# Elasticity of floppy and stiff random networks

Figure 1. A two-dimensional visualization of a fibrous network undergoing shear strain (green arrows). The cross-link density is such that it is approximately an isostatic material.

The authors examine the elastic properties of amorphous materials with a numerical simulation of a random network of harmonic springs as a model system in two dimensions. They show mathematically and through the simulation results that the behavior of the system can be characterized in terms of only a few key parameters, the most important of which is the network coordination, or the average number of springs per node in the network, relative to a specific case that they describe as a "marginally rigid network". They call this parameter $\delta z$. It is found that a critical point in the elastic response of the material occurs for both floppy ($\delta z < 0$) and stiff ($\delta z > 0$) networks when they approach $\delta z = 0$. They also explain how these results can be applied to materials that are characterized as glassy or fibrous networks.

## General Information

Authors: M. Wyart, H. Liang, A. Kabla, and L. Mahadevan.

Date: November 19, 2008.

School of Engineering and Applied Sciences, Harvard University, Cambridge, Massachusetts 02138, USA

Physical Review Letters 101, 215501 (2008). [1]

## Summary

Figure 2. (a) A chain of connected springs can rotate freely without any energy input until they are all aligned. (b) The authors simulate a random network of 10,000 particles with coordination z = 4.0 subjected to shear strain 0.005, and show that it responds heterogeneously. (c) Zooming in shows that there are large regions of rotational deformation within the network.

In this article, Wyart and colleagues attempt to characterize amorphous materials in general using a few key parameters. They first introduce the concept of the coordination number $z$ of a network of points that are connected to each other through ideal spring-like bonds:

$z = \frac{2 N_c}{N}$

where $N$ is the number of points in the network and $N_c$ is the number of constraints. One special case is a system which is isostatic, when the number of constraints exactly equals the number of degrees of freedom. If the system exists in d spatial dimensions, this corresponds to the case $z = 2d$, or $z = z_c = 4$ in the case of the authors' two-dimensional simulated network. The results of the study are applicable to three dimensions as well. The relative coordination parameter thus can be defined as $\delta z = z - z_c$.

One property that floppy networks have in common is that they are free to bend or stretch without any energy input, up to some threshold amount of bending and stretching. This is illustrated with a simple system of coupled springs in Figure 2(a). It is shown through simulation that this simple model in fact generalizes to large two-dimensional networks of springs. This case, $\delta z < 0$, gives a fundamentally different elastic response than those corresponding to a stiff system ($\delta z > 0$) or an isostatic system ($\delta z = 0$). An isostatic system responds as soon as there is a nonzero strain $\gamma$ applied, and the stress $\sigma \propto \gamma^2$. A stiff network, in contrast, will yield a stress $\sigma \propto \gamma$. The case $\delta z = 0$ can therefore be thought of as a critical point.

By modeling and analyzing the displacement field in the floppy network, it can be shown that the network amplifies shear with larger and larger displacements as $\delta z \to 0$. In other words, the amplification factor diverges as the network becomes rigid. Moreover, the authors demonstrate that the relative coordination of the network, with the two exponents corresponding to nonaffine displacements and the shear modulus, completely characterizes the system. The nature of the interactions between points in the network, as well as the distribution in types and sizes of interactions, determines the effective coordination parameter. If the relative network coordination $\delta z$ can be determined for a material, they argue, it can be used to classify the material relative to others.

## Connection to soft matter

This study examines the elastic properties of glasses, fibrous structures, and polymer networks more generally in terms of the "relative network coordination" $\delta z$, which can be thought of as a measure of the density of cross-links and the distribution of cross-links within a polymer network. If a rubbery material has cross-links that are sufficiently dense and overlapping such that the number of elastic constraints is large compared with the number of polymer strands, the material may surpass the critical isostatic condition $\delta z = 0$ and it will exhibit a rigid response to strain.

The authors offer an attractively simple way of thinking about what causes a network to be rigid and what enables a material to be floppy, or flexible up to a particular point without any energy input. However, while $\delta z$, their key parameter, is easy to control and measure in a two-dimensional network that they simulate, it is harder to characterize in any given material whose molecular structure and cross-linking pattern may not be immediately obvious or measurable.