materials failure patterns and avalance statistics

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1 Eindhoven University of Technology MASTER Emergent plasticity in glassy and polymer-like materials failure patterns and avalance statistics de Caluwe, E.E.K. Award date: 28 Link to publication Disclaimer This document contains a student thesis (bachelor's or master's), as authored by a student at Eindhoven University of Technology. Student theses are made available in the TU/e repository upon obtaining the required degree. The grade received is not published on the document as presented in the repository. The required complexity or quality of research of student theses may vary by program, and the required minimum study period may vary in duration. General rights Copyright and moral rights for the publications made accessible in the public portal are retained by the authors and/or other copyright owners and it is a condition of accessing publications that users recognise and abide by the legal requirements associated with these rights. Users may download and print one copy of any publication from the public portal for the purpose of private study or research. You may not further distribute the material or use it for any profit-making activity or commercial gain

2 Emergent plasticity in glassy and polymer-like materials Failure patterns and avalanche statistics Thesis to obtain the degree of Master of Science in Applied Physics at Technische Universiteit Eindhoven Faculteit Technische Natuurkunde about the research conducted at Technische Universiteit Eindhoven Faculteit Technische Natuurkunde University of Virginia Department of Mechanical and Aerospace Engineering Author: E.E.K. de Caluwé Supervisors: Prof.Dr. M.A.J. Michels Prof.Dr. M. Utz Dr. A. Lyulin 22nd February 28

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4 Abstract The microscopic mechanical behaviour of amorphous disordered materials in the glass phase with or without deformation is ill-understood in the literature. Given the disordered nature of these materials most theories deal with statistical properties of molecular motion in these materials. Cooperative motion of molecules appears to be a very important trait in the dynamics associated with the glass transition, the irreversible deformation and emergent plasticity. This thesis is an addition to the research being performed on this kind of microscopic dynamics. Two models have been developed to study different aspects of emergent plasticity in disordered media. The first model is a finite-element type of simulation of a lattice consisting of linear springs with a certain threshold at which the individual springs break. The threshold is randomly distributed, with different types of statistical distributions. It has been shown that this simple model can display tough or brittle behaviour just as real polymers. Different statistical properties of the lattice under mechanical load are studied. It is shown that Random Percolation theory can be applied in some regimes. The scaling behaviour of the system failure is studied and it is shown that cascade-like avalanches of local ruptures can be observed at all length scales up to the system size. This is a trademark of Self Organized Criticality. The second model is an atomistic model of a binary Lennard-Jones system both in liquid and glass phase. It is shown that below the glass transition temperature cooperative particle motion can be observed with avalanche-like statistics similar to the first model. Cooperative clusters of moving particles are studied for different temperatures in the glass phase. Furthermore in the glassy state also the avalanche statistics under shear are looked at. This makes a comparison of the statistics of the finite-element type of simulations and the binary Lennard-Jones system possible.

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6 CONTENTS 3 Contents 1 Introduction Historical Background Organization of this Research Research Goal Simulation Scales Simulated Properties Definitions and Background Mathematical Notations Mechanical Definitions Strain and Shear Simulation Box Scaling Phenomena Random Percolation Self Organized Criticality Computer Simulations Finite Element Simulations The Spring Network System Equations Non-Ideal Springs Iterative Breaking Rules Macroscopic Stress-Strain Cholesky Factorization Binary Lennard-Jones Glass Simulations Simulation Details Thermodynamical Ensembles Computer Code Finite-Element Modeling: Results and Discussion Comparison to Previous Work Influence of the Averaging Procedure Influence of α and β on the Macroscopic Stress-Strain Relation System Size Dependence of Yield Point Evolution of Damage as a Function of Strain Avalanche Statistics SOC: Post-Yield Behaviour

7 4 CONTENTS 4.8 Connected Avalanches: Fractal Dimension Energy Drops Binary LJ Glass: Results and Discussion Glass Transition Equilibrium Dynamics: Thermal Motion Non-Equilibrium Dynamics: Shear Simulation protocol Deformation events Simulation results Recommendations and future work Conclusions and Recommendations Conclusions Recommendations Bibliography 77

8 5 Chapter 1 Introduction Polymers and polymer-based technologies are becoming increasingly more important in the scientific community and in the industry. Scientifically there is still a lot unknown about the microscopic phenomena in polymers and in general disordered materials. Although the macroscopic dynamical behaviour, i.e. stress-strain relationships, is well studied and described, the molecular origin is still the object of much research. Materials, not only polymers, can display totally different mechanical yielding and failure. In general all materials show an initial linear elastic response to applied stress. Materials differ mainly in the non-linear deviation and the eventual failure. Two general mechanical classes of materials can be discerned in terms of deformation and failure: brittle and tough materials. Brittle materials display a relatively short non-linear regime after which they show failure, i.e. they break. In contrast, tough materials have a relatively long non-linear inelastic regime. Usually there is a yield peak, after which a strain softening regime can be observed, i.e. the decrease of stress when strain increases. For some materials, most notably rubber-like materials, before failure a strain hardening regime can be observed, i.e. an increase in stress when strain is increased. 1.1 Historical Background Ordered or crystalline materials have always been better understood in literature. Those materials are easier to describe in a top-down generic formalism. On the other hand the study of the microscopic behaviour of disordered media is still at the cutting edge of science. In different parts of science, e.g. mathematics, physics and material science, new mathematical formalisms like Random Percolation theories, Self Organized Criticality and Scaling have emerged. These are all members of the family of advanced statistical theories. They usually deal with statistical distributions of properties instead of just meaningful average values. It is realized now that glassy polymers show cooperative behaviour at the molecular level. For a single molecule to move through the material it is in general required that large clusters of neighbouring clusters move simultaneously. These clusters form causally connected avalanches of deformation. The study

