[论文解读] Stiffening of semiflexible biopolymers and cross-linked networks

T. van Dillen, Patrick R. Onck+1 more|ArXiv.org|Nov 23, 2006

Advanced Materials and Mechanics参考文献 1被引用 3

一句话总结

本文研究了剪切作用下二维交联半柔性生物聚合物网络的机械硬化行为，比较了纤维波动的静态与动态模型。结果表明，与静态直线化相比，动态热波动可使轴向刚度提高2至4倍，且硬化效应源于纤维层面的熵弹性硬化与网络层面的非协调重取向，其中在大应变下拉伸占主导地位。

ABSTRACT

We study the mechanical stiffening behavior in two-dimensional (2D) cross-linked networks of semiflexible biopolymer filaments under simple shear. Filamental constituents immersed in a fluid undergo thermally excited bending motions. Pulling out these undulations results in an increase in the axial stiffness. We analyze this stiffening behavior of 2D semiflexible filaments in detail: we first investigate the average, {static} force-extension relation by considering the initially present undulated configuration that is pulled straight under a tensile force, and compare this result with the average response in which undulation dynamics is allowed during pulling, as derived earlier by MacKintosh and coworkers. We will show that the resulting mechanical behavior is rather similar, but with the axial stiffness being a factor 2 to 4 larger in the dynamic model. Furthermore, we study the stretching contribution in case of extensible filaments and show that, for 2D filaments, the mechanical response is dominated by {enthalpic} stretching. Based on the single-filament mechanics, we develop a 2D analytical model describing the mechanical behavior of biopolymer networks under simple shear, adopting the affine deformation assumption. These results are compared with discrete, finite-element (FE) calculations of a network consisting of semiflexible filaments. The FE calculations show that local, nonaffine filament reorientations occur that induce a transition from a bending-dominated response at small strains to a stretching-dominated response at larger strains. Stiffening in biopolymer networks thus results from a combination of stiffening in individual filaments and changes in the network topography.

研究动机与目标

• 理解剪切变形下半柔性生物聚合物网络中应变硬化的起源。
• 比较纤维响应的静态（波动冻结）与动态（拉伸过程中热涨落）模型。
• 量化单根纤维硬化与网络重排对整体力学响应的相对贡献。
• 基于仿射变形假设，建立二维网络的解析模型，并通过有限元模拟进行验证。
• 确定在二维半柔性纤维中，熵弹性还是焓弹性贡献在力学响应中占主导地位。

提出的方法

• 利用统计力学建模半柔性纤维的平衡波动形态，通过路径积分和抛物柱面函数推导松弛分布。
• 计算不可伸长与可伸长纤维在拉伸载荷下的力-伸长关系，纳入热涨落效应。
• 基于仿射变形假设与单根纤维的力-伸长关系，构建二维解析网络模型。
• 对初始呈波状的二维网络进行离散有限元（FE）模拟，施加单向剪切载荷。
• 将解析仿射模型与FE结果进行比较，评估仿射假设的有效性并量化非仿射效应。
• 通过标度分析推导剪切刚度与纤维密度和刚度的关系：$ G \propto \rho \mu \frac{\Gamma^2}{(1+\Gamma^2)^{3/2}} $。

实验结果

研究问题

• RQ1 与静态直线化相比，拉伸过程中动态热波动如何影响半柔性纤维的轴向刚度？
• RQ2 单根纤维硬化与网络层面重排对整体应变硬化的相对贡献是什么？
• RQ3 在二维半柔性纤维中，主导力学响应的机制是熵弹性（弯曲涨落）还是焓弹性（拉伸）？
• RQ4 在离散网络模拟中，仿射变形假设与实际的非仿射纤维重取向相比如何？
• RQ5 在二维生物聚合物网络中，剪切刚度如何随应变、纤维密度和刚度变化？

主要发现

• 考虑拉伸过程中热波动的动态模型预测的轴向刚度比波动被冻结的静态模型高出2至4倍。
• 对于二维半柔性纤维，力学响应主要由焓弹性拉伸贡献，而非熵弹性弯曲效应。
• 有限元模拟揭示，由于非仿射纤维重取向，在大应变下响应由弯曲主导转变为拉伸主导。
• 生物聚合物网络的应变硬化源于纤维层面的硬化与网络层面拓扑变化的共同作用，而不仅仅是组分性质的体现。
• 解析仿射模型能定性捕捉行为，但在大应变下低估刚度，原因在于忽略了非仿射效应。
• 剪切刚度满足关系 $ G \propto \rho \mu \frac{\Gamma^2}{(1+\Gamma^2)^{3/2}} $，证实了模拟与实验中观察到的应变依赖性硬化行为。