Polydisperse polymer networks with irregular topologies I: Mechanics of cross-link distributions

Abstract

The structure of polymer networks, defined by chain lengths and connectivity patterns, fundamentally influences their bulk properties. While existing polymer network models connect chain properties to emergent network behavior, they are often limited to monodisperse networks with regular connectivities. In this work, we introduce a novel modeling framework that shifts the focus from individual polymer chains to cross-links and their connected chains as the fundamental unit of analysis. The key features of this framework are the relaxation of the cross-link junction position to satisfy local force balance, and physically intuitive means for satisfying material frame indifference. We explore two distinct limiting behaviors for the orientation of the frame of a cross-link: (1) the free rotation limit, which assumes the cross-link rotates to minimize free energy, and (2) the frame averaging limit, which incorporates structural heterogeneity by averaging over all possible cross-link orientations. It is found that an increase in variance in monomer numbers generally leads to network softening, while in bimodal networks, the onset of strain stiffening is controlled by shorter chains and the stiffening response is modulated by the ratio of short to long chains. By deriving closed-form approximations for both limits valid in the regimes of small deformation or small polydispersity, we offer an efficient computational approach to modeling the mechanics of complex, polydisperse networks. An aim of this framework is to take a step toward the rational modeling and design of heterogeneous polymer networks with structures tailored for specific properties.

Figures (10)

• Figure 1: Computational rendering of a polydisperse polymer network microstructure. Chains differ in monomer number, meet at cross-link junctions with differing number of chains, and have varying local connectivities. The polydisperse polymer network microstructure displayed here is generated via end-linking coarse-grained precursor chains in the LAMMPS molecular dynamics simulation package plimpton1995fastthompson2022lammps. These simulations are modified from those developed by Riggleman and colleagues ye2020molecularbarney2022fracturezhang2024predicting. In the figure, the red beads represent cross-linker sites, yellow beads represent precursor chain end monomers with the capability of bonding to cross-linker sites, and blue beads represent inner-chain monomers.
• Figure 2: Thermal fluctuations of cross-links. Thermal fluctuations cause a) rigid rotations of the end positions of the cross-link, $\bm{X} \rightarrow \bm{Q} \bm{X}$ and b) perturbations of the junction position, $\bm{y}$.
• Figure 3: a) The polydisperse $4$-chain RVE gets mapped under $\bm{F} \bm{Q}^*$ for the free rotation limit. In the deformed RVE, the origin and cross-link position are denoted by a red ball and green ball, respectively. b) In the frame averaging limit, the RVE response to $\bm{F}$ is averaged over all possible frames, represented by $\bm{Q} \in SO\left(3\right)$. The chain deformations, stretches, and cross-link position are, in general, different for each $\bm{Q}$.
• Figure 4: Agreement of closed-form approximation for free rotation RVEs with various degrees of polydispersity undergoing uniaxial a) and simple shear b) deformations.
• Figure 6: Free energy response for free rotation RVEs with various degrees of polydispersity for a) uniaxial and b) simple shear deformations. c) $\left(40, 120, 120, 120\right)$ free rotation RVEs under uniaxial deformations of $\lambda = 1.25, 1.5, 1.75,$ and $2$. Cross-link displacements for polydisperse free rotation RVEs under d) uniaxial and e) simple shear deformations.

Theorems & Definitions (15)

• Remark 1
• Remark 2
• Remark 3
• Proposition 1
• proof
• proof
• Proposition 2: Known solution for inner minimization and unification of discrete, monodisperse polymer network models
• Lemma : Equipartition property
• proof
• proof