9 6 Introduction of these clusters and avalanches exploits the analogy with Random Percolation and Self Organized Criticality respectively. 1.2 Organization of this Research Research Goal This project is intended to be on the one hand a contribution to the understanding of the dynamics of disordered materials and on the other hand the statistics involved, which appear to be closely related to more general physical phenomena as Random Percolation [3] and Self Organized Criticality, [1, 2]. The ultimate goal would be to understand yielding and emergent plasticity in a real practically useful polymer. This is a very hard problem, because a real polymeric material is very complex and difficult to study in detail on very small timescales. Furthermore the complexity of the system introduces a lot of features that should be considered as noise for the purpose of this research. The goals of this report are twofold. First to give insight into the yield of polymerlike materials using statistical theories and Finite-Element Modeling. Second to understand the cooperativity of glassy dynamics and emergent plasticity by atomistic molecular dynamics and Monte-Carlo simulations Simulation Scales The problem of emergent plasticity in polymers and the microscopic dynamics involved makes a bottom up approach - in the sense that macroscopic phenomena will be explained based on microscopic mechanisms - very useful. Therefore computer models will be developed and applied, and a multi-scale approach will be used. Three simulation scales are defined. First at the highest mesoscopic level a finite element type of toy model is used as a simple linear way to capture the essence of a mechanically disordered material. The model is equivalent to a network of springs on a disordered lattice. The spring lattice can be loaded and the individual springs can break according to specific rules [4]. Significant improvements have been made as compared to previous studies [6, 7, 8] in terms of size, size dependence and statistics. Since the essential assumptions of this model involve spring breaking rules, this part is of most importance for yield and failure in polymer-like materials. The second step will be to study the cooperative dynamics of a binary Lennard Jones liquid via an atomistic molecular dynamics simulation. This step is based on algorithms and approaches developed initially by Utz et al. [9, 1, 11, 12]. The general framework of the existing molecular dynamics and Monte Carlo code is used to design a specific model, tailored for the study of cooperative motion in binary Lennard-Jones glasses. In short this means defining a suitable glass forming system and defining and implementing a method to identify microscopic plastic events during deformation and to analyze the simulation results statistically. This second approach will focus on emergent plasticity and cooperative clusters of mobile particles in a simple glass forming system.

10 1.2. ORGANIZATION OF THIS RESEARCH 7 The third simulation scale would be a full atomistic molecular dynamics simulation of a chemically realistic polymer. Using the results of the first two scales it should be possible to also understand emergent plasticity in full molecular dynamics simulation of a polymer. Given the limited time available for this project this third step will be a recommendation for future research Simulated Properties The main results to be gained from the two parts of the project are physical characteristics that deal with plastic events and with cluster and avalanche formation. In all simulations plastic deformation or failure events are defined. Based on this definition correlations in space (clusters) and time (avalanches) can be defined. Numerous properties can be calculated with regard to these events, inspired by Random Percolation theory, Self Organized Criticality and fractal geometry. These terms will be explained later in this report. In general all these properties are functions of the applied strain or stress. Conclusions will be drawn based on these quantitative properties with respect to the underlying mechanism of macroscopic features such as emergent plasticity, deformation and failure of polymer-like materials.

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12 9 Chapter 2 Definitions and Background 2.1 Mathematical Notations The mathematical notation in this thesis will be introduced below. Especially for vector and matrix equations this could be useful. A vector is distinguished from a scalar by an overline as in: X = the vector labeled X (2.1) Furthermore a matrix is distinguished from a scalar and vector identity as: The norm of a vector X has the notation: X = the matrix labeled X (2.2) X (2.3) This is to be distinguished from the determinant of a matrix X: A normalized vector X is given by: X (2.4) X X X (2.5) The inner product of two matrices or vectors X and Y is given by: The real part of a complex number z is given by: X Y (2.6) R(z) = the real part of z (2.7)

