Perkolation und Elastizität von Netzwerken - Von Zellulären Strukturen zu Faserbündeln

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.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 Perkolationseigenschaften 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 Radius 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, 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

Beyond the percolation universality class: the vertex split model for tetravalent lattices

Wepropose a statistical model defined on tetravalent three-dimensional lattices in general and the three-dimensional diamond network in particular where the splitting of randomly selected nodes leads to a spatially disordered network, with decreasing degree of connectivity. The terminal state, that is reached when all nodes have been split, is a dense configuration of self-avoiding walks on the diamond network. Starting from the crystallographic diamond network, each of the four-coordinated nodes is replaced with probability p by a pair of two edges, each connecting a pair of the adjacent vertices. For all values 0 ⩽ p ⩽ 1the network percolates, yet the fraction fp of the systemthat belongs to a percolating cluster drops sharply at pc = 1 to a finite value fp c . This transition is reminiscent of a percolation transition yet with distinct differences to standard percolation behaviour, including a finitemass f > 0 p c of the percolating clusters at the critical point. Application of finite size scaling approach for standard percolation yields scaling exponents for p → pc that are different from the critical exponents of the second-order phase transition of standard percolation models. This transition significantly affects the mechanical properties of linear-elastic realizations (e.g. as custom-fabricated models for artificial bone scaffolds), obtained by replacing edges with solid circular struts to give an effective density ϕ. Finite element methods demonstrate that, as a low-density cellular structure, the bulkmodulus Kshows a cross-over froma compression-dominated behaviour, K (ϕ) ∝ ϕκ with κ ≈ 1, at p = 0 to a bending-dominated behaviour with κ ≈ 2 at p=1

Morphology and linear-elastic moduli of random network solids

The effective linear-elastic moduli of disordered network solids are analyzed by voxel-based finite element calculations. We analyze network solids given by Poisson-Voronoi processes and by the structure of collagen fiber networks imaged by confocal microscopy. The solid volume fraction φ is varied by adjusting the fiber radius, while keeping the structural mesh or pore size of the underlying network fixed. For intermediate φ, the bulk and shear modulus are approximated by empirical power-laws K(φ) α φn and G(φ) α φm with n≈ 1.4 and m≈ 1.7. The exponents for the collagen and the Poisson-Voronoi network solids are similar, and are close to the values n = 1.22 and m = 2.11 found in a previous voxel-based finite element study of Poisson-Voronoi systems with different boundary conditions. However, the exponents of these empirical power-laws are at odds with the analytic values of n = 1 and m= 2, valid for low-density cellular structures in the limit of thin beams. We propose a functional form for K(φ) that models the cross-over from a power-law at low densities to a porous solid at high densities; a fit of the data to this functional form yields the asymptotic exponent n≈ 1.00, as expected. Further, both the intensity of the Poisson-Voronoi process and the collagen concentration in the samples, both of which alter the typical pore or mesh size, affect the effective moduli only by the resulting change of the solid volume fraction. These findings suggest that a network solid with the structure of the collagen networks can be modeled in quantitative agreement by a Poisson-Voronoi process. The dependence of linear-elastic properties on effective density is studied for porous network solids, by voxel-based finite element methods. The same dependence is found for solid structures derived from Poisson-Voronoi processes and from confocal microscopy images of collagen scaffolds. We recover the power-law for the bulk modulus for low densities and suggest a functional form for the cross-over to a high-density porous solid

Beyond the percolation universality class: the vertex split model for tetravalent lattices

Wepropose a statistical model defined on tetravalent three-dimensional lattices in general and the three-dimensional diamond network in particular where the splitting of randomly selected nodes leads to a spatially disordered network, with decreasing degree of connectivity. The terminal state, that is reached when all nodes have been split, is a dense configuration of self-avoiding walks on the diamond network. Starting from the crystallographic diamond network, each of the four-coordinated nodes is replaced with probability p by a pair of two edges, each connecting a pair of the adjacent vertices. For all values 0 ⩽ p ⩽ 1the network percolates, yet the fraction fp of the systemthat belongs to a percolating cluster drops sharply at pc = 1 to a finite value fp c . This transition is reminiscent of a percolation transition yet with distinct differences to standard percolation behaviour, including a finitemass f > 0 p c of the percolating clusters at the critical point. Application of finite size scaling approach for standard percolation yields scaling exponents for p → pc that are different from the critical exponents of the second-order phase transition of standard percolation models. This transition significantly affects the mechanical properties of linear-elastic realizations (e.g. as custom-fabricated models for artificial bone scaffolds), obtained by replacing edges with solid circular struts to give an effective density ϕ. Finite element methods demonstrate that, as a low-density cellular structure, the bulkmodulus Kshows a cross-over froma compression-dominated behaviour, K (ϕ) ∝ ϕκ with κ ≈ 1, at p = 0 to a bending-dominated behaviour with κ ≈ 2 at p=1

Tuning elasticity of open-cell solid foams and bone scaffolds via randomized vertex connectivity

Tuning mechanical properties of and fluid flow through open-cell solid structures is a challenge for material science, in particular for the design of porous structures used as artificial bone scaffolds in tissue engineering. We present a method to tune the effective elastic properties of custom-designed open-cell solid foams and bone scaffold geometries by almost an order of magnitude while approximately preserving the pore space geometry and hence fluid transport properties. This strong response is achieved by a change of topology and node coordination of a network-like geometry underlying the scaffold design. Each node of a four-coordinated network is disconnected with probability p into two two-coordinated nodes, yielding network geometries that change continuously from foam- or network-like cellular structures to entangled fiber bundles. We demonstrate that increasing p leads to a strong, approximately exponential decay of mechanical stiffness while leaving the pore space geometry largely unchanged. This result is obtained by both voxel-based finite element methods and compression experiments on laser sintered models. The physical effects of randomizing network topology suggest a new design paradigm for solid foams, with adjustable mechanical properties