Summary of fiber model and interactions.
Illustration of the 3D semiflexible polymer network model. (a) The network initially consists of straight rods with random positions and orientations. The beads making up the rods are shown. (b) Cross-links are assigned when beads from different fibers are located within δ cl from each other. The network is then energy-minimized based on the total potential U, resulting in relaxed network with local heterogeneities (c). The effective cylindrical topology of the fibers is shown. During the shear deformation of the network, beads within δ w from the bottom wall are fixed, while beads within δ w from the top wall are moved according to the imposed shear strain γ. An example of strained network (γ = 0.5) is shown in (d). The arrow indicates the direction of shear strain application. The sample network shown here has fiber volume fraction of Φ = 7.4%, consists of fibers with aspect ratio L/d = 20, and is cross-linked with δ cl /d = 0.7. Local heterogeneities are intrinsically introduced through self-assembly of the cross-linked network.
Typical overall response of cross-linked semiflexible polymer networks. Shear stress τ is shown as a function of shear strain γ for networks of different volume fractions (Δ, Φ = 9.3%; □, Φ = 7.4%; ○, Φ = 5.6%). Three distinct regimes can be observed: low network stiffness G 0 at small strains, high network stiffness G L at large strains, and the nonlinear transition between the two strain regimes, starting at the critical strain γ c .
The relation of input δ cl and Φ values to output R cl of the network [●, R cl (δ cl /d) for fixed Φ = 5.6%; ▲, R cl (Φ) for fixed δ cl = d]. The error bars are based on 200 different random network realizations. The mean of R cl (δ cl /d, ϕ) is shown in the inset.
Influence of network structure on the network response at small strain. Small-strain stiffness G 0 is plotted in (a) against R cl and in (b) against R e . The data are obtained from networks with fixed fiber dimensions but various Φ (▲, Φ = 9.3%; ■, Φ = 7.4%; ●, Φ = 5.6%) and cross-link densities, 0.5 < δ cl /d < 1.5, resulting in networks with varying connectivity. The relation between the averaged macroscopic quantity R cl and the averaged microscopic quantity R e is shown in (c). The dashed line is a guide for the eye.
Influence of network structure on the network response at intermediate and large strains. Both the critical strain γ c and the large-strain stiffness G L , shown in (a) and (b) respectively, exhibit scaling relations with R cl at all volume fractions tested (▲, Φ = 9.3%; ■, Φ = 7.4%; ●, Φ = 5.6%), demonstrating the importance of network structure parameters even beyond the linear elastic regime shown in Figure 5.
Illustration of nonaffine deformation. The initial configurations of the fibers are shown in the top panel, with the red dots indicating the location of cross-links. After shear deformation, the final configurations of the fibers (black lines) and cross-link locations differ from the expected configurations were the deformation affine (gray lines and dots).
Quantification of network affinity and rearrangement at different strain levels. The dimensionless parameters for nonaffinity A, as defined in previous work,17 and cumulative rearrangement V vary with γ. A measures the deviation of the actual position vector of the cross-links from the expected affine position upon strain increment δγ, , while V monitors the absolute deviation from u aff (γ) of all beads in the network up to γ. The vertical dashed line indicates γ c .
List of independent variable parameters for simulating semiflexible polymer network model.
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