13 1 Definitions and Background 2.2 Mechanical Definitions Strain and Shear Definitions of Strain In different disciplines of science and engineering different definitions for strain are used. For clarity the three most common ones will be described below. The first definition is the engineering strain, ɛ eng. Let us consider uniaxial deformation for simplicity. The engineering strain would then be defined as: ɛ eng = L L (2.8) Here L is the initial length of the sample projected on the axis of deformation and L the difference in the projected length before and after deformation. This definition is therefore irrespective of how the sample was deformed. Another definition of strain is true strain, ɛ true. Again for uniaxial deformation this can be defined by: ɛ true = L1 L dl L (2.9) with L and L 1 the projected length along the axis of deformation before and after deformation respectively. For a constant uniaxial strain rate this would give: ɛ true = ln( L 1 L ) (2.1) In the limit of small deformation, true strain is equal to engineering strain as shown by: ɛ true = ln( L 1 L ) L 1 L 1 = ɛ eng (2.11) In general the true strain and engineering strain are related by: Finally the draw ratio is defined as: ɛ true = ln(1 + ɛ eng ) (2.12) λ = ɛ eng + 1 (2.13) Simulation Box All simulations in this work have periodic boundary conditions. This means that it is enough to define one simulation box to run a simulation that extends to infinity without artificial boundaries, but evidently with an information content only equal to the information content of the simulation box itself. A simulation box is in general defined by three principal axes, that span the three dimensional space as a basis set. With sufficient generality for this purpose the box can thus be naturally represented by a 3x3 matrix as:

14 2.3. SCALING PHENOMENA 11 h = k x l x m x k y l y m y k z l z m z (2.14) Here h represents the simulation box and k, l, and m are the principle axes of the box. For example a cubic lattice would require orthogonal principal axes of the same length: k = l = m (2.15) k l = k m = l m = (2.16) The volume of a box is given by the determinant of the matrix h: V = h (2.17) The advantage of this representation is that a deformation, i.e. strain or shear, can be described by a transformation matrix: ɛ 11 ɛ ɛ = ɛ 21 ɛ (2.18)..... Using this deformation matrix ɛ, any strain or shear can in principle be described, since in principle any new full rank matrix can be generated by multiplying an existing full rank matrix by a transformation matrix. The rank of a matrix is here the usual definition of rank in linear algebra, i.e. the number of linear independent columns or rows of a matrix. The generalized engineering strain can now be defined as: h 1 = ɛ eng h (2.19) with h representing a simulation box before deformation and h 1 the simulation box after deformation. This is the most natural property to use in computer simulations. In order to be able to relate the imposed strain in the computer simulations to true strain, the true strain matrix defined in equation 2.9 corresponding to the engineering strain of equation 2.18 is given by: ɛ true = ln(1 + ɛ 11) ɛ 12 ɛ 13 ɛ 21 ln(1 + ɛ 22 ) ɛ 23 (2.2) ɛ 31 ɛ 32 ln(1 + ɛ 33 ) 2.3 Scaling Phenomena Many physical phenomena are invariant under scaling transformations. In reality these phenomena are not fully scale invariant but only within certain bounds. Usually the upper boundary of scale invariance has a direct relation to for example the finite amount of energy available in a physical system or its finite system size. On the other side of the spectrum the scale invariance is usually bounded by the smallest physical scales where the phenomena still makes sense.

15 12 Definitions and Background As an example consider the well known scale invariant Richter law for earthquakes. The distribution of earthquakes over particular intensity within a given time interval obeys a power law: n(i) I γ f(i) (2.21) with n(i) the distribution of earthquakes of a certain intensity I, γ a constant exponent and f(i) a cutoff function that is a constant for I within the scale invariant bounds and drops to zero faster than the power law function itself, i.e. the cutoff function goes typically exponentially to zero. The qualitative behaviour of the upper cutoff function f is shown in figure 2.1. This type of equation is a trademark of scale free or scale invariant phenomena, i.e. for I = αi and n = α γ n (with I and I below the cutoff) the exact same relation as in 2.21 is found for I := I and n := n. The cutoff function f cuts off this behaviour for intensities where the energy involved in an earthquake is unavailable and for intensities below a minimum energy required for an earthquake to take place. 2.4 Random Percolation Random Percolation (RP) can in one sentence be described as a mechanism of randomly adding connective elements on a discrete or continuum lattice until eventually connective paths emerge with length scales of the lattice itself. The element density for which macroscopic connectivity occurs is the critical point of a RP system: the percolation threshold p c. Upon approaching the critical point the connectivity of the system diverges. An example of discrete RP can be seen in figure 2.2. The figure shows that after the random addition of connective elements to the lattice, eventually a connective path is formed from the left to the right thereby establishing macroscopic connectivity. In the literature usually two different kinds of percolation criteria are used. One is connectivity percolation, the kind that has been mentioned above. The second is rigidity percolation. A set of connected elements is only considered a rigid cluster if the elements have no degrees of freedom with respect to each other under the assumption of fixed bond lengths, but free bond angles. Percolation is then the formation of one infinite rigid cluster. Random Percolation deals with clusters, regardless of the definition used. The maximum cluster size increases as more elements are added to the system and the maximum cluster size diverges when approaching the critical point of the system. Therefore the statistics associated with RP has to do with clusters. These clusters are dependent on the history of the process of RP. Typically the following statistical properties are interesting. The probability for any lattice site to belong to a cluster of size s 1 is given by p, the occupational ratio of the lattice, which is defined as the fraction of sites in the lattice that are occupied by connective elements. The cluster size can be defined in terms of the number of elements, s, or the spatial extent of a cluster: X = max i, j i x i x j (2.22)

