Elastic response of filamentous networks with compliant crosslinks

Experiments have shown that elasticity of disordered filamentous networks with compliant crosslinks is very different from networks with rigid crosslinks. Here, we model and analyze filamentous networks as a collection of randomly oriented rigid filaments connected to each other by flexible crosslinks that are modeled as wormlike chains. For relatively large extensions we allow for enthalpic stretching of crosslink backbones. We show that for sufficiently high crosslink density, the network linear elastic response is affine on the scale of the filaments’ length. The nonlinear regime can become highly nonaffine and is characterized by a divergence of the elastic modulus at finite strain. In contrast to the prior predictions, we do not find an asymptotic regime in which the differential elastic modulus scales linearly with the stress, although an approximate linear dependence can be seen in a transition from entropic to enthalpic regimes. We discuss our results in light of recent experiments.

Figure 1

Schematic of a Mikado filamentous network. A crosslink is inserted at every intersection of any two filaments. A crosslink connecting two filaments is shown as a wiggly red line in the zoomed-up intersection of two filaments.

Figure 2

Force-extension curve of a wormlike chain crosslink, including enthalpic stretching. Crosslink becomes a Hookean spring when $u\gg l_{\mathrm{ee}}$, with stiffness equal to the differential stiffness of the WLC curve at $u=l_{\mathrm{ee}}$. Different curves correspond to different values of $l_{\mathrm{ee}}/l_0$. The lowermost curve corresponds to $l_{\mathrm{ee}}/l_0=0.90$, and successive curves lying above this one correspond to increasing values of $l_{\mathrm{ee}}/l_0$, with the diverging one corresponding to the WLC curve. The dashed line marks the divergence of force when a crosslink is stretched to its contour length. Inset: In the rest frame of the filament, the deformation appears as pure rotation indicated by the dashed arc. The crosslink is shown as a filled red circle at distance $x$ from the center of the filament stretches by $x\gamma \sin \left(2\theta \right)/2$.

Figure 3

Linear modulus as a function of the number of crosslinks per filament (symbols). Dotted line corresponds to the analytical prediction of Eq. (3). The curve connecting the symbols corresponds to the empirical correction to Eq. (3) in which $n$ is replaced by $n-n_c$, where $n_c=4.93$. Inset shows $G_0/(\rho k_{\mathrm{cl}}L/96)$ obtained from simulations (symbols) and the empirical fit (line) plotted with respect to the reduced crosslink density.

Figure 4

(a) Differential modulus in units of linear modulus $G_0$ for $n=70$ versus $\gamma$. The small vertical lines mark the critical strain ${\gamma }_c$. (b) Same as (a) but now strain is expressed in units of ${\gamma }_c$. Different curves correspond to different values of $n$. The inset shows the critical strain ${\gamma }_c$ for different values of $l_0$.

Figure 5

Affine calculation. Differential modulus in units of linear modulus $G_0$ versus $\sigma /{\sigma }_c$ for $l_0/L=0.1$ for one- and two-dimensional filamentous networks with compliant crosslinks.

Figure 6

(a) Differential modulus in units of linear modulus $G_0$ for $n=70$ versus $\sigma$. (b) Same as (a) but now stress is in units of ${\sigma }_c$. Different curves correspond to different values of $n$. The modulus scales approximately as $K\sim {\sigma }^2$ before it enters the power-law regime $K\sim {\sigma }^{3/2}$.

Figure 7

Differential nonaffinity as a function of strain for different crosslink densities. Strain is expressed in units of the critical strain. The maximum in the differential nonaffinity shifts towards the left with increasing crosslink density.

Figure 8

Average stress per filament as a function of the orientation angle of the filaments. The orientation angle is expressed in units of the angle ${\theta }_{\gamma }={\tan }^{-1}[1/(1+\gamma )]$ corresponding to the strain. The peak corresponds to the filaments oriented along the direction of the applied strain. The secondary smaller peak corresponds to the filaments oriented against the direction of the applied strain. Black lines correspond to the affine approximation. The inset shows the angular distribution of filaments. Besides a small increase in the number of filaments along the direction of strain, the distribution is practically uniform.

Figure 9

(a) Differential modulus in units of linear modulus $G_0$ for $n=70$ versus $\gamma /{\gamma }_c$ for different values of $l_{\mathrm{ee}}/l_0$. The modulus converges to a constant at high values of strain. The inset shows the differential modulus when filaments are compliant. (b) Differential modulus plotted with respect to stress in units of critical stress. Black solid line indicates the approximate $K\sim {\sigma }^2$. The dashed line with unit slope is a guide to the eye to show that the crossover regime approximately follows $K\sim \sigma$.

Figure 10

(a) Differential modulus in units of linear modulus $G_0$ for $n=10$ versus $\gamma /{\gamma }_c$. The inset shows the linear modulus $G_0$ at the rigidity percolation point. If the rotation of filaments around the crosslink costs energy, $G_0$ depends anomalously on the torsion constant $k_{\tau }$ and $k_{cl}$. (b) Differential modulus plotted with respect to stress. Black solid line depicts $K\sim {\sigma }^{3/2}$, whereas the dashed line indicates the approximate $K\sim \sigma$. Filled circles ($l_{\mathrm{ee}}/l_0=0.92$) and stars ($l_{\mathrm{ee}}/l_0=0.96$) correspond to enthalpic stretching of the crosslinks.

Figure 11

Network configuration at $\gamma =2.5{\gamma }_c$ for $n=10$ and $l_0/L=0.1$. Red symbols indicate the set of stretched crosslinks that sustain 90% of the total stress in the network.