Effective-medium approach for stiff polymer networks with flexible cross-links

Recent experiments have demonstrated that the nonlinear elasticity of in vitro networks of the biopolymer actin is dramatically altered in the presence of a flexible cross-linker such as the abundant cytoskeletal protein filamin. The basic principles of such networks remain poorly understood. Here we describe an effective-medium theory of flexibly cross-linked stiff polymer networks. We argue that the response of the cross-links can be fully attributed to entropic stiffening, while softening due to domain unfolding can be ignored. The network is modeled as a collection of randomly oriented rods connected by flexible cross-links to an elastic continuum. This effective medium is treated in a linear elastic limit as well as in a more general framework, in which the medium self-consistently represents the nonlinear network behavior. This model predicts that the nonlinear elastic response sets in at strains proportional to cross-linker length and inversely proportional to filament length. Furthermore, we find that the differential modulus scales linearly with the stress in the stiffening regime. These results are in excellent agreement with bulk rheology data.

Figure 1

(Color online) Schematic figure of an isotropic stiff polymer network with highly compliant cross-linkers.

Figure 2

(a) A single filament connected by $n$ flexible cross-links to the surrounding network, which we model as an effective elastic continuum (shown here as a gray background) and (b) illustrates the proposed nonuniform deformation of the cross-linkers on a single filament in a sheared background medium. (c) Force-extension curve of a finitely extensible Hookean (FEH) cross-linker (dashed curve) and of a WLC cross-linker (solid curve).

Figure 3

(Color online) (a) The modulus $G_{1\text{D}}={\tau }_0/ϵ$ for the 1D representation of the linear medium model with FEH cross-links with $K_{EM}=10k_{cl}$ (blue dash-dotted curve) and $K_{EM}=100k_{cl}$ (red dotted curve). We also show $G_{1\text{D}}$ for the model with WLC cross-links with $K_{EM}=10k_{cl}$ (blue dashed curve) and $K_{EM}=100k_{cl}$ (red solid curve). The inset shows the ratio of the average tension $\overline{\tau }$ and the midpoint tension ${\tau }_0$.

Figure 4

(Color online) The 1D differential modulus $d{\tau }_0/dϵ$ of the rod with WLC cross-linkers as a function of the extensional strain $ϵ$ imposed on the medium parallel to the orientation of the rod. The red dashed and black dotted curves show exact calculations using Eqs. (1, 2) and the solid blue and green dash-dotted curves show approximate calculations using Eq. (7).

Figure 5

(Color online) The differential modulus $K=d\sigma /d\gamma$ normalized by the linear modulus $G_0$ as a function of strain normalized by the critical strain ${\gamma }_c$. The universal curve for a semiflexible polymer network with rigid cross-links is shown as a black dashed curve. We also show the results of the self-consistent model with WLC cross-links (red solid curve) and simple cross-links (blue dash-dotted curve), the linear medium model with WLC cross-links with $K_{EM}=100k_{cl}$ (green dotted curve).

Figure 6

(Color online) The differential modulus $K=d\sigma /d\gamma$ normalized by the linear modulus $G_0$ as a function of stress normalized by the critical stress ${\sigma }_c$ for the self-consistent model with WLC cross-links (red solid curve), FEH cross-links (blue dash-dotted curve), and the linear medium model with WLC cross-links with $K_{EM}=100k_{cl}$ (green dotted curve). The inset shows the rigid linker model together with the self-consistent model for a network with flexible cross-links. In this case the stress is normalized by a stress ${\sigma }_0$, which marks the knee of the curve.

Figure 7

(Color online) The reduced tension profile along the rod, normalized by the midpoint tension ${\tau }_0$. This profile is calculated with the self-consistent model with WLC cross-links.

Figure 8

(Color online) The differential stiffness $df/du$ as a function of force $f$ for a filamin cross-linker (solid line) and for several F-actin polymer segment lengths.