Khandagale Pratik, Breitzman Timothy, Majidi Carmel, Dayal Kaushik
Department of Mechanical Engineering, Carnegie Mellon University, Pittsburgh, Pennsylvania 15213, USA.
Air Force Research Laboratory, Wright-Patterson Air Force Base, Ohio 45433, USA.
Phys Rev E. 2023 Jun;107(6-1):064501. doi: 10.1103/PhysRevE.107.064501.
Polymer networks formed by cross linking flexible polymer chains are ubiquitous in many natural and synthetic soft-matter systems. Current micromechanics models generally do not account for excluded volume interactions except, for instance, through imposing a phenomenological incompressibility constraint at the continuum scale. This work aims to examine the role of excluded volume interactions on the mechanical response. The approach is based on the framework of the self-consistent statistical field theory of polymers, which provides an efficient mesoscale approach that enables the accounting of excluded volume effects without the expense of large-scale molecular modeling. A mesoscale representative volume element is populated with multiple interacting chains, and the macroscale nonlinear elastic deformation is imposed by mapping the end-to-end vectors of the chains by this deformation. In the absence of excluded volume interactions, it recovers the closed-form results of the classical theory of rubber elasticity. With excluded volume interactions, the model is solved numerically in three dimensions using a finite element method to obtain the energy, stresses, and linearized moduli under imposed macroscale deformation. Highlights of the numerical study include: (i) the linearized Poisson's ratio is very close to the incompressible limit without a phenomenological imposition of incompressibility; (ii) despite the harmonic Gaussian chain as a starting point, there is an emergent strain-softening and strain-stiffening response that is characteristic of real polymer networks, driven by the interplay between the entropy and the excluded volume interactions; and (iii) the emergence of a deformation-sensitive localization instability at large excluded volumes.
由交联柔性聚合物链形成的聚合物网络在许多天然和合成软物质系统中普遍存在。当前的微观力学模型通常不考虑排除体积相互作用，例如，除了通过在连续尺度上施加唯象的不可压缩性约束之外。这项工作旨在研究排除体积相互作用对力学响应的作用。该方法基于聚合物自洽统计场理论的框架，它提供了一种有效的中尺度方法，能够在不进行大规模分子建模的情况下考虑排除体积效应。一个中尺度代表性体积单元填充有多个相互作用的链，并且通过这种变形映射链的端到端向量来施加宏观尺度的非线性弹性变形。在没有排除体积相互作用的情况下，它恢复了经典橡胶弹性理论的封闭形式结果。对于有排除体积相互作用的情况，使用有限元方法在三维中对模型进行数值求解，以获得在施加的宏观尺度变形下的能量、应力和线性化模量。数值研究的亮点包括：（i）线性化泊松比非常接近不可压缩极限，而无需唯象地施加不可压缩性；（ii）尽管以调和高斯链为起点，但由于熵和排除体积相互作用之间的相互作用，出现了真实聚合物网络特有的应变软化和应变硬化响应；以及（iii）在大排除体积下出现变形敏感的局部化不稳定性。