16 2.4. RANDOM PERCOLATION 13 f(x) 1 cutoff x Figure 2.1: The qualitative behaviour of the cutoff function f(x) with argument x a physical property such as a length scale. The cutoff function is constant for x < 1 and drops to zero for x > 1. with X the size of a cluster in a particular dimension and i and j indices of all elements in a cluster, with the coordinates x i and x j respectively. The average size S of all clusters is defined as: S = s P s (2.23) with P the probability for a random site to belong to a cluster of size s, as given by: P = n ss s n ss (2.24) and with n s the statistical probability distribution, normalized to unity, to find a cluster of size s. The center of mass r CM of a cluster is given by the average of the centers of all the elements forming the cluster: r CM = s j r j s (2.25) The center of mass can then be used to conveniently express the gyration radius r g of a cluster of size s as another measure for the spatial extent: r g 2 = s (r i r CM ) 2 i s (2.26) Alternatively this can also be defined as a function of the distance between all the primitive elements in a cluster:

17 14 Definitions and Background Figure 2.2: Example of Random Percolation on a discrete lattice. One by one connective elements are added at random lattice sites to the system. Eventually a connective path is formed (in red) from the left to the right side spanning the entire system and establishing macroscopic connectivity. r g 2 = 1 2 s (r i r j ) 2 i,j s (2.27) The correlation length, ξ is an important property in RP theory. It is the average distance between two primitive elements of the same cluster calculated over clusters of all sizes s: ξ 2 = 2 s r g 2 n s s 2 s n ss 2 (2.28) In a RP process the probability distribution n s of cluster sizes s for p approaching p c is typically given by: n s (p) = s τ f(s σ (p p c )) (2.29) where f is a cutoff function that is not universal and is to be determined numerically. This cutoff function, introduced earlier in section 2.3,is a constant for its argument well below unity and drops to zero for values of the argument larger than the unity: f(x) constant, x (2.3) f(x), x (2.31)

18 2.5. SELF ORGANIZED CRITICALITY 15 Combining equation 2.29 with the definition of the average cluster size S in equation 2.23 gives: S p p c γ (2.32) showing the divergence of S when approaching the critical percolation threshold p c. The scaling exponent γ is given by: p c : γ = 3 τ (2.33) σ The correlation length ξ has also an algebraic divergence upon approaching ξ p p c ν (2.34) In general if clusters have a well defined fractal dimension, then the number of elements in a cluster and the spatial size (r g ) are related through a power law for sufficiently large cluster sizes: s r D f g (2.35) with D f the scaling exponent which is by then by definition the fractal dimension. In the framework of Random Percolation the fractal dimension D f can be related to the other exponents σ and ν: D f = 1 (2.36) σν The scaling exponents are considered to be universal when Random Percolation theory applies to a system. Universal in the sense that they do not depend on the topology of a system or on the physical properties involved. The literature values of these exponents are given in table 2.1 for both 2-D and 3-D systems. 2.5 Self Organized Criticality A physical system that is said to be critical is a system that is unstable and has perturbations on all length scales that bring it back to equilibrium. For a Self Organized Critical (SOC) system this process of perturbations Table 2.1: Literature values of the universal scaling exponents in systems that can be described by the Random Percolation framework for both two and three dimensional problems. D γ τ ν D f

19 16 Definitions and Background brings a system from one critical point to the next, leading to a cascade of perturbations over all length scales, i.e. scale invariant perturbations. Therefore typical statistics associated with this kind of phenomena is avalanche statistics. The example given in section 2.3 of the scale invariance of the intensity of earthquakes is an example of a critical system. Since one earthquake brings the system, i.e. the shell of the earth consisting of sliding plates, to a new critical point and triggers the next earthquake this is a manifestation of Self Organized Criticality. Also plasticity, deformation and fracture in materials can be looked at within the framework of SOC. A macroscopic rearrangement in a material is nothing more than an avalanche, i.e. a cooperative motion, of individual irreversible microscopic events in a material. Empirically it is known that macroscopic behaviour of a material changes as stress is applied. Stress is therefore the driving force that drives the system towards criticality. Let an avalanche have a size a, i.e. consist of a causally correlated events such as local failures. The scale-free distribution of avalanche sizes as a function of the stress σ applied on a material is then: ( ) γ a n a (σ) f(a/a (σ)) (2.37) a with n a the normalized distribution of avalanches of size a as a function of the applied stress. The function f is a cutoff function that expresses that the maximum avalanche size is limited by the external stimulus given. Therefore a is a function of σ. The scaling exponent is indicated by γ. In the theory of SOC a lot of exponents have the same names as in the theory of RP. They appear in a similar context, but values for the exponents found in RP are not necessarily applicable to the corresponding SOC exponents. Obviously a has as lower bound: σ a a since in equilibrium without external stimulus a material is at rest so no avalanches can occur. The upper bound for a is given by the size of the system. Meaning that at this critical point avalanches comparable to the size of the system can occur which will lead to mechanical failure. Since in between a and a no other typical scales are available, the cut-off diverges algebraically on the approach of the critical stress σ c : a (σ) σ σ c 1/δ (2.38) The m-th moment of the distribution n a is given by the standard statistical definition: < a m >= a m γ f(a/a )da a m γ+1 σ σ c γ m 1 δ (2.39) where the right part is obtained by substitution the divergence of a according to equation For all moments to yield an integrable result, the condition γ < 2 must hold. The distribution 2.37 as a function of σ is defined on each infinitesimal interval around σ. Integrating this expression over a stress interval gives the cumulative distribution of avalanche sizes from σ = to a given stress σ:

20 2.5. SELF ORGANIZED CRITICALITY 17 ( ) τ a C a (σ) G(a/a (σ)) (2.4) a with G a cut-off function and τ given by τ = γ + δ. The damage p accumulated in a material is the sum of all elements that have participated in an avalanche, or the sum of all avalanche sizes. The damage dp accumulated in an infinitesimal stress interval dσ is therefore: dp < a > dσ (2.41) with the average avalanche size < a > given by equation 2.39 the damage rate can be related to the stress σ by: dp dσ σ σ c 2 γ δ (2.42) As mentioned before the upper bound for a will be governed by the system size. For a finite system as is the case in a simulation, the divergence of a given by equation 2.38 should be corrected for the system size L and should approach this value of L: a a (σ) = σ σ c 1/δ +CL D f (2.43) with D f defined to be the fractal dimension of the avalanches assuming there is one, that relates the number of elements a to a geometrical size to be compared to the system size.

21 18 Definitions and Background

22 19 Chapter 3 Computer Simulations In order to get insight in the microscopic origin of the mechanical behaviour of disordered materials, and especially of polymers, this research makes use of computer models. Such an approach has, of course, its advantages and disadvantages. Disadvantages are that the computer models can only describe a very much simplified version of reality, both in terms of physical mechanisms and in terms of the size of the systems studied and the available simulation time scales. On the other hand there are also clear advantages. The simplifications of reality that are necessary to make implementation of a computer model feasible also increase the potential insight gained. It is on the one hand much easier to discern the relevant data through all the noise of an experiment, since the simulations are tailored to the specific problem. On the other hand it is also possible to focus specifically or exclusively on the influence of certain parameters or mechanisms independently, which is often impossible in reality. Also certain combinations or ranges of parameters might be impossible to achieve in reality but perfectly tunable in a model. The two models used in this research each have their own merits and drawbacks. The first is a very simple spring lattice toy model and the second is a continuum binary Lennard-Jones molecular system. 3.1 Finite Element Simulations The spring network toy model is used to capture the essence of linear elastic behaviour of a material and the deviation from linear behaviour as caused by microscopic failure in a material. The model is equivalent to a network of springs on a disordered lattice. The spring lattice can be loaded and the individual springs can break according to specific rules The Spring Network The system is made of a network or lattice of linear elastic elements. In essence the model is a form of a finite element type of simulation. The network can be both two or three dimensional. Our test results show that the two dimensional network captures the essence of the dynamics just as well as a three dimensional one, at the same time requiring much less computational power. For these two

23 2 Computer Simulations reasons the focus of the present thesis is on two dimensional networks. An example of such a network can be seen in figure 3.1. The two dimensional simulation box is randomly filled with N = L 2 points according to a uniform distribution over a 1 1 normalized lattice. The points are then connected by bonds that represent springs (or rather elastic rods of finite equilibrium length) along the edges of a Delaunay triangulation of the set of points. This triangulation ensures that no points lie within the circumference of each triangle, that all vertices are part of at least one triangle and that all triangles fill the entire simulation box System Equations In this section the equations that need to be solved to find the equilibrium configuration of a spring network under strain are discussed. In this section the springs are assumed to be perfectly linear. In later sections spring breaking rules will be added. The Individual Springs The finite element type of geometries that are used in the spring lattice simulations consist of springs forming a random triangulated Delaunay lattice. All individual springs are Hookian springs upon extension from their non-zero equilibrium value. The mechanical response of one spring is given by: F = k η U + U U + U (3.1) with F the force exerted by the spring in response to a deformation away from its equilibrium vector U. The force is expressed in terms of the Hookian spring constant k and the extension relative to the equilibrium position η, which is defined as: η = U + U U 1 (3.2) The force is a vectorial property parallel to the sum of the initial spring vector, U and the displacement U + U, according to figure 3.2. The vectorial nature of this spring response that is necessary to express the force in terms of the displacement makes the equation non-linear with respect to the displacement. However as illustrated in figure 3.2 the problem can be linearized for small angles between U and U + U. When linearized, the displacement should be substituted by the displacement projected on the original spring vector as in figure 3.3, or U cos(α). The requirement of a small angle between the initial spring vector and the spring vector after displacement for a valid approximation is satisfied for small displacement U compared to U. During the simulation we are interested in relatively small macroscopic strain values. Locally there can be in principle much larger deformations, but only for broken springs that have a negligible spring constant and therefore a negligible influence on the total configuration. Therefore it is valid to approximate the individual springs with the linearized displacement. The relative extension η is then given by:

24 3.1. FINITE ELEMENT SIMULATIONS 21 Figure 3.1: Example of a two dimensional network used in the spring lattice type of simulations. The simulation box is normalized to have a length of 1. A simulation is characterized by a size L that determines the number of points N = L 2 that are randomly distributed over a discrete lattice. The points are connected by bonds that represent springs along the edges of the Delaunay triangulation of the set of points. All springs together form network of springs. η = U Ũ U (3.3) and the direction of the force is approximated by: U + U U + U Ũ (3.4) Substituting these two equations into the non-linear spring response in equation 3.1 gives the linear response: F = k U ( U Ũ) Ũ (3.5) This can be written in a more convenient matrix form: F = K U, K := k U UŨ (3.6) with K containing the spring stiffness and initial spring geometry. For the system of one spring only K is a 2 2 matrix for the two dimensional case.

25 22 Computer Simulations Figure 3.2: Example of an isolated spring, meant to illustrate the different geometrical properties. Since the displacement is not parallel to the original spring U, the force is non-linear to the displacement U. In the limit for small angle between U and U + U, the problem can be linearized. A System of N connected Springs The equation 3.6 can be generalized to describe a system of springs connecting the N points. The system equation, i.e. the equation that expresses all displacements as a function of the boundary conditions in equilibrium, has then the same form as equation 3.6, however K is now a 2N 2N matrix for the two dimensional case: F = K X (3.7) The 2N 1 vector F contains the forces on each point with two elements for each point, for a two dimensional lattice as in figure 3.1. Note that in equilibrium the sum of all forces on each point not subject to a boundary condition must be zero. The system equation is now expressed in terms of the 2N 1 vector X of the displacements of all points. This is possible since the displacement U ij of the spring connecting the points i and j can be expressed in terms of the displacement of the end points: U ij = X j X i (3.8) Boundary Conditions The force at each point is now expressed by the system equation 3.7 as a function of all the displacements of the individual points. The sum of forces on all springs that are not on a lattice edge must be zero in equilibrium. In order to have a fully determined system of equations, for which there is a unique solution, for each point on the edges additional boundary conditions must be defined.

26 3.1. FINITE ELEMENT SIMULATIONS 23 Figure 3.3: For small angle between U and U + U, the displacement can be linearized by U cos(α) and decomposed into x and y components. Macroscopic strain is applied by imposing the extension through the boundary conditions. In one direction the two opposing edges have a constant distance 1 and in the other direction the opposing edges have a constant distance 1 + ɛ, with ɛ the macroscopic strain. To reduce the influence of arbitrary boundary conditions for a relatively small simulation model, periodic boundary conditions will be assumed in both dimensions. This means that the system can in fact be regarded as a toroïdal construct that can be inflated 3.4. Only springs on a lattice edge have a non-zero corresponding element in the F vector, since the boundary conditions implicitly impose an external force. For larger systems the majority (1 4/N) of the points lies not on a boundary. Therefore the F vector contains mostly zero elements and is thus very sparse. The system matrix K contains only non-zero elements where points are connected to each other. The coordination number is the average number of neighbours each point has. Since this number approaches a constant value for larger systems, each point is connected on average to a constant number of neighbouring points. Therefore the number of springs scales as O(N) but the number of possible connections between all points, and the size of the system matrix, as O(N 2 ). Therefore the system matrix is very sparse for larger systems. The sparse nature of the equations involved makes it possible to significantly reduce the computational memory and CPU cost of an algorithm. The computational cost of solving the system 3.7 of 2N coupled equations scales with (2N) 2. This holds both for memory and flops (floating point operations) count Non-Ideal Springs The previous sections have dealt with ideal linear springs. However we are mainly interested in the behaviour of the network of springs when springs are

27 24 Computer Simulations Figure 3.4: The spring lattice network symmetry can be seen as a toroïdal structure. allowed to break. The mechanical response of a non-ideal spring under load is schematically drawn in figure 3.5. The force, F exerted on a spring is proportional to the relative extension η until a certain threshold strain, t. At this point the individual springs irreversibly break. This means in numerical practice that the springs are given a negligibly small Hookian spring constant: η t : F = k η (3.9) η > t : F = k η (3.1) k ( k ) (3.11) A certain fraction, 1 β, of all springs are given an infinite threshold strain value. This means that they are unbreakable. They should mimick the effect of covalent polymer bonds, whereas the breakable ones account for the cohesive interactions. The property β can be interpreted as the fraction of breakable bonds with a finite threshold. The finite threshold values for the individual springs are distributed according to a power-law distribution: P (t) = (1 α)w α 1 t α, t [, w] (3.12) This distribution is normalized to unity on the interval for < t w. The constant w is the maximum value of t and is given by w 2 in this project. For a given w distribution has one variable parameter, α. The threshold distribution is schematically drawn in figure 3.5 for some values of α. For α = the distribution P (t) is a constant or uniform distribution. The value of α should be less than α = 1 because then (for t ) the distribution is no longer integrable. For α 1 a divergent tail of weak bonds emerges. The distribution is no longer uniform, but has an overpopulation of very weak bonds. The parameter α describes therefore the type of disorder in the system with respect to the thresholds. The disorder is defined to be weak for (near) uniform distributions with α and to be strong for α.

28 3.1. FINITE ELEMENT SIMULATIONS 25 Figure 3.5: The spring threshold distribution as a function of α for several values of α. In summary a particular spring network is characterized by 3 parameters. These parameters are the lattice size, L 2 N, and two parameters that define the mechanical behaviour of individual springs, α and β Iterative Breaking Rules In the previous sections it was explained how to calculate the equilibrium configuration for a network of springs with a given macroscopic strain imposed via the boundary conditions. However the essential element of the spring lattice simulation in this work is studying the non-linear effect of breaking springs. The way this is solved is an iterative application of the procedure for ideal springs. The equilibrium configuration of the system and the specific boundary conditions are calculated by solving the system matrix equation 3.7. For each spring the relative extension η is then calculated and compared to the threshold of each spring that is assigned when the simulation is initialized. The spring with the highest ratio of r = η/t is determined. If r > 1 then this spring is broken and its stiffness is set to a negligible value. In this way only one spring can break at a time or if r 1 the equilibrium configuration is the final equilibrium configuration for the non-linear breakable system. When a spring is broken, the system matrix K must be updated at all elements that are connected to the single spring that breaks. This means that only 2 2 matrix elements change. This is a rank-2 update to the system matrix, equivalent to two rank-1 updates. A rank-n update means that a rank- N matrix is added to the matrix to be updated. For this updated system matrix K that differs only in 4 elements, i.e. the 2 two-dimensional end points of a spring, the same equation 3.7 has to be solved until r 1. This is equivalent to the physical experiment where the system is loaded until a spring breaks after which it is unloaded and slowly reloaded again until the next event.

29 26 Computer Simulations Macroscopic Stress-Strain Since all springs are linear (up to the threshold) it is possible to calculate the macroscopic strain and stress at which the spring would have broken would the strain have been continuously increasing as in a real experiment. A sequence of breaking events forms a stress strain curve. It is now possible that a spring breaks at a strain that is smaller than the strain at which the previous spring broke. This is physically impossible in a real experiment; then these springs would have been broken at the same time. A sequence of such causally connected breaking events is defined in the simulation as a causally connected avalanche Cholesky Factorization Since almost the same equation has to be solved over and over when springs break, algorithms that can store intermediate calculations which can be easily updated are very efficient. Therefore in this work we make use of an algorithm of solving the system equations that makes use of Cholesky factorization and Cholesky updates with sparse intermediate results. This kind of algorithm is also found to be the most efficient for two-dimensional lattice type of simulations in the literature [13]. A Cholesky factorization is possible for symmetric positive definite matrices. The factorization gives for any symmetric positive definite matrix A: A = L L (3.13) with L a lower triangular matrix. matrix A = A ij has the properties: A matrix A is positive definite if the i : R(A ii ) = A ii > (3.14) i, j i : A ij A ii A jj A ii + A jj 2 (3.15) If the system of springs has a stable equilibrium solution, then a mechanical system matrix is always positive definite. Therefore our system of equations can be factorized and rewritten to: F = L L X (3.16) Solving a coupled linear set of N equations of this form, equation 3.16 is much faster than the corresponding equation 3.7; it is in fact an order O(N) algorithm instead of O(N 2 ), and makes use of a forward or backward substitution method. In other words, once we know the Cholesky factorization of the system matrix, the computation is an order O(N) algorithm instead of O(N 2 ). The factorization itself is however computationally expensive. But it has the convenient property that the Cholesky factorization of a matrix A can be updated very efficiently to the Cholesky factorization of the matrix A + rank-1 update. This is exactly the property that makes this the most efficient algorithm to compute the system of linear springs including breaking events.

30 3.2. BINARY LENNARD-JONES GLASS 27 The entire algorithm is implemented in Matlab given that software s convenient vector oriented nature for this problem. Matlab does not have a standard method for the sparse factorization and update of symmetric positive definite matrices. Therefore the external package Cholmod has been imported in Matlab with considerable effort. This package is based on the work of Davis et al. [31, 32, 33, 34]. The implementation of the new algorithm in the existing program required a total rewrite of the simulation code of Malakhovski that paid back in about a 1 fold speed increase and about 1 times larger systems that can be simulated. 3.2 Binary Lennard-Jones Glass The second part of this research is focused on an atomistic model of a binary Lennard-Jones glass. This is a system consisting of two types of atoms (A and B), which experience only inter atomic Lennard-Jones interactions: V αβ (r αβ ) = 4ɛ αβ [ ( σαβ r αβ ) 12 ( σαβ r αβ ) 6 ] (3.17) α, β {A, B} (3.18) In this equation V αβ is the inter atomic potential. This potential is a function of r only, which is the distance between two atoms in question. The potential is a binary interaction type. The functional form of V depends on the typical Lennard-Jones parameters σ αβ and ɛ αβ, which define the identity of and thereby the difference between the particles A and B. The reason to study a binary Lennard-Jones system is that a unary Lennard- Jones system is very prone to crystallize when the system is cooled down. A binary system however, with a suitable difference in the LJ parameters, will form a glass when cooled down Simulations There are different ways to model an atomic system with a binary Lennard-Jones interaction. Two very common methods are Monte Carlo type of simulations and molecular dynamics type of simulations. These are the methods that will be used in this research. The (Metropolis) Monte Carlo scheme iterates through phase space on a path of optimal probability. Propagation in this scheme means trying small perturbations, trial moves, to the configuration. A probability is assigned to the new system based on the energy difference associated with the perturbation. The trial move is accepted with a probability proportional to the Boltzmann factor of this energy. A decrease in energy is always accepted. The key strength of Monte Carlo is its ability to efficiently explore the phase space of a particular thermodynamic ensemble. This is particularly useful in determining equilibrium configurations. Despite the efficiency of a Monte Carlo simulation in calculating and comparing different independent configurations of the same system, it is not suitable for

31 28 Computer Simulations studying the dynamics of a system. An iterative sequence of Monte Carlo generated systems is merely statistically correlated, since only small perturbations of a system are used in order to generate a new system. In contrast, classical molecular dynamics methods are based on solving the Newtonian equations of motion of all atoms in a system. An arbitrary number and variety of interactions can be modeled by assuming a functional form of potentials or forcefields associated with all independent degrees of freedom. A configuration of a system is propagated by solving the equations of motion for the next time step. The result of this exercise is a sequence of systems that represent the evolution in time of the system. This method is therefore very useful to study the dynamics of a system and gives results that have a meaningful concept of time Simulation Details For the binary Lennard-Jones system the degrees of freedom are the distances between all particles. These form however an over constrained system and are in fact not independent. An independent set of configurational degrees of freedom is in this case the positions of the particles. In three dimensions the number of configurational degrees of freedom is thus 3(A + B). Now a method of time propagation and of solving the equations of motion has to be chosen. In the literature a widely used method is the discretization of the equations of motion in time using a Verlet-equivalent algorithm. The scheme used in this research is the velocity Verlet algorithm, which has the following discretization form: r(t + δt) = r(t) + δtv(t) δt2 a(t) (3.19) v(t + δt) = v(t) + 1 δta(t + δt) (3.2) 2 with r the position of a particle, δt the timestep of the scheme, v the velocity of a particle and a the acceleration of a particle as defined by the Newton equation: F = ma (3.21) The velocity Verlet algorithm is frequently used in the literature, partly because it only requires storage of r, v, and a and because the velocity is explicitly calculated to be used for monitoring the kinetic energy of the system at each time step. The two different particles A and B are modeled after Ni 8 P 2 because this system is known to have a deep eutectic point in the phase diagram for the chosen composition ratio and is therefore very prone to form a glass; this is also one of the reasons why it has been studied extensively in the literature [16, 17, 18, 19, 2, 27, 28]. The LJ parameters of A and B are taken from the articles of Kob and Andersen and given in table 3.1 for the different combinations of two particles. All units in the simulations are expressed in the Lennard-Jones units ɛ AA and σ AA. The Lennard-Jones potential is implemented with a cutoff, where the

32 3.2. BINARY LENNARD-JONES GLASS 29 Table 3.1: The Lennard-Jones parameters for the model system in units of ɛ AA and σ AA. αβ ɛ σ AA AB BB V αβ 2 ε αβ σ αβ r cutoff r αβ Figure 3.6: The Lennard-Jones potential is implemented with a cutoff (red line), and is shifted and corrected with a quadratic term (dashed line) to make the potential continuous and continuously differentiable at the cutoff, which has the value of 2 here for illustration purposes only. potential is shifted and corrected with a quadratic term to make the potential continuous and continuously differentiable at the cutoff value (typically 2.5 in the literature as well as in this project). This is done mainly to improve the stability of the integration scheme. These corrections are schematically drawn in figure 3.6. The actual potential that is simulated is of the form of equation V αβ (r αβ ) = 4ɛ αβ [ ( σαβ r αβ ) 12 ( σαβ r αβ a αβ = 6 b αβ = 7 ) 6 + a αβ ( σαβ r c ( σαβ r c ( rαβ r c ) 12 3 ) ) 2 + b αβ] (3.22) ( σαβ r c ( σαβ r c ) 6 (3.23) ) 6 (3.24)