A Unified Thermo-Chemo-Mechanical Framework for Bulk and Frontal Polymerization: Coupled Kinetics and Front Stability

TL;DR

This work develops a unified, thermodynamically consistent framework for polymerization that simultaneