Percolation and Elasticity of Networks

From Cellular Structures to Fibre Bundles

Perkolation und Elastizität von Netzwerken

Von Zellulären Strukturen zu Faserbündeln

Der Naturwissenschaftlichen Fakultät der Friedrich-Alexander-Universität Erlangen-Nürnberg zur Erlangung des Doktorgrades Dr. rer. nat.

vorgelegt von Susan Nachtrab

aus Rudolstadt Als Dissertation genehmigt von der Naturwissen- schaftlichen Fakultät der Universität Erlangen-Nürnberg.

Tag der mündlichen Prüfung: 9. Dezember 2011

Vorsitzender der Promotionskommission: Prof. Dr. Rainer Fink

Erstberichterstatter: Prof. Dr. Klaus Mecke

Zweitberichterstatter: Prof. Dr. Ben Fabry Abstract

A material’s microstructure is a principal determinant of its effective physical properties. Structure-property relationships that provide a functional form for the dependence of a physical property (e.g. elasticity) on the microstucture’s morphology are essential for the physical understanding and also the practical application in material design. This work focuses on materials with spatial mesoscopic network structure. A new model with adjustable network topology is introduced and its percolation properties and effective elastic properties are examined. In an initially four-coordinated network, ordered or disordered, each vertex is separated with probability p to form two two-coordinated vertices, yielding network geometries that change continuously from network structures to bundles of unbranched, interwoven fibres. The percolation properties of this so-called vertex model are studied for a two- dimensional square lattice and a three-dimensional diamond network, revealing a percolation transition at p = 1 in both cases. The analysis of the pair-connectedness function and finite size scaling exhibits critical behaviour with critical exponents β = 0.32 ± 0.02, ν = 1.29 ± 0.04 in two dimensions and β = 0.0021 ± 0.0004, ν = 0.54 ± 0.01 in three dimensions. The values of the exponents differ from those of conventional site and bond percolation, indicating that this vertex model belongs to a different universality class. In addition, in three dimensions the critical exponents do not obey the hyperscaling relations of bond percolation, but a heuristic new hyperscaling relation is found. After inflating the network edges to circular cylinders of finite radius, the resulting structure is interpreted as solid material in network shape, henceforth called network solid. Changes of probability p strongly affect the mechanical properties of such network solids. This is demonstrated by calculating the effective linear-elastic bulk and shear moduli using a finite element method based on voxel representations of the structures. Separating a fraction of the network nodes leads to a strong decay of the effective moduli whose functional dependence can be approximated, for p < 0.5, by an exponential decay for both, fixed and periodic boundary conditions. This is verified for ordered (diamond and nbo) as well as for irregular (foam) initial structures. Compression experiments on laser-sintered models based on diamond network solids confirm these results. In case of periodic boundary conditions, a cross-over from an exponential to a power-law decay in (1 − p) close to the critical point at p = 1 is observed. From this, the elastic critical exponent fc can be estimated as fc = 3.0 ± 0.1, which also differs from the site and bond percolation exponent. The morphological analysis of this work has several applications. For linear-elastic solids, it suggests that the network connectivity can be used as design parameter, for example for open-cell metal foams or bone scaffolds, as the elastic properties can be adjusted to a given value while keeping the pore space geometry and thus transport properties almost constant. The results of the percolation analysis are especially relevant for network models of biological or synthetic polymers with varying degree of cross-linking.

iii

Zusammenfassung

Die Mikrostruktur eines Materials hat großen Einfluß auf seine effektiven physikalischen Eigenschaften. Funktionale Abhängigkeiten physikalischer Größen (z.B. der Elastizität) von der Morphologie der Mikrostruktur sind essentiell für das physikalische Verständnis und die praktische Anwendung im Materialdesign. Der Fokus dieser Arbeit liegt auf Materialien, deren Mikrostruktur durch ein Netzwerk dargestellt werden kann. Ein neues Modell mit einstellbarer Netzwerktopologie wird vorgestellt, dessen Perkolationseigen- schaften und effektive elastische Eigenschaften untersucht werden. Ausgehend von einem ursprünglich vier-verbundenen, geordneten oder ungeordneten Netzwerk wird jeder Vertex mit Wahrscheinlichkeit p in zwei zwei-verbundene Vertizes getrennt. Netzwerkgeometrien werden dadurch kontinuierlich zu Bündeln verschlungener Fasern. Somit wird über den Parameter p die mittlere Verbundenheit der Vertizes von vier im ursprünglichen Netzwerk auf zwei bei p = 1 herabgesetzt. Die Perkolationseigenschaften dieses sogenannten Vertex-Modells werden auf dem zweidimensionalen Quadratgitter und dem dreidimensionalen Diamantnetz untersucht. Beide Modelle zeigen einen Perkolationsübergang bei p = 1. Die Analyse der Paar- Verbundenheitsfunktion und das Finite-Size-Scaling offenbaren kritisches Verhalten mit kritischen Exponenten β = 0.32 ± 0.02, ν = 1.29 ± 0.04 in zwei Dimensionen und β = 0.0021 ± 0.0004, ν = 0.54 ± 0.01 in drei Dimensionen. Die Werte der Exponenten unterscheiden sich von konventioneller Site- und Bond-Perkolation, was darauf hindeutet, dass das Vertex-Modell zu einer neuen Universalitätsklasse gehört. In drei Dimensionen werden außerdem neue Hyperskalen-Beziehungen für das Vertex-Modell vorgeschlagen. Werden die Kanten der Netzwerkstrukturen durch Kreiszylindern mit endlichem Ra- dius ersetzt, können sie als Festkörper in Form eines Netzwerkes interpretiert werden. Nachfolgend werden diese Strukturen Netzwerkkörper genannt. Mithilfe einer Finite- Elemente-Methode, die auf voxelierten Repräsentationen der Strukturen basiert, werden die effektiven, linear-elastischen Eigenschaften berechnet, wobei eine Veränderung von p großen Einfluß auf die mechanischen Eigenschaften solcher Netzwerkkörper hat. Die Trennung von Netzwerkknoten führt für p < 0.5 zu einem starken Abfall der effektiven linear-elastischen Moduln, deren funktionale Abhängigkeit von p sowohl für periodische als auch nicht-periodische Randbedingungen durch einen exponentiellen Abfall approximiert werden kann. Dies kann für geordnete (Diamant und nbo) und für ungeordnete (Schaum) Ausgangsstrukturen gezeigt werden. Kompressionsexperimente an Laser-gesinterten, auf der Diamantstruktur basierenden Modellen bestätigen dieses Ergebnis. Im Falle periodischer Randbedingungen kann ein Cross-over Verhalten von einem exponentiellen zu einem algebraischen Abfall in (1 − p) nah am kritischen Punkt p = 1 beobachtet werden. Daraus kann der elastische kritische Exponent fc zu fc = 3.0 ± 0.1 abgeschätzt werden, der sich ebenfalls von dem elastischen kritischen Exponenten der Site- und Bondperkolation unterscheidet. Für die morphologische Analyse dieser Arbeit gibt es verschiedene Anwendungen. Für linear-elastische Festkörper legt sie nahe, die Netzwerkverbundenheit als Designparameter,

v z.B. für oﬀenzellige Metallschäume oder Knochengerüste, zu verwenden, da die elastischen Eigenschaften auf einen bestimmten Wert eingestellt werden können, während die Porenraumgeometrie und somit die Transporteigenschaften kaum verändert werden. Die Ergebnisse der Perkolationsanalyse sind im Speziellen für Netzwerkmodelle biologischer und synthetischer Polymere mit variablem Vernetzungsgrad relevant.

vi Parts of the results of the research described in this thesis have been published or accepted for publication in the following journal articles:

Susan Nachtrab, Sebastian C. Kapfer, Christoph H. Arns, Mahyar Madadi, Klaus Mecke, Gerd E. Schröder-Turk, Morphology and Linear-Elastic Moduli of Random Network Solids, Advanced Materials, Volume 23, Issue 22-23, pages 2633–2637, 2011

Susan Nachtrab, Sebastian C. Kapfer, Dominik Rietzel, Dietmar Drummer, Mahyar Madadi, Christoph H. Arns, Andrew M. Kraynik, Gerd E. Schröder-Turk and Klaus Mecke, Tuning Elasticity of Open-Cell Solid Foams and Bone Scaﬀolds via Randomized Vertex Connectivity, Advanced Engineering Materials, 21 September 2011, DOI = 10.1002/adem.201100145

vii

Contents

1 Introduction: Networks and Foams1 1.1 Network Structures for Tissue Engineering...... 2 1.2 Solid Open-Cell Foams...... 3 1.3 Network Percolation and Elastic Properties...... 5

2 Network models with topology changes: From Open Cells to Fibre Bundles7 2.1 Graph Representation of Networks...... 7 2.2 Ordered Four-Coordinated Networks...... 8 2.2.1 Diamond Network...... 8 2.2.2 NbO Network...... 9 2.3 Disordered Four-Coordinated Networks...... 10 2.3.1 Voronoi Diagrams...... 10 2.3.2 Liquid Foams...... 11 2.4 Disconnection Mechanism: Node Separation...... 14 2.4.1 Shifting Nodes When Separating?...... 15 2.4.2 Periodic Boundary Conditions...... 21 2.5 Topological Characterisation...... 21

3 Percolation Properties 27 3.1 Cluster Characteristics...... 30 3.2 Scaling Theory...... 33 3.2.1 Universality Classes...... 34 3.2.2 Fractal Dimension Df of a Cluster...... 35 3.2.3 Finite Size Scaling...... 36 3.3 Implementation of the Vertex Model...... 40 3.4 Percolation Properties of the 2D Vertex Model...... 41 3.4.1 Fraction of Edges in Percolating Clusters...... 42 3.4.2 Number of Clusters...... 44 3.4.3 Pair-Connectedness Function G(r) ...... 45 3.4.4 Fractal Dimension Df ...... 47 3.4.5 Finite Size Scaling...... 47 3.4.6 Hyperscaling Relations...... 47 3.5 Percolation Properties of the 3D Vertex Model...... 50 3.5.1 Fraction of Edges in Percolating Clusters...... 50 3.5.2 Cluster Size Distribution ns ...... 53 3.5.3 Pair-Connectedness Function G(r) ...... 54 3.5.4 Number of Clusters...... 54 3.5.5 Fractal Dimension Df ...... 56 3.5.6 Finite Size Scaling...... 56

ix Contents

3.6 A Percolation Model in a New Universality Class?...... 58 3.6.1 2D Vertex Model...... 58 3.6.2 3D Vertex Model...... 59 3.6.3 Conclusion...... 63

4 Effective Linear-Elastic Properties of Network Solids 65 4.1 Theory of Linear Elasticity...... 65 4.2 Effective Medium Theory...... 70 4.3 Finite Element Method for Voxelised Structures...... 73 4.3.1 Sources of Error...... 75 4.4 Effective Elastic Properties of Network Solids...... 77 4.4.1 Voxelised Network Solids from Network Graphs...... 77 4.4.2 Elastic Moduli of Poisson-Voronoi and Collagen Network Solids. 78

5 Topology Dependence of Effective Linear-Elastic Properties 87 5.1 Numerical Results for Linear-Elastic Moduli of Network Solids...... 87 5.1.1 Effective Moduli as Function of Disconnection Probability p ... 89 5.1.2 Effective Moduli as Function of Volume Fraction φ: Probability p affects Power-Law Exponent...... 91 5.1.3 Influence of Boundary Conditions and System Size...... 92 5.1.4 Effect of Varying the Ratio of the Microscopic Moduli Gs/Ks .. 95 5.2 Relationship to Percolation...... 95 5.3 Compression Testing of Laser-Sintered Diamond Network Solids..... 102 5.3.1 Specimens Production and Experimental Set-up...... 102 5.3.2 Effective Young’s Modulus of Laser-Sintered Diamond Network Solids...... 106 5.4 A New Design Principle for Bone Scaffolds?...... 109

6 The Role of Network Topology in Material Design and Physics 111 6.1 Design of Bone Scaﬀolds and Metal Structures...... 112 6.2 Relevance for Cross-linked Bio-polymer Networks...... 112 6.3 A Percolation Model in a New Universality Class?...... 113

Acknowledgements 115

List of Abbreviations 117

Bibliography 119

x 1 Introduction: Networks and Foams

Microstructured materials are of interest not only in material science, but also in physics, biology, geology, chemistry and engineering. Famous natural examples are wood [271, 244, 78, 70, 173], cork [77], nutshells [119, 32], rock [253, 91, 146], foam [258, 59], bone [185, 154, 70, 36, 40] and tissue [128, 167]; man-made microstructured materials are, for instance, paper [3], steel [263, 101] or foams made of ceramic [192, 150], metal [13, 17] or polymers [170, 171, 181]. Fig. 1.1 shows some examples. The underlying microstructure is a major determinant of the physical properties of a material such as its mechanical response, electromagnetic properties and transport capabilities. Techniques to deliver insight into microstructure have strongly advanced over the last decades. Microscopy and X-ray imaging techniques have been giving two- dimensional information for a long time, but the development of atomic force microscopy [25, 216], confocal microscopy [172, 51], computed tomography [52, 236] and scanning electron microscopy [207, 198] made it possible to get three-dimensional real space information (in contrast to three-dimensional reciprocal space information obtained by X-ray scattering) which brought great insights and new impulses in research. The microstructural influence on crack formation [141] and failure [28], optical effects [176, 107] and charge transport in semiconductors [213] or transport through microstructured media [211] are of current scientific interest. The morphology, i.e. the shape or distribution in space, of many materials can be described by ordered or disordered network structures. Examples are synthetic [181, 195]

(a) Coconut shell (b) Berea sandstone (c) Polymer foam

Figure 1.1: Examples of natural and man-made microstructured materials. (a) Highly connected pore space of a coconut shell. (175µm)3 subsample reproduced from Ref. [32] (b) Pore space of Berea sandstone with porosity φ = 0.121. (1.08mm)3 subsample reproduced from Ref. [8] (c) Silicone resin foam: prece- ramic polymer foam that may be converted into a ceramic foam by pyrolysis. Ca. (800µm)3 subsample. Courtesy of Friedrich Wolﬀ, Lehrstuhl für Polymer- werkstoﬀe, Universität Erlangen-Nürnberg

1 1 Introduction: Networks and Foams

and biological polymer networks [224, 197], bone [76], hydrogels [7] or open-cell foams [79, 13]. Several interesting physical effects result from the network-like structure such as strain-stiffening and other non-linear behaviour in cross-linked bio-polymer networks [74, 238, 190], negative normal stress in networks consisting of semiflexible or stiff polymers [117, 49], auxetic behaviour of foam structures [142], rigidity in string networks [56] and spring networks [265] and floppy modes in fibre networks [97]. Modifications in the geometry, i.e. the spatial structure, and topology, i.e. the connectivity, of the underlying network structure may cause changes in physical properties. Especially local variations in connectivity may have great impact when resulting in a so-called percolation transition from macroscopically connected to macroscopically unconnected systems or vice versa. Percolation is an important structural feature as it is essential for physical properties such as conductivity, elasticity or transport through a medium. A percolation transition might result in a phase transition of the material from non-conducting to conducting or from no elastic resistance to rigidity. Material design aims at performance optimisation for a specific application. With the help of various structure-property relations, a structure can be designed optimising the demands while matching certain constraints. Control over the material properties is not limited to the macroscopic scale but can rather be extended to the microscopic and the nano-scale.

The key result of this thesis is the change of morphological and physical properties of network structures when decreasing the average vertex coordination or, in biological terms, the degree of cross-linking. In the following, two examples of material design are given where the network connectivity can be used as design parameter. Firstly, section 1.1 demonstrates that network structures are of special interest for artificial tissue growth using bone scaffolds. Requirements and constraints to scaffold geometries are specified and first achievements described. Secondly, solid open-cell foams, ordered or disordered, which may be represented by network structures, are depicted in section 1.2. Different manifestations and applications are mentioned and a short overview is given over the literature numerically investigating the influence of foam structure, cell shape and strut cross-section on the linear-elastic properties of open-cell foams.

1.1 Network Structures for Tissue Engineering

Due to its diversity and complexity, mimicking natural tissue is a challenge for structural design. Apart from biocompatibility, the basic functions a structure has to fulfil when replacing a certain bone or tissue with the objective of tissue regrowth are temporary mechanical support while at the same time sustaining transport of nutrients and waste product evacuation, cell migration and cell attachment. Since these are conflicting requirements, the design of the scaffold’s microstructure aims at an optimal compromise between mechanical stiffness, transport properties or permeability through its pores and structural commensurability with the length scales of the native tissue [109, 104]. Porous ceramics [270], salt-leached scaffolds [106] and scaffolds obtained from bio-mineralisation [133, 241] are used as bone substitutes. Since the development of additive manufacturing methods such as 3D printing [80], fused deposition modelling [110], precise extrusion [266], stereolithography [50, 166] and selective laser-sintering [242, 262], biodegradable

2 1.2 Solid Open-Cell Foams

(a) (b) (c) (d)

Figure 1.2: Bone scaffolds produced by rapid prototyping are often based on simple regular structures. (a) Anatomically shaped scaffold produced by selective laser-sintering. µ-CT images of possible scaffold structures prepared by stereolithography: (b) primitive cubic, (c) diamond, (d) gyroid architecture. (Image (a) reprinted by permission from Macmillan Publishers Ltd: Na- ture Materials, Vol. 4, Scott J. Hollister, Porous scaffold design for tissue engineering, 518-524, copyright (2005), Ref. [104]. Images (b),(c) and (d) reprinted from Biomaterials, Vol. 31, F.P.W. Melchels et al., Mathematically defined tissue engineering scaffold architectures prepared by stereolithography, 6909-6916, copyright (2010), with permission from Elsevier, Ref. [166].) polymer scaffolds with three-dimensional, complex microstructure, or architecture, that facilitate regrowth can be generated. Astonishingly, the repository of structural motifs for the described design problem used by additive manufacturing methods so far is modest. Simple regular network geometries as shown in Fig. 1.2 are employed [109]; smoother periodic geometries based on minimal surfaces [196, 166, 125] or with topologically optimised unit cells [53, 105] are only beginning to emerge, as are irregular structures [136]. For all of these, the dominant shape parameter that can be adjusted to tune mechanical properties and transport properties is the volume fraction φ of the solid material; with increasing φ the system becomes simultaneously less permeable but mechanically stiffer [104]. The work presented in this thesis suggests a new design paradigm for bone scaffolds that is based on randomizing network topology. The use of irregular or randomised geometries is intuitive because the material being mimicked, natural bone, is also irregular [126, 121]. By changing its topology, the effective elastic properties of a scaffold appropriate for bone regrowth can be adjusted by almost an order of magnitude while preserving the pore space geometry and hence fluid transport properties through the pore space.

1.2 Solid Open-Cell Foams

Open-cell solid foams are strut networks consisting of polyhedral cells that ﬁll space. These foams can be made of, for example, polymers, metal or ceramics. Apart from applications in ﬁltration [178], sound [123, 131] or energy absorption [13, 170, 131], or

3 1 Introduction: Networks and Foams

(a) Polymer foam (b) Metal foam (c) Ceramic foam

Figure 1.3: Different applications of open-cell solid foams. (a) Polymer foam for drug release: Biodegradable polyurethane foam incorporating the anti-cancer com- pounds DB-67 and doxorubicin. (b) Metal foam as catalyst support for platinum for fuel cell applications: (left) original metal foam (right) metal foam washcoated with platinum at 100, 000× magnification. (c) Ceramic foam filter: (left) original ceramic foam (right) ash-loaded foam after long-term test in a firewood heat-producing plant. (Image (a) reprinted from Acta Biomaterialia, Vol. 5, W.N. Sivak et al., Simultaneous drug release at different rates from biodegradable polyurethane foams, 2398-2408, copyright (2009), with permission from Elsevier, Ref. [226]. Image (b) reprinted from Applied Catalysis A: General, Vol. 281, Sirijaruphan et al., Metal foam supported Pt catalysts for the selective oxidation of CO in hydrogen, 1-9, copyright (2005), with permission from Elsevier, Ref. [225]. Image (c) reprinted from Ref. [2], courtesy of J. Adler.)

thermal insulation [14, 131, 264] (see Fig. 1.3), the most striking feature is their high stiffness at extremely low density which makes them best candidates for lightweight materials [13, 17]. This gives open-cell solid foams an utmost importance in car industry and aerospace applications. Numerous analytical [255, 256, 84], numerical [250, 272, 204, 84, 116, 145] and experimental studies [79, 13] investigate the effective linear elastic properties of ordered and disordered open-cell solid foams as a function of the solid volume fraction φ. A commonly observed feature are power-laws for the linear-elastic moduli as function of φ, as also found for simple regular network models [79], regardless of cell-shape irregularities and variations in strut cross-sectional area [145, 116]. All of these studies investigate the influence of different geometrical parameters on the linear-elastic mechanical properties, but the topology of the network structure is not deliberately changed. Changes may occur due to closure of cavities when increasing the solid volume fraction, but in general the topology of the network structure is kept constant. In this thesis, we focus on the influence of a topology change. A strong dependence of the linear-elastic properties of open-cell solid foams on the foam topology is shown. This suggests a new production process for open-cell metal foams with adjustable elastic properties.

4 1.3 Network Percolation and Elastic Properties

1.3 Network Percolation and Elastic Properties

In order to study how changes of the topology on the microscale affect the macroscopic elastic properties, a statistical network model is introduced. This model has one parameter, the probability p, to tune the network topology. Each network vertex is separated into two vertices with probability p. By increasing p from 0 to 1 an initially four-coordinated, foam-like network is continuously transformed into a bundle of two-coordinated, entangled fibres. The network edges are inflated to a certain radius r and the resulting structure is interpreted as network solid. The linear-elastic properties of this network solid are investigated as function of parameter p. The main feature of this model is that it allows to adjust topology and relative density independently.

The initially four-coordinated, ordered or disordered network structures are introduced in section 2.2 and section 2.3. The topology of these structures is randomised by incidentally separating network nodes as described in section 2.4. As percolation is crucial for mechanical rigidity, the percolation properties of this model are studied first in chapter 3. A percolation transition is found at p = 1, but differing critical exponents than for usual percolation suggest that our model belongs to another universality class. In chapter 4 and chapter 5, the network structures are interpreted as network solids with finite radius r. Chapter 4 introduces the finite element method used to investigate the linear-elastic properties of these network solids. The method is first applied to fibre networks given by Poisson-Voronoi processes and by the structure of collagen fibre networks imaged by confocal microscopy. If treated as a network solid, the same dependence of linear-elastic moduli on the effective density is found for both network types. Chapter 5 concentrates on the calculation of the effective elastic properties of network solids as a function of parameter p using the finite element method described in chapter 4. The influence of the boundary conditions, the finite system size and the local elastic moduli as well as the scaling exponents of the elastic moduli are discussed. Finally, compression tests of laser-sintered network realisations were performed. As described in section 5.3, the effective Young’s modulus E showed a similar dependence of p as was found in the finite element study.

2 Network models with topology changes: From Open Cells to Fibre Bundles

This chapter introduces the network models used for the following analyses and describes the stochastic process that is used to decrease the network’s connectivity. The study is restricted to networks where all vertices initially are four-coordinated, i.e. have four network edges per vertex. This comprises regular networks, namely the diamond network and the so-called nbo network which are described in section 2.2, and disordered networks (see section 2.3) derived from the edge network of a liquid foam simulated by Surface Evolver (data kindly provided by Andrew Kraynik). The stochastic process used to tune the average vertex coordination is introduced in section 2.4. It is based on local changes to the coordination of randomly selected vertices; With probability p each four-coordinated vertex is disconnected, i.e. replaced by two two-coordinated vertices that connect pairs of its four neighbour vertices. As p increases, the network changes structurally from a fully four-coordinated network at p = 0 to an ensemble of intergrown self-avoiding random walks without any branching at p = 1. To avoid confusion, it is important to note that this process is distinctly different from the conventional bond percolation approach, where randomly chosen edges are deleted rather than redirected. The parameter p, which is very specific to the stochastic process model, is shown to be equivalent to the Euler characteristic χ. The network models and the disconnection process derived here are used in chapter 3 to study percolation properties and in chapter 5 to study the effective linear-elastic mechanical behaviour.

2.1 Graph Representation of Networks

For all network structures, it is assumed that the network struts are solid cylinders with equal radius throughout the network. The geometry of a network is given by a line 3 representation which can be described by a graph G := (V,E) with V ⊂ R the set of network vertices and E := {{v1, v2} : v1, v2 ∈ V, v1 =6 v2} the set of all edges or links between two vertices specified as unordered pairs of vertices. One-edge-loops, i.e. edges from one vertex to itself, and more than one edge between a given pair of vertices shall be excluded. Both V and E are assumed to have finite numbers of elements. Thus, in the terminology of graph theory, the network graphs are simple, undirected, finite graphs [27, 42] which are embedded in three-dimensional space. If e = {u, v} ∈ E with u, v ∈ V is an edge of graph G, then u and v are adjacent vertices; u and e as well as v and e are called incident. The degree or coordination number c of a vertex v ∈ V gives the number of edges incident with v. If c = 2, for example, we call the vertex two-coordinated and respectively for different values of c. Vertices with coordination number c > 2 are in the following referred to as nodes. When inserting n ≥ 0 vertices with degree c = 2 dividing

7 2 Network models with topology changes: From Open Cells to Fibre Bundles

(a) Diamond translational unit (b) Diamond network structure cell in a cubic base

Figure 2.1: Graph representation of the diamond structure: the blue spheres symbolise the nodes, the magenta lines the network edges. In the diamond structure, each node has coordination number four. The four equivalent edges connect a node to its four nearest neighbours located at the corners of a tetrahedron. Four tetrahedra are necessary to build up the cubic unit cell (a) which can be periodically continued in all three directions of space to construct a diamond network structure (b).

edges of G into two parts, the resulting graph is called subdivision S(G). The insertion of such vertices, which also causes additional edges, has no inﬂuence on the geometrical and topological properties of the graph [27]. In this thesis, only periodic structures are considered that solely contain nodes with coordination number c = 4. Their structure is described in the following sections starting with ordered networks followed by a random foam representing disordered structures.

2.2 Ordered Four-Coordinated Networks

The variety of ordered networks is enormous [188]. When searching the database ”Reticular Chemistry Structure Resource” (RCSR) for networks comprising only four-coordinated vertices, 568 structures are displayed. We picked two representatives from this group of networks: the diamond network (dia) and the nbo network (nbo), as they are the only two regular nets, i.e. they consist of just one kind of vertex and one kind of edge [57]. Both are nets of cubic symmetry, but diﬀer in their node constitution and symmetry group.

2.2.1 Diamond Network

Due to its high stability and yet simple composition, the diamond structure became famous. Its high symmetry is probably the reason why it has often been used in modelling, for instance for protein folding [227], spin exchange in the kinetic Ising model [58], formation of polymer electrolyte membranes [231] or band gap simulations for photonic crystals [147] — only to mention a few.

8 2.2 Ordered Four-Coordinated Networks

(a) In the NaCl structure, all sites are oc- (b) The NbO structure has vacancies at cupied by an atom resulting in nodes the corners and in the centre yielding with coordination number six. nodes with planar coordination.

Figure 2.2: Sodium chloride in comparison to niobium monoxide. The solid circles denote sodium/niobium atoms, the open circles depict chlorine/oxygen atoms.

To construct a diamond crystal, an fcc-lattice is decorated with two atoms at the basis points b1 = (0, 0, 0) and b2 = (0.25, 0.25, 0.25)afcc, with afcc being the linear size of the fcc unit cell [130]. This results in a tetrahedral structure for all nodes, i.e. all nodes are in the centre of a regular tetrahedron and are equally connected to their four nearest neighbours sitting in the corners of the tetrahedron. The bond angle equals 109.5◦, the tetrahedron angle. A cubic translational unit cell can be build up of four tetrahedra and contains eight vertices. Fig. 2.1 shows a cubic translational unit cell and a diamond network consisting of 23 unit cells.

2.2.2 NbO Network

Named after niobium monoxide (NbO), the nodes of the nbo network are located at the positions of the niobium (Nb) and the oxygen (O) atoms in the chemical compound NbO. The NbO lattice is shown in Fig. 2.2. Its structure is reminiscent of sodium chloride (NaCl) but with some vacancies:

(i) The eight corners of the translational unit cell are empty.

(ii) The octahedral composition of the Nb atoms lacks an oxygen atom in its centre.

In contrast to NaCl, both Nb andO atoms have four coplanar bonds instead of being six-fold coordinated [88]. For the nbo network structure, it is not distinguished between Nb andO atoms but all sites are decorated with equal network nodes. A nbo network consisting of 23 unit cells is displayed in Fig. 2.3. The main features of this structure are its cubic symmetry and the four-coordinated, planar nodes.

9 2 Network models with topology changes: From Open Cells to Fibre Bundles

Figure 2.3: Graph representation of the nbo network structure: the blue spheres symbolise the nodes, the magenta lines the network edges. The network structure is based on the crystal structure of niobium monoxide (NbO). The coordination of all network nodes is planar.

2.3 Disordered Four-Coordinated Networks

Generally, nature is not perfectly ordered. The degree of disorder ranges from a small number of defects in an otherwise periodic crystal structure [112], via foam structures that are random but fulﬁl certain rules [258] to completely random structures as, for instance, porous rocks [11, 9]. In order to extend the analysis of the mechanical properties described in chapter 5 from completely regular to irregular structures, random network models are considered. The condition that all network nodes shall be of degree four excludes models such as the Delauney triangulation [187, 54] or subgraphs of it such as the Gabriel graph, the relative neighbourhood graph or the minimum spanning tree [5, 85] since they consist of vertices with varying degrees. Random plane processes [169] are not suitable either as they generally result in six-coordinated nodes. Voronoi diagrams [187, 54] and foam structures [258], on the other hand, comprise only four-coordinated nodes and are considered more in detail.

2.3.1 Voronoi Diagrams m Let P = {p1,..., pn} ⊂ R be a set of points, or sites, with pi 6= pj for i =6 j, i, j ∈ {1, . . . , n}. The m-dimensional Voronoi cell of site pi [187] is deﬁned as

V (pi) := {x | ||x − pi|| ≤ ||x − pj|| for j 6= i ∈ {1, . . . , n}} (2.1) and contains all points that are closer to pi than to any other site. Voronoi cells are polyhedra [187]. Hence, they are bounded by polygons which are in turn bounded by edges. The end points of an edge are called vertices. The set V = {V (p1),...,V (pn)} is called the m-dimensional Voronoi diagram and tessellates space. A two-dimensional example of a Voronoi diagram is shown in Fig. 2.4.

