^{1,2,a)}, Søren Enemark

^{3}and Raj Rajagopalan

^{1,2,3}

### Abstract

The microstructural basis of the characteristic nonlinear mechanics of biopolymernetworks remains unclear. We present a 3D network model of realistic, cross-linked semiflexible fibers to study strain-stiffening and the effect of fiber volume-occupancy. We identify two structural parameters, namely, network connectivity and fiber entanglements, that fully govern the nonlinear response from small to large strains. The results also reveal distinct deformation mechanisms at different length scales and, in particular, the contributions of heterogeneity at short length scales.

This work was supported by the National University of Singapore Grants R279-000-214-133 and R279-000-214-731, and the Chemical and Pharmaceutical Engineering Program of the Singapore-MIT Alliance. N.A.K. was supported by NUS Graduate School for Integrative Sciences and Engineering (NGS) and the Global Enterprise for Micro-Mechanics and Molecular Medicine (GEM4). We thank the anonymous reviewers for their constructive comments on the manuscript.

I. INTRODUCTION

II. METHODS

A. Network model

B. Network generation and deformation

III. RESULTS

A. Parameterization of network structural heterogeneities

B. Length-scale-dependent network mechanics at small strains

C. Nonlinear strain-dependent network mechanics

D. Networkdeformation mechanism

IV. DISCUSSION

V. CONCLUSIONS

### Key Topics

- Networks
- 18.0
- Biopolymers
- 15.0
- Cell mechanics
- 13.0
- Elasticity
- 12.0
- Cell filament networks
- 11.0

## Figures

Summary of fiber model and interactions.

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.

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 }.

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.

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 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.

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).

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 }.

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 }.

## Tables

List of independent variable parameters for simulating semiflexible polymer network model.

List of independent variable parameters for simulating semiflexible polymer network model.

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