Elasticity of Stiff Polymer Networks

We study the elasticity of a two-dimensional random network of rigid rods (“Mikado model”). The essential features incorporated into the model are the anisotropic elasticity of the rods and the random geometry of the network. We show that there are three distinct scaling regimes, characterized by two distinct length scales on the elastic backbone. In addition to a critical rigidity percolation region and a homogeneously elastic regime we find a novel intermediate scaling regime, where the elasticity is dominated by bending deformations.

Figure 1

Typical networks at low and high density. Dangling bonds, not contributing to the elasticity, have been cut off. The stress distribution is shown in false colors; the load on a filament increases from blue to red. The left picture is for $\rho =10$, system size $L=10$, and an aspect ratio $\alpha =0.0001$. $99.99\%$ of the strain energy is stored in bending modes. In contrast, the right picture shows a network for $\rho =50$, $L=2$, and $\alpha =0.01$, where only $5\%$ of the strain energy is in bending modes; the remainder is stored in compression modes. For the choice of units see the main text.

Figure 2

Double logarithmic plot of the scaling function $h(x)$ for the shear modulus of the “Mikado model” with free hinges as a function of $x=L|\delta \rho {|}^{\nu }$ with $\delta \rho =\rho -{\rho }_c$ for a series of densities $\rho$ indicated in the graph. Note that for finite systems the shear modulus is also nonzero below ${\rho }_c$ (lower branch in the plot). The corresponding plot for stiff hinges is similar, but exhibits a different ${\rho }_c=5.71$ (see text).

Figure 3

Double logarithmic plot of the shear modulus $G$ as a function of $\alpha$ for fixed $k_{\mathrm{b}\mathrm{e}\mathrm{n}\mathrm{d}}(\ell )$. Data shown for free hinges.

Figure 4

Scaling plot of the shear modulus for free hinges for a series of densities above $\rho =15$ indicated in the graph (same data as in Fig. 3). Data collapse to the crossover scaling form, Eq. (2), is obtained with ${\nu }^{\prime }=2.83$. Note that here and in all other figures the unit of length is $\ell$ and the unit of the shear modulus $G$ is $\kappa /{\ell }^3$.