10 2.3 Disordered Four-Coordinated Networks

Figure 2.4: Two-dimensional Voronoi diagram: space is tessellated into polyhedral cells that contain all points that are closer to the site located in this cell than to any other site.

Depending on the set P of sites, the Voronoi diagram V may be ordered or disordered. If P is sufficiently disordered, the resulting Voronoi diagram is non-degenerate [187]. In particular, the set P has to fulfil two prerequisites: (i) Non-collinearity: No k +1 points of P must lie on a (k − 1)-dimensional hyperplane m of R , k = 2, . . . , m. m (ii) Non-cosphericity: No m + 2 points must lie on the boundary of a sphere in R . A non-degenerate Voronoi diagram in two dimensions contains only vertices that have exactly three edges, in three dimensions each vertex is of degree four. The Poisson point process is one important point process that fulfils non-collinearity and non-cosphericity. The probability to find a given number of points N(A) in a volume m A ⊂ R is given by λA e−λAx P (N(A) = x) = , x = 0, 1, 2,... (2.2) x! where the intensity λ is a measure for the density of points. The resulting Voronoi diagram is called Poisson-Voronoi diagram and is non-degenerate.

2.3.2 Liquid Foams A liquid foam is a two-phase system consisting of gas cells enclosed by liquid cell walls [258]. When talking about liquid foams, one distinguishes between wet foams and dry foams depending on the volume fraction of liquid contained in the borders and films between the gas-filled cells. For dry foams, the films separating different cells contain a negligible amount of liquid, thus, the film thickness is infinitesimally small. In three- dimensional dry foams at equilibrium, the cells were found to be polyhedral with curved facet surfaces of constant mean curvature. They meet at borders obeying Plateau’s law:

11 2 Network models with topology changes: From Open Cells to Fibre Bundles

(a) (b)

Figure 2.5: (a) Distribution ρ(λ) of edge length λ for diﬀerent network structures. For networks obtained from Voronoi diagrams, ρ(λ) follows a broad distribution with large probabilities for both very small and very large edges. In contrast, foam structures, including monodisperse foams, have an explicit length scale and a narrow distribution of edge lengths. (b) Visualisation of Plateau’s law: Plateau border and tetrahedral foam vertex in a dry foam. (Image (a) reprinted from Advanced Engineering Materials, Vol. 8, A.M. Kraynik, The Structure of Random Foam, 900-906, copyright (2006), with permission from John Wiley and Sons, Ref. [137].)

(i) Each foam edge is the common line of three intersecting cell faces. Surface area minimisation forces the angle between them to be 120◦. The resulting edge is called a Plateau border.

(ii) At each vertex, exactly 4 foam edges — or equivalently 6 cell facets — meet yielding a vertex with tetrahedral edge angles of 109.5◦ as shown in Fig. 2.5 (b).

Yet the particular size and shape of individual foam cells vary from cell to cell and their statistics have been investigated analytically [98], experimentally [199, 217] and numerically [139, 137].

Fig. 2.5 (a) shows the distribution ρ(λ) of edge length λ of diﬀerent structures. The probability of edges of length l in networks obtained from Voronoi diagrams follows a broad distribution with large probabilities for both very small and very large edges. This is problematic for the mechanical analysis, as will be described in more detail below. Therefore, rather than studying Voronoi diagrams, periodic foam structures are investigated as they have an explicit length scale and a narrow distribution of edge lengths as also shown in Fig. 2.5 (a).

Dry liquid foams, more precisely their network of Plateau borders, are used as a model for solid open-cell foams (see e.g. Refs. [258, 17] for a general discussion of solid foams). We focus on periodic open-cell foams with monodisperse cell volumes, i.e. all foam cells have equal volume, generated by Andrew Kraynik using Brakke’s Surface Evolver [30]. Starting from a periodic, three-dimensional Voronoi diagram [187] of loose sphere packings

12 2.3 Disordered Four-Coordinated Networks

(a) Surface evolver foam (b) Foam network structure

Figure 2.6: Graph representation of a foam network structure: A periodic, monodisperse, open-cell foam structure was evolved from a Voronoi tessellation of a loose sphere conﬁguration by surface energy minimisation using Brakke’s Surface Evolver [30]. The foam vertices are taken to be the network nodes, the Plateau borders are the network edges. (a) A foam structure output from Surface Evolver. (b) A foam network structure: the blue spheres symbolise the nodes, the magenta lines the network edges.

generated by random sequential adsorption [138], the structure is obtained by surface area and hence surface energy minimisation under the constraint that all cells have the same volume. When the local geometry of the structure obeys Plateau’s laws, it can be assumed to be a foam in mechanical equilibrium. Note that in the minimisation process only energies due to the Laplace pressure diﬀerence between the cells, but not due to gravity, were taken into account. A geometrical characterisation of the resulting random monodisperse foam structures can be found in Refs. [138, 139, 137]. The vertices of the foam are taken to be the network nodes, the Plateau borders are the network edges. Fig. 2.6 shows a foam structure output from Surface Evolver and part of the resulting foam network.

In the next section, an algorithm is described which splits up graph nodes and thus converts a network structure into unbranched but interwoven strings. This procedure can be applied to the graph representation of any four-coordinated network. For the analysis of the elastic properties of these structures in chapter 5, the network graph edges are cylindrically inﬂated to a given radius and interpreted as a network solid with a given volume fraction of solid material. In order to obtain a similar topology change of the network solid to that caused by the disconnection process in the graph representation, the disconnected parts of the structure have to be separated by a distance greater than twice the inﬂation radius 2r. This only works reliably if all edges of the structure are

13 2 Network models with topology changes: From Open Cells to Fibre Bundles

longer than 2r. Otherwise, a displacement of shorter edges may result in a coalescence with other parts of the network, causing the topology of the network to change in a diﬀerent way than intended. Hence, network models with minimal edge length greater than twice the maximal inﬂation radius 2rmax are needed. This excludes Poisson-Voronoi or Laguerre structures1 since their edge length distribution ρ(λ) increases with decreasing edge length λ, as can be seen in Fig. 2.5 (a). In contrast, the edge length distribution of the monodisperse foam structures described above is peaked around a mean edge length λ¯ and vanishes quickly away from λ¯ (see Fig. 2.5 (a)), making them ideal structure models for the topological transformations described below.

2.4 Disconnection Mechanism: Node Separation

In order to study the influence of the topology on various physical properties, an algorithm is introduced which randomly separates network nodes and thus changes a network’s topological structure. The separation of more and more nodes converts an initially totally connected cellular or network structure into a bundle of loose, unbranched entangled fibres. The following algorithm can be applied to the graph representation of any four-coordinated network. Let the initial graph contain Ns ∈ N nodes with the same coordination number c = 4 for all nodes. The network connectivity is randomised by the following approach: The four-coordinated nodes are picked in random order and are incidentially separated with probability p, p ∈ [0, 1]. Separating a node means replacing the four-coordinated node with two two-coordinated vertices. In two dimensions, this results in two different ways of separation: horizontally and vertically, as illustrated in Fig. 2.7. In three dimensions, there are three possibilities to separate a node of coordination number c = 4. Fig. 2.8 shows all possible configurations for a diamond and a nbo vertex. When a node is separated, the direction of separation is chosen at random with equal probability for the different variants. The two-dimensional (2D) separation algorithm is summarised in a decision tree that is applied to each network node: 1/2 p 1/2 1−p

First, it is decided whether the node is separated with probability p or stays intact with probability 1 − p. In case of a separation, it is decided if the node is separated vertically or horizontally, where the probability for both is 1/2.

1Laguerre structures [187], also called Laguerre diagrams, are weighted Voronoi diagrams, more precisely m power Voronoi diagrams. Let P = {p1, . . . , pn} ⊂ R be a set of points and W = {w1, . . . , wn} ⊂ R the set of weights assigned to the points. Instead of the Euclidean distance between point pi and point p, a weighted distance d(pi, p) is used for the construction of the Voronoi cells which is deﬁned 2 as d(pi, p) := ||xi − x|| − wi where xi and x are the position vectors of pi and p, respectively. With 2 wi = ri , Laguerre diagrams can be regarded as Voronoi diagrams of circles with radii ri.

14 2.4 Disconnection Mechanism: Node Separation

(a) Initial conﬁguration (b) Vertical separation (c) Horizontal separation

Figure 2.7: Separating a four-coordinated node basically means inserting one additional vertex, rearranging the edges that belonged to the former node but keeping the number of edges constant. When embedding a regular graph with vertices of degree four in two-dimensional space, there are two possibilities for the separation of an initially four-coordinated vertex (a): vertically (b) and horizontally (c).

PNs(p) Let c¯(p) := 1/Ns(p) i=1 ci be the average coordination number of a given realisation of a graph at probability p. Ns(p) ∈ N is the number of graph vertices at p including the remaining four-coordinated nodes of the initial network and the two-coordinated nodes originating from a node separation. hc(p)i denotes the ensemble average of the coordination number of a certain network node. In the limit Ns(p) → ∞ of networks of infinite size, c¯ → hci. By increasing probability p, the average coordination number hc(p)i of the vertices continuously decreases from hc(p = 0)i = 4 for the initial network to hc(p = 1)i = 2 where all vertices are separated. The initially cellular network decomposes into different clusters, i.e. connected components, and for p = 1 it is degraded to a bundle of interwoven fibres. The fibres at p = 1 consist of simple chains of nodes that do not cross each other. Fig. 2.9 illustrates this process in two dimensions. An example of a partially decomposed diamond network is shown in Fig. 2.10.

2.4.1 Shifting Nodes When Separating?

When separating a four-coordinated node, the two two-coordinated vertices may or may not by shifted with respect to the position of the original node. In this section, it is discussed that shifting is not necessary for the percolation analysis, but crucial when analysing a network’s elastic properties with a voxel-based ﬁnite element scheme. Details about the applied shifting method are given.

In the graph representation, separating a four-coordinated node effectively means adding a vertex with a new label and amending the list of edges, i.e. replacing the label of the original four-coordinated vertex by the new label in two of its original connections as shown in Fig. 2.11 (b). Although both two-coordinated vertices have identical coordinates, namely the same position as the four-coordinated node they are replacing, they can be told apart by their label and their connections. This is sufficient to distinguish between two possibly different clusters that contain one of the new vertices each and, thus, touch each other. Hence, shifting is not necessary for the percolation analysis.

15 2 Network models with topology changes: From Open Cells to Fibre Bundles

(a) A diamond node or a node in a foam structure. (b) A nbo node.

Figure 2.8: Node separation in three dimensions: A node with coordination number c = 4 can be divided into two vertices with coordination number c = 2 in three diﬀerent ways. The original vertex connects an arbitrary edge to one of the three remaining edges. The two edges that are left over are connected by the second, newly inserted vertex. The process is illustrated for a tetrahedral node as occurring in a diamond or a foam structure and a planar node as can be found in nbo network structures. The shift of the two two-coordinated vertices with respect to the original position of the four-coordinated node is not necessary for percolation analysis, but crucial when analysing the networks’ elastic properties with a voxel-based ﬁnite element scheme.

16 2.4 Disconnection Mechanism: Node Separation

Figure 2.9: The process of separating nodes of coordination number c = 4 into two nodes of coordination number 2 reduces the connectivity of the sample as a whole. Initially, all nodes belong to the same component, but at some point, parts of the network will be disconnected. If the fraction of separated nodes is increased, the sample decomposes into several individual fragments which become more and more fibre-like as more and more nodes are separated. When all nodes are separated (p = 1), the network consists of nodes of coordination number c = 2 only, i.e. without any branching. The edges form closed loops or curves ("fibres") that terminate on the sample boundaries. Thus, there is a transition from a cellular structure at p = 0 to a configuration of entangled fibres at p = 1. Note that separated nodes have been slightly displaced for enhanced clarity.

17 2 Network models with topology changes: From Open Cells to Fibre Bundles

Figure 2.10: A realisation of a diamond network with a fraction p = 3/4 of separated nodes. Each connected component is shown in a diﬀerent colour.

For the evaluation of the effective linear-elastic properties of these network structures (see chapter 5) using the voxel-based finite element method described in chapter 4, the network graphs are interpreted as network solids consisting of homogeneous, cylindrical struts with a finite radius r which are rigidly connected at the network nodes (for details see section 4.3). In this model, two parts of a structure have to be separated in space to be mechanically detached. In order to achieve an equivalent topology change of the network solid to that caused by the separation process, a shift of the two-coordinated vertices is essential. Fig. 2.8 illustrates this schematically for a tetrahedral node as occurring in a diamond or a foam structure and a planar node as can be found in nbo network structures. The used algorithm of node separation and vertex allocation was chosen as it is simple to compute. In principle, it consists of three steps: (i) Insert three new graph vertices with the coordinates of the initial node and two new edges connecting (a) the original node with one of the three new vertices, (b) the remaining two new vertices with each other. (ii) Redirect three edges from the original node to one of the new vertices each. (iii) Pull the four vertices away from each other along the edges. The process is illustrated in Fig. 2.11. The distance a the vertices are shifted along the edges is determined by the volume fraction of the network solid and the angle α between two incident edges. In case of the diamond network, α = 109.47◦ regardless of which possibility of separation is chosen. Let D be the distance the vertices shall be separated by with D > r, r the cylinder radius of the network struts yielding a certain volume fraction of solid, then distance a is calculated as a = D/ cos(α/2) . (2.3)

18 2.4 Disconnection Mechanism: Node Separation

2 3 2 3 2 3

I II I II I II

I II 1 7

1 1 6 V VI

IV III 6 8

IV III IV III IV III 5 4 5 4 5 4

vertices edges vertices edges vertices edges 1 ( 0, 0) I (1,2) 1 ( 0, 0) I (1,2) 1 (-a, a) I (1,2) 2 (-1, 1) II (1,3) 2 (-1, 1) II (6,3) 2 (-1, 1) II (7,3) 3 ( 1, 1) III (1,4) 3 ( 1, 1) III (6,4) 3 ( 1, 1) III (8,4) 4 ( 1,-1) IV (1,5) 4 ( 1,-1) IV (1,5) 4 ( 1,-1) IV (6,5) 5 (-1,-1) 5 (-1,-1) 5 (-1,-1) V (1,6) 6 ( 0, 0) 6 (-a,-a) VI (7,8) 7 ( a, a) 8 ( a,-a)

(a) Four-coordinated node (b) Node separation in graph representa- (c) Mechanical separation tion of a node

Figure 2.11: Node separation: (a) Original unseparated node. (b) Node separation in graph representation: A new vertex is added to the graph. In two of the four connections of the originally four-coordinated node the new label replaces the original node’s label. The two now two-coordinated vertices are uniquely identiﬁed by diﬀerent labels, even though their positions are both identical to the position of the original, four-coordinated node. (c) Node separation and vertex shifting to obtain elastic separation: Three new vertices and two new edges are inserted, three of the four original edges are redirected to the new vertices; all vertices are pulled a distance a apart (see Fig. 2.12 (b)) along the original edges, which results in a spatial separation of the two parts.

19 2 Network models with topology changes: From Open Cells to Fibre Bundles

r α D D a

(a) (b)

Figure 2.12: (a) In order to mechanically separate two vertices that result from the separation of a node, they must be spatially separated by a distance D > r, r being the inﬂation radius of the network struts. (b) A spatial separation of the disconnected parts is achieved by pulling the vertices, which all have the same coordinates as the node they replace, a distance a along the edges of the original node. a = D/ cos(α/2) is a function of the edge angle α and distance D, which in turn is determined by the inﬂation radius r or, in other words, the volume fraction φ.

The calculation is illustrated in Fig. 2.12. Since the nbo network has planar nodes, this algorithm fails if the node is separated such that the straight fibres remain intact (see Fig. 2.8 (b), last possibility) as α then equals π, resulting in cos(α/2) = 0. In that case, the fibres are shifted in opposite directions perpendicular to the plane the fibres lie in. The edges are deformed such that the resulting fibres become Gauss-shaped curves. Fig. 2.13 shows a nbo node separated in this way. For that, the edges representing the fibres are subdivided into polygons by inserting additional vertices of degree 2. A subdivision does not change the properties of the graph. Thus, apart from limited computer memory, there are no restrictions to the number of inserted vertices. The number was chosen large enough to give satisfyingly smooth results independent of the discretisation used for the mechanical analysis. The vertices of the two fibres are pulled apart in opposite directions. To keep the fibres smooth, the amount of displacement a of each vertex depends on the distance x of that vertex to the position of the original node with a maximal around the position of the node and a minimal away from it. For each vertex in a sphere with radius R around the original node, the individual shift is calculated using 2 2 the Gauss function a(x) = exp −Ax /2 σ with the boundary values a(x = 0) = D and a(x = R) = 1 where 1 refers to the size of one finite element in the discretised representation of the network solid (for details see section 4.3 and section 4.4) and is hence the smallest shift possible. This choice implies that the absolute value of the shift depends on the discretisation: the finer the discretisation, the smaller the size of the elements. An example of such a node separation is shown in Fig. 2.13. In the diamond and the nbo network all edges are of the same length, thus the maximal inflation radius, or the maximum volume fraction respectively, is determined by the lattice constant. On the contrary, the edge lengths λ of a monodisperse foam obey a continuous distribution ρ(λ) which peaks around an average value λ¯ and decays to zero away from λ¯ as can be seen in Fig. 2.5 (a). It is impossible to appoint a minimum edge

20 2.5 Topological Characterisation Figure 2.13: The normal separation algorithm fails for the planar nodes of the nbo structure if the node is separated in such a way that the straight fibres stay intact. Alter- natively, they are separated by shifting the fibres in opposite directions perpendicular to the fibre plane. The edges are deformed so that the resulting fibres become Gauss-shaped curves. length λ > 0 which results in a maximum inflation radius r = 0. Yet the number of short edges is small, causing only a few possible defective separations. This discussion shows why the foam models with a well-defined typical edge length and very small probability for small edges are more useful for our analysis than e.g. Voronoi diagrams. To ensure that the separation yields the desired topology change, i.e. the volume fraction is low enough to avoid unwanted edge coalescence, the Euler characteristic is used to monitor sufficient node separation as described in section 2.5.

2.4.2 Periodic Boundary Conditions

For all of the structures introduced above, periodic boundary conditions apply in all three coordinate directions. This allows us to construct networks of different sizes by taking a varying number of translational unit cells and merge them to form a new, larger sample. Thus, the influence of finite size effects can be studied (see section 3.4, section 3.5 and section 5.1.3). A point on the border of a translational unit cell always has an equivalent at the opposite border. When using periodic boundary conditions, one of the points is erased and the edges of this point are connected to its periodic copy. Hence, the separation algorithm maintains periodicity even if there are vertices lying directly on the boundary. Fig. 2.14 sketches the issue in two dimensions.

2.5 Topological Characterisation

The Euler characteristic is used in order to monitor the topology change of a network structure caused by the disconnection algorithm described in section 2.4. This is especially important for the inflated network structures as a sufficient vertex separation has to be guaranteed for the analysis of the mechanical properties as a function of probability p (see chapter 5). The topology of an object can be characterised by topological invariants, i.e. quantities that have equal values for objects with the same topological structure. Important topological invariants are the dimension, orientability, number of boundary components and the Euler characteristic χ, sometimes also called Euler number [200]. The most important of these is the Euler characteristic which is widely used in different fields of physics. As one of the additive, motion-invariant integral measures called Minkowski functionals [93, 220, 219, 163], it is used to quantify the morphology of spatial patterns

21 2 Network models with topology changes: From Open Cells to Fibre Bundles

(a) (b) (c) (d)

Figure 2.14: Boundary conditions of a network on a 2D square lattice: (a) Open boundary conditions (b) Periodic boundary conditions in x-direction. In 2D, there are two possibilities of separating a four-coordinated node: (c) vertically and (d) horizontally. Here, both variants are shown for a vertex sitting exactly on the boundary of a system with periodic boundary conditions.

including random point sets [162], galaxy distributions [161, 218], disordered porous media [12] and reaction-diﬀusion systems [159]. The Hamiltonian of complex ﬂuids [164] and composite media [160] can be expressed in terms of the Minkowski functionals and a thermodynamic phase diagram can be calculated. Furthermore, the Euler characteristic is applied to estimate percolation transitions [240, 182], to compute the magnetisation transition in the Potts model [26] and to calculate gravitational corrections to the fermion masses m1/2 in broken supersymmetries. Finally, the Euler number is most appropriate for characterising the connectivity of random networks [158]. Hence, the topology change caused by the separation algorithm described in section 2.4 is described by the Euler characteristic. Let G be the graph representation of a given network. The Euler characteristic χ(G) is an integer that describes the connectivity of graph G and is calculated via

χ(G) := V − E (2.4) with the number of graph vertices V and the number of graph edges E [158]. Eq. (2.4) can be interpreted as a special case of the deﬁnition of the Euler characteristic for a surface S partitioned into V vertices and E edges,

χ(S) := V − E + F, (2.5) where F is the number of resulting faces or plaquettes. According to Euler’s formula, hollow convex polyhedra all have χ = 2. For arbitrary geometric objects, χ can take any (integer) number [200]. For hollow objects, the Euler characteristic χ is connected to the genus g of the structure by χ = 2 (1 − g) [200]. The genus g is the maximum number of non-intersecting closed curves along which the object can be cut without dividing it into two parts. For solid structures, χ = 1 − g as the factor 2 for hollow objects results from taking the inner and the outer side of a surface into account. Descriptively, the Euler characteristic for solid objects is the alternating sum of the number of parts a structure consists of, the number of handles and the number of cavities [163].

22 2.5 Topological Characterisation

An important feature of the Euler characteristic χ is its additivity in the sense that

χ(G1 ∪ G2) = χ(G1) + χ(G2) − χ(G1 ∩ G2) (2.6) with G1 and G2 being two diﬀerent network graphs. This can be used to calculate χ for complex graphs by dividing it into smaller parts and calculating their Euler characteristics separately. A highly positive χ indicates that the object consists of numerous individual components whereas one-component structures with many holes have a highly negative χ. Changes of the structure of a graph may entail a topology change and thus a change of its Euler characteristic χ. In the separation mechanism described in section 2.4, increasing the disconnection probability p from 0 to 1 leads to a decomposition of the formerly connected graph into several separate components. To characterise this topological change with the help of the Euler characteristic χ, a single disconnection step is considered. Let Gn = (Vn, En) be the graph representation of a network with n separated nodes where Vn is the set of vertices, Vn = |Vn| is the number of vertices (including both four-coordinated and two-coordinated vertices), En is the set of edges, En = |En| is the number of edges and χ(Gn) is the Euler characteristic of graph Gn. By separating a further node, one four-coordinated node is replaced by two two-coordinated ones as shown in Fig. 2.7, eﬀectively increasing the number of graph vertices Vn by one. The edges that belonged to the former node are reallocated without changing their number En. Hence, the Euler characteristic χ(Gn+1) of the graph Gn+1 = (Vn+1,En+1) with n + 1 separated nodes can be calculated as

χ(Gn+1) = Vn+1 − En+1 = Vn + 1 − En = χ(Gn) + 1 . (2.7)

Hence, with every node separation, the Euler characteristic increases by 1, which results in

χ(Gn) = χ0 + n . (2.8)

χ0 = χ(G0) is the Euler characteristic of the initial network with p = 0. Note that the existence of the two two-coordinated vertices is not necessary as they are only subdividing the two edges that connect the neighbouring nodes of the separated node. They can also be deleted as subdivisions S(G) of a graph G have the same Euler characteristic χ(G) = χ(S(G)), which is illustrated using the simple example of two vertices connected by one edge in Fig. 2.15. With the total number N of nodes in the original network, (χ(Gn) − χ0)/N = n/N yields the fraction of separated nodes. For a given value of disconnection probability p, the number of separated nodes varies from realisation to realisation with hni = pN where h·i denotes the ensemble average. Thus,

hχ(Gn) − χ0i hχ(Gn)i − χ0 hχ(p)i − χ0 p = = = , (2.9) N N N as is evidenced for all three structures in Fig. 2.16. Note that the values for (χ(p) − χ0)/N are calculated on a discretisation of the network structures, the so-called voxel- representation (see section 4.4). The algorithm for the calculation of the Euler characteristic from three-dimensional digital images is based on counting the occurrences of special local patterns as described in Refs. [111] and [12].

23 2 Network models with topology changes: From Open Cells to Fibre Bundles

χ = V − E χ = V − E χ = V − E = 2 − 1 = 3 − 2 = 6 − 5 = 1 = 1 = 1

(a) (b) (c)

Figure 2.15: The Euler characteristics χ(G) of graph G and χ(S(G)) of a subdivision S(G) of graph G are equal, χ(G) = χ(S(G)): Graph G (a) and two subdivisions (b) and (c) are shown and their Euler characteristics are calculated. V is the number of vertices, E is the number of edges.

For the speciﬁc disconnection model used here, the Euler characteristic χ and the disconnection probability p are equivalent quantities. However, χ is deﬁned more generally than p, in particular for network geometries that do not result from a disconnection process. The Euler characteristic χ may therefore be useful to generalise the investigations, results and predictions of chapter 3 and chapter 5 to more general models and to experimental image data.

24 2.5 Topological Characterisation

1.0 diamond 0.8 nbo N foam ) / 0.6 0 χ )-

p 0.4 ( χ ( 0.2 φ=0.06 0.0 0 0.2 0.4 0.6 0.8 1 p

Figure 2.16: Euler characteristic χ(p) as function of disconnection probability p for all network structures. The values for (χ(p) − χ0)/N, with the number N of nodes in the initial, fully connected network and χ0 the associated Euler characteristic, are calculated on the voxel-representation of the network structures (see section 4.4) and averaged over 5 diﬀerent network realisations. For p = 0, all nodes are intact whereas for p = 1, all nodes are separated. In contrast, if 0 < p < 1, the number of separated nodes in each realisation is a random variable. Hence, the Euler characteristic of each realisation with PM 0 < p < 1 is a random variable, too. The data are averages χ¯ = i=1 χi over M samples, error bars represent the standard deviation σ = (1/(M − PM 2 1/2 1) i=1(χi − χ¯) ) . The diamond and the nbo network used for this analysis are composed of 43 translational unit cells, the monodisperse open- cell foam consists of 125 cells. For all structures, open boundaries applied. The shown data was obtained at a solid volume fraction φ = 0.06. Diamond: N = 427, χ0 = −407. Nbo: N = 384, χ0 = −288. Foam: N = 742, χ0 = −598.

3 Percolation Properties

The node separation algorithm introduced in chapter 2 is interpreted as an alternative percolation model on lattices which is hereafter referred to as vertex model. Varying the disconnection probability p from 0 to 1 causes more and more four-coordinated nodes to be replaced by two two-coordinated vertices. This process reduces the connectivity of the network and may change a percolating system into a non-percolating one at a critical value pc, the so-called percolation threshold,. This chapter shows that the vertex model belongs to a different universality class than standard bond or site percolation. For the planar square lattice and the spatial diamond network, the percolation threshold pc is numerically determined as pc = 1; the fractal dimension Df is ascertained and the critical exponents β and ν are determined by finite size scaling. In the following, the basics of percolation theory and scaling theory relevant for the percolation analysis are explained. A broader introduction can be found in the textbooks of Stauffer and Aharony [233] and Sahimi [210]. Subsequently, the two-dimensional (2D) and three-dimensional (3D) results are presented. Knowledge of the percolation threshold pc is important for the elasticity analysis in chapter 5, as a percolating structure is a prerequisite for mechanical stiffness.

Percolation tells us if a system is macroscopically connected, that is percolating, or not. The concept was ﬁrst used by Flory [67] and Stockmayer [237] to describe polymerisation and gelation before its application was extended to transport through disordered media [33, 174, 113], spreading phenomena of, e.g., ﬁres [60, 233, 193] or epidemics [87, 177, 215], fracture processes [212, 83, 210, 120], oil production [129, 6] and star formation [221],

(a) site percolation model (b) bond percolation model

Figure 3.1: Site and bond percolation on a 2D simple square lattice.

27 3 Percolation Properties

just to mention a few. Percolation models are based on random graphs or a random process and their percolating properties are studied as function of the underlying random variable. The two classical percolation problems are site percolation and bond percolation on regular lattices as, for example, the primitive cubic lattice or the triangular lattice. In site percolation, a lattice site is either occupied with probability p˜, p˜ ∈ [0, 1], or vacant with probability 1 − p˜, independent of the occupational state of its neighbours. The underlying random process is a Bernoulli process with parameter p˜ [132]. Two neighbour sites are considered connected if they are both occupied. Bond percolation, on the other hand, assumes all sites to be occupied and connected by lines, so-called bonds. Equivalently to site percolation, a bond can be set with probability p˜ or vacant with probability 1 − p˜. Two sites are connected if there is at least one path of bonds between them. A set of connected sites is called a cluster [233, 210, 248]. Fig. 3.1 shows an example of site and bond percolation on a 2D simple square lattice. If p˜ is close to one, the system is almost completely connected, whereas at p˜ close to zero it consists of individual sites or bonds and maybe some small clusters. Hence, a change in p˜ from 0 to 1 results in a topology change of the network, with a transition from an unconnected to a connected state of the network at some value 0 < p˜c < 1. Tab. 3.1 shows a comparison of the site and the bond percolation model with the vertex model introduced in the last chapter concerning the topology change in one disconnection step. Removing a site from a network graph causes the Euler characteristic to change by ∆χ = c − 1 with c the coordination number of the removed site. In contrast, the deletion of a bond results in a change ∆χ = 1. In case of the vertex model, the separation of a node yields the Euler characteristic to change by ∆χ = 1.

Percolation theory studies the geometrical phase transition from a non-percolating to a percolating system when changing parameter p˜. For p˜ < p˜c with p˜c being the percolation threshold, only non-percolating clusters exist. This phase, called subcritical phase, is separated by p˜c from a so-called supercritical phase for p˜ > p˜c which has at least one percolating cluster [82, 248]. Tab. 3.2 summarises percolation thresholds for 3D bond and site percolation on different lattices. Analytically, percolation is studied in the thermodynamic limit of infinite lattice sizes. Hence, a percolating cluster is necessarily infinite and the terms “percolating cluster” and “infinite cluster” are used synonymously in literature. Since computer resources and simulation time are finite, only finite systems can be modelled numerically. From the analysis of systems with different sizes one attempts to extract the properties of the system in the limit of infinite system size. The so-called finite size scaling is explained in section 3.2.3. Our model presented in section 2.4 is neither a site nor a bond percolation problem despite the fact that bond percolation and our model have the same change in Euler number. We start from a lattice with nodes of coordination number c = 4. With probability p, each four-coordinated node is replaced by two two-coordinated vertices. From three possible connection configurations, a random configuration is chosen with equal probability. Note that p is the probability that a node is separated, which is contrary to the site and bond percolation definition where p is the probability to find an occupied site or a set bond. At p = 0, the system definitely percolates as no node is separated. Thus, we are in the supercritical phase. By increasing p, we approach

28 Table 3.1: Change in Euler characteristic ∆χ = χ1 − χ0 for a single disconnection step. χ0 = −3 is the Euler characteristic of the initial network which consists of V0 = 13 vertices with coordination number c and E0 = 16 edges. χ1 denotes the Euler characteristic of the graph after the degradation. The processes are illustrated for two dimensions, but the graphs can easily be embedded in three-dimensional space with the same results.

χ0 = V0 − E0 χ1 ∆χ = χ1 − χ0

χ1 = (V0 + c − 1) − E0 ∆χ = c − 1 site

χ1 = V0 − (E0 − 1) ∆χ = 1 bond

χ1 = (V0 + 1) − E0 ∆χ = 1 vertex the critical point, however, not as usual from below but from above. This may seem unconventional, but it turns out that the percolation threshold is pc = 1 making the subcritical phase of the vertex model unaccessible since p can only be varied from 0 to 1.

Boundary conditions Boundary conditions have a strong influence in finite size systems, in particular for percolation problems. In a non-periodic finite system, a percolating or sample-spanning cluster is a cluster that connects two opposite sites of a sample. One may distinguish between percolation in specific directions, e.g. in x-direction, if the system shows anisotropy. For infinite systems, spanning the whole system implies infinity of the percolating cluster. Hence, percolation analysis in infinite systems is equivalent to proving the existence of an infinite cluster. For site and bond percolation, it was shown that there exists exactly one unique infinite cluster [90]. For numerical percolation studies, system sizes are always finite. Applying periodic

Table 3.2: Percolation thresholds for site and bond percolation on some regular lattices in three dimensions, taken from [233] and [210]. site bond cubic primitive 0.3116 0.2488 diamond 0.43 [233], 0.4299 [210] 0.388 [233], 0.3886 [210] bcc 0.246 [233], 0.2464 [210] 0.1803 [233], 0.1795 [210] fcc 0.198 0.119

29 3 Percolation Properties

boundary conditions in one or more directions to reduce boundary effects results, on the one hand, in a potential increase of percolation compared to non-periodic boundary conditions as formerly disconnected clusters can be connected across the sample boundary. On the other hand, it raises a conceptual problem in defining percolation in the direction of the periodic boundary condition. One may call a cluster ‘percolating’ if it extends for more than one unit cell length in the direction in question. We decided to only apply periodic boundary conditions in all but one direction and consider percolation in the non-periodic direction to avoid a redefinition of percolation. In the following, if not further mentioned, 2D networks with periodic boundary conditions in x-direction and 3D networks with periodic boundaries in x- and y-direction are examined. As mentioned, only percolation in y-direction (2D) or z-direction (3D) is considered in those networks.

3.1 Cluster Characteristics

When increasing the disconnection probability p, i.e. randomly separating more and more nodes, the graph that initially consists of one connected piece decomposes into separate connected components called clusters. The number and size of these clusters vary from realisation to realisation. To characterise the percolation process, the statistics of the clusters are studied. Main cluster characteristics are the size s of a cluster, the cluster size distribution ns, the cluster radius Rs and the pair-connectedness function G(r) where r is the distance between two lattice points.

Size and mass of a cluster Assume a cluster to have v vertices (or sites in terms of site percolation) and let ci be the coordination number of vertex i. The size s of cluster j is deﬁned as v X sj := ci (3.1) i=1

and gives twice the number of edges belonging to cluster j. Note that the cluster size is usually defined as the number of sites belonging to a cluster [233]. For the vertex model, this definition causes a change of the system size as a function of disconnection probability p. As the number of edges is independent of p, we chose the edges for the characterisation of the cluster sizes and the total size of the network. However, since we also need the number of vertices of a cluster, we will refer to it as the mass v of a cluster. By summing over all network clusters, the total size of the network is obtained by PNcl stotal := j=1 sj, with Ncl being the total number of clusters. In the vertex model, separating a four-coordinated vertex increases the number of vertices in the network. However, as the connections of the initial vertex are apportioned between the two new vertices, the total size stotal, which equals twice the number of all network edges, stays constant. Hence, the total size stotal is independent of the fraction of separated vertices and a constant characteristic of the network. In contrast, for site and bond percolation, the total number of edges depends on probability p. P The size of the percolating clusters sperc := j∈P sj, P the set of percolating clusters, is of special interest. To facilitate comparison between different networks, this quantity is

30 3.1 Cluster Characteristics normalised by dividing it by the system size yielding the fraction fperc of network edges belonging to one of the percolating clusters

fperc := sperc/stotal . (3.2)

For site and bond percolation, there exists only one unique percolating cluster in a percolating system in thermodynamic limit, as shown in Ref. [90]. In ﬁnite systems of size L, several sample-spanning clusters may exist that grow together to form a single cluster in the limit L → ∞. This does not hold for our model, as will be shown later.

Mean cluster size and cluster size distribution If ns is the number of clusters with size s, the product s ns gives the number of edges contained in all clusters with size s. Pstotal The summation over all possible cluster sizes s=1 ns = Ncl yields the total number of clusters Ncl. The mean cluster size s¯ can be deﬁned in several ways [233]:

• When selecting one of the network edges with equal probability, the probability that it belongs to a cluster of size s is s n s n w := s = s s Pstotal (3.3) stotal s=1 s ns

P Pstotal 2 yielding a mean cluster size s¯ := s s ws = s=1 s ns/stotal.

• An alternative way of averaging refers to the probability w˜s of getting a cluster of size s when arbitrarily1 picking a cluster n n w˜ := s = s . s Pstotal (3.4) Ncl s=1 ns

P Pstotal The mean cluster size is then s¯ := s s w˜s = s=1 s ns/Ncl. Note that the total number of clusters Ncl depends on probability p and may diﬀer in diﬀerent realisations.

In the following, we will refer to the first definition in Eq. (3.3) for the mean cluster size s¯ as the normalisation is independent of probability p and the individual realisation of the network. As ns gives the frequency of clusters of size s, it is also referred to as cluster size distribution. Equivalently, the cluster mass distribution mv gives the number of clusters with mass v. Similar to the mean cluster size s¯, a mean cluster mass m¯ can be defined. Since the cluster size s and the cluster mass v are connected by Eq. (3.1), it is reasonable to assume that the distributions ns and mv show similar characteristics, in particular with respect to their scaling behaviour.

Cluster radius The volume of a cluster consisting of v vertices can be quantiﬁed by the cluster radius Rv v 1 X Rv := |ri − r0| (3.5) v i=1

1Arbitrarily is here used to mean with equal probability for every cluster.

31 3 Percolation Properties

3 where ri ∈ R are the coordinates of the ith vertex and v 1 X r0 := ri (3.6) v i=1

is the cluster’s centre of mass. The cluster radius Rv quantiﬁes the density of a cluster; however, it does not say anything about its topological structure.

Pair-connectedness function and correlation length The pair-connectedness func- + tion G(r) [248], which is defined for r ∈ R , is the probability that two vertices at distance r from each other belong to the same cluster. It describes the connectivity of the clusters and is thus also referred to as two-point connectivity function [90]. G(r) resembles the radial distribution function ρ(r) in fluid theory [94] describing how the particle density varies as a function of the distance r from one particular fluid particle, but with the crucial difference that in case of the pair-connectedness function, there has to be a path between the two points considered whereas in the radial distribution function of liquid particles, two particles at distance r are regarded independently of the rest of the fluid. The pair-connectedness function G(r) is defined as

G(r) : = hδ (|ri − rj| − r) g(ri, rj)i (3.7)

P P r r δ (|ri − rj| − r) g(ri, rj) = i j (3.8) P P δ (|r − r | − r) ri rj i j where h·i is the average over all pairs of vertices i and j with a distance |ri − rj| = r 3 3 to each other; i, j ∈ [1,N] where N is the number of vertices, ri ∈ R and rj ∈ R are the positions of vertex i and j, respectively. δ denotes the δ-distribution and function g(ri, rj) is deﬁned as 1 if vertices ri and rj belong to the same cluster g(ri, rj) := (3.9) 0 otherwise indicating whether or not vertex j is on the same cluster as vertex i. Note that, for site and bond percolation, g(ri, ri) = 1 as all sites can be interpreted to be connected to themselves. Hence, the pair-connectedness function G(r) is always equal to 1 for r = 0. In the vertex model at p = 0, G(0) = 1, too. However, for p > 0, two-coordinated vertices that were generated by separating a node share the same position. Thus, g(ri, ri) may be 0 or 1 which results in 0 < G(0) < 1. Since all vertices are located on a lattice, the values distance r can take are discrete. P Summing over all possible distances r G(r) yields the average number of connected vertices.

For site and bond percolation in the subcritical regime p˜ < p˜c, G(r) decays exponentially in the asymptotic limit r → ∞ [90] r G(r) ∼ exp − for p˜ ∈ / {0, p˜c} , (3.10) ξ(˜p)

32 3.2 Scaling Theory with the occupation probability p˜. The length scale ξ(p˜) characterising the decay is called correlation length. It is the length over which fluctuations are correlated. For p˜ < p˜c, the correlation length ξ is a measure for the average size of the finite clusters. Hence, ξ(˜p) → 0 for p˜ → 0. In the supercritical regime p˜ > p˜c, the probability of finding two points at distance r 2 belonging to the same cluster is dominated by the probability [fperc(p˜)] that both points are part of the infinite cluster. In order to obtain information about the fluctuations, the pair-connectedness function G(r) is replaced by the so-called truncated pair-connectedness function Gf(r) [90], giving the probability of finding two points at distance r that belong to the same finite cluster. It can be shown that the truncated pair-connectedness function Gf(r) also decays exponentially with r for r → ∞, similar to Eq. (3.10)[ 90]. As for p˜ < p˜c, the correlation length ξ(p˜) describes only finite clusters and, hence, can be taken as measure of the size of the holes in the infinite cluster with ξ → 0 for p˜ → 1 [45]. When approaching the critical point p˜c, the order in the system extends more and more, leading to long order correlations and a diverging correlation length ξ(p˜) → ∞ for p˜ → p˜c. Hence, ξ(p˜) can no longer be used as length scale. The system becomes scale invariant which demands algebraic dependencies. Experimental findings for many systems and exactly solvable models suggest a power-law decay for the correlation function G(r) 1 G(r) ∼ for p˜ → p˜c (3.11) rd−2+η where d is the system dimension and η a system-dependent critical exponent [66, 232, 90, 269]. The pair-connectedness function G(r) as defined in Eq. (3.7) takes all clusters into account, but it could also be defined for clusters of mass v only. Gv(r) is the conditional probability that assuming that vertex i belongs to a cluster of mass v, a vertex j at distance r is part of the same cluster. Infinite clusters are considered separately; their pair-connectedness function is denoted G∞(r). G(r) is sometimes called the averaged pair-connectedness function and is related to Gv(r) and G∞(r) by [103]

Nperc X X G(r) = v mv Gv(r) + Ppcl,l G∞(r) (3.12) v l=1 with Nperc the number of percolating clusters and Ppcl,l the probability of randomly choosing an edge belonging to the percolating cluster l. In other physical systems, such as the Ising model or ﬂuid systems, the equivalent quantity to the pair-connectedness function is the pair-correlation function. Exponential decays of the pair-correlation function with distance r are common for systems with a second order phase transition away from the critical point, as are the diverging correlation length ξ and a power-law decay at criticality [232]. If the vertex model shows a second order transition, a power-law behaviour can be expected.

3.2 Scaling Theory

When a system exhibits a continuous phase transition, such as ﬂuid and magnetic systems, critical phenomena can be observed [232]. In the immediate vicinity of the

33 3 Percolation Properties

critical point, in the so-called critical domain, many quantities vanish or diverge exhibiting a power-law scaling behaviour. There are different critical exponents characterising this scaling for different quantities. These exponents are not independent from each other but can be associated by so-called scaling relations. There are different sets of values for these exponents defining different universality classes. Two systems with the same critical exponents belong to the same universality class. If two physical systems are in different universality classes, they are governed by fundamentally different physical laws [233, 210, 31, 45].

3.2.1 Universality Classes The universality hypothesis states that there is only a small number of different classes of continuous phase transitions. Each of these universality classes is characterised by a set of critical exponents. These classes are universal in the sense that they only depend on the dimensionality of the system and system symmetries, but are independent of the interaction details of the systems [31]. In contrast, the prefactors in the scaling laws do depend on the interaction details and are hence non-universal [248]. Exact values for the percolation critical exponents are known for one and two dimensions and for the mean-field theory [45]. Mean field theory can be applied in dimensions higher than the upper critical dimension du of a system where fluctuations become irrelevant and can be neglected as they are much smaller than the mean-field energy [269]. In case of percolation, the upper critical dimension equals du = 6; The mean field is realised on the so-called Bethe lattice [45]. This tree structure corresponds to an infinite dimensional lattice as no loops may occur [45]. The set of critical exponents that characterises this class is given in Tab. 3.3. Site and bond percolation belong to another important universality class. Continuum percolation models of differently shaped overlapping particles are geometrically in the same universality class as lattice percolation [248] implying that the critical exponents concerning percolation are the same. Elastic and transport critical exponents, on the other hand, differ in general from their lattice counterparts [248]. In the case of invasion percolation, i.e. one fluid in a porous medium replaces another along the path of least resistance, there are indications that the model belongs to the same class as site and bond percolation, too [261]. In contrast, in directed percolation where there are preferred directions of motion, there are two correlation lengths: one along the favoured direction and one perpendicular to it. Thus, an additional critical exponent ν0 =6 ν exists for the additional correlation length which results in changes in the scaling relations [233, 100]. Interestingly, the random walk also fits into this scheme of universality classes. The 2 mean squared displacement hRN i of a random walk on a lattice scales as 2 2ν hRN i ∼ N (3.13) where N is the number of steps and ν the universal critical exponent. In the classical case of a non-interacting random walk ν = 1/2 [55], which is the mean-field exponent. In contrast, for an interacting or self-avoiding random walk, the Flory mean field model predicts the so-called Flory exponent ν = νF = 3/(d + 2) [68, 55] which depends on the dimension d of the system. This results in νF = 3/4 for two dimensions and νF = 3/5 for three-dimensional systems. The numerically calculated values by Li et al. [144] confirm the two-dimensional result ν = 3/4, for a three-dimensional self-avoiding

34 3.2 Scaling Theory

Table 3.3: Critical exponents for site percolation in two and three dimensions [233, 248, 45].

d = 2 d = 3 mean ﬁeld β edges in inﬁnite cluster fperc ∼ (p − pc) β 5/36 0.41 1 ν correlation length ξ ∼ |p − pc| ν 4/3 0.88 1/2 pair-connectedness G(r) ∼ r−(d−2+η) η 5/24 −0.068 0 γ mean cluster size s¯ ∼ |p − pc| γ 43/18 1.80 [233] 1 −τ cluster mass distribution mv ∼ v τ 187/91 2.18906[45] 5/2 fractal dimension Df 91/48 2.52 4

walk they obtained ν = 0.5877 ± 0.0006, which is close to the value derived by Flory [68, 55]. Therefore, non-interacting random walks and Flory random walks are in diﬀerent 2 universality classes. Note that hRN i can also be interpreted as the end-to-end distance of a freely-jointed polymer chain with N non-interacting or interacting monomers. Exceptions from the universality hypothesis are the eight-vertex model and the Potts model. The critical exponents of these models vary continuously with the interaction parameters and cannot be put into a universality class [18, 20, 19]. Tab. 3.3 lists the set of exponents of the universality class that site and bond percolation belong to. For comparison, the mean-ﬁeld exponents are given, too.

3.2.2 Fractal Dimension Df of a Cluster Diﬀerent systems showing a continuous phase transition have in common that, at the critical point, the correlation length ξ diverges and the system is assumed to become fractal, i.e. self-similar or scale-invariant. For percolation models at the percolation threshold, the clusters show fractal behaviour. The fractal dimension Df indicates how the cluster mass M(R), i.e. the number of vertices belonging to one cluster, scales when changing the spatial resolution R. If the exponent Df in the scaling rule

M(R) ∝ RDf (3.14) is a fractional number (i.e. non-integer), the object is called a fractal with fractal dimension Df. In a strict mathematical sense, fractals are infinitely scalable, i.e. look the same on every zoom level. Natural objects such as ferns or crystals, or finite computer generated patterns, are always approximations to fractals. That means the object is statistically self-similar over a large but finite range of scales and a characteristic smallest length scale exists [64, 228]. A standard technique to determine the fractal dimension Df is the cumulative mass method [64, 37]. In a sampling region, all points, e.g. white pixels of a black and white image or nodes of a cluster, are counted as a function of the sampling region’s volume (see Fig. 3.2). This so-called mass is averaged over several sampling regions with randomly located centres. From the scaling behaviour of the mass with the volume the mass

35 3 Percolation Properties

Figure 3.2: Cumulative mass method for the determination of the fractal dimension Df: The mass M(R) of a percolating cluster is determined by counting the vertices contained in a box of size R. The scaling of M(R) with increasing box size R gives the fractal dimension Df of the percolating cluster, see Eq. (3.14). Here, one percolating cluster from a conﬁguration at p = 1 is shown. The vertices are illustrated by spheres that are coloured depending on their position: the inner cube contains just yellow spheres, the middle cube the yellow and the red ones and the biggest box contains all spheres.

dimension or cluster fractal dimension is determined. The cluster fractal dimension was used, for example, to describe protein aggregation in time [64], to show that colloid aggregates are fractals [64], to analyse the shape of neurons [229, 38] or the structure of trabecular bone [41]. In section 3.4 and section 3.5, the cumulative mass method is used to evaluate the cluster fractal dimension Df of the percolating clusters in the vertex percolation model at probability p = 1, i.e. all four-coordinated nodes were divided into two vertices with coordination number c = 2. A cluster does not necessarily have to be a fractal. For a cluster to be a fractal, it is necessary that the density decreases as the cluster size increases. The density of the cluster is only constant if Df equals the Euclidean dimension. Thus, the cluster fractal dimension Df is a measure of how densely a cluster ﬁlls space [64].

3.2.3 Finite Size Scaling In theory, the percolation problem is defined on an infinite lattice. Unfortunately, there are only exact results in one dimension and for a few cases in two dimensions. All other results were achieved by numerical calculations or experiments [232], both of which are limited in system size due to limited resources. Sufficiently far away from the critical point, finite system size is not a limitation as the correlation length ξ is relatively small and systems of linear size L ξ can be considered infinite. However, when looking at quantities in the vicinity of or at the critical point, it gets harder and harder to do (numerical) experiments in sufficiently large systems since ξ increases when approaching

36 3.2 Scaling Theory the critical point until it finally diverges. Especially at the critical point p = pc, all measurements will be carried out at L ξ which will always entail finite size effects such as shifted thresholds and rounded exponents. In order to obtain information about the infinite system at criticality, the scaling of quantities as function of the system size L is investigated, which allows extrapolation to the limit L → ∞. This method is called finite size scaling and is based on the scaling hypothesis [232, 233, 45] which assumes two things: (i) The correlation length ξ diverges with the critical exponent ν as −ν ξ ∼ (p − pc) (3.15)

when approaching the critical point.

(ii) If a general quantity X(p) of an inﬁnite system scales near its critical point as

δ X(p) ∼ (p − pc) (3.16)

where δ is the associated critical exponent, then X(p, L) in a ﬁnite system of size L is a generalized homogeneous function [233, 232]

X(p, L) = λ−1X(λap, λbL) (3.17)

−δ with λ = (p − pc) , a = 1/δ and b = −ν/δ and can be written as

δ 1/ν X(p, L) = (p − pc) X 1,L (p − pc) . (3.18)

In the following, the scaling behaviour of diﬀerent quantities will be examined and relations between the critical exponents will be derived.

Fraction fperc of edges belonging to percolating clusters The fraction fperc(p) of edges that belong to an inﬁnite cluster can be considered as the order parameter of the percolation phase transition. It is assumed to obey

β fperc(p) ∼ (p − pc) (3.19)

−1/ν with its critical exponent β. From Eq. (3.15), (p − pc) ∼ ξ can be concluded giving

− β fperc(p) ∼ ξ ν . (3.20)

This result is valid for inﬁnite systems and also holds if L ξ. If, however, L ξ, the correlation length ξ is bounded by the system size. Thus, [45]

−β/ν ξ for L ξ fperc(p, L) ∼ (3.21) L−β/ν for L ξ .

As ξ diverges at p = pc, L ξ will always be true at the critical point, no matter what the system size L is. Thus fperc(pc,L) decays as

−β/ν fperc(pc,L) ∼ L (3.22)

37 3 Percolation Properties which can be used (see Fig. 3.6 and Fig. 3.11) to determine the ratio β/ν. The fraction fperc(p, L) can be written in the shape of Eq. (3.18) which yields β ˆ 1 fperc(p, L) = (p − pc) f L ν (p − pc) (3.23)

ˆ 1 ν ν −ν where f(L ν (p−pc)) is the universal scaling function. Let x := L (p−pc) = L/(p−pc) . For ν > 0 and x 1, it follows that 1 xν ∼ L/ξ, i.e. the system size L is much smaller −β/ν than the correlation length ξ. Hence, according to Eq. (3.21), fperc(p, L) ∼ L . On −β/ν the other hand, if x 1, the system is much larger than ξ and thus fperc(p, L) ∼ ξ . This requires x 1 fˆ(x) = const x−β x 1 . (3.24)

Mass M(l) of a percolating cluster As mentioned before, percolating clusters are fractals at p = pc [233, 45]. The mass of a percolating cluster M(l) within a window D of size l scales with its fractal dimension Df as M(l) ∼ l f, cf. Eq. (3.14). In the supercritical phase for p > pc, there is generally at least one percolating cluster. Let Npcl(p, l) be the number of percolating clusters in a window of size l. In case of site and bond percolation, Npcl(p, l) = 1. Yet this does not hold for the vertex model where Npcl(p, l) ≥ 1, as will be shown in sections 3.4.2 and 3.5.4. Let us pick one of these clusters. On scales much shorter than the correlation length ξ, this cluster still appears to be fractal, that is, its density decreases with lDf−d. For length scales much larger than ξ, the cluster has a constant density given by the probability Ppcl(p, l) that an arbitrary edge belongs to the percolating cluster under consideration. Assuming that the percolating clusters cannot be distinguished from each other as all are equivalent, the probability Ppcl(p, l) = fperc(p, l)/Npcl(p, l) and all of them should have the same fractal dimension Df. Hence, D l f for l ξ M(l) ∼ d (3.25) l fperc(p, l)/Npcl(p, l) for l ξ . The function M(l) shall be continuous at l = ξ resulting in

D d l f ∼ l fperc(p, l)/Npcl(p, l) . (3.26) l=ξ

Applying Npcl(p, l) = 1 for site and bond percolation and using l = ξ and Eq. (3.21) results in the hyperscaling relation2 β d − Df = . (3.27) ν

Pair-connectedness function G(r) For site and bond percolation, the mass of the percolating cluster M(l) can also be calculated from the pair-connectedness function G∞(r) for the percolating cluster [103] Z l d D M(l) ∝ d r G∞(r) ∼ l f . (3.28) r=0 2An equation connecting diﬀerent critical exponents is called a scaling relation. If such an equation additionally contains the system dimension d, it is referred to as a hyperscaling relation [233].

38 3.2 Scaling Theory

Hence, G∞(r) scales as

D −d G∞(r) ∼ r f . (3.29)

At criticality, self-similarity applies [233, 45], i.e. the system looks the same at all length scales. Thus, when looking at a ﬁnite system of size L at the critical point, one cannot diﬀerentiate between a percolating cluster and a large, i.e. not completely contained in the system of size L, non-percolating cluster. Consequently, for large clusters the same scaling relations as for percolating clusters must apply, for instance, their mass v Df scales with the cluster radius Rv as v ∼ Rv . The connectedness function Gv(r, L) is the conditional probability that, given that a vertex belongs to a cluster of mass v, a vertex at distance r is part of the same cluster [103]. For p → pc,

−(d−D ) Gv(r, L) ∼ r f (3.30) is assumed.

Equivalently to Eq. (3.3), the probability of picking a vertex that belongs to a cluster of mass v is v mv, where mv is the cluster mass distribution. Provided a constant density d of vertices, the probability to ﬁnd a cluster of mass v in a box of size L is v mv L . At p = pc, the distribution mv of the number of clusters with mass v conforms to the −τ power-law mv ∼ v with the Fisher exponent τ [233]. Self-similarity at criticality implies scale-invariance. Hence, rescaling by a factor λ, i.e. L by L 7→ λL and Rv 7→ λRv, which yields v ∼ λDfv, results in

d D 1−τ d v mv L = (λ fv) (λL) (3.31) giving the hyperscaling relation [103] d τ = 1 + . (3.32) Df In case of site or bond percolation, the pair-connectedness function G(r, L) can be described as the cluster average over the pair-connectedness functions Gv(r, L) of all ﬁnite clusters and, in the supercritical regime, G∞(r, L) of the unique inﬁnite cluster [103],

X v G(r, L) = v mv Gv(r, L) + fperc G∞(r, L) . (3.33) v

v fperc is the fraction of vertices belonging to the percolating cluster and is deﬁned similarly to fperc (cf. Eq. (3.2)). As the number of vertices and the number of edges of a cluster are v coupled through Eq. (3.1), the scaling behaviour of fperc equals that of fperc (cf. Eq. (3.21) and Eq. (3.22)). Let us consider the scaling behaviour of G(r, L) at the critical point p = pc. For this, the two terms of Eq. (3.33) are considered separately. The sum of Gv(r, L) over all ﬁnite cluster masses v scales as

X −(d−D ) X −(d−D ) 2−τ v mvGv(r, L) ∼ r f v mv ∼ r f s (3.34) v v

39 3 Percolation Properties

Df which can be rewritten by replacing v by Rv , by using the hyperscaling relation Eq. (3.32) and by substituting Rv by r as for large r and large Rv they are interchangeable:

X −2(d−D ) v mv Gv(r, L) ∼ r f . (3.35) v

The second term in Eq. (3.33), which quantiﬁes the contribution of the inﬁnite cluster to the cluster average of G(r, L), scales as

v −β/ν −(d−Df) fperc(pc,L)G∞(r, L) ∼ L r (3.36) where Eq. (3.21) and Eq. (3.29) were used. G∞(r, L) is considered for large r which means, in a system of size L, that r and L are of the same order of magnitude and can thus be replaced by each other. With the hyperscaling relation Eq. (3.27), Eq. (3.36) can be written as

v −2(d−Df) fperc(pc,L)G∞(r, L) ∼ r . (3.37)

From Eq. (3.35) and Eq. (3.37), it can be concluded that [103]

G(r, L) ∼ r−2(d−Df) . (3.38)

By comparison of the exponents of Eq. (3.11) and Eq. (3.38) and by application of Eq. (3.27), one obtains [233]

d − 2 + η = 2(d − Df) = 2β/ν . (3.39)

This hyperscaling relation relates the decay of the pair-connectedness function G(r, L) at the critical point p = pc to the ratio β/ν of the critical exponents ν and β of the correlation length ξ and the order parameter fperc, respectively. Eq. (3.39) can be used to determine β/ν.

3.3 Implementation of the Vertex Model

The 2D percolation program was implemented by Matthias Hoﬀmann. Details about the implementation can be found in Ref. [102].

For the 3D analysis, two different codes were developed. In the first program, the entire network structure, i.e. the coordinates of all vertices and a connection list containing the labels of all vertices a given vertex is connected to by an edge, is stored in memory during the entire computation. This is computationally rather expensive and quickly reaches the boundaries of main memory (several ten gigabytes) for network sizes not larger than 120 unit cells, but has the advantage that a comprehensive set of structural properties, such as cluster statistics, percolation properties and connectedness functions, can be evaluated on the whole set of clusters. However, for the calculation of the critical exponents using the power-law exponents associated with the fraction fperc(1/L) of edges in percolating clusters and the pair- connectedness function G(r), it is important to simulate significantly larger systems. For this purpose, a separate and more specific program was developed, allowing larger systems

40 3.4 Percolation Properties of the 2D Vertex Model

Figure 3.3: Shifted diamond unit cell: The cubic unit cell of diamond (see Fig. 2.1 (a)) is shifted by (1/8, 1/8, 1/8)a, where a is the linear size of the unit cell. This is done in order to avoid nodes to lie on the boundary of a system. Each face of the shifted cell contains four open ends.

to be analysed. The structure is built up layer by layer, but rather than storing the whole network configuration, only two layers are stored at any time. Additionally, a relatively small cluster map is kept in memory that allows the computation of the percolating fraction fperc and sufficient information is stored to reconstruct the cluster through the centre of the lattice on which the pair-connectedness function G(r) is evaluated. With this approach, G(r) could be calculated on systems up to size L = 1000 unit cells, and fperc on systems up to size L = 1500 unit cells, at the expense of worse statistics. In both programs, nodes that are located on a system boundary are difficult to handle — especially when it comes to differentiating between vertices with a coordination number lower than four and nodes on a boundary. In order to avoid this problem, the cubic unit cell of diamond, which is shown in Fig. 2.1 (a) in section 2.2, is shifted by (1/8, 1/8, 1/8)a, where a is the linear size of the unit cell. The resulting, shifted unit cell is illustrated in Fig. 3.3. All nodes are located clearly inside the unit cell and are not shared between adjacent cells.

3.4 Percolation Properties of the 2D Vertex Model

This section studies the percolation properties of the 2D vertex model and compares them to 2D site percolation. It is shown that the 2D vertex model exhibits a percolation transition at p = 1 instead of pc = 0.407254 [183], the site percolation threshold. For both models, the critical exponents are determined. For this, the power-law decays of the fraction fperc of edges belonging to a percolating cluster and the pair-connectedness function G(r) are used as is finite size scaling. Besides, the fractal dimension Df is compared. Both, the critical exponents and the fractal dimension clearly differ for the two models indicating that the vertex model belongs to a different universality class than site percolation.

The percolation behaviour of a 2D square lattice (see Fig. 2.7) with periodic boundary conditions in x-direction is investigated when separating the nodes with coordination number c = 4 into two vertices with coordination number c = 2 as a function of disconnection probability p, as described in section 2.4. The system size is given by L, the number of unit cells in x-direction. The unit cell is chosen in such a way that no

41 3 Percolation Properties

Figure 3.4: Square lattice consisting of 2 × 2 unit cells. The unit cell is chosen such that no nodes are located on the border. There are two open ends on the top and on the bottom of each unit cell.

nodes are located on its border as shown in Fig. 3.4. In order to avoid boundary eﬀects in y-direction, the dimension of the system is three times the dimension of the system in x-direction.3 When separating more and more nodes, the system disintegrates into individual clusters. Considering a 2D system with periodic boundary conditions in x-direction, three kinds of clusters may occur: (i) percolating clusters entirely spanning the system in y-direction, (ii) clusters starting at one boundary and returning to it, henceforth called uturn clusters, (iii) and inner loops that do not touch the boundaries at all. At p = 1, all clusters are line-like objects without any branches, which in two dimensions anticipates crossing of diﬀerent clusters. Thus, each percolating cluster divides the plane into half-planes left and right of it. For loops, there are two possibilities: either they are empty or they contain other loops. If uturn clusters do not return to a boundary point right next to their starting point, they must have an even number of boundary points between their ends as they enclose at least one other uturn cluster.

3.4.1 Fraction of Edges in Percolating Clusters

We first study the fraction fperc of edges that pertain to a percolating cluster which was defined in Eq. (3.2). In Fig. 3.5, fperc is plotted as a function of p for the site percolation and the vertex percolation model for different system sizes L. Note that, for site percolation, parameter p is related to the occupation probability p˜ by p = 1 − p˜. Hence, p is the probability that a lattice site is vacant. At p = 0, both systems consist of a single percolating cluster that contains all vertices. Hence, fperc is equal to 1, for site percolation and the vertex model. Increasing p in the site percolation model is equivalent to removing sites from the system. The network decomposes into different clusters causing fperc to decrease. At pc = 1 − p˜c, where p˜c is the known percolation threshold for 2D site percolation, fperc drops to some L-dependent, finite value greater than 0. For p > pc, fperc equals 0. When increasing p in the vertex model, nodes are separated rather than removed. The connectivity of the network decreases as p increases, also yielding a decomposition into different network parts. Thus, fperc decreases with p, similar to site percolation. A similar drop is observed, too, however at p = 1 instead of p = pc.

3All 2D simulations were performed by Matthias Hoﬀmann, see Ref. [102].

42 3.4 Percolation Properties of the 2D Vertex Model

1 1

pc = 0.407254 pc 0.8 0.8 2D site 2D vertex 0.6 0.6 perc perc f 0.4 f L = 10 0.4 L = 31 L = 10 0.2 L = 100 0.2 L = 100 L = 316 L = 1000 L = 1000 L = 10000 0 0 0 0.1 0.2 0.3 0.4 0.5 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 p p (a) 2D site percolation (b) 2D vertex model

Figure 3.5: Fraction fperc of edges belonging to one of the percolating clusters as function of p in 2D: For p = 0, fperc = 1 as all vertices belong to a single percolating cluster. With increasing p, sites are removed from the site percolation model and nodes are separated in the vertex model. Both results in a decrease of fperc with p. At p = pc and p = 1, for site percolation and the vertex model, respectively, fperc drops to some size-dependent, ﬁnite value greater than 0. The system size L represents the number of unit cells in x-direction. Note that parameter p = 1 − p˜ for site percolation. (For a deﬁnition of fperc see Eq. (3.2).)

For better visualisation, the same data is replotted in Fig. 3.6 as function of 1/L for different probabilities p close to pc and close to 1, for site percolation and for the vertex model, respectively. Let us first consider site percolation (Fig. 3.6 (a)) at p < pc. With increasing system size L, fperc saturates to some positive constant, the exact value being a function of p. Hence, infinitely large systems always percolate for p < pc. On the contrary, at the critical point p = pc, the fraction fperc of edges belonging to a percolating cluster decreases with increasing system size. As scaling theory applies, this decay conforms to a power-law as described in Eq. (3.22) with fperc → 0 in the limit L → ∞. The ratio of the critical exponents β/ν can be computed by fitting the power-law exponent. Our simulation yields β/ν = 0.104 ± 0.003 and thus reproduces the theoretical value β/ν = 5/48 ≈ 0.104167 [233]. The graph of the vertex model (Fig. 3.6 (b)) looks alike. In the limit L → ∞, fperc approaches some positive, p-dependent constant for p < 1 whereas at p = 1 it appears to decay to 0. This means that either the number of vertices in a percolating cluster is negligible compared to the number of vertices in non-percolating clusters or that no percolating cluster exists anymore. The latter case would define a percolation transition at p = 1. In that case, scaling theory would apply and critical exponents could be determined. Using Eq. (3.22), the power-law exponent of fperc at p = 1 is fitted giving β/ν = 0.249 ± 0.002. This clearly differs from the site percolation value.

43 3 Percolation Properties

1 100 2D site 2D vertex 0.7 0.5

-1 0.3 10 p = 1.000 perc perc

f p = 0.4 f p = 0.999 p = 0.403 p = 0.9975 0.2 ± p = 0.406 p = 0.995 ∝ 1/L0.104 0.003 ± p = 0.407 ∝ 1/L0.249 0.002 p = 0.99 p = 0.4072 p = 0.98 p = 0.40725 p = 0.9 0.1 10-2 10-4 10-3 10-2 10-1 100 10-4 10-3 10-2 10-1 100 1/L 1/L (a) 2D site percolation (b) 2D vertex model

Figure 3.6: Fraction fperc of edges in a percolating cluster as function of inverse system size 1/L in 2D: If p < pc for site percolation and p < 1 for the vertex model, fperc saturates to some positive constant. For p = 1 (p = pc for site percolation), fperc decreases conforming to a power-law for increasing system size L. (For a deﬁnition of fperc see Eq. (3.2).)

3.4.2 Number of Clusters

What do we expect for the numbers of the diﬀerent cluster types when increasing the number of unit cells in the system? First, the number of open ends in the surface layers of the system, hereafter referred to as boundary points, is proportional to the system size L. Each unit cell contains two open ends on each side, thus, the number of open ends in one layer is 2L. As periodic boundary conditions in x-direction apply, the top and the bottom of a system are its only surface layers. Hence, there are 4L open ends that are boundary points. In a vertex model system at p = 1, all nodes are separated. Therefore, the clusters cannot have any crossing points but they are either loops or ‘occupy’ two of the boundary points. In particular, each percolating cluster ‘occupies’ exactly one boundary point at the top and one at the bottom whereas a uturn cluster ‘occupies’ two boundary points on the same surface. All boundary points belong to either a percolating or a uturn cluster which yields three conclusions:

(i) The sum of the number of uturn clusters and percolating clusters is 2L.

(ii) There are always as many uturn clusters starting and ending at the top as at the bottom of the system.

(iii) Since the uturn clusters always take two boundary points, the remaining number of boundary points at the top or the bottom, respectively, is even for any number of uturn clusters. Thus, there is always an even number of percolating clusters in the system — which includes zero in a non-percolating system.

Fig. 3.7 displays the frequency of occurrence of the diﬀerent kinds of clusters as function of the system size L. Data was obtained from systems with periodic boundary conditions in x-direction but open boundary conditions in y-direction. For site percolation, the

44 3.4 Percolation Properties of the 2D Vertex Model

107 106 6 loops loops 10 uturn 105 uturn ∝L2.024 ± 0.002 5 perc 1.99 ± 0.04 perc ± 10 P ∝ L 104 P ∝L1.005 0.001 1.0023 ± 0.0008 104 ∝ L 103 103 2 2 10 10 N ± N ± 1 perc = 0.728 0.003 1 perc = 0.635 0.002 10 10 2D site 2D vertex P = 0.362 ± 0.002 P = 0.633 ± 0.002 0 100 10 number of clusters number of clusters 10-1 10-1 101 102 103 104 101 102 103 104 L L (a) 2D site percolation (b) 2D vertex model

Figure 3.7: Number of clusters as function of system size L for site percolation at the percolation threshold pc = 0.407254 and for the vertex model at p = 1 in two dimensions. ‘loops’ is the number of inner loops, ‘uturn’ the number of uturn clusters and ‘perc’ the number of percolating clusters. The constant number Nperc of percolating clusters and the constant percolation probability P (see page 45) are determined by a least squares ﬁt. Data was obtained from systems with periodic boundary conditions in x-direction but open boundaries in y-direction.

critical percolation threshold is pc = 0.407254 [183]; for the vertex model it is p = 1. In both models, the number of loops scales with two, indicating that there is a certain mean size of loops which is independent of the system size. The number of uturn clusters increases linearly whereas the number of percolating clusters stays constant. Since the number of boundary points scales as L, this means that the probability to form a uturn cluster is constant as function of L whereas the probability to form a percolating cluster scales with 1/L. The percolation probability P that a certain realisation of a system of size L percolates is also shown. The value of P was extracted by least squares ﬁtting of a constant. For site percolation we get P = 0.633 ± 0.002 and for the vertex model P = 0.362 ± 0.001. If P was equal to 0 or 1, the systems would be in the non-percolating or percolating state, respectively. Values of P between 0 and 1 hint at that both systems are at their critical point. A remarkable diﬀerence is that, for site percolation, the average number of percolating clusters equals the percolation probability approving that, if the system percolates, there is only one percolating cluster. For the vertex model, the average number of percolating clusters is Nperc = 0.728 ± 0.002 which is twice the percolation probability P . This results from the fact that there must always be an even number of percolating clusters. The system either does not percolate or it has not only one but at least two percolating clusters.

3.4.3 Pair-Connectedness Function G(r)

Fig. 3.8 shows the pair-connectedness function G(r) as a function of the distance r between two vertices. For large distances r, G(r) is expected to decay exponentially as described by Eq. (3.10). Only at criticality, the decay becomes a power-law as in Eq. (3.11)

45 3 Percolation Properties

Lx = 2000 vertex 2D L 1 x = 1000 vertex 10 Lx = 500 vertex L -0.480 ± 0.001 x = 2000 site ∝ r Lx = 1000 site L ) 0 x = 500 site r 10 ( G

-1 ± 10 ∝ r-0.2055 0.0003

10-2 10-1 100 101 102 103 r (a) Pair-connectedness function G(r)

105 vertex 104 site 2D 103 102 L = 200 101 x M(R) 100 site: Df = 1.861 ± 0.007 10-1 vertex: Df = 1.702 ± 0.007 10-2 10-1 100 101 102 R (b) Fractal dimension Df

Figure 3.8: (a) Pair-connectedness function G(r) (defined in Eq. (3.7)) for a 2D vertex model system compared to 2D site percolation. For both models, G(r) exhibits a power-law, but with different exponents. This suggests that the two models belong to different universality classes. The exponents of the power-laws were determined by linear least squares fitting in the log-log plot in the range 2 [1, 10 ] in systems of linear dimension Lx = 2000. (b) Mass M(R) of a single percolating cluster in a sphere with radius R. The log-log slope, computed by 2 linear least squares fitting in the range [1, 10 ], equals the fractal dimension Df of a single percolating cluster (see Eq. (3.14)). For both, the vertex model and site percolation, Df is determined in a system of linear size Lx = 200.

46 3.4 Percolation Properties of the 2D Vertex Model as the system turns to be self-similar without any representative length scale. For site percolation at the critical point, the slope in the log-log plot presented in Fig. 3.8 (a) is determined as −0.2055 ± 0.0003. The theoretical value is −(d − 2 + η) = −2β/ν = 0.2083¯, which is close to the numerical value but outside the stated statistical error range. The pair-connectedness function of the vertex model at p = 1 also shows a power-law decay. Thus, the system is at its critical point since an algebraic behaviour can only be observed if the correlation length ξ is infinite. The slope of −(d − 2 + η) = −0.480 ± 0.001 clearly differs from the usual percolation model, suggesting that the vertex model belongs to a different universality class than site percolation.

3.4.4 Fractal Dimension Df The fractal dimension, giving information about the structure and space filling properties of the percolating clusters, is determined by counting the mass M(R) ∼ RDf of one percolating cluster in a sphere with radius R when sitting on one of the nodes of that cluster. The results for the 2D site percolation and the 2D vertex model are shown in Fig. 3.8 (b). The slope in the log-log plot gives a fractal dimension of Df,site = 1.861±0.007 for site percolation and Df,vertex = 1.702 ± 0.007 for the vertex model, both in a 200 × 200 system. The value of the fractal dimension Df,site for site percolation is close to the theoretical value of 91/48 ≈ 1.896, but differs more than the quoted error range. This suggests that the error range, which is the statistical error from the linear least squares fit, underestimates the real error. Possible additional sources of error may be systematic errors, as finite size effects, which need to be studied further.

3.4.5 Finite Size Scaling Fig. 3.9 shows a finite size scaling analysis which was performed in order to verify the hypothesis that the vertex model has different critical exponents than site percolation. The fraction fperc of edges in a percolating cluster is a function of probability p and −β the linear system size L. Consistent with Eq. (3.23), fperc(p − pc) for site percolation 1/ν collapses onto a single master curve when plotted against (p−pc)L using the theoretical critical exponents β = 5/36 and ν = 4/3 (see Fig. 3.9 (a)). For small values of x, the remaining scaling function Pˆ(x) scales as x−β. For large x it stays constant, as expected from Eq. (3.24). Fig. 3.9 (b) shows that using the same exponents for the vertex model is not as successful. A data collapse is achieved for β = 0.32 ± 0.02 and ν = 1.29 ± 0.04 (see Fig. 3.9 (c)). The individual extraction of β and ν (rather than β(ν)) is possible because of the specific way of plotting that distributes the β- and ν-dependence to the two different axes of the plot.

3.4.6 Hyperscaling Relations

With the numerically determined critical exponents β and ν and the fractal dimension Df, the validity of the hyperscaling relations Eq. (3.27) and Eq. (3.39) is tested. As the numerical values for the site percolation quantities are all close to the theoretical values, the hyperscaling relations are considered fulﬁlled. On the basis of the available data, it is hard to decide whether the critical exponents of the vertex model satisfy the hyperscaling relations or not. Inserting the numerical results into the ﬁrst relation yields Df − d + β/ν = −0.093 ± 0.007, where the errors

47 3 Percolation Properties

2.5 101 p = pc-0.00001 2D vertex 2.0 p = pc-0.0001 p = 0.999 β β p = 0.995 - p = pc-0.001 - ) p = p -0.01 ) p = 0.99 p c p - 1.5 p = p -0.1 - p = 0.98

c c c 0 p = 0.9 p p -5/36 10 ( ∝ x ( 1.0 perc perc ν=4/3, β=5/36 f 2D site f pc=0.407254 0.7 10-1 10-5 10-4 10-3 10-2 10-1 100 101 102 10-3 10-2 10-1 100 101 102 103 1/ν 1/ν x = (pc-p) L x = (pc-p) L (a) 2D site percolation (b) 2D vertex model, site percolation exponents

101 2D vertex ν = 1.29 ± 0.04, β

- β = 0.322 ± 0.018 ) p - c 0 p 10

( p = 0.999 p = 0.995 p = 0.99 perc

f p = 0.98 p = 0.9

10-1 10-3 10-2 10-1 100 101 102 103 1/ν x = (pc-p) L

Figure 3.9: Finite size scaling for the fraction fperc of edges in a percolating cluster as described in Eq. (3.23). (a) For 2D site percolation, all curves collapse on one master curve when using the known critical exponents β = 5/36 and 1/ν ν = 4/3. The scaling function becomes constant for x = (p−pc)L 1 and decreases as x−β for x 1, as expected from Eq. (3.24). (b) The use of the 2D site percolation critical exponents does not collapse the 2D vertex model data. (c) A collapse onto a master curve is achieved with β = 0.322 ± 0.018 and ν = 1.29 ± 0.04.

48 3.4 Percolation Properties of the 2D Vertex Model

Table 3.4: 2D results for critical exponents and hyperscaling relations: For site percolation and the vertex model, the numerically determined values are compared to theoretical values [233]. Periodic boundary conditions in x-direction applied. 2D site vertex theoretical numerical β 5/36 ≈ 0.1389 5/361 0.32 ± 0.02 ν 4/3 ≈ 1.33 4/31 1.29 ± 0.04 β/ν 5/48 ≈ 0.1042 0.104 ± 0.003 0.249 ± 0.002 Df 91/48 ≈ 1.896 1.861 ± 0.007 1.702 ± 0.007 d − 2 + η 5/24 ≈ 0.2083 0.2055 ± 0.0003 0.480 ± 0.001

0 = Df − d + β/ν 0 " −0.04 ± 0.01 ? −0.049 ± 0.009 ? 0 = d − 2 + η − 2β/ν 0 " −0.003 ± 0.007 " −0.018 ± 0.005 ? 1 Data collapsed using these exponents

49 3 Percolation Properties

of the two numerical values are simply summed up. Clearly, according to the error range, this value is unequal to 0. However, as described above, the error ranges appear to be too small as only statistical errors are taken into account. The same is true for d − 2 + η − 2β/ν = −0.006 ± 0.005. Since the discrepancy is small in the latter case, relation Eq. (3.39) is considered fulﬁlled whereas relation Eq. (3.27) is interpreted as not satisﬁed.

3.5 Percolation Properties of the 3D Vertex Model

This section studies the percolation properties of the 3D vertex model and compares them to 3D site percolation. The 3D vertex model exhibits a percolation transition at p = 1, which clearly differs from the 3D site percolation threshold pc = 0.56988 [251]. The critical exponents are computed using the fraction fperc of edges belonging to a percolating cluster, the pair-connectedness function G(r) and finite size scaling. Similar to the 2D case, the two models differ in their critical exponents and fractal dimensions. Thus, the 3D vertex model and the 3D site percolation model appear to belong to different universality classes. Another important observation is that, contrary to site percolation, in the 3D vertex model the number of percolating clusters is greater than one and increases with system size L.

A diamond lattice consisting of L3 unit cells is investigated. With probability p, the initially four-coordinated nodes are replaced by two two-coordinated vertices whereupon there are three possibilities of connecting the resulting edges (see Fig. 2.8 in section 2.4).

3.5.1 Fraction of Edges in Percolating Clusters

In Fig. 3.10 (a), the fraction fperc of edges belonging to a percolating cluster is plotted as a function of p for systems with open boundary conditions and different system sizes L. In a fully connected network at p = 0, fperc is equal to 1. Raising p decreases the connectivity of the network consecutively, which does not necessarily mean that the network decomposes into different parts. Over a certain range, all nodes stay connected to the one and only percolating cluster and a deviation from fperc = 1 is seen not until larger values of p. Close to p = 1, fperc drops dramatically, but to some value greater than 0 that depends on the system size L. Thus, the behaviour in two and three dimensions is alike, only the drop for p close to 1 is much more abrupt in 3D; see Fig. 3.5 for comparison. x y z Fig. 3.10 (b) shows the fractions fperc, fperc and fperc of edges in clusters percolating in a certain direction, i.e. x-, y- or z-direction, compared to fperc. For small p, all numbers coincide since only a few nodes are separated and there is only one percolating cluster x y z touching all boundaries. With increasing p, fperc, fperc and fperc decrease faster than fperc, revealing that the huge percolating cluster splits into several smaller clusters that do x y z not percolate in all directions anymore. Due to symmetry reasons, fperc = fperc = fperc. x y z Note that fperc, fperc and fperc are not necessarily 1/3fperc. This is only a lower bound as a cluster may well percolate in more than one direction and thus its edges contribute x y z to more than one of the fractions fperc,fperc,fperc. Fig. 3.11 shows fperc plotted against the inverse system size 1/L. For p < 1, a saturation to some positive constant is observed for L → ∞. At p = 1, the data can be fitted with a power-law. The exponent is very small, though, and yields a critical exponent ratio close

50 3.5 Percolation Properties of the 3D Vertex Model

1 100 3D vertex 3D vertex 0.8

0.6 x,y,z 10-1 perc f 0.4 perc L = 5 f L = 20, xyz 0.2 L = 10 L = 20, x L = 30 L = 20, y L = 120 L = 20, z 0 10-2 0 0.2 0.4 0.6 0.8 1 10-3 10-2 10-1 100 p 1 - p (a) percolation in any direction (b) percolation in speciﬁc directions

Figure 3.10: (a) Fraction fperc of edges belonging to any percolating cluster as a function of probability p in a system with open boundary conditions. At p = 0, all nodes belong to the same cluster which, of course, percolates. With increasing p, fperc ﬁrst decreases slowly, but drops rapidly for large p. (b) For open boundary systems, percolation may be considered for all three directions in space independently. In principle, the behaviour is the same for all three directions. Merely the total number of edges belonging to a percolating cluster is slightly lower when only taking one direction into account. For a deﬁnition of fperc see Eq. (3.2).

51 3 Percolation Properties

1.0 100 periodic bc open bc 0.7 3D vertex 3D vertex 0.5 0.4 10-1 p = 0.999 0.3 p = 0.9995 perc perc f β/ν = 0.0038 ± 0.0007 f p = 0.99985 p = 0.999 p = 0.99999 0.2 p = 0.9995 p = 1 p = 0.99985 p = 0.99999 10-2 p = 1 β/ν = 0.013 ± 0.003 0.1 10-4 10-3 10-2 10-1 100 10-4 10-3 10-2 10-1 1/L 1/L (a) 3D vertex model, periodic boundary conditions (b) 3D vertex model, open boundary conditions

Figure 3.11: Fraction fperc of edges belonging to percolating clusters as function of inverse system size 1/L. The behaviour of fperc changes dramatically with disconnection probability p. For p < 1, fperc asymptotically approaches some positive constant. In contrast, for p = 1 a power-law decay can be fitted to the data, though the exponent is very small. The behaviour of systems with periodic boundary conditions in x- and y-direction (a) and open boundary conditions (b) is qualitatively the same, but the quantitative difference is substantial. For p = 1, fperc is in the range of 0.3 for periodic boundaries, but over an order of magnitude lower for open boundary conditions. For p < 1, fperc is close to 1 for L → ∞ in both cases. For a definition of fperc see Eq. (3.2).

52 3.5 Percolation Properties of the 3D Vertex Model to 0 (β/ν = 0.0038 ± 0.0006 for a system with periodic boundary conditions in x- and y-direction). β/ν = 0 would result in fperc(1/L) = const. for all L. As for the 2D vertex model in the previous section, the stated error range appears to be too small. Hence, on the basis of the available data, it is not possible to decide whether the exponent is equals 0 or not. Independent of whether fperc is constant with L or decays slowly at p = 1, there is a drastic change in behaviour of fperc from p < 1 to p = 1. This signiﬁes that there is some sort of transition. Yet, future investigations have to clarify what kind of transition is observed.

Inﬂuence of Boundary Conditions Comparison between systems with periodic boundary conditions in x- and y-direction and systems with open boundary conditions exhibited no essential qualitative diﬀerence, see Fig. 3.11 (a) and Fig. 3.11 (b). However, the fraction fperc for periodic boundary conditions is substantially higher than for open boundary systems because non-percolating clusters may become percolating via the periodic boundary. The investigations are henceforth restricted to systems with periodic boundary conditions.

3.5.2 Cluster Size Distribution ns

Fig. 3.12 shows the cluster size distribution ns for p = 1 in systems of diﬀerent sizes L. τ ns decays according to the power-law ns ∼ s ; the critical exponent τ is computed as 1.60 ± 0.02. For 3D site percolation, the literature value is ≈ 2.19 [45], which is signiﬁcantly higher.

105 L = 10 104 L = 20 L = 40 103 L = 80 2 〉

10 s

n 1 10 〈 -1.60±0.02 100 ∝ s/L -1 10 3D vertex 10-2 10-1 100 101 102 103 104 s / L

Figure 3.12: Cluster size distribution ns of the 3D vertex model at p = 1: ns decays τ according to a power-law ns ∼ s with the cluster size s and the Fisher exponent τ = 1.60 ± 0.02. The data was averaged over 100 realisations for systems of linear system size L = 10, 20, 40 and over 10 realisations for L = 80.

53 3 Percolation Properties

3.5.3 Pair-Connectedness Function G(r)

To investigate in more detail what happens at p = 1, the averaged pair-connectedness function G(r) is studied. Fig. 3.13 (b) shows G(r) for the vertex model with different linear size L. As was the case in two dimensions, a power-law decay is observed. The exponent is determined as −1.07 ± 0.02 which is significantly different from −(d − 2 + η) = −0.9318, η = −0.068 [248] being the value for the critical exponent for site percolation in d = 3 dimensions. In Fig. 3.13 (a), our numerical result for site percolation is displayed showing that we reproduce the theoretical value within the error range.

100 100 p = pc 3D site 3D vertex 10-1 = 1 2 β/ν = -0.90 ± 0.04 p 10-1 10-2 L = 20 -0.9318 -0.90 ± 0.04 ∝ r ∝ r for L = 200 L = 40 G(r) G(r) 10-3 = 80 -2 L 10 L = 50 L = 200 = 100 = 500 L -0.9318 10-4 L L = 150 ∝ r L = 800 -1.07 ± 0.02 ∝ r for L = 800 L = 200 L = 1000 10-3 10-5 100 101 102 10-1 100 101 102 103 r r (a) 3D site percolation (b) 3D vertex model

Figure 3.13: The averaged pair-connectedness function G(r), as defined in Eq. (3.7), for systems of different linear extension L with periodic boundary conditions in x- and y-direction. (a) For site percolation at criticality, the theoretical power-law exponent d − 2 − η = −0.9318 is reproduced. The value for the percolation threshold pc = 0.56988 is taken from Ref. [251]. (b) By contrast, the exponent of the vertex model at p = 1 is −1.07 ± 0.01 which significantly differs from the site percolation value.

3.5.4 Number of Clusters

As for the 2D model, we analyse the diﬀerent types of clusters that may occur and their numbers as a function of the system size L. Let us focus on vertex model systems with periodic boundary conditions in x- and y-direction. The number of boundary points in the surface layers at the top and at the bottom of the system is proportional to L2. As each shifted diamond unit cell contains four open ends on each side of the surface (see Fig. 3.3), the number of boundary points in one of the layers is 4L2 and hence there are 8L2 boundary points in total. At p = 1, all nodes are separated, thus the clusters are simple chains of nodes. As in two dimensions, they are either loops, percolating clusters or uturn clusters when assuming periodic boundary conditions in x- and y- direction. The ﬁrst of the three conclusions in 2D has to be adapted, the second and the third are the same:

(i) The sum of the number of uturn-clusters and percolating clusters is 4L2.

54 3.5 Percolation Properties of the 3D Vertex Model

106 3D vertex all 5 10 = 1 loops p uturn 104 perc P 103 ± 2 L2.292 0.032 10 ± L3.029 0.002 1 ± 10 L2.016 0.003 ± L1.034 0.015 100

number of clusters P (p=1)=1 10-1 101 102 103 L

Figure 3.14: Number of clusters in systems at disconnection probability p = 1 as function of system size L. The number of all cluster types, i.e. inner loops, uturn clusters and percolating clusters, increases with a constant power of L, even the number of percolating clusters increases approximately linearly with L. Also shown is the percolation probability P (p = 1) being 1 for all L.

(ii) There are always as many uturn clusters starting and ending at the top as at the bottom of the system. (iii) Since the uturn clusters always take two boundary points, the remaining number of boundary points at the top or the bottom, respectively, is even for any number of uturn clusters. Thus, there is always an even number of percolating clusters in the system — which includes zero in a non-percolating system. The total number of clusters N(L) is

N(L) = Nloops(L) + Nperc(L) + Nuturn(L) (3.40) with Nloops(L) the number of loops, Nperc(L) the number of percolating clusters and Nuturn(L) the number of uturn clusters. Fig. 3.14 shows the occurrence of the different cluster types in systems at p = 1 as a function of the system size L. The most important point to note is that, contrary to the common percolation models and to the 2D model, the number of percolating clusters Nperc(L) increases almost linearly with L. Consequently, the probability to form a percolating cluster is approximately 1/L. The number of 3.029 loops Nloops(L) scales with L indicating that there is a finite density of loops of a given mean size which is no function of the system size but rather predefined by the separation process and the lattice structure. The number of uturn clusters Nuturn(L) is approximately proportional to L2 (exact value: 2.016 ± 0.003) as is the number of boundary points in the surface layers. To sum up, 3.029 1.034 2.016 N(L) = AloopsL + ApercL + AuturnL (3.41) | {z } =4L2

55 3 Percolation Properties with proportionality constants Aloops, Aperc and Auturn. Due to the different exponents for different cluster types, the total number N(L) of clusters appears to scale with an effective exponent of 2.29. Fig. 3.14 also comprises P (p = 1), the probability that a system at p = 1 percolates. P (p = 1) equals 1 for all system sizes. Thus, in every system, at least one percolating cluster exists. This suggests that, if there is a percolation transition at p = 1, there is an infinite cluster at the critical point.

3.5.5 Fractal Dimension Df The fractal dimension is computed by counting the mass M(R) of each percolating cluster in a sphere of radius R with the centre of the sphere belonging to this percolating cluster. Fig. 3.15 (a) shows M(R) for different percolating clusters in a system of size L = 80 at probability p = 1 and the average over all percolating clusters of 100 realisations. From the slope in the log-log plot of the average data as a function of the system size L (see Fig. 3.15 (b)), the fractal dimension is extrapolated to 1.965 ± 0.003. At p = 1, all clusters inherently are line elements. A fractal dimension different from 1 means that the percolating clusters fill space in a different way than lines do. Consequently, the clusters are fractals which only occurs at the critical point.

105 2.15 105 3D vertex 2.10 3D vertex 4 L=80 L 4 10 =40 10 L = 80 ) 3 L=20 2.05 R 10 L

( =10 2 ) ∝ R1.95 2.00 M 10 f R 3 1 ( 10 10 D 0 M 1.95 10 - 0 1 2 1.90 10 1 10 R 10 10 102 1.85 (-1.5 ± 0.1)1/L + (1.965 ± 0.003) average 101 1.80 100 101 102 0 0.02 0.04 0.06 0.08 0.1 R 1/L (a) (b)

Figure 3.15: (a) Mass M(R) of diﬀerent percolating clusters and the averaged mass over all percolating clusters of 100 realisations of a system with L = 80 translational unit cells at probability p = 1. (b) The fractal dimension Df of a percolating cluster is determined as a function of system size L and extrapolated to L → ∞.

3.5.6 Finite Size Scaling

Fig. 3.16 shows the finite size scaling of the fraction fperc(p, L) of edges belonging to a percolating cluster. Using Eq. (3.23), fperc(p, L) can be scaled such that the data for different system sizes L and probability p 6= 1 collapses onto one master curve. For 3D site percolation, a data collapse is achieved with the critical exponents β = 0.41 and ν = 0.88 [233], as shown in Fig. 3.16 (a). For the vertex model, inserting the site percolation exponents does not yield a data collapse, but the exponents must be identified.

56 3.5 Percolation Properties of the 3D Vertex Model

101 pc = 0.57 3D site −β ) p -

c 0 p 10 ( p=0.5690 x-β p=0.5695 perc

f p=0.5696 p=0.5697 ν=0.88, β=0.41 p=0.5698 10-1 10-3 10-2 10-1 100 101 1/ν x = (pc-p) L (a) site percolation

101 L = 5 p = 0.99999 L = 8 p = 0.9999 L = 10 p = 0.9995 −β

) L = 20 p = 0.999 p

- L = 30

c 0 L = 40 p 10 L = 60 ( L = 120 3D vertex perc f -β ν=0.54 ± 0.01, β=0.0021 ± 0.0004 x 10-1 10-5 10-4 10-3 10-2 10-1 100 101 102 103 104 1/ν x = (pc-p) L (b) vertex model

Figure 3.16: Finite size scaling for the fraction fperc (see Eq. (3.2)) of edges belonging to 1/ν a percolating cluster as described in Eq. (3.23): For x = (p − pc)L 1, the scaling function is expected to become constant. For x 1 it should decrease as x−β. (a) For 3D site percolation, using the critical exponents β = 0.41 and ν = 0.88 [233] yields a collapse of all data onto one master curve. (b) The same critical exponents do not collapse the 3D vertex model data though. A collapse onto a master curve is achieved with β = 0.0021 ± 0.0004 and ν = 0.54 ± 0.01.

57 3 Percolation Properties

In Fig. 3.11, the ratio β/ν was determined as 0.0038. By varying β and ν while keeping their ratio ﬁxed, we found the data to collapse onto a master curve for β = 0.0021±0.0004 and ν = 0.54 ± 0.01, see Fig. 3.16 (b).

3.6 A Percolation Model in a New Universality Class?

We investigated if there is a percolation transition in a 2D square lattice and a 3D diamond lattice when replacing each four-coordinated node by two two-coordinated vertices with probability p. This model was discussed before as the vertex model.

3.6.1 2D Vertex Model

In 2D, the algebraic decay of the fraction fperc(p = 1, 1/L) and the pair-connectedness function G(r) at p = 1 indicates the absence of a typical length scale which usually is the correlation length ξ. In case of a second order phase transition, ξ diverges at the critical point and the system becomes self-similar. Thus, the scaling behaviour of fperc(p = 1, 1/L) and G(r) support the hypothesis of a percolation transition at p = 1. The fractal dimension Df = 1.702 ± 0.007 and the critical exponents β = 0.322 ± 0.018 and ν = 1.29 ± 0.04 differ significantly from the known values for site percolation and the mean field model, suggesting that the 2D vertex model belongs to a another universality class. By applying conformal field theory, Klaus Mecke [179] derived values for the critical exponents and the fractal dimension of the described vertex model on a 2D square lattice. The numerical results for ν and β agree with the analytical values ν = 4/3 and β = 1/3 within the error range; the value for the fractal dimension Df = 1.702 ± 0.007 is close to the theoretical value Df = 79/48 ≈ 1.646. From the comparison of the numerical with the theoretical site percolation values we conclude that the stated error range is too small. Especially the differences between the expected and the achieved numerical values for the fractal dimension Df and the power-law exponent d − 2 + η of the pair-connectedness function G(r) are an order of magnitude higher than the given error. This indicates that, apart from the stated statistical error, systematic errors are present, probably due to the finite size of the systems. This can be solved by either simulating much bigger systems (which is limited by computational resources) or by taking corrections to scaling into account [157]. Without knowing the exact errors, the verification of the hyperscaling relations is a delicate problem. As can be seen for the relation 0 = Df − d + β/ν, which is definitely fulfilled for site percolation in two dimensions, the numerical data suggests the opposite as Df − d + β/ν = −0.04 ± 0.01 (see table Tab. 3.4). Hence, on the basis of the presented data, it is hard to decide whether or not the hyperscaling relations Eq. (3.27) and Eq. (3.39) are fulfilled.

Asymmetric Node Separation

In the diploma thesis of Matthias Hoﬀmann [102], a generalisation of the vertex model is developed by introducing an asymmetry parameter x ∈ [0, 1] determining the direction of separation at a node. The vertex model studied here is the symmetric case x = 1/2 where a separation in horizontal and in vertical direction are equally probable. Ref. [102] shows that for x 6= 1/2 the system shows a phase transition at some critical point

58 3.6 A Percolation Model in a New Universality Class?

4 0 < pc(x) < 1 with the bond percolation critical exponents. For the derivation of the 0 threshold pc(x), the dual lattice L of the square lattice L of the vertex model is divided into two sub-lattices Li with i = 1, 2 as shown in Fig. 3.17. The sites of L are grouped into two diﬀerent types: sites s1 ∈ L on horizontal lines of L1, henceforth called type-1-sites, and sites s2 ∈ L on vertical lines of L1, hereafter referred to as type-2-sites (and vice versa when referring to sub-lattice L2). Two neighbouring sites of Li are connected by a bond if the vertex between them is connecting, i.e. it is either a non-separated, four-coordinated node or the separation is such that the edges are parallel to the bond rather than crossing it. Parameter x is the probability that two sites are connected by a bond. The probability that a node remains intact is 1 − p. With probability (x p) a node at a type-1-site s1 ∈ L is separated horizontally, whereas the probability for a vertical separation equals (1 − x) p. On the other hand, nodes at type-2-sites s2 ∈ L are vertically separated with (x p) and horizontally with (1 − x) p. At p = 1, x = 0 and x = 1 result in conﬁgurations exclusively consisting of loops with their centres being the sites of lattice L1 and lattice L2 respectively. Adding up the probability qi that a bond is set in Li yields q1 = (1 − p) + x p and q2 = (1 − p) + (1 − x) p. Let S1 and S2 denote the sets of set bonds on lattice L1 and L2. In Ref. [102], it is shown that the vertex model percolates if and only if both S1 and S2 percolate. In particular, this includes the percolation threshold pc(x). For both sub-lattices, the probability qi to set a bond has to be greater than the bond percolation threshold pc,b = 1/2 [233] of a 2D square lattice yielding: 1 1 q1 = (1 − p) + px ≥ ⇐⇒ p ≤ (3.42) 2 2(1 − x) 1 1 q2 = (1 − p) + p(1 − x) ≥ ⇐⇒ p ≤ . (3.43) 2 2x Hence, to ensure that both lattices are percolating, the probability p has to comply with both requirements which results in 1 1 p ≤ min , =: pc(x) . (3.44) 2(1 − x) 2x

The functional dependence of the percolation threshold pc(x) is illustrated in Fig. 3.18. There are two critical lines, one for x < 1/2 and one for x > 1/2, separating the non- percolating from the percolating phase that meet in one point at (x = 1/2, pc(x) = 1). Since this point belongs to two critical branches, it is double critical making it a distinct point with critical properties diﬀerent from the rest of the points along the critical lines. Simulations of the enhanced vertex model [102] conﬁrm this functional dependence of the percolation threshold pc(x). For the vertex model at criticality, i.e. x = 1/2 and p = 1, the critical exponents of a probably new universality class were found (see section 3.4). In contrast, for x 6= 1/2, the phase transition is found to have bond percolation critical exponents [102].

3.6.2 3D Vertex Model In the 3D vertex model, there are several hints for a phase transition at p = 1, too. Firstly, the percolating fraction of edges fperc(p = 1,L) very slowly decays to 0 for L → ∞ 4This model arose from a cooperation with Klaus Mecke, Gerd E. Schröder-Turk and Matthias Hoﬀmann. Hence, there may be similarities between passages in Ref. [102] and this section.

59 3 Percolation Properties

(a) Square lattice L (small grey circles, diagonal solid lines) and its dual lattice L0 which is divided into two sub-lattices L1 (white circles, dotted line) and L2 (black circles, dashed line)

(b) x = 0.7, p = 0.95

Figure 3.17: (a) Square lattice L and its dual lattice L0 that is divided into the two sub- lattices L1 and L2. (b) When separating a node in the vertex model, a bond is removed in one of the sub-lattices such that no bond is crossed by a vertex model cluster. (c) In a fully separated system at disconnection probability p = 1, the bonds of lattice L1 and lattice L2 do not cross anymore.

60 3.6 A Percolation Model in a New Universality Class?

1 non-percolating 0.8

0.6 c p 0.4 percolating

0.2

0 0 0.2 0.4 0.6 0.8 1 x

Figure 3.18: The percolation threshold pc(x) as a function of the asymmetry parameter x. If the disconnection probability p < pc(x), the system percolates, else there are only non-percolating clusters. The two branches of the critical line meet in exactly one point: the symmetric case x = 1/2.

according to a power-law with exponent β/ν = 0.0038 ± 0.0007, whereas fperc(p < 1,L) for L → ∞ saturates to some positive value. Secondly, the percolating clusters at p = 1 are found to be fractal with fractal dimension Df = 1.965 ± 0.003. Thirdly, the averaged pair-connectedness function G(r) exhibits a power-law decay. This leads to the conclusion that the transition at p = 1 is of second order since in a first order transition, the correlation length ξ stays finite [24] and does not allow for a self-similar system with fractal clusters and a power-law behaviour. The decay exponent of G(r) is −1.07 ± 0.02, that is, on the one hand, clearly different from the site percolation value of −0.9318 and, on the other hand, relatively close to −1 which results in η close to the mean field value η = 0. An important point to emphasise is the existence of more than one percolating cluster in the vertex model which is contrary to conventional site and bond percolation. More precisely, at criticality, the number of percolating clusters Nperc(pc,L) increases linearly with the system size L. Hence, in an infinitely large system at p = pc, there are infinitely many percolating clusters. A consequence of this observation is a change of the hyperscaling relations Eq. (3.27) and Eq. (3.39) as their derivation was based on the assumption of a unique percolating cluster. For the derivation of the new hyperscaling relations, the cluster number Nperc(pc,L) is assumed to scale with the system size L with θ some arbitrary exponent θ. Inserting Nperc(pc,L) ∼ L into Eq. (3.25) yields

D d − β θ L f ∝ L L ν /L (3.45)

which results in the relation

β d − Df = + θ , (3.46) ν

61 3 Percolation Properties which is the counterpart of Eq. (3.27). Assuming all percolating clusters to be equivalent, the averaged pair-connectedness function G(r, L) can be written as

Nperc(pc,L) X X v G(r, L) = v mv Gv(r, L) + fperc(pc,L)/Nperc(pc,L)G∞(r, L) v l=1 X v = v mv Gv(r, L) + fperc(pc,L)G∞(r, L) . (3.47) v

The ﬁrst term is not inﬂuenced by the number of percolating clusters, hence the scaling is the same as in Eq. (3.35)

X −2(d−D ) v mv Gv(r, L) ∼ r f . v

The number of percolating clusters Nperc(pc,L) cancels in the second term of Eq. (3.47) due to the similarity of the percolating clusters. Yet, it enters the equation indirectly via the hyperscaling relation in Eq. (3.46):

β v − ν −(d−Df) fperc(pc,L) G∞(r, L) ∼ L r ∼ r−(d−Df−θ) r−(d−Df) = r−(2 (d−Df)−θ) . (3.48)

v Here, we used again that fperc scales as fperc (cf. Eq. (3.22)) and that r is large, namely r ≈ L, and can thus be substituted. Assuming θ ≥ 0, the fraction of the ﬁnite clusters decays faster than that of the percolating clusters and the ﬁnal scaling law for G(r, L) reads

−(2 (d−D )−θ) G(r, L) ∼ r f for p → pc . (3.49)

Comparing the exponent of Eq. (3.49) in the limit L → ∞ with that of Eq. (3.11) and using Eq. (3.46) results in the hyperscaling relation β d − 2 + η = 2 (d − Df) − θ = 2 + θ . (3.50) ν For the vertex model, θ ≈ 1 (cf. Fig. 3.14). All numerical exponents and the hyperscaling relations for three dimensions are summarised in Tab. 3.5. Inserting the values for Df, η, τ, β and ν in Eq. (3.32), Eq. (3.46) and Eq. (3.50) results in

d − Df = 1.035 ± 0.003 ≈ β/ν + 1 = 1.0038 ± 0.0007 (3.51)

d − 2 + η = 1.07 ± 0.02 ≈ 2 (d − Df) − 1 = 1.035 ± 0.006 (3.52) d − 2 + η = 1.07 ± 0.02 ≈ 2 β/ν + 1 = 1.0076 ± 0.002 (3.53)

τ = 1.60 ± 0.02 6= 1 + d/Df = 2.527 ± 0.003 . (3.54)

Eq. (3.51), Eq. (3.52) and Eq. (3.53) show that the values for the hyperscaling relations Eq. (3.46) and Eq. (3.50) ﬁt quite well, although based on these values it is not possible to decide whether the relations are fulﬁlled or not. As mentioned before for the 2D case,

62 3.6 A Percolation Model in a New Universality Class? the quoted statistical error seems to be too small due to finite size effects. In order to verify the hyperscaling relations, further investigations are necessary. By contrast, Eq. (3.32), which did not need to be adjusted by the exponent θ, is surely not fulfilled, see Eq. (3.54). There is no obvious reason for this deviation. For an explanation, more critical exponents should be determined such as γ, the exponent of the mean cluster size, and further scaling and hyperscaling relations containing τ should be studied.

As the uncertainty of the numerical values is larger than the stated statistical error, especially small values such as the critical exponent β = 0.0021 ± 0.0004 must be looked at carefully. It cannot be excluded that β = 0. Let us assume β = 0. From the hyperscaling relations Eq. (3.46) and Eq. (3.50) it can be concluded that Df = 2 and η = 0, respectively. This set of values is close to the numerical values and fulﬁls both new hyperscaling relations. At disconnection probability p = 1, all clusters are line-like objects that wriggle through the system and twine around each other. This could be interpreted as a solution of ﬂexible polymers with interacting polymer chains. Since the vertex model is critical at p = 1, one may assume that the polymer solution is at its critical point, i.e. at the so-called Θ-temperature [267], too. At the Θ-temperature, the interacting random walks of the line-like polymers with an end-to-end distance RN , which usually scale as [267]

ν hRN i ∼ N F (3.55) with νF the Flory-Exponent, appear as classic random walks with

1/2 hRN i ∼ N . (3.56)

Thus, the exponent ν would take its mean ﬁeld value 1/2. Note that the numerical value of the 3D vertex model 0.54 ± 0.01 is not that far oﬀ.

3.6.3 Conclusion It could be shown that the 3D vertex model shows a percolation transition at disconnection probability p = 1 with critical exponents clearly different from conventional site and bond percolation. Although the exponents ν and η are close to their mean field values, the transition does not belong to the mean field universality class as β distinctly differs from its mean field value 1. To understand the nature of the transition and to prove that the model belongs to a new universality class, more investigation is necessary. We propose that, as for the 2D case, asymmetry is introduced in the separation process. Equivalent to the two critical lines in Fig. 3.18, three critical surfaces are expected that meet not in one point but maybe in a plateau. It is also anticipated that the transition at the critical surfaces, similar to the transition at the critical lines in 2D, belongs to the universality class of bond percolation.

63 3 Percolation Properties

Table 3.5: 3D results for critical exponents and hyperscaling relations: For site percolation and the vertex model, the numerically determined values are compared to (numerically determined) literature values [233, 45]. Periodic boundary conditions in x- and y-direction applied. 3D site vertex theoretical numerical β 0.41 0.411 0.0021 ± 0.0004 ν 0.88 0.881 0.54 ± 0.01 β/ν ≈ 0.47 — 2 0.0038 ± 0.0007 τ 2.18906 — 2 1.60 ± 0.02 2 Df 2.52 — 1.965 ± 0.003 d − 2 + η ≈ 0.93 0.90 ± 0.04 1.07 ± 0.02

0 = Df − d + β/ν ≈ 0 " — −1.03 ± 0.004 0 = d − 2 + η − 2β/ν ≈ 0 " −0.03 ± 0.04 " 1.06 ± 0.03

0 = Df − d + β/ν + 1 — −0.03 ± 0.004 ? 0 = d − 3 + η − 2β/ν 0.06 ± 0.03 ?

0 = τ − 1 − d/Df ≈ 0 " — −0.93 ± 0.03 1 Data collapsed using these exponents 2 This value was not determined independently.

64 4 Eﬀective Linear-Elastic Properties of Network Solids

In this and the following chapter, the network structures, which have up to now been represented by embedded network graphs with infinitely thin edges, are interpreted as monolithic solids in network shape with a finite volume fraction; they are referred to as network solids. These network solids are microstructured materials. Their effective linear-elastic properties, that is the homogenised elastic moduli of the material at length scales much larger than the typical microstructural feature, are calculated using a Finite Element (FE) method based on a voxelised representation of the network solids. To voxelise a network solid means to discretise it into small cubic elements, so-called voxels. This chapter studies the influence of the volume fraction on the effective elastic properties of the crystalline network solids. The same method will be applied in chapter 5 in order to study the influence of topology on the effective linear-elastic properties. In this chapter, basic continuum mechanics are covered, the elastic moduli are defined and the concept of effective medium theory is discussed. The voxel-basedFE method is described. A first application of theFE method to Poisson-Voronoi networks and collagen scaffolds reveals that the intensity of the Poisson-Voronoi process and the collagen concentration in the samples, both of which alter the typical pore or mesh size of the network, affect the effective moduli only by the resulting change of the solid volume fraction φ. These findings suggest that collagen networks interpreted as network solids, neglecting all thermal fluctuations and floppiness, can be modeled in quantitative agreement by a Poisson-Voronoi network. Further, a cross-over from a power-law at low densities to a porous solid at high densities is observed and a functional form for the bulk modulus K(φ) is proposed.

4.1 Theory of Linear Elasticity

Continuum mechanics studies the response of materials to external forces, assuming the material to be continuously distributed in space rather than consisting of individual atoms. This, of course, is an approximation as all information of the molecular or atomic structure of the material is ignored. But for length scales larger than the atom size or the distances between atoms, continuum theory yields good results. In the following, an introduction to continuum mechanics is given. Special attention is paid to linear elasticity since it is the basis for the finite element method described afterwards. Definitions and notations are based on Ogden’s book [186]. 3 Let body B occupy domain B ⊂ R , a compact subset of three-dimensional, Euclidean space. Suppose that some given material is continuously distributed in B. Now, boundary conditions representing an external strain field are set. One is interested in the resulting deformations of the body. A deformation causes a change of the domain occupied by B. 3 The domain covered by B at time t is denoted Bt, Bt ⊂ R . Let us choose a configuration

65 4 Eﬀective Linear-Elastic Properties of Network Solids

B0 Bt

x X χ X 3 X 2 X x O 1 3 x 2 x o 1

Figure 4.1: Body B in the reference conﬁguration B0 and an arbitrary other conﬁguration Bt after deforming B. The position X is changed to x by the transformation χ.

of B serving as reference for all deformations and call it B0. The positions of all points of B0 are referred to by capital X, X ∈ B0. A new conﬁguration can be linked to the reference conﬁguration by a one-to-one mapping

χ : B0 → Bt . (4.1) χ and its inverse χ−1 — both assumed to be twice continuously differentiable — can be used to transform points between the current and the reference configuration x = χ(X), X = χ−1(x) (4.2) where x ∈ Bt is the position in the current configuration corresponding to X ∈ B0 in the reference configuration. Assuming deformations to be continuous, B0 and Bt are connected by a deformation path x = χ(X, t). For the study of linear elasticity and infinitesimal deformations, one is not interested in the time evolution of states of the material, but only in a comparison of the initial state with the final state. Thus it is not necessary to take the time dependence into account. The differential of x can be written as dx = A dX (4.3) with the deformation gradient A ∂x A := = Grad χ(X) (4.4) ∂X where “Grad” in contrast to “grad” is the differentiation with respect to X instead of x. A is a second-order tensor with Cartesian components

∂xi Aiα = . ∂Xα

66 4.1 Theory of Linear Elasticity dx and dX are called line elements in the current and the reference conﬁguration, respectively. Three linearly independent line elements dX(i) and dx(i), i = 1, 2, 3 form a volume element dV := det dX(1)dX(2)dX(3) dv := det dx(1)dx(2)dx(3) . (4.5)

Using Eq. (4.3) yields dv = det AdX(1)AdX(2)AdX(3) = (det A) dV := JdV. (4.6)

J is the Jacobi determinant of the transformation Eq. (4.3). It describes the local change in volume during the deformation. By convention, volume elements are always positive and consequently J = det A > 0. Let us consider the effect of a deformation on a line element |dX| starting at point X. The Euclidean norm | · | measures the distance between two points. The difference of squared lengths of a line element in the current and the reference configuration is

2 2 T |dx| − |dX| = dX A A − 1 dX . (4.7)

If ATA = 1, there is no change in length of any line element dX and the material is said to be unstrained at X. This is true, for instance, for a rigid body rotation described by orthogonal tensors. For ATA 6= 1, body B is deformed or strained. ATA is the Cauchy-Green deformation tensor which is used to deﬁne the Green strain tensor E

1 T E := A A − 1 (4.8) 2 or, in Cartesian components, 1 1 ∂xi ∂xi Eαβ = (AiαAiβ − δαβ) = − δαβ (4.9) 2 2 ∂Xα ∂Xβ which is a measure for the change in length of an arbitrary line element. δαβ is the Kronecker-Delta. Note that the summation convention is used and will be followed throughout chapter 4 and chapter 5 if not stated diﬀerently. When deforming, points of the body have to be moved to some other position. Let

u(X) := χ(X) − X (4.10) be the displacement of point X. Then

D := Grad u(X) ≡ A − 1 (4.11) is the displacement gradient and Eq. (4.9) can be rewritten 1 E = D + DT + DTD 2 1 ∂uα ∂uβ ∂ui ∂ui Eαβ = + + . (4.12) 2 ∂Xβ ∂Xα ∂Xα ∂Xβ

67 4 Eﬀective Linear-Elastic Properties of Network Solids

Homogeneous deformations If the deformation gradient A(X) is a function of the coordinate X, the deformation is called inhomogeneous.A homogeneous deformation can, in general, be described by

x = AX + c (4.13) with the second-order deformation gradient tensor A and the rigid translation c, both independent of X. An example for a homogeneous deformation is a pure dilation. If a cube of size L is stretched or compressed to a cube with edge length l, the deformation is a pure dilation and the deformation gradient is given by

 l  L 0 0 l A =  0 L 0  . (4.14) l 0 0 L

Mass conservation Let m(B) be the mass of body B, Z m(B) := ρ(x, t) dv , (4.15) B with ρ(x, t) the mass density at position x and time t and dv the volume element in B. Mass is an intrinsic measure1 of body B meaning that it is independent of any motion and has to be conserved during the deformation process at every step in time. Thus, Eq. (4.15) can be written as Z Z ρ(x, t) dv = ρ0(X) dV (4.16) B B0

and, using Eq. (4.6), one can derive the familiar form of the mass conservation equation

ρ˙ + ρ div χ˙(X) = 0 (4.17) which in mechanical equilibrium, where there is no time dependence, simpliﬁes to −1 ρ = J ρ0.

Balance equations For the deformation, an external force has to be applied to body B giving rise to a body or volume force Z ρ(x, t) b(x, t) dv (4.18) B and a contact force Z t(x, n) da (4.19) ∂B

1 A real-valued function µ : A → R deﬁned on a σ-algebra A is called a measure if (i) it is positive µ(A ∈ A) ≥ 0, S∞ P∞ (ii) it fulﬁls σ-additivity µ n=1 An = n=1 µ (An) for every sequence (An) ∈ A of pairwise disjoint sets Ai ∩ Aj = ∅ if i 6= j.[239, 132]

68 4.1 Theory of Linear Elasticity with b(x, t) the body force density, t(x, n) load or traction ﬁeld per unit area of surface, n the unit normal vector at x and ∂B the border of domain B. By virtue of Newton’s second law, an external force evokes a change of linear momentum in time

Z Z d Z ρ(x, t)b(x, t) dv + t(x, n) da = ρ(x, t)χ˙ (x, t) dv . (4.20) B ∂B dt B

This equation is called the balance of linear momentum.2 Using Cauchy’s Theorem stating that the vector t(x, n) can be written as t(x, n) = T(x)n, the Cauchy stress tensor T is introduced. It is a second-order tensor and can be shown to be symmetric T = TT. The configuration in which T vanishes is called natural configuration. Let us assume that B0 is a natural configuration. For a body at rest, the right-hand side of Eq. (4.20) obviously equals zero. Using the generalised Gauss’s theorem, the following differential equation can be derived to solve the boundary value problem for deformation χ of a body at rest

div T + ρ b = 0

χ(x) = χB(x) for x ∈ BD ⊂ ∂B T(x) n = TB(x) for x ∈ BN ⊂ ∂B . (4.21)

∂B is the border of domain B. BD denotes the part of ∂B with Dirichlet boundary conditions, BN the part of ∂B with Neumann boundary conditions. χB(x) and TB(x) are the boundary values of the transformation χ and the Cauchy stress tensor T, respectively. These equations are not sufficient to determine all unknowns as the theory so far does not take the material properties into account. This is accomplished by adding a constitutive law which describes the nature of the material. More precisely, it specifies the stress response of the material to an arbitrary motion. In general, a constitutive law gives T as a function of t, x, x˙ and spatial derivatives of x. This function has to obey certain assumptions, for instance, that the material properties should not be changed by a rigid body motion and should reflect material symmetries such as isotropy. The constitutive law completes the set of equations. If the Cauchy stress T in the current configuration only depends on the state of deformation compared to an arbitrary reference configuration, but not on the deformation path taken to get from the reference to the current configuration, the material is called Cauchy elastic. If, in addition, there exists a scalar energy function W (A), the material is referred to as hyperelastic or Green elastic. This is what is usually meant by elastic. W (A) is a measure for the elastic energy stored in the body due to the deformation. It is called elastic potential energy function or strain-energy function. Integration over the whole body Z dVW (A) (4.22) B0 gives the total elastic energy stored in B.

2Similarly, the torque balance equation resulting from the applied force and the rate of change of the rotational momentum balance can be deduced.

69 4 Eﬀective Linear-Elastic Properties of Network Solids

Linearisation of the Elastic Theory

Linear elastic theory applies to small deformations only as it neglects quadratic and higher terms in the displacement. In linear theory, the strain tensor E (Eq. (4.12)) simpliﬁes to 1 ∂uα ∂uβ Eαβ = + . (4.23) 2 ∂Xβ ∂Xα

When the reference conﬁguration is a natural conﬁguration (see page 69), one does not need to distinguish between x and X anymore. The linear constitutive law is the generalised Hooke’s law

T = CE,Tij = Cijkl Ekl (4.24) where Cijkl is the elastic modulus tensor. The elastic energy is given by 1 1 W = tr {(CE)E} = CijklEijEkl . (4.25) 2 2 Using the symmetries of the stress tensor T and the strain tensor E and applying that the elastic energy W is a scalar, it can be shown that Cijkl satisﬁes the following symmetries

Cijkl = Cjikl = Cijlk = Cklij (4.26)

reducing its 81 entries to 21 independent ones. Material symmetries can be used for further reduction. In case of isotropic materials, Cijkl can be written as

2 Cijkl = K − G δijδkl + G(δikδjl + δilδjk) (4.27) 3 with the bulk modulus K and the shear modulus G. These two elastic moduli completely characterise an isotropic material in the regime of linear-elastic deformations. For uniaxial compression, it is convenient to use Young’s modulus Ei, which is sometimes just referred to as the elastic modulus. It is deﬁned as

Tii := EiEii (4.28) where Ei determines the resulting stress Tii when a compression in direction i is applied. Here, the summation convention does not apply. In general, a material has an inﬁnite number of Young’s moduli, one for each possible direction in space. However, for isotropic materials, all moduli are equal, i.e. Ei = E, and Young’s modulus is related to the bulk modulus K and the shear modulus G of the material by 9KG E = . (4.29) 3K + G

4.2 Eﬀective Medium Theory

So far, we assumed that a homogeneous material was uniformly distributed in the domain B. A closer look at the material may reveal that, on a much smaller length scale

70 4.2 Eﬀective Medium Theory B B P 5 P 6 P 4 l P 3 P C 7 eff P 8 P 2 P 1 L

Figure 4.2: (Left) Microstructured material: domain B contains eight subdomains Pi which consist of different materials with different physical properties, represented by different patterns. L and l are the macroscopic and the microscopic length scale, respectively. (Right) If L l, homogenisation can be applied. This approach averages over the heterogeneous microstructure and allows the treatment of a material as being macroscopically homogeneous with effective elastic properties, such as the effective elastic tensor Ceff. than the sample size (but still much larger than atomic or molecular distances), it consists of several different phases. More precisely, the domain B contains N subdomains Pi, SN i ∈ [1,...,N], of different materials which obey B = i=1 Pi. The subdomains Pi and their composition is called the microstructure of B. This heterogeneous microstructure may be ordered or random. A schematic sketch is shown in Fig. 4.2, examples of real microstructured materials are displayed in Fig. 1.1. Generally, it is far too complex to analyse the microstructure in detail and apply continuum mechanics to it in order to calculate the elastic properties of a macroscopic sample. We are rather interested in a characterisation by so-called effective elastic properties reflecting the macroscopic, spatially averaged behaviour of the structure. The calculation of the effective elastic properties, also known as homogenisation [43], of both ordered and disordered composite materials has been of interest for decades. Analytical calculations for regular networks and cellular solids were performed [79, 46, 256, 89], yet a universal solution for disordered or random materials does not exist. Early analytical mean-field theories, so-called effective medium theories, started from a single spherical [34] or ellipsoidal [63] inclusion in an otherwise homogeneous matrix. By different approaches as differential schemes [156, 184, 16, 81] or self-consistent theories [99, 35, 22], generalisations concerning the number/concentration, shape, composition and number of phases of the inclusions were accomplished [62, 122, 249, 247]. Due to the complexity of the task especially for disordered materials, numerical simulations became more and more

71 4 Eﬀective Linear-Elastic Properties of Network Solids

important over the years, and a variety of model systems such as networks consisting of springs [114, 127, 189, 265], strings [56] or elastic Euler beams or rods [1, 97, 95, 108, 49] were proposed. Different random models were suggested for the finite element calculation of the effective elastic properties of porous media [204, 203, 10]. Alternatively, structures obtained by 3D imaging techniques were used to compute the properties of real samples and to compare them to experimental values [11, 133, 152]. In the following, a two-phase composite material is considered consisting of one solid phase and one phase that may be another solid, a fluid or void [248]. Each phase is characterised by a homogeneous, isotropic elastic tensor Ci, i = 1, 2. A prerequisite for homogenisation is a separation in length scales: the macroscopic scale L, which is of the order of the system size and the microscopic scale l characterising the inhomogeneity of the microstructure. If L l, the effective elastic properties can be determined by spatial averages of the local strain and stress fields resulting in an averaged Hooke’s law

hT i = CeffhEi , (4.30) with T the stress tensor, E the strain tensor and Ceff the effective elastic moduli tensor. h.i denotes the volume average over the whole sample of volume V 1 Z hf(x)i := d3x f(x) . (4.31) V

It can be proven [248, 21, 214] that the averages of the local stress T and the local strain E equal the macroscopic stress Teﬀ and strain Eeﬀ, respectively,

Eeﬀ = hE(r)i and T eﬀ = hT(r)i . (4.32)

The total energy w per unit volume can then be computed from the macroscopic quantities as 1 w = EeffCeff Eeff . (4.33) 2 ij ijkl kl For a material consisting of two isotropic phases, Ceff is a function of the bulk moduli K1 and K2 and the shear moduli G1 and G2 of phase 1 and phase 2, respectively. Note eff eff that C = C (K1,G1,K2,G2) is homogeneous of degree one

eﬀ eﬀ C (αK1, αG1, αK2, αG2) = αC (K1,G1,K2,G2) . (4.34)

Choosing α = 1/K1 results in Ceff (K ,G ,K ,G ) ijkl 1 1 2 2 eff = Cijkl(1,G1/K1,K2/K1,G2/K1) , (4.35) K1 thus the number of independent variables is reduced from four to three without loss of generality [248]. For heterogeneous materials consisting of a solid phase with elastic moduli K1 and G1 and a void phase with K2 = 0 and G2 = 0, this implies that the effective elastic properties are in principle determined by the ratio G1/K1 and scale linearly with the bulk modulus K1. The effective modulus tensor Ceff has, in general, 21 independent entries. This number is reduced if the material’s microstructure exhibits any symmetries. In case of a material

72 4.3 Finite Element Method for Voxelised Structures with isotropic microstructure, the elastic behaviour is determined by just two moduli, the bulk modulus K and the shear modulus G. Materials with cubic symmetry are characterised by three “principal elasticities” [245] which are the bulk modulus K and the two shear moduli G and G2. The ratio of the two shear moduli G2/G is an appropriate measure for the anisotropy of a sample. For isotropic samples, G2/G = 1; the more anisotropic the considered sample, the more G2/G diﬀers from 1. The response of a material with cubic symmetry to uniaxial compression is described by two Young’s moduli [254] which can be calculated from G and G2 according to Eq. (4.29).

4.3 Finite Element Method for Voxelised Structures

The Finite Element Method (FEM) is a standard method for the numerical solution of physical and engineering problems based on partial differential equations as, for example, the analysis of stability, stress or natural frequencies of, temperature distribution in and flow through arbitrarily shaped bodies [4, 274, 29]. In the following, only the basic ideas of this technique are explained. For a detailed description the reader is referred toFE textbooks, e.g. Zienkiewicz [274] and Braess [29]. The fundamental principle of the Finite Element Method is the division of body B into i SN i a number N of finite subdomains B called elements with the condition B = i=1 B , i.e. body B is completely partitioned without any gaps between the elements. This division, also referred to as mesh, can be characterised by a discretisation parameter h which corresponds to the size of the elements: Large h stands for a coarse division consisting of only a few subdomains, small h for a fine division of the domain in many small parts. The differential equation describing the problem in question is approximately solved on the discretised domain. For h → 0 the solution of the discretised problem is assumed to converge to that of the continuous problem [274, 29].

Depending on the physical model, its degrees of freedom and the spatial geometry of the system, different types of elements can be used. Since network structures consisting of rather long, cylindrical struts are particularly investigated (see chapter 2), beam elements are an appropriate choice. Beam elements [4] are bar-like elements that are rigidly joined at connection points. In contrast to truss elements, they resist not only axial tension or compression but also bending and torsion when based on Euler-Bernoulli beam theory or bending, torsion and transversal shear when based on Timoshenko beam theory. The use of beam elements is quite limited. They are designed for the limit of cellular structures with long thin edges [272, 84, 145, 116] and allow good access to the very low density regime but are less suitable for the high-density regime where the distinction between edges and joints is less clear. Instead of using beam elements, we decided onFE calculations on a cubic (or hexahedral) mesh. A composite structure comprising several different materials is discretised in small cubic elements, henceforth called voxels. Each voxel is filled with exactly one material. All materials are assumed to be homogeneous isotropic linear-elastic materials. This voxel-based Finite Element Method was suggested by Garboczi and Day [72, 73]. The major advantage of this approach is its simplicity and universality in application. It allows the analysis of both medium- and high-density geometries. Besides, the voxel-based FEM can easily be applied to disordered composite materials, both random model materials

73 4 Eﬀective Linear-Elastic Properties of Network Solids

[202, 204, 203, 71, 165] and digital images of real structures gained from 3D imaging techniques [11, 208, 133, 134]. In this study, linear-elastic properties of composite structures with two constituents, one solid and one void phase, are considered. The samples are divided into cubic subdomains by superimposing the grid points of a cubic lattice onto them and identifying a grid point as “solid” if it lies within the solid domain and as “void” otherwise. The grid points are then interpreted as the centre points of the cubic voxels with homogeneous isotropic linear-elastic properties. For the computation of the eﬀective elastic properties of a heterogeneous material consisting of two phases, the equilibrium balance equation

div T + ρ b = 0 (4.36) with boundary values

χ(x) = χ0(x) x ∈ BD ⊂ ∂B (4.37) T(x) n = T0(x) x ∈ BN ⊂ ∂B (4.38)

is used where T is the Cauchy stress tensor, ρ is the mass density, b is the body force density, BD is the part of the border ∂B for which the displacement is set to the boundary value χ0(x), BN is the part of the boundary for which the Cauchy stress tensor is set to the boundary value T0(x). For Green elastic materials, this boundary value problem can be formulated as a variational principle and the solutions are stationary points of the total energy [29]. This comprises the energy to be extreme and, thus, partial derivatives of it to be 0 or smaller than a certain value depending on the wanted (or possible) accuracy. The total elastic energy W stored in the body is Z 1 3 X W = d x Eij(x)Cijkl(x)Ekl(x) (4.39) 2 V ijkl where V is the volume of the sample and

1 1 2 2 Cijkl(x) = F (x)Cijkl + F (x)Cijkl (4.40)

is the elastic moduli tensor with the elastic moduli tensors of the isotropic bulk materials 1,2 2 C = k1,2 − g1,2 δijδkl + g1,2 (δikδjl + δilδjk) (4.41) ijkl 3

and the indicator functions F 1(x) and F 2(x) which equal 1 if the voxel containing point x is part of phase 1 and phase 2, respectively, and 0 otherwise. The effective elastic moduli tensor is computed by minimising the elastic energy W stored in the sample using a conjugate gradient method [194, 223]. The degrees of freedom of the minimisation are the locations of the vertices of the voxels, i.e. the eight corners of each cubic voxel. The minimisation can be done with different boundary conditions. In the following analysis, both fixed and periodic boundary conditions are applied. When using fixed boundary conditions, the outer voxels are displaced according to the macroscopic strain, but their coordinates are kept fixed during the minimisation afterwards. Hence, the degrees of freedom of the minimisation are reduced. The fixed boundary problem, also

74 4.3 Finite Element Method for Voxelised Structures referred to as Dirichlet problem, equates to the boundary values in Eq. (4.37). For periodic boundaries, a macroscopic strain is applied by enforcing the size of the periodic unit cell. The positions of the outer voxels are not fixed but the displacement of their vertices are constrained to displacements that keep the initial periodicity of the sample. Thus, there is also a slight reduction of degrees of freedom, albeit not as much as for fixed boundary conditions. Cubic symmetry of our initial systems reduces the number of independent components of the elastic moduli tensor C from 21 to 3: the bulk modulus K and two shear moduli G and G2. Thus, three different displacement conditions are enough to gather the three independent elastic moduli. The bulk modulus is determined by hydrostatic compression 3 + 3 + of the sample to size l , l ∈ R , where the original volume had been L with L > l, L ∈ R . This deformation is a simple scaling of coordinates with the compression factor l/L. The displacement u is the difference in coordinates in the current and the reference configuration, u = x − X = (l/L − 1)X, which, by applying Eq. (4.11) and Eq. (4.23), leads to the linear strain tensor E with components l Eij = − 1 δij =: δij . (4.42) L

The shear moduli are determined by (i) a pure shear and (ii) shear along a face diagonal of the object.

(i) A pure shear deformation can be accomplished by keeping one coordinate, say X3, constant while scaling X1 with factor λ1 and X2 with factor λ2. Since a shear is an isochoric deformation, λ2 = 1/λ1 (compare Eq. (4.6)) giving the linear strain tensor E     λ1 − 1 0 0 0 0 E = 0 1 − 1 0 =: 0 − 0 .  λ1   +1  (4.43) 0 0 0 0 0 0

is the so-called engineering strain. If X1 is stretched, X2 is compressed and vice versa. In the linear elastic regime, deformations must be small, so 1 and −/( + 1) can be approximated by −/( + 1) ≈ −.

(ii) As for a pure shear, X3 is kept constant. X1 and X2 transform as xi = (1 + ) Xi, i = 1, 2 resulting in the linear strain tensor

 0 0  E =  0 0  . (4.44) 0 0 0

All 3 deformations are shown in Fig. 4.3.

4.3.1 Sources of Error

In general, numerical modelling of composite materials is aﬀected by diﬀerent sources of error. An error analysis of the voxel-based Finite Element Method permits to evaluate the quality of the numerical solution, see for instance Refs. [72, 205, 206].

75 4 Eﬀective Linear-Elastic Properties of Network Solids

y y y L L

ε ε L ε z z

z L− l L L

ε L x x ε x l L+ ε L (a) hydrostatic compression (b) pure shear (c) shear along a face diagonal

Figure 4.3: Test strains applied to a cubical symmetric sample to calculate the three independent elastic moduli.

An important issue when using the voxel-based FEM are discretisation errors and the convergence of the discretised, approximate solution to the exact solution for inﬁnitesimal element size h → 0. The resolution of a sample, which is directly proportional to the number n3 of voxels and inversely proportional to the element size h, has to be high enough to resolve all morphological features of the microstructure of the material. However, since the computational time scales with the number of elements, the division is kept as coarse as possible. It was empirically found that eﬀective properties P , including elastic moduli, depend linearly on the inverse linear number of voxels n−1 [205, 206,8, 155]

−1 P (n) = P0 + a n , (4.45) where a is a fit parameter and P0 is the value for the continuum limit n → ∞. Thus, the continuum value P0 can be extrapolated from simulations of the same sample at different discretisations.3 To avoid finite size errors, the size of the samples has to be chosen carefully. The system size has to be much larger than the length scale of the microstructure in order to be representative. Whether a sample is big enough can be tested by computing the effective elastic properties for different system sizes. If the effective properties are size-dependent, the system has to be enlarged. When working with random material models, another source of error is the statistical error due to fluctuations in the model structures. Two samples of a random material are two realisations of the same stochastic process which differ in their structure and effective properties. By means of averaging over N different realisations, the statistical error of the effective elastic properties, which is given by σ/N, σ the standard deviation, is reduced to an acceptable level. Finally, the limited numerical accuracy of a computer results in round-off errors in the solution of the differential equations. However, as usually a large number of significant digits is carried in the computation, this error is often small compared to the three sources of error described above [274].

3Attention has to be paid when considering properties as function of the volume fraction φ as φ varies slightly when changing the discretisation. This can be accounted for by interpolating between diﬀerent volume fractions [8].

76 4.4 Eﬀective Elastic Properties of Network Solids

4.4 Eﬀective Elastic Properties of Network Solids

At first, the generation of a porous solid material, henceforth called network solid, from the graph representation of a network is described. Subsequently, before investigating the effective elastic moduli for network solids obtained from the vertex model (which is described in section 2.4), the FEM is tested to work for network structures by analysing the linear-elastic behaviour of disordered network solids obtained from Poisson-Voronoi processes (see section 2.3.1) with different intensities and from collagen scaffold geometries for different collagen concentrations.

4.4.1 Voxelised Network Solids from Network Graphs

A network given by a graph representation, i.e. a graph embedded in three-dimensional space, consists of points in the Euclidean space connected by straight lines and has, thus, volume zero. In order to convert the network graphs into network solids, the graph edges are first voxelised and then inflated to cylinders of radius r. The resulting solid component is a monolithic solid of homogeneous material, in contrast to a strut framework that consists of edges hinged by a specific mechanism at vertices. The solid volume fraction φ is related to the fibre radius r.

Transformation from graph representation to voxelised data set

A network graph G as introduced in chapter 2, which is embedded in three-dimensional space, is transferred onto a three-dimensional, discretised data set consisting of cubic voxels. The process is illustrated in Fig. 4.4. Depending on the size of a single voxel, graph G is subdivided such that the distance between all connected vertices in the subdivision S(G) is less than the edge length of a voxel. A voxel of the data set is set to 1 if it contains a graph vertex, it is set to 0 otherwise. Although it is only checked which voxels contain a graph vertex, the topology of the network remains unchanged as the distance between connected vertices is chosen small enough that two vertices are located either in the same or in neighbouring voxels. Hence, their connectivity is kept. As network graphs are inﬁnitely thin, the resulting structure is a one-voxel thick representation of the network.

Dilation to a given volume fraction

If V denotes the set of all voxels in a sample and S is the set of voxels set to 1, then F := V\S is the set of voxels set to 0. Taking phase 1 to be solid and phase 0 to be void, the volume fraction of solid is φ := |S|/|V| with |S| the number of phase 1 voxels and |V| the number of all voxels. The network structures are cylindrically inﬂated to the prescribed volume fraction φ by dilating the one-voxel thick network representation to a certain radius r. Dilation — the dual operation to erosion with respect to set completion — is a basic morphological operator. When applying it to binary images, it is deﬁned as the Minkowski sum of body X with a structural element B [230, 149]

δB(X) := X ⊕ B (4.46)

77 4 Eﬀective Linear-Elastic Properties of Network Solids where ⊕ denotes the Minkowski sum

A ⊕ B := {x + y ∈ R : x ∈ A, y ∈ B} (4.47)

d d 4 of two subsets of Euclidean space A ⊂ R and B ⊂ R . The result of the dilation operation depends on the choice of the structural element B. In three-dimensional morphology, usually a compact ball B with radius centred at the origin,

d B = x ∈ R : |x| ≤ , (4.48)

is used. Because of the associativity of the Minkowski sum, n dilations with a ball of radius are equivalent to a dilation with a ball of radius n. A dilation always enlarges body X which may cause initially existing cavities to be closed, loose parts to be connected or insections to be filled. The process of dilation is equivalent to thresholding of a distance transform. Using an Euclidean distance transform complies with a sphere-shaped structuring element where the sphere’s radius equals the final threshold [149]. The one voxel thick network representation is dilated to a certain volume fraction φ of solid using the Euclidean distance map (EDM) of the 0-phase. The EDM of phase 0 specifies the distance of each centre of a voxel to the next centre of a voxel of phase 1. Starting from the voxels nearest to the 1 phase, the voxels of phase 0 representing fluid, air or vacuum are iteratively set to 1 until the desired volume fraction φ is reached. Because of the finite number of voxels in a sample, the accessible values for the volume fraction φ are discrete (with larger data sets implying finer discretisation). The algorithm returns the configuration closest to the specified volume fraction φ. Since we assume all network struts to be of the same thickness, all voxels with the same distance are set to 1 at the same time, yielding — depending on the discretisation resolution — smaller or larger deviations from the wanted volume fraction. For the dilation of periodic structures, the initial graph is continued in all directions and transferred to a proportionally larger segmented data set as shown in Fig. 4.4. The dilation is then carried out on the extended segmented image only counting those 1-voxels to the solid volume fraction that are located in the original voxel set. From the final image, the original subset is cut yielding a periodic segmented image of the original periodic graph.

4.4.2 Elastic Moduli of Poisson-Voronoi and Collagen Network Solids

The eﬀective linear-elastic properties of network solids obtained from Poisson-Voronoi diagrams and collagen scaﬀolds are calculated on voxelised representations of the networks applying theFE scheme used by Garboczi and co-workers [ 72, 202, 165].5 As a function of the volume fraction φ, the data for both structures collapses onto one curve. For small volume fractions φ, power-laws are found for the bulk and the shear modulus with the expected exponent 1 for the bulk modulus K and 1.69 for the shear modulus G, which

4 Sometimes, dilation is deﬁned as the Minkowski sum δB (X) := X ⊕ Bˇ of body X with the point d reﬂected structural element Bˇ := {−x : x ∈ B} for B ⊂ R [239]. 5All data in this chapter was gained using aFE implementation by Christoph H. Arns [ 8] based on the FE scheme suggested by Garboczi [72]. In chapter 5, an implementation by Sebastian C. Kapfer of the sameFE method is used. Both codes were tested to give the same results.

78 4.4 Eﬀective Elastic Properties of Network Solids

(a) Periodic unit cell graph (b) Extended dilated image

Figure 4.4: Dilation of a periodic graph: In order to avoid boundary eﬀects, the graph is periodically continued in all directions, dilated to a certain volume fraction, and the original subset is cut out afterwards again. Note that for the volume fraction only ﬁlled voxels in the inner subset are counted.

slightly diﬀers from the known analytical behaviour G ∝ φ2.

The most common model for disordered networks is the so-called Poisson-Voronoi process (cf. section 2.3.1) where the Voronoi diagram [187] of points with random uncorrelated coordinates is computed. The network is given by the edges of the Poisson- Voronoi diagram. In the strict mathematical model [175], random uncorrelated points are placed with intensity λ, implying that the point density fluctuates between different realisations around its average value λ. An advantage of this model is the availability of analytical expressions for fundamental morphological characteristics. A two-dimensional illustration of a Voronoi diagram and the corresponding, discretised network solid is given in Fig. 4.5. Numerous studies have addressed the linear-elastic properties of Voronoi network solids, with different degree of disorder, based onFE calculations. For instance, Van der Burg [250] studied Voronoi network solids of seeds with a minimum distance, Zhu et al. [272] and Li et al. [145] addressed Voronoi diagrams with adjustable irregularity, Roberts and Garboczi [204] analysed Voronoi tessellations of equi-sized hard sphere packings. All of them find power-laws for the effective Young’s modulus as function of the solid volume fraction φ as observed for regular solid network models [79], but with exponents slightly differing from the theoretical value 2 for open cell foams [79]. A second class of disordered network solids is obtained from the structure of collagen scaffolds extracted from experimental data. Reconstituted collagen networks from calf- skin collagen and rat tail tendon were fluorescently labelled and imaged by confocal microscopy. Three different collagen concentrations c = 1.2, 1.6, 2.4 mg/ml were analysed, corresponding to different mesh sizes [168]. The resulting 3D grey-scale data sets were transformed into binary data sets using standard threshold-segmentation after smoothing by anisotropic diffusion [69]. A line representing the collagen fibres was extracted from the binary data sets as the medial axis of the collagen phase, providing the random networks (see Ref. [168] for details of the morphology and structural properties of the

79 4 Eﬀective Linear-Elastic Properties of Network Solids

(a) Poisson-Voronoi network (b) Network solid

Figure 4.5: (a) 2D Poisson-Voronoi construction. (b) Discretisation of the corresponding network solid that results by voxelisation and inﬂation of the network struts radius r.

collagen networks). Fig. 4.6 shows the line representation of a collagen network with c = 1.6 mg/ml. These 1D networks are inflated and converted to discretised network solids by the procedure sketched in section 4.4.1. Note that the concentration c of collagen determines the mesh or pore size of the network structure. A Poisson-Voronoi diagram is isotropic on average, thus its elastic properties can be characterised by only two moduli. The same is assumed for the medial axis scaffolds of collagen [168]. Fig. 4.7 shows the effective bulk and shear moduli K(φ) and G(φ) for network solids obtained by the edges of Poisson-Voronoi networks (based on Voronoi diagrams of a fixed number N of seeds in a unit cube rather than intensity λ) and collagen scaffolds. The network structures are inflated to the prescribed volume fraction φ by the algorithm described in section 4.4.1. The φ-dependence studied is the dependence on the strut or fibre radius the networks are inflated to for a fixed network structure. Effective linear-elastic moduli are computed assuming microscopic elastic moduli Ks = Es/[3(1−2νs)] = 0.69 GPa and Gs = Es/[2(1+ν)] = 0.18 GPa corresponding to collagen. These values are estimated from measurements of Young’s modulus Es = 0.5 GPa and Poisson’s ratio νs = 0.38 [115]. While this is in agreement with other measurements, e.g. in aqueous media [252], estimates from atomic force microscopy (AFM) measurements on electrospun collagen fibres for Ks and Gs are an order of magnitude lower than the values used in our calculations [268]. The Finite Element calculations have been performed with fixed boundary conditions as the Poisson-Voronoi diagrams and the collagen scaffolds, in general, are not periodic.

When numerically investigating random material models, three sources of error must be considered. First, due to the random nature of the model, different samples exhibit statistical fluctuations in structure and hence in effective properties. To reduce this error, numerical measurements are usually averaged over a certain number of independent realisations of the model. The self-assembly of collagen networks from a solution of

80 4.4 Eﬀective Elastic Properties of Network Solids

Figure 4.6: A perspective view of the medial axis representing the collagen network with concentration 1.6 mg/ml and typical pore size hDf i ≈ 1.9 µm [168]. collagen monomers is a random process. As we have only one collagen network realisation for each concentration, averaging is not possible and the amount of statistical error cannot be specified. For the Poisson-Voronoi network solids, network solids with different numbers of sites were studied. One realisation for each number of sites was generated and inflated to different radii in order to obtain samples with different volume fractions. It would be desirable to extend the analysis to several realisations per number of sites. Secondly, uncertainties arise by the discretisation of the network solids into cubic voxels. The influence of the discretisation on the effective elastic moduli is studied at the example of the collagen scaffolds. Increasing the number of voxels the sample is divided into while keeping the sample size constant results in smaller voxels and hence in a finer discretisation. Fig. 4.8 shows the effective bulk modulus K and the effective shear modulus G as function of the inverse linear voxel size n−1. As predicted by Eq. (4.45), a linear dependence is found. In order to find an estimate for the continuum value of the effective elastic moduli, an extrapolation to n → ∞, or 1/n → 0, is necessary. The calculation of the effective elastic properties for several different discretisations is very time consuming. Especially the analysis of the mechanical properties of the randomised network solids as function of the disconnection probability p in chapter 5 would exceed time and computer resources. We restricted the computations in chapter 4 and chapter 5 to one discretisation for each parameter set. Certainly, the results are not the continuum values of the elastic properties, but are considered as estimates which indicate how the elastic properties change as function of different parameters. The third source of error are finite size errors which may occur if the sample is too small compared to the microstructural features it contains. According to Ref. [204], about

81 4 Eﬀective Linear-Elastic Properties of Network Solids

0 0 Poisson-Voronoi -1 -1 Collagen -2 / GPa) / GPa) 1.69 G K ( -2 ( 1.00 -3 10 10 1

log 1 -3 log -4 -5 -2 -1.5 -1 -0.5 0 -2 -1.5 -1 -0.5 0

log10(ρ/ρs) log10(ρ/ρs)

Figure 4.7: Elastic moduli of linear-elastic solids obtained by inflating Poisson-Voronoi networks and collagen scaffolds to network solids of a given volume fraction φ = ρ/ρs. Different values of φ correspond to different inflation radii. Squares represent Poisson-Voronoi networks and include configurations with different numbers N = 100, 500, 1000, 2000 of seed points. Circles represent the collagen structure including three different collagen concentrations c = 1.2, 1.6, 2.4 mg/m`. For the effective bulk modulus K, the curved line represents a best fit of all data (Poisson-Voronoi and collagen) to the functional form of Eq. (4.50) and the straight line its asymptotic power-law in the limit φ → 0. For the effective shear modulus G, the deviations from the effective power-law behaviour are very small, even for φ close to unity.

82 4.4 Eﬀective Elastic Properties of Network Solids

0.010 0.006 1.2 mg/ml (Gel 1) 1.2 mg/ml (Gel 1) 0.008 1.6 mg/ml (Gel 2) 0.005 1.6 mg/ml (Gel 2) 2.4 mg/ml (Gel 3) 0.004 2.4 mg/ml (Gel 3) 0.006 0.003 0.004

K [GPa] G [GPa] 0.002 0.002 0.001 0.000 0.000 0 0.05 0.1 0.15 0 0.05 0.1 0.15 linear voxel size [µm] linear voxel size [µm]

Figure 4.8: Linear dependence of effective elastic moduli on discretisation: As described in Eq. (4.45), the effective elastic moduli of network solids depend linearly on the inverse voxel size. Data was obtained from the structure of collagen scaffolds with different collagen concentrations assuming a microscopic bulk modulus Ks = 0.69 GPa and shear modulus Gs = 0.18 GPa corresponding to collagen.

100 cells are necessary (which corresponds to about five cells in each direction) when considering random open cell foams to keep the finite size errors of the same order as the discretisation errors. As Poisson-Voronoi network solids are foam-like structures, Poisson-Voronoi diagrams with 100 or more sites, giving rise to 100 or more cells, are investigated. Collagen scaffolds are not foam-like but network structures. However, they have a relatively low solid volume fraction and their pore space is interconnected, as it is the case for open-cell foams. Thus, assuming the collagen scaffolds to be comparable to foam structures with respect to finite size effects, a sample size of 20 µm was chosen which is 6.67 times the typical pore size of the collagen gel with concentration c = 1.2 mg/ml with a pore size of 3 µm (1.9 µm for c = 1.6 mg/ml, 1.28 µm for c = 2.4 mg/ml)[168]. In Fig. 4.6, one collagen sample is shown. The linear-elastic constitutive equations are dimensionless, and the effect of the intensity λ in the Poisson-Voronoi process corresponds to a change of length scale. Therefore, at least in a system not influenced by the global boundaries, a change of λ (or N) does not affect K and G if the fibre radius r is adjusted to maintain constant volume fraction φ (and if discretisation effects can be neglected). The Poisson-Voronoi data in Fig. 4.7 for different numbers N of seeds collapse onto a common curve, conforming to this expectation despite the influence of the fixed boundary conditions. For the collagen network solids, a dependence of K or G on the collagen concentration c is not observed either, as shown in Fig. 4.7. A change in the line or collagen density c can also be compensated by changing the fibre inflation radius r. This implies that changes in collagen concentration affect the geometry of the networks in a similar way as the intensity λ in the Poisson-Voronoi process, i.e. effectively only by rescaling the length scale, manifested for example in the typical pore size. For small to intermediate volume fractions, the available data for both the Poisson- Voronoi and the collagen network solids are well approximated by empirical power-laws

n m K ∼ K0φ and G ∼ G0φ for φ → 0 (4.49)

83 4 Effective Linear-Elastic Properties of Network Solids with scaling exponents n and m and apparent moduli K0 and G0 for φ = 1. Linear regression to data points with φ < 0.35 yields n ≈ 1.46 and m ≈ 1.79 for the Poisson- Voronoi network solid, and n ≈ 1.42 and m ≈ 1.69 for the collagen network solid. The similarity of the respective exponents for the Poisson-Voronoi networks and the collagen networks is further evidence for structural and mechanical similarity of these two geometries. The values for the exponents are also close to the values previously reported for Poisson-Voronoi network solids: The finite element analysis in Ref. [204] gives m ≈ 2.12 and n ≈ 1.22 for Poisson-Voronoi network solids with periodic boundary conditions; the difference gives an estimate of the possible relevance of the imposed boundary conditions. An evaluation of the influence of the slightly differing initial point distribution (uncorrelated positions compared to hard sphere packings) on the exponents is not possible on the basis of the available data.6 The validity of empirical power-laws as those in Eq. (5.3), extracted from voxel-based finite element simulation that is unable to access the φ → 0 limit, is difficult to ascertain. Unless K0 equals the microscopic bulk modulus Ks, deviations from the power-law are necessary to ensure the correct limit K(φ) → Ks for the high-density limit φ → 1. The crossover from the power-law behaviour to K(φ = 1) = Ks makes the extraction of reliable estimates for the correct power-law exponents difficult. A functional form for K(φ) valid for all φ ∈ [0, 1] is proposed fulfilling the requirement that for φ = 1 the homogeneous solid is recovered, hence K(φ = 1) = Ks. For values of φ close to unity, the network solid is essentially a solid block with small isolated cavities 2 (“hollow inclusions”), which lead to a linear correction K(φ) ∼ Ks −(1−φ)δK+O((1−φ) ). This can be demonstrated by effective medium approximations for several distinct shapes of the cavity, see e.g. Eq. 19.70 in Ref. [248] or Refs. [34, 44, 81]. Requiring further that n0 a power-law K(φ) ∼ K0φ is valid for small φ → 0 and with an exponential ansatz for the crossover, one obtains φ n0 Ks K(φ) = K0 φ (4.50) K0

0 with δK = Ks log(Ks/K0). This form has two free parameters, the exponent n and K0. An analogous form is proposed for the shear modulus G(φ). The solid curve in Fig. 4.7 shows the functional form of Eq. (4.50), with n0 ≈ 1.0 and K0 ≈ 0.044 GPa ≈ 0.06Ks determined by linear least squares fitting to all data points of both the collagen and the Poisson-Voronoi network solids. The exponent n0 of the asymptotic power-law in the φ → 0 limit corresponds to the exponent 1, established as the correct exponent for cellular structures in the limit of thin beams [256, 273, 255, 79]. This form is a further indication that the empirical exponents obtained by fitting with Eq. (4.49) discussed above do not represent the asymptotic behaviour for φ → 0. It is interesting to note that for the shear modulus G the deviations from the empirical power-law are significantly less pronounced than for the bulk modulus, yielding m0 ≈ 1.69 and G0 ≈ 0.10 GPa ≈ 0.57Ks. However, the known analytic behaviour for the shear modulus G(φ) ∝ φ2 in the limit of thin beams, φ → 0, is not recovered by our finite element analysis, possibly due to the use of fixed boundary conditions and relatively small system sizes compared to the mesh sizes of the networks.

6The results of this paragraph, in particular the cross-over behaviour of K as function of volume fraction φ, were obtained in collaboration with the co-authors of Ref. [180].

84 4.4 Eﬀective Elastic Properties of Network Solids

Summary We have compared the structure of experimental collagen fibre networks with random networks generated by Poisson-Voronoi processes. If treated as a network solid, we find the same dependency of linear elastic moduli on effective density or fibre radius for both network types. The microscopic details of bio-polymer networks are undoubtedly poorly represented by the simple linear-elastic model applied here. Bio-polymers are thermal systems, the microscopic elastic properties are anisotropic and differ in fibre tangent and normal directions, and points where fibres touch may or may not be rigidly cross-linked. As a technical point, typical collagen fibre radii, approximately 10 − 250 nm [197, 234, 252, 47] with approximate network mesh sizes of the order of µm [168], are thinner than the minimal network solid radius r that can be computed by voxelised finite element methods. A simple extrapolation of G(r) to a realistic value of G ≈ 100 Pa for the effective shear modulus of collagen observed in experiments [15] yields fibre radii between approximately 30 nm (c = 2.4 mg/m`) and 50 nm (c = 1.2 mg/m`), and the expected trend with collagen concentration. Note that this extrapolation is quite sensitive because of the power-law dependence of K on r. Given the caveats listed above it is interesting to note that these extrapolated estimates are of the same order of magnitude as experimental values. This is an indication that morphology and structure of the network is an important determinant of mechanical properties that leads to similarities between systems with very different microscopic details. Steady improvements in 3D imaging methods, in particular also for biological materials [23], highlight the importance of methods that deduce effective properties from structural and morphological analyses.

5 Topology Changes: Strong Eﬀect on Elastic Properties of Network Solids

This chapter is dedicated to the mechanical behaviour of network solids derived from the network model with topology changes described in chapter 2. The connectivity of initially four-coordinated networks is reduced by subjecting them to local disconnection operations with probability p. The resulting structures with randomised topology are interpreted as network solids. Their effective elastic properties are examined using the Finite Element method introduced in section 4.3 and compression experiments on laser-sintered models. The main result is a strong reduction of all elastic moduli with disconnection probability p. For volume fractions φ < 0.5, the functional dependence can be approximated by an exponential decay. The compression experiments conform to these numerical results. As expected, the study of the effective elastic properties as function of volume fraction φ at constant p reveals power-laws for all elastic moduli. Remarkably, the exponent for the effective bulk modulus K changes with p, signaling a change of the deformation mode. Furthermore, the influence of boundary conditions, the system size and a change of the ratio of the microscopic elastic moduli Gs/Ks is studied. Finally, a close relationship to the percolation properties analysed in chapter 3 is demonstrated by the scaling behaviour of the bulk modulus K(p) close to the percolation critical point.

Initially four-coordinated network graphs are subsequently degraded to graphs with an average coordination number c = 2 by randomly separating network nodes as described in section 2.4. When converting the network graphs into network solids as explained in section 4.4.1, the graph vertices are not only separated but also slightly shifted in opposite directions since they have to be separated in space to be mechanically separated in the voxel-based Finite Element model used for the computation of the effective elastic properties. Details about the shifting can be found in section 2.4.1. The discretised data set is inflated to a certain volume fraction φ as in section 4.4. The maximal inflation radius is chosen such that the separated parts stay disconnected. The whole process of unlinking, transferring to a voxelised data set and inflation is summarised in Fig. 5.1. Note that the edges of the foam network introduced in section 2.3 are, in fact, slightly curved but straightened for the finite element analysis described here.

5.1 Numerical Results for Linear-Elastic Moduli of Network Solids

The eﬀective linear-elastic properties of random network solids obtained from diamond, nbo and foam structures are calculated using a voxel-basedFE scheme based on Garboczi’s algorithm using a simple cubic mesh [73, 72]. The program used was written by Sebastian C. Kapfer. Initially, the samples have cubic symmetry, thus three eﬀective elastic moduli,

87 5 Topology Dependence of Eﬀective Linear-Elastic Properties

(a) (b)

Figure 5.1: Separating and inflating a graph to a network solid with cylindrical edges of finite radius demonstrated for a diamond node. The maximal inflation radius is chosen such that the separated parts stay separated. (a) subset of a connected diamond graph, (b) the same subset after separating one node (c) cubic discretisation of the connected subset, (d) cubic discretisation of the separated diamond graph.

88 5.1 Numerical Results for Linear-Elastic Moduli of Network Solids the bulk modulus and two shear moduli, are computed. In case of an isotropic sample, the two shear moduli would be equal. The network solids consist of a solid and a void phase with locally, i.e. in each voxel, isotropic and linear-elastic material properties. Since the elastic modulus tensor Ceff is homogeneous of degree one as shown in Eq. (4.34), it scales linearly with the bulk modulus Ks and only the ratio of the microscopic bulk modulus Ks to shear modulus Gs is important. Hence, without loss of generality, Ks is set to 1. The ratio Ks/Gs is arbitrarily chosen to be Ks/Gs = 2 for the following analyses except for section 5.1.4 where the impact of Ks/Gs on the effective elastic properties is examined. The void phase is modelled as vacuum, i.e. kv = 0 and gv = 0. For each unlink probability p, the effective properties are averaged over 5 independent realisations generated with different sets of random numbers. The average value of M realisations of a quantity X is calculated using

M 1 X X¯ = Xm . (5.1) M m=1

The given error bars are the standard deviation σ v u M u 1 X 2 σ = t Xm − X¯ (5.2) M − 1 m=1 unless otherwise stated.

5.1.1 Effective Moduli as Function of Disconnection Probability p Fig. 5.2 (a) shows the effective Young’s modulus E of the network solids as function of probability p for a fixed volume fraction φ = 0.14, including results for several network geometries. The diamond and the nbo network solid consist of 43 cubic unit cells (corresponding to 512 nodes in case of diamond and 384 nodes for nbo), the considered foam structure had 125 polyhedral cells. The key result is that the effective Young’s modulus E depends very sensitively on p and the functional dependence E(p) is approximated by an exponential decay for small p (although the possibility of an algebraic behaviour in (1 − p) due to critical morphological changes at p = 1 is discussed in section 5.2). Fig. 5.2 (b) displays the effective bulk modulus K and shear modulus G of diamond network solids with different volume fractions. This demonstrates that the results are not specific to the particular value φ = 0.14 and that the approximately exponential change of effective properties also holds for the bulk and shear moduli up to p ≈ 0.5. The initial diamond network at p = 0 has cubic symmetry, which implies a certain degree of anisotropy. Warren and Kraynik showed in Ref. [138] that foam structures, which may by thought to be elastically isotropic, exhibit slight anisotropy, too. As mentioned in section 4.2, the amount of anisotropy can be measured by the ratio of the two effective shear moduli G2/G [254], where G is the resistance against pure shear and G2 the resistance against shear along a face diagonal, see Fig. 4.3. The inset of Fig. 5.2 shows G2/G for diamond and foam network solids at different p. The effective elastic properties of the diamond network solid at p = 0 are much more anisotropic than that of foam at p = 0. Furthermore, it can be seen that the diamond sample becomes

89 5 Topology Dependence of Eﬀective Linear-Elastic Properties

-1 10 100 φ = 0.14 φ = 0.14 10-1 φ = 0.10 φ = 0.06 -2 [GPa] 10 K 10-3

10 s dia E 10-2 10-4 100 G /

/ 1 2 E G foam 10-1 0 0 0.1 0.2 0.3 0.4 0.5 -2 dia p 10 nbo -3 foam 10

-3 -4 [GPa]

10 10 G 0 0.1 0.2 0.3 0.4 0.5 0 0.1 0.2 0.3 0.4 0.5 p p (a) (b)

Figure 5.2: Strong decay of elastic moduli with increasing disconnection probability p: Effective elastic moduli are obtained by finite element calculations with fixed boundaries on samples with 2003 voxels. (a) Effective Young’s modulus E for the diamond, the nbo and the foam network with a solid volume fraction of φ = 0.14. The decrease with p can be approximately modelled as an exponential decay. (b) Effective bulk and shear modulus K and G of the randomised diamond networks with different solid volume fractions φ as a function of the disconnection probability p. The inset shows the ratio of the two shear moduli G and G2 that are almost identical for the isotropic foam model, but differ for structures such as the diamond lattice with cubic symmetry.

90 5.1 Numerical Results for Linear-Elastic Moduli of Network Solids increasingly more isotropic with larger p as the separation of nodes disturbs the initial order, resulting in an almost isotropic structure for p = 1. Note that the shown results were obtained with aFE scheme with fixed boundary conditions, i.e. the displacement of the outer voxels is initially set according to the macroscopic strain but is fixed during the minimisation of the elastic energy. As computational resources are limited, the system size was first constrained to 43 unit cells, i.e. a relatively small number. The influence of the boundary conditions and the system size are studied in more detail in section 5.1.3.

5.1.2 Eﬀective Moduli as Function of Volume Fraction φ: Probability p aﬀects Power-Law Exponent

The eﬀective bulk and shear moduli of composite materials are strongly aﬀected by the volume fractions of its constituents. Here, cellular and network solids consisting of a single solid component of homogeneous material and a void complement are considered. As reported in numerous studies, the linear-elastic moduli of such structures obey power-laws in the volume fraction φ of solid

κ γ γ K ∼ K0 φ ,G ∼ G0 φ ,G2 ∼ G2,0 φ 2,E ∼ E0 φ for φ → 0 (5.3) with exponents κ, γ, γ2 and which are between 1 and 4 [79, 202] and are strongly influenced by the cell geometry and the strut deformation mechanisms. When deforming a macroscopic sample, the predominant deformation mechanisms may be strut stretching (or likewise compression) or strut bending, depending on the arrangement of the struts on the micro-scale. If a structure contains elements that connect two opposite sites of a sample along a straight line, so-called straight through elements, stretching will be the dominating deformation. In contrast, if the microstructure is composed of long, thin but not sample-spanning struts, bending is favoured. In general, if a deformation is bending-controlled, the resulting exponent equals 2, whereas it equals 1 for stretching dominated strains, independent of the considered modulus. These results are the bottom line of numerous analytical studies on different models as, for instance, the thread model [75], the hexagonal and face-centred model [135], the randomly oriented fibre model [46], the tetrahedral joint model [255], the open cell foam based on a cubic unit cell [79], and the Kelvin foam [256, 273]. Remarkably, for the bulk modulus K, only K ∼ φ is found. For Young’s modulus and the shear modulus,FE analyses of Voronoi models [250, 272, 202, 145], a node-bond model and a Gaussian random field model [202], the Kelvin foam [84, 116] and 3D images of synthetic foams [84, 116] yield exponents between 1 and 4. Experimental values [79, 13] support these findings. A couple of numerical findings shall be presented here as they are connected to the results of this chapter:

(i) For low density foams, Young’s modulus and the shear modulus increase with increasing irregularity of the foam structure, whereas the bulk modulus decreases [250, 272, 145].

(ii) Interestingly, the foam is stiﬀer for non-uniform cross-sections than uniform cross- sections at constant volume fraction φ. Note that the bulk modulus K only depends on stretching in the isotropic case (K ∝ φ) but depends also on bending when anisotropy comes into play (K ∝ φ2 )[84].

91 5 Topology Dependence of Eﬀective Linear-Elastic Properties

Fig. 5.3 shows effective elastic moduli of disconnected diamond network solids as function of the volume fraction φ, at different values of disconnection probability p. As expected, we observe power-laws for all effective elastic moduli at all p. The exponents were determined by least squares fitting in the log-log plots. For the shear moduli G, G2 and Young’s modulus E, the power-law exponents are independent of p and take the values γ = 2.180 ± 0.003, γ2 = 2.101 ± 0.008 and = 2.148 ± 0.003 (compare the upper right graph in Fig. 5.3). From the fact that all three exponents are close to 2 it can be concluded that uniaxial deformations and shear deformations are dominated by bending rather than by stretching or compression. In contrast, the exponent of the bulk modulus K is not constant but increases with p from κ = 1.225 ± 0.009 at p = 0 up to κ = 2.3 ± 0.1 at p = 0.8. The value of κ close to 1 for p = 0 — the discrepancy to the analytic value κ = 1 for a diamond lattice at φ 1 [256] may be due to φ not small enough or discretisation effects, see section 4.3.1 — suggests that the hydrostatic compression of the four-coordinated diamond network is dominated by strut compression. As there are no straight through elements in the diamond structure, this must be due to the tetrahedral connectivity at the nodes that locks the network struts to their positions. Free translations or rotations of parts of the network are prohibited and thus, alignment in the direction of the acting force or rearrangement in a stress-avoiding configuration is not possible. This results in a macroscopic resistance controlled by stretching of the microscopic material. However, with increasing p, tetrahedral configurations are separated and thus constraints confining the struts are lifted, making bending more favourable and resulting in an increase of the exponent with p. This finding is similar to that of Ref. [84] where increasing anisotropy yields stronger bending effects.

5.1.3 Inﬂuence of Boundary Conditions and System Size

At p = 0, the structures of the diamond, nbo and foam networks are periodic in x-, y- and z-direction, i.e. they may be periodically continued in all three directions of space. The separation algorithm (see section 2.4) keeps this periodicity of the initial networks. This enables us to not only calculate the effective elastic properties with fixed but also with periodic boundary conditions. Fig. 5.4 shows a comparison of the elastic moduli calculated with fixed and periodic boundary conditions. Young’s modulus E, the bulk modulus K and the shear moduli G and G2 are shown for diamond, nbo and foam both for periodic and fixed boundary conditions at a volume fraction φ = 0.1. With periodic boundary conditions, all moduli are even more sensitive to a change in p manifested in a much faster drop for increasing p as the approximately exponential decay for fixed boundaries. Note that the calculated moduli always have lower values for periodic than for fixed boundary conditions, even at p = 0. The reason are the degrees of freedom in the energy minimisation: By locking the positions of the outer layer of voxels as is done for fixed boundary conditions, the degrees of freedom are reduced much more than by enforcing the size of the unit cell and maintaining periodicity. With more degrees of freedom available or, in other words, less constraints, the energy may reach a lower value giving lower elastic moduli. Fig. 5.5 and Fig. 5.6 show the effective elastic moduli for varying system sizes, for periodic and fixed boundary conditions, respectively. In order to study finite size effects, calculations for different system sizes were performed. For the diamond network, systems with 43, 63, 83 and 163 translational unit cells (nbo: 43, 63 and 83 unit cells, foam:

92 5.1 Numerical Results for Linear-Elastic Moduli of Network Solids

10-1 p = 0.00, ∝ φ1.225 ± 0.009 p=0.00, ∝ φ2.185 ± 0.004 p = 0.20, ∝ φ1.91 ± 0.03 p=0.20, ∝ φ2.17 ± 0.01 2.09 ± 0.03 2.17 ± 0.03 -1 p = 0.40, ∝ φ p=0.40, ∝ φ 10 p = 0.60, ∝ φ2.2 ± 0.1 p=0.60, ∝ φ2.2 ± 0.1 p = 0.80, ∝ φ2.3 ± 0.1 p=0.80, ∝ φ2.3 ± 0.1 10-2

10-2

-3 K G 10 10-3

10-4 10-4

10-5 10-5 0.06 0.08 0.1 0.15 0.2 0.06 0.08 0.1 0.15 0.2 φ φ

10-1 2.6 p=0.00, ∝ φ2.079 ± 0.014 p=0.20, ∝ φ2.10 ± 0.02 p=0.40, ∝ φ2.13 ± 0.03 2.4 p=0.60, ∝ φ2.2 ± 0.1 p=0.80, ∝ φ2.2 ± 0.1 10-2 2.2

-3

E 10 1.8

exponent 1.6

κ -4 10 1.4 γ = 2.180 ± 0.003 γ2 = 2.101 ± 0.008 1.2 ε = 2.148 ± 0.003

10-5 1 0.06 0.08 0.1 0.15 0.2 0 0.2 0.4 0.6 0.8 1 φ p

Figure 5.3: Changing exponent with p in the φκ-power-law for the bulk modulus: The effective bulk and shear moduli K and G and the effective Young’s modulus E of randomised diamond networks with 83 unit cells are shown for different disconnection probabilities p as function of the volume fraction φ; data was obtained byFE calculations with periodic boundaries on network solids discretised by 4003 voxels. For all p, the moduli obey power-laws in φ: κ γ γ K ∼ φ , G ∼ φ , G2 ∼ φ 2 and E ∼ φ . For each p, the exponents were determined by least squares fitting in the log-log plots and plotted as a function of p (bottom right). The exponents γ, γ2 and remain more or less constant. By fitting the data with a constant, we obtained γ = 2.180 ± 0.003, γ2 = 2.101 ± 0.008 and = 2.148 ± 0.003.(G2(x) is not shown.) In contrast, the exponent for the bulk modulus K increases from κ = 1.225 ± 0.009 to κ = 2.3 ± 0.1 as bending modes become more important with increasing p. The error bars in the exponent plot are the fit errors resulting from the least squares fit. 93 5 Topology Dependence of Effective Linear-Elastic Properties

10-1 10-1 φ = 0.1 -2 10-2 10 10-3 10-3 10-4 [GPa] -4 dia pbc [GPa] E K 10 dia fbc 10-5 nbo pbc -5 10 nbo fbc -6 foam pbc 10 foam fbc 10-6 10-7 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 p p

10-1 10-2

10-2 10-3

10-3 10-4 [GPa] [GPa] -4 2 G

10 G

10-5 10-5

10-6 10-6 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 p p

Figure 5.4: Comparison of fixed and periodic boundary conditions for the effective elastic moduli of diamond, nbo and foam. For periodic boundary conditions, the decrease of the effective elastic moduli with disconnection probability p is much faster than the approximately exponential drop for fixed boundaries. For diamond and nbo, the network solids consisted of 43 unit cells, the foam structure contained 125 cells. These results were obtained byFE calculations on voxel representations with 2003 voxels; each data point is the average PN x¯ = 1/N i=1 xi over N = 5 realisations; error bars denote the standard PN 2 1/2 deviation σ = (1/(N − 1) i=1(xi − x¯) ) .

94 5.2 Relationship to Percolation

125, 216, 521 cells) with 503 voxels per unit cell were studied. (For the foam, a similar discretisation with respect to cell size was chosen.) In case of periodic boundaries, the results of differently sized systems coincide for all geometries, thus, it can be concluded that finite size effects are small and the simulated system sizes are big enough for periodic boundary conditions. On the other hand, for fixed boundary conditions, finite size effects are observed (see Fig. 5.6). With increasing size, the slopes of the elastic moduli gets steeper suggesting that in the limit of infinitely many (translational unit) cells ntucs → ∞ the values approach the periodic result. Hence, we conclude that in order to make predictions for macroscopic properties by applying FEM with fixed boundaries, the used system sizes are too small. Therefore, we use periodic boundary conditions in all of the following calculations.

5.1.4 Eﬀect of Varying the Ratio of the Microscopic Moduli Gs/Ks

Thus far, the ratio of the microscopic bulk and shear modulus Ks and Gs was kept constant at Ks/Gs = 2, or written differently Gs/Ks = 0.5. To investigate the influence of this ratio, Gs/Ks is changed from 0.0 to 2.0. Due to the homogeneity of the elastic modulus tensor (cf. Eq. (4.34)), it is sufficient to keep Ks = 1 while changing Gs from 0.0 to 2.0. Fig. 5.7 shows the effective elastic properties as function of the ratio Gs/Ks. TheseFE calculations were performed with periodic boundary conditions for diamond network solids with 43 unit cells. Starting from voxels that have no bending resistance but only a stiffness related to stretching or compression, the microscopic shear resistance is increased continuously. The principal dependence of the moduli on the disconnection probability p remains unchanged for all moduli. The fully connected network at p = 0 is affected most by the ratio change since there are the most constraints acting on the struts avoiding free translations or rotations of parts of the network. Thus, force-alignment or stress-avoiding reorganisation is prohibited which makes compression, shearing and torsion of the microstructural material necessary. For Gs = 0, all effective elastic moduli equal 0 for all p, even the bulk modulus K. As shearing does not cost any energy, it is energetically more favourable than stretching or compression of struts and even a hydrostatic compression is realised by shear deformations only. With increasing Gs, the struts get stiffer and stiffer against shearing and bending, naturally increasing the networks effective shear modulus G, but also enhancing the amount of compression/stretching in the network and hence, its effective bulk modulus K.

5.2 Relationship to Percolation

Evidently, percolation is intimately linked to mechanical stability, and hence linear-elastic moduli. At the percolation threshold pc, the system looses the ability to resist forces, i.e. the elastic moduli drop to zero. This can be made more precise by the study of the bulk modulus K(p) as a function of p in the vicinity of pc.

In chapter 3, the percolation transition of site and bond network models was described. When interpreting the network edges of a site or bond network as elastic springs, the elastic properties of these networks can be studied. They become critical at or close to the percolation threshold p˜c, too, depending on the interaction modes taken into account. In the vertex model, a transition from an elastic network to a network without elastic

95 5 Topology Dependence of Eﬀective Linear-Elastic Properties

10-1 10-2 10-2 10-2 4 tucs 6 tucs -2 10 8 tucs -3 -3 -3 16 tucs 10 10 10 10-3 10-4 10-4 10-4 [GPa] [GPa] [GPa] [GPa] -4 2 E K G

10 G 10-5 10-5 10-5 10-5 φ = 0.1 10-6 10-6 10-6 10-6 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 p p p p (a) diamond, periodic boundary conditions

10-1 10-1 10-2 10-1 4 tucs 8 tucs -2 10-2 10-2 10 10-3 10-3 10-3 10-3 10-4 10-4 [GPa] [GPa] [GPa] [GPa] -4 -4 2 E K G

10 10 G 10-5 -5 -5 -5 10 10 10 10-6 φ = 0.1 10-6 10-6 10-6 10-7 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 p p p p (b) nbo, periodic boundary conditions

10-1 10-2 10-2 10-1 125 cells 216 cells 10-2 512 cells 10-2 10-3 10-3 10-3 10-3 10-4 10-4 [GPa] [GPa] [GPa] [GPa] -4 2 -4 E K G

10 G 10 10-5 10-5 10-5 10-5 φ = 0.1 10-6 10-6 10-6 10-6 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 p p p p (c) foam, periodic boundary conditions

Figure 5.5: Eﬀective elastic moduli for diﬀerent system sizes and periodic boundary conditions: The bulk modulus K, the shear moduli G and G2 and Young’s modulus E were obtained byFE calculations on voxel representations with a volume fraction of φ = 0.1 for all structures. For diamond and nbo, the network solids were discretised with 50 voxels per unit cell. The foam structure containing 125 cells was represented by a 2003 voxel set; a similar discretisation with respect to the cell size was chosen for the larger foams. Each data point was averaged over 5 realisations; for K, G and G2, error bars denote the standard deviation σ, for E = f(K,G) error bars result from the quadratic propagation of error.

96 5.2 Relationship to Percolation

10-1 10-2 10-2 10-1 φ = 0.1 10-2 10-2 10-3 10-3

10-3 10-3 10-4 10-4 [GPa] [GPa] [GPa] [GPa] -4 2 -4 E K G

10 G 10 4 tucs 6 tucs 10-5 10-5 10-5 8 tucs 10-5 periodic 10-6 10-6 10-6 10-6 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 p p p p (a) diamond, ﬁxed boundary conditions

10-1 10-1 10-2 10-1 φ = 0.1 -2 10-2 10-2 10 10-3 10-3 10-3 10-3 10-4 10-4 [GPa] [GPa] [GPa] [GPa] -4 -4 2 E K G

10 10 G 10-5 4 tucs -5 -5 -5 10 periodic -6 10 10 10

10-6 10-6 10-6 10-7 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 p p p p (b) nbo, ﬁxed boundary conditions

10-1 10-2 10-2 10-1 φ = 0.1

10-2 10-2 10-3 10-3

10-3 10-3 [GPa] [GPa] [GPa] [GPa] 2 E K G G 10-4 10-4 10-4 125 cells 10-4 216 cells 512 cells periodic 10-5 10-5 10-5 10-5 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 p p p p (c) foam, ﬁxed boundary conditions

Figure 5.6: Effective elastic moduli for different system sizes and fixed boundary conditions: The bulk modulus K, the shear moduli G and G2 and Young’s modulus E clearly indicate a finite size effect whereupon for increasing system size, the values approach those of the periodic systems. These results were obtained by FE calculations on voxel representations with a volume fraction of φ = 0.1 for all structures. For diamond and nbo, the network solids were discretised with 50 voxels per unit cell. The foam structure containing 125 cells was represented by a 2003 voxel set; a similar discretisation with respect to the cell size was chosen for the larger foams. Each data point was averaged over 5 realisations; for K, G and G2, error bars denote the standard deviation σ, for E = f(K,G) error bars result from the quadratic propagation of error.

97 5 Topology Dependence of Eﬀective Linear-Elastic Properties

10-1 10-2

10-2 10-3

10-3 s s K K -4 / / 10 K G 10-4 Gs/Ks = 0.2 G /K = 0.4 -5 -5 s s 10 10 Gs/Ks = 0.6 Gs/Ks = 1.0 Gs/Ks = 2.0 10-6 10-6 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 p p 0.050 0.008 p = 0.00 0.045 0.007 p = 0.10 0.040 p = 0.25 0.006 p = 0.50 0.035 p = 0.90 0.030 0.005 s s K K / / 0.025 0.004 K G 0.020 0.003 0.015 0.002 0.010 0.005 0.001 0.000 0.000 0 0.5 1 1.5 2 0 0.5 1 1.5 2

gs/Ks gs/Ks

Figure 5.7: Varying the ratio Gs/Ks for fixed Ks = 1 GPa does not change the strong decay of the effective elastic moduli with disconnection probability p (top). Bulk modulus K and shear modulus G of network solids based on a diamond 3 structure with 4 translational unit cells both increase with increasing Gs/Ks as the material gets more resistant against shear yielding more energetic expenses when deforming the material (bottom). Varying Ks for fixed Gs would yield the same functional dependencies as, due to the homogeneity of the elastic modulus tensor (see Eq. (4.34)), only the ratio Gs/Ks is important. Data was obtained byFE calculations with periodic boundary conditions on voxel representations with 2003 voxels and a volume fraction of φ = 0.1. Each data point was averaged over 5 realisations; for K and G error bars denote the standard deviation σ.

98 5.2 Relationship to Percolation resistance is not found for p < 1. According to the results of section 3.5, the percolation threshold of the vertex model is pc = 1. The elastic scaling exponent fc, which describes the scaling behaviour of Young’s modulus E in the critical domain, is determined for the site percolation model and the vertex model.

Kantor and Webman [124] and Feng and Sen [65] studied the macroscopic elastic moduli of bond percolation network systems by considering the network edges to be elastic springs that can be stretched and bent. In these cases, the deformation energy H is

c1 X 2 c2 X 2 H = [(ui − uj) · Rij] eij + (δθijk) eijeik (5.4) 2 2 i,j ijk with two constants c1 and c2, the displacements ui and uj of node i and j, the unit vector Rij from node i to node j, the spring constant eij between i and j and the change of angle δθijk between node i, node j and node k. In this model, also called bond bending model, the elastic properties become critical at the percolation threshold p˜c. Without bending, i.e. c2 = 0, the elastic behaviour becomes critical at some other point p˜e, sometimes referred to as rigidity percolation threshold obeying p˜c ≤ p˜e [210, 233]. The eﬀective Young’s modulus E(˜p) = 0 for p˜ < p˜c,e.

Scaling close to the critical point

When approaching the bond percolation critical point p˜c from above, i.e. while the structure is percolating, E(˜p) is expected to obey the scaling laws [210, 248]

fc fe E(˜p) ∼ (˜p − p˜c) or E(˜p) ∼ (˜p − p˜e) (5.5) with the Young’s modulus critical exponents fc and fe for the model with and without bending, respectively. Note that the Young’s modulus exponent is not completely universal as its value depends on microscopic details of the model as, for instance, whether or not bending forces are taken into account. For the bond bending model, Sahimi [209] proposed the scaling relation

fc = µ + 2ν (5.6) where µ is the conductivity critical exponent and ν the critical exponent of the correlation length, see Eq. (3.15). In 3D, µ = 2.0 [210] and fc was numerically determined to fc ≈ 3.75 [210, 248]. Usually, other elastic moduli such as the bulk and shear modulus scale with the same exponent fc [248]. We have seen in section 5.1.1 that the effective elastic properties of the vertex model depend sensitively on the disconnection probability p. In contrast to elastic site or bond percolation networks, we do not find a transition from an elastic network to a network without elastic resistance for p < 1. Fig. 5.8 displays the effective bulk modulus K as function of p normalised to its value at p = 0 for the site percolation model and the vertex model. The modulus of the site percolation model drops to 0 for some value pc clearly lower than 1, whereas in the vertex model, the bulk modulus has a finite value over the whole interval. This is not surprising as we showed in section 3.5 that for the vertex model pc = 1. Hence, one may expect the effective elastic properties of the vertex

99 5 Topology Dependence of Eﬀective Linear-Elastic Properties

1.0 1.0

100 100

0.8 (0) 0.8 (0) K 10-2 K 10-2 ) / ) /

p -4 p -4 ( 10 ( 10 K K (0) 0.6 -6 (0) 0.6 -6

K 10 K 10 0.01 0.1 1 0.01 0.1 1 p - p ) / c ) / 1 - p p p

( 0.4 ( 0.4 K K 3.6±0.1 ± ∝ (p -p) ∝ (1-p)3.0 0.1 0.2 c 0.2 site vertex 0.0 0.0 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 p p

Figure 5.8: Power-law for bulk modulus K(p) close to criticality: Bulk modulus ratio K(p)/K(0) as function of probability p for the site percolation model (Left) and the vertex model (Right). The scaling exponents were determined by a linear least squares ﬁt of the log-log data shown in the inset. Due to huge error bars for (pc − p) < 0.1, the ﬁt range was restricted to (pc − p) ∈ [0.1, 0.5]. The shown data is based on diamond structures with 43 translational unit cells that were degraded by deleting and separating network nodes for site percolation and vertex percolation, respectively. The bulk modulus K was computed on voxel representations with 2003 voxels using aFE scheme with periodic boundary conditions. Each data point was averaged over 5 realisations; error bars denote the normalised standard deviation σ/K(0).

100 5.2 Relationship to Percolation

100

10-1 (0) K 10-2 ) / p ( K 10-3 φ = 0.1 10-4 0 0.2 0.4 0.6 0.8 1 p

Figure 5.9: Crossover of the bulk modulus from exponential to power-law decay with disconnection probability p: For p < 0.5, the decay of the bulk modulus ratio K(p)/K(0) is well approximated by an exponential decay. However, at p ≈ 0.5, a crossover to an algebraic decay is observed. The data shown is based on diamond structures with 43 translational unit cells that were degraded by separating network nodes. The bulk modulus K was computed on voxel representations with 2003 voxels using aFE scheme with periodic boundary conditions. Each data point was averaged over 5 realisations; error bars denote the normalised standard deviation σ/K(0).

model to scale as in Eq. (5.5) for p close to 1. Plotting the data over (pc − p) shows a power-law for both models close to their critical points and the elastic critical exponents are determined by a linear least squares fit in the log-log plot (see inset of Fig. 5.8). For site percolation, fc = 3.6 ± 0.1 which is in good agreement with the literature value of 3.75 for three dimensions. The vertex model exponent, on the other hand, is fc = 3.0±0.1 and is thus considerably lower. This supports the hypothesis that our model belongs to a different universality class than site and bond percolation. Apparently, close to the critical point the elastic moduli data can be explained by a power-law as in Eq. (5.5). For the site percolation model, this scaling law also fits to the data away from the critical point (see Fig. 5.8 (left)). In contrast, in section 5.1.1 it was demonstrated for the vertex model that the strong decay of the elastic moduli can be approximated by an exponential decay. Note though, that this data was obtained with fixed boundaries, whereas the scaling data was sustained with periodic boundary conditions. Fig. 5.9 shows the bulk modulus K gained with periodic boundary conditions in a linear-log plot revealing that for small p the functional dependence can be modelled as exponential decay, too. Thus, there has to be a crossover from exponential to algebraic decay with the elasticity scaling exponent fc. This crossover can be observed in Fig. 5.9 at p ≈ 0.5.

101 5 Topology Dependence of Eﬀective Linear-Elastic Properties

5.3 Compression Testing of Laser-Sintered Diamond Network Solids

In order to support the numerical results by experiments,1 realisations of diamond network solids were produced by selective laser-sintering for diﬀerent values of disconnection probability p and their elastic moduli were extracted by compression testing. First, the selective laser-sintering technique is introduced before describing the samples and the experimental setup more in detail. Finally, the results are presented and compared to the simulations. As shown in Fig. 5.15, the experimental values also show a strong decay of Young’s modulus E with p. Within the error bars, the functional dependence E(p) can also be approximated by an exponential decay with a slope similar to that of the simulated diamond network solids.

5.3.1 Specimens Production and Experimental Set-up Selective Laser Sintering

Selective laser-sintering (SLS) is an additive manufacturing technique, where a three- dimensional, complex geometrical structure is built layer-by-layer from a computer model (see Fig. 5.10). For this purpose, the structure to be built is sliced into successive, equally thick layers. Unlike stereolithography where a photo-polymer solution is used, SLS is powder-based. Possible base materials are plastic, metal or ceramics powders.

Figure 5.10: Additive manufacturing: A 3D object is built up layer-by-layer.

Fig. 5.11 schematically shows the principle of a SLS machine. A thin layer of powder is levelled by a roller before a CO2 laser beam scans the layer and locally melts the powder, thereby solidifying the cross-sections of the object. By using a laser, the heat is deposited very locally and causes a coalescence of powder particles in the selected region. After completing one layer, the platform is lowered by a certain height (0.1 − 0.2 mm). Preheated powder is provided by a reservoir and applied on top by a roller or a wiper blade before the next cross-section is retraced by the laser. Each layer is thus unsolvably fused to the underlying, previously solidiﬁed fabric [259]. As the unmelted powder supports the melt structure during the fabrication, no support structures are needed. The processing of polymers such as polyamide needs to be done at a constant ambient temperature. As mentioned before, the melting energy is brought in locally by the laser, without increasing the temperature of the surrounding powder [61]. Unwanted fusion of particles adjacent to the wanted structure, e.g. by thermally conducted energy, is avoided by the selective energy induction by the laser, the high melting enthalpy and the low thermal conductivity of the polymers used. At the end of the production process, the

1The compression testing experiments were performed in cooperation with the Institute for Polymer Technology (Erlangen), the FIT GmbH (Parsberg) and the NetFabb GmbH (Parsberg).

102 5.3 Compression Testing of Laser-Sintered Diamond Network Solids

Figure 5.11: Process of selective laser-sintering: (bottom) a thin layer of powder is levelled, (left) a CO2 laser scans the layer and melts the powder in the cross-section of the object, (right) the object is lowered by the thickness of one layer. Image reprinted from Ref. [201], courtesy of D. Rietzel.

unmelted but thermally degraded powder is separated from the laser-sintered component and is recycled [140]. Since it is possible to create almost any shape or geometric feature using SLS— apart from structures including closed cavities since unused powder cannot be removed afterwards — it is a fundamental technique to generate scaﬀolds not only for mechanical testing but also tissue engineering [104].

Polyamide Diamond Network Solids

The set of specimens is based on the diamond network model. For six diﬀerent values of the disconnection probability p, eight samples of one realisation with a unit cell length of 7.5 mm and a volume fraction of φ = 0.14 were manufactured. Cylindrical subsections of theses networks with 60 mm in height and diameter were laser-sintered on an EOSINT P360 machine using polyamide 12 (PA2200, EOS GmbH, Krailing). Literature values for the microscopic Young’s modulus Es of polyamide 12 are only available from tensile tests. The results vary from around 550 N/mm2 up to 1700 N/mm2, depending on processing strategy and positioning of the samples in the building chamber [257, 39]. Fig. 5.12 shows photographs of one sample.

103 5 Topology Dependence of Eﬀective Linear-Elastic Properties

(a) (b) (c)

Figure 5.12: (a) Photograph of one specimen based on the diamond network model with unlink probability p = 0.2. (b) View from top showing passing instead of crossing struts. (c) In the close-up view, unlinked nodes are shown. It can be seen that not all nodes are cleanly disjoint; there is still some material between two disconnected parts. Note the surface roughness of the structure resulting from the production from powder.

A principle issue of selective laser-sintering is the anisotropy in z-direction caused by the layer-by-layer build-up. In order to be able to compare mechanical properties of different specimens, their orientation during the fabrication has to be the same. Here, all samples were built in z-direction. In addition, inhomogeneities in mechanical strength may occur due to shortcomings of the production method such as only partial fusion of layers to the underlying structure, inhomogeneously distributed powder causing deflections from the strut thickness of the model or imprecise melting of the powder yielding filled holes or missing connections, see Fig. 5.12(c). In most of the cases, this results in weak spots. Hence, theoretically identical samples vary in their properties due to statistical variations in melting the powder and dependent on where in the building chamber the sample was placed. Therefore, eight identical samples of one random network realisation were produced for each value of probability p. The samples were placed in two lines at different heights in the building chamber (4 specimens in each line) to get independent data points and an error estimate.

Since the mechanical properties of polyamide are sensitive to environmental conditions, it is most important that the set of specimens is in the same initial condition prior to measurement to avoid another change in parameter. For this reason, they underwent the same preparatory treatment of storing at 70◦ in a vacuum furnace until constant weight was achieved.

104 5.3 Compression Testing of Laser-Sintered Diamond Network Solids

Figure 5.13: Experimental set-up for uniaxial compression: A Zwick/Roell type 1484 and a 50 kN measuring box were used to record force-displacement curves.

Uniaxial compression testing

A Zwick/Roell universal testing machine type 1484 was used for the uniaxial compression along the cylinder axis of the laser-sintered specimens. Fig. 5.13 shows a photograph of the experimental set-up. The cylindrical object is aligned in z-direction, the building direction of the production process. At the top and at the bottom, the cylinder is confined by the loading platens, the sides of the specimens are free. By lowering the upper loading platen at a speed of 10 mm/min, the sample is deformed and its responding force is quantified by a 50 kN measuring box. The displacement was measured using the crosshead displacement. A typical force-displacement diagram is shown in Fig. 5.14. It consists of a linear regime and a plateau before breakage. In the linear regime, Young’s modulus Ez is defined as

Tzz Ez := (5.7) Ezz where Tzz is the stress and Ezz is the strain in z-direction. The summation convention does not apply here. The applied force F acts uniformly on the entire surface A of the specimen. A force per unit area can be deﬁned as P := F/A. From the boundary conditions, we get Tzz = P [143]. Using Eq. (4.10), the compression displacement can be written as function of the z-coordinate Z in the reference conﬁguration (see section 4.1)

u = (ux, uy, uz) = (0, 0, λZ − Z) (5.8)

105 5 Topology Dependence of Eﬀective Linear-Elastic Properties with the compression factor λ = h/h0 where h and h0 are the height of the sample after compression and its initial height, respectively. As we only consider the linear regime, Eq. (4.23) can be used and gives

∂uz h − h0 ∆h Ezz = = λ − 1 = =: . (5.9) ∂Z h0 h0

Thus, Eq. (5.7) can be rewritten to P F/A E = = . (5.10) ∂uz/∂Z ∆h/h0

In the experiment, the force F and the change in height ∆h are directly measured. Young’s modulus can thus be determined by a linear ﬁt to the measured force-displacement diagram using A F (∆h) = E ∆h . (5.11) h0

For all specimens, the cross-sectional area A of the samples equals A = πd2/4 = 2 2827.4 mm and the height of the cylinders is h0 = 60 mm, except for statistical variations due to the production process.

5.3.2 Eﬀective Young’s Modulus of Laser-Sintered Diamond Network Solids

Tab. 5.1 summarises the data gained by uniaxial compression experiments. In Fig. 5.14 (centre), force-displacement curves for several values of p are shown. According to Eq. (5.11), Young’s modulus E is given by the slope k of the force- displacement curves via E = k h0/A where A is the cross-sectional area and h0 is the initial height of the sample. Slope k is computed by linear least squares fitting of the data by a straight line F (∆h) = k ∆h + b with force F , displacement ∆h and the fit parameters k and b. The linear regime of the force-displacement curves is divided into two regions, see Fig. 5.14 (centre). The analysis is performed separately in region I and region II. The results are given in Tab. 5.1; In Fig. 5.14 (right) Young’s modulus E is plotted as a function of the disconnection probability p. In both regions, the data can be approximated by an exponential decay in p. However, the values in Region II are slightly lower than in Region I, as is the slope of the exponential fit. For each probability p, two series of structures were analysed that were built by the same method, but at different heights in the building chamber. As reported in Ref. [86], for instance, the mechanical properties of laser-sintered polymer samples vary with their position in the building chamber. One may thus expect that the two series can be told apart by means of their Young’s modulus. Especially for low p < 0.4, the data clearly separates into two lines where one is considerably stiffer than the other. This is ascribed to the difference in the production process between the two series and each line is interpreted as one series. It is peculiar that the data points for p = 0.2 do not fit to this assumption which raises the suspicion of a swap of these two samples after production. Since this cannot be ascertained, it is averaged over both series in order to compare the experimental values for Young’s modulus E to those obtained by simulation. This gives

106 5.3 Compression Testing of Laser-Sintered Diamond Network Solids

1500 102 p=0.0 p=0.2 Region I ] I II 2 1 p=0.3 10 [N/mm E 1000 100 0 0.1 0.2 0.3 0.4 0.5 p [N]

F 2 p=0.5 10 500 Region II ] 2

101 [N/mm E

0 100 0 2 4 6 8 0 0.1 0.2 0.3 0.4 0.5 ∆h [mm] p

Figure 5.14: (left) On top: Cylindrical sample of a laser-sintered model of the diamond network with a solid volume fraction φ = 0.14 and p = 0.2. Below: A close-up view of a sample with φ = 0.14 and p = 0.5 shows both connected and separated nodes. (centre) Force-displacement diagram obtained by compression testing, for several values of p. ∆h is the compression applied vertically along the cylinder axis to a cylinder of initial height h0, F is the measured resulting force caused by the elastic response of the material. The linear region in the force-displacement diagram is divided into Region I and Region II in which the slopes are calculated independently. (right) Young’s modulus E is given by E = k h0/A where k is computed by linear least squares ﬁtting of F (∆h) by a straight line F (∆h) = k ∆h + b (compare Eq. (5.11)), in the regions I and II individually. All specimens have an initial height h0 = 60 mm and diameter D = 60 mm resulting in a cross-sectional area A = π D2/4 = 2827.4 mm2. Both, the data in Region I and in Region II can be approximated by an exponential decay in p. Note, that the values in Region II are slightly lower as in Region I and that the exponential decay is not as steep. For each p, two series of structures were analysed (built by the same method, but at slightly diﬀerent positions in the construction chamber). Between 1 and 4 specimens were analysed for each of the two series. The results of series 1 and 2 are displayed in red and blue, respectively. The ¯ P data points are averages k = i ki/M over the M specimens (not averaged over the two distinct series), and error bars are the error of the average P ¯ 2 1/2 ∆k = (1/(M(M − 1)) i(ki − k) ) of the distribution of k over the M samples.

107 5 Topology Dependence of Eﬀective Linear-Elastic Properties

10-1

s E / E 10-2

dia 10-3 nbo foam 4 dia experiment 3

2 [mm] Rp p

1 R 0 0.1 0.2 0.3 0.4 0.5 p

Figure 5.15: Effective Young’s modulus E for the diamond, nbo and foam networks with a volume fraction of φ = 0.14, obtained by finite element analysis, and for the selective laser-sintered diamond network by uniaxial compression. The material Young’s modulus Es of the laser-sintered model is not known. 2 Data is shown assuming Es = 900 Nm/mm (which is well within published estimates for polyamide 12 [257, 39]) chosen to give approximately similar absolute values to the corresponding finite element data; see also Fig. 5.14 for details of the experimental data. Also shown (on the right-hand y-axis) is the average pore size Rp of the diamond network that remains almost constant as function of p.

rise to large error bars as the error ∆E is calculated√ as half the difference√ between the two series plus the maximum of the errors ∆E1 = σ1/ M1 and ∆E2 = σ2/ M2, σ1 and σ2 being the standard deviations and M1 and M2 being the number of analysed samples of series 1 and 2, respectively. The total error is thus ∆E = |E1 − E2|/2 + max(∆E1, ∆E2). Fig. 5.15 shows both numerical and experimental results in comparison for different geometries at constant volume fraction φ = 0.14. The ratio E/Es is plotted as a function of the disconnection probability p, where E is the effective Young’s modulus of the network structure and Es the microscopic Young’s modulus of the solid material. As the exact value of the material Young’s modulus Es of the laser-sintered model is not 2 known, Es = 900 Nm/mm is chosen as it gives approximately similar absolute values compared to the corresponding finite element data. The assumed value lies well within 2 2 published estimates of 550 N/mm up to 1700 N/mm for polyamide 12 [257, 39]. The experimental data, now averaged over both series, supports the sensitive dependence of Young’s modulus E on p found by the finite element analysis. Within the error bars, the

108 5.4 A New Design Principle for Bone Scaﬀolds? functional dependence E(p) can also be approximated by an exponential decay with a slope similar to that of the simulated diamond network structure.

5.4 A New Design Principle for Bone Scaﬀolds?

The effective elastic properties of random network solids with decreasing connectivity were determined numerically using the Finite Element (FE) method and experimentally by uniaxial compression tests. Both results show a strong decay of Young’s modulus E with increasing disconnection probability p. For samples much larger than the length scale of the microstructure, varying p from 0 to 1 changes the mechanical properties by several orders of magnitude as shown in Fig. 5.4. Hence, p can be used as design parameter, for example, for bone scaffolds. The stiffness of the scaffold can be adjusted, more precisely lowered, to a required value by increasing p. Fig. 5.15 also demonstrates that the shape of the void or pore space, quantified by the pore size Rp which is determined using the covering radius transform [246, 168], changes far less with increasing disconnection probability p than the topology of the solid domain. The solid and the void volume fractions and structural length scales remain constant, and pore size distributions change only marginally. This is an attractive feature for the design of artificial scaffolds as pore transport properties (predominantly determined by pore geometry) can be held fixed while mechanical properties can be adjusted to the desired value.

109 5 Topology Dependence of Eﬀective Linear-Elastic Properties

Table 5.1: The data obtained by uniaxial compression of laser-sintered network models with different values of p is plotted in a force-displacement diagram F (∆h) (see Fig. 5.14). In the linear region, F (∆h) = k ∆h + b is fitted with fit parameters k and b using the linear least squares method. For each probability p, two series of structures were analysed that were built at slightly different positions in the building chamber. For each series, M specimens were investigated. The ¯ P data points are averages k = i ki/M over the M specimens (not averaged over the two distinct series), the stated error is the error of the mean value P ¯ 2 1/2 ∆k = (1/(M(M − 1)) i(ki − k) ) .

Region I Region II series 1 series 2 series 1 series 2

348.2 557.1 280.5 492.6 p = 0.0 ki 385.4 464.6 334.3 409.4 531.2 468.4 k¯ 367 518 307 457 ∆k 27 48 39 5 219.6 449.6 183.0 368.0 p = 0.1 ki 246.9 455.9 202.1 365.6 210.1 169.7 k¯ 226 453 185 367 ∆k 20 5 17 2 348.5 166.6 283.6 151.1 p = 0.2 ki 386.8 172.6 317.8 149.1 276.7 234.3 k¯ 337 170 279 150 ∆k 56 5 42 2 144.0 235.3 143.9 231.8 p = 0.3 ki 154.7 169.2 155.8 167.1 154.6 141.5 k¯ 151 202 147 199 ∆k 7 47 8 46 139.5 156.1 176.6 192.8 p = 0.4 ki 138.8 174.7 140.1 180.0 111.9 129.0 k¯ 133 156.1 165 192.8 ∆k 14 — 25 — 81.5 55.4 131.4 86.3 p = 0.5 ki 72.0 67.4 121.2 89.6 87.6 120.9 k¯ 80 61 125 88 ∆k 8 9 6 3 110 6 The Role of Network Topology in Material Design and Physics

In material design, the material’s microstructure can be used to adjust or optimise a material’s performance. A prerequisite is the understanding of the influence of microstructural changes on the physical properties of the material which requires the knowledge of structure-property relationships. The physical properties of network-like systems (such as cross-linked bio-polymer networks or metal foams) are largely determined by the system’s geometry or spatial structure. This thesis has studied the effect of topology changes while keeping the volume fraction and the pore geometry approximately constant. A new network model was introduced with a continuous topology change controlled by the node disconnection probability p. An initially four-coordinated network node is replaced by two two-coordinated vertices with probability p. Varying p from 0 to 1 turns a network with originally exclusively four-coordinated nodes into a random bundle of unbranched fibres. Using the example of the square and the diamond lattice, an investigation of the model’s percolation properties in two and three dimensions showed a percolation transition at p = 1 if all possible disconnection configurations of a node are equally probable. A remarkable feature of the model is that there is always an even number of percolating clusters which in three dimensions even increases with system size. This clearly separates our model from usual site or bond percolation for which in the thermodynamic limit there is exactly one percolating cluster. It seems that the model even belongs to a different universality class than normal percolation since the pair-connectedness function and finite size scaling yield critical exponents ν = 1.29 ± 0.04 and β = 0.32 ± 0.02 (ν = 0.54 ± 0.01, β = 0.0021 ± 0.0004 in 3D) that differ considerably from the critical exponents ν = 4/3 and β = 5/36 (ν = 0.88, β = 0.41 in 3D) for site and bond percolation. By inflating the network structures to a certain radius, so-called network solids are obtained. A sensitive structure-property relationship between mechanical properties and the average connectivity is revealed by Finite Element analysis of the effective linear-elastic properties of network solids with disconnected four-coordinated vertices. Separating a fraction of the network nodes leads to a strong decay of the effective linear- elastic moduli, verified for ordered and irregular initial structures. Whether or not the network represents the edges of a cellular structure (as does the foam sample) or those of a periodic network (diamond and nbo) does not affect this behavior. Uniaxial compression experiments on laser-sintered realisations of disconnected diamond structures confirmed the numerical findings. This model achieves a separation of geometry and topology, in the sense that the shape of the void or pore space changes far less with increasing disconnection probability p than the topology of the solid domain. The solid and void volume fractions and the structural length scales remain constant, and pore-size distributions change only marginally. We emphasise that the structural changes of the networks are distinctly different from

111 6 The Role of Network Topology in Material Design and Physics

those of a standard network percolation model. In standard percolation models, random deletion or cutting of an increasing number of edges or vertices allows stresses to relax locally. Deletion of a sufficient fraction of edges results in a break up of the network into connected components none of which span the entire system. This leads to a mechanical failure of the resulting network solid at a finite model-specific value of p that for the diamond network is 0.57 [233]. In contrast, our network model is not based on cutting edges and local relaxation of stress, but on redirecting the force lines at vertices. The network remains percolating for all values of p < 1, but the change in topology changes the way stress is transmitted through the network.

6.1 Design of Bone Scaﬀolds and Metal Structures

Two applications of the investigated randomised networks are bone scaffold design and an alternative production method for solid metal foams. The two main requirements for an artificial bone scaffold are temporary mechanical support of the order of the original tissue and pore sizes that facilitate cell attachment and enable flow through the scaffold [104]. The vertex model features a separation of geometry and topology which can be used for the production of custom-designed scaffolds. First, a structure appropriate for tissue regrowth is produced. Then, by topology changes, the mechanical properties of a scaffold may be adjusted by orders of magnitudes while preserving the pore space geometry and hence the flow properties. Beyond bone scaffolds, the proposed model suggests that network connectivity is also an important parameter for open-cell metal foams. Assume we do not start from a four-coordinated network but from a convoluted ravel of metal fibres. The random insertion of mechanical connections between the long unconnected fibres, e.g. by sintering [153, 48, 148] , may yield a production process for porous metal fibre structures with a sensitive structure-property relationship similar to that described in chapter 5. This method would be capable of producing random foam-like structures with adjustable mechanical properties.

6.2 Relevance for Cross-linked Bio-polymer Networks

During the last decade, the mechanical response of bio-polymer networks has been in the focus of theoretical and experimental investigations. Both single fibre properties and network response have been studied in detail [252, 222, 268, 191, 197, 235]. A characteristic feature of bio-polymers is the non-linear strain-stiffening effect making the network stiffer for increasing strains. Although only dealing with the static linear-elastic response of networks, the 3D model presented in this work is of relevance as most theoretical studies are so far only for 2D systems, e.g. [151, 260, 95, 97, 265, 49]. Apart from the polymer concentration, the parameters determining the network architecture are the cross-linker density which is related to the distance between cross-links lc and the connectivity of the network. Separating a node in the vertex model corresponds to decreasing the cross-linker density which results in an increased lc. For disconnection probability p > 0.5, we find a power-law decay of the shear modulus with the number of disconnected nodes which conforms to [74, 243, 108].

112 6.3 A Percolation Model in a New Universality Class?

When dealing with floppy polymers, i.e. polymers with a persistence length lp of about the mesh size or smaller as, for example, fibrin [118] or vimentin [92], thermal fluctuations have to be taken into account. The fluctuations cause fibre undulations resulting in a reduced stretching resistance [151]. With straight segments between cross-links, our model can only be applied to stiff polymer networks such as collagen and F-actin or flexible polymers at T = 0. It remains to be tested if and to what extent fluctuations can be included in the model by using a reduced effective microscopic bulk modulus for the voxels. Head et al. [95, 96] and Wilhelm and Frey [260] found the stiffness exponent fc of 2D random networks consisting of stiff fibres to be fc ≈ 3 which appears to be non-universal. It is speculated that this holds also in 3D which could be tested using, for instance, the here presented model. Concerning the elasticity results described in chapter 5, a comparison of the calculated power-law exponents for the elastic moduli with those in Refs. [151] and [108] would be interesting. For this purpose, an analytical calculation of the mean distance between four-connected nodes — which is consistent with the distance between cross-links lc in the bio-polymer models — as function of the disconnection probability p is necessary.

6.3 A Percolation Model in a New Universality Class?

The percolation properties of the vertex model (chapter 3), and the suggestion that it belongs to a new universality class, suggest a whole list of new interesting issues. As the percolation model was only considered on the diamond lattice, a confirmation of the results on different lattices is an important next step. Furthermore, in a side project of this thesis (Matthias Hoffmann’s diploma thesis, Ref. [102]), a local asymmetry parameter x is introduced in the two-dimensional disconnection mechanism. It is shown that the system has a phase transition at some critical point 0 < pc(x) < 1 with the bond percolation critical exponents. A generalisation of this model to three dimensions would be extremely helpful in order to determine the nature of the transition at p = 1 in the symmetric vertex model. Finally, an analytical model similar to Baxter’s eight-vertex model, a six-vertex model in the case of the 2D square lattice, explaining the difference of the found critical exponents from normal percolation exponents is desirable, and appears possible.

113

Acknowledgements

First of all, I would like to thank Prof. Dr. Klaus Mecke for the opportunity to do a PhD at his chair.

I thank my advisers Dr. Gerd Schröder-Turk and Prof. Dr. Klaus Mecke for the supervision of my research work and also for suggesting the topic of the work. I am grateful to Gerd for his help and ideas and the many fruitful discussions during the work. Thanks, Gerd, for your positive attitude and your steady encouragement and motivation, and particularly for your support to finally finish this work. I thank Klaus for his scientific support especially in interpreting the data, his literature suggestions and discussions. I am also grateful for the opportunity to perform parts of this work at the Australian National University (ANU) in Canberra/Australia, and acknowledge support by the German academic exchange service (DAAD) through a joint PPP project with the "Group of Eight" universities for this visit.

I would like to acknowledge Prof. Dr. Christoph Arns (School of Petroleum Engineering, The University of New South Wales, Sydney/Australia) and Dr. Mahyar Madadi (Applied Mathematics, Australian National University, Canberra/Australia) who introduced me to the finite element method and elasticity during my stay at the ANU. I owe Sebastian Kapfer a debt of gratitude for writing the finite element code used for the elasticity analyses in this thesis, for adjusting the code to my needs and for always answering my questions concerning the usage, error messages, compilation problems ... Many thanks to Matthias Hoffmann for programming the two-dimensional version of the network percolation which turned out to be really exciting. Also, I owe thanks to Andrew Kraynik (Sandia National Laboratories, Albuquerque, New Mexico/USA) for providing foam structures of different sizes generated using Brakke’s Surface Evolver and to Ben Fabry (Zentrum für Medizinische Physik und Technik, Friedrich-Alexander-Universität Erlangen-Nürnberg) and Stefan Münster (Zentrum für Medizinische Physik und Technik, Friedrich-Alexander-Universität Erlangen-Nürnberg) for the confocal microscopy data sets of the collagen networks. I thank Prof. Dr. Thomas Franosch for fruitful discussions about percolation and scaling. Furthermore, I would like to show my gratitude to the Regionales Rechenzentrum Erlangen (RRZE) for letting me run my jobs on the computer cluster “lima” before the official opening of the new cluster. Thanks also for the help with the compilation and all other problems. I am grateful to Carl Fruth (FIT GmbH, Parsberg) and Alexander Oster (NetFabb GmbH, Parsberg) for laser-sintering multiple diamond network solids, free of charge. Many thanks to the Institute of Polymer Technology chaired by Prof. Dr.-Ing. Dietmar Drummer, especially to Dominik Rietzel and Jürgen Karsten for the mechanical testing

115 Acknowledgements

and to Holmger Ullrich for photography of the samples.

I claim to be no “DAU”, i.e. “dümmster anzunehmender User”, yet I had many questions and problems concerning the computer, or even more often, the printer! Sebastian Kapfer, Johannes Reinhard, Markus Spanner, Christian Goll and Matthias Hoffmann supported me whenever I had an IT problem. Many thanks for your kind and immediate help. Finally, I would like to thank Margret Heinen-Krumreich for her help with all adminis- trative issues. It was a pleasure to share my office with Kerstin Falk, Anatoly Danilevich and Matthias Hoffmann. Special thanks to Tolik for the regular tea supply. My special thanks to all people who were associated to ”Die Weihnachtsvorlesung - Sandmännchens Reise durch die Welt der granularen Materie” (2008). This event was a lot of work, but when looking at it now, it was worth it. Thanks to everybody who put his heart and soul into it. It was a great show. I would like to thank Katrin Kania, Alexander Mattausch and Anatoly Danilevich for reading this manuscript with great care.

I gratefully acknowledge the support of the Cluster of Excellence “Engineering of Advanced Materials” at the University of Erlangen-Nuremberg, which is funded by the German Research Foundation (DFG) within the framework of its “Excellence Initiative”.

116 List of Abbreviations

2D two-dimensional

3D three-dimensional

Nb niobium

NbO niobium monoxide

O oxygen

EDM Euclidean distance map

FE Finite Element

FEM Finite Element Method

SLS Selective laser-sintering

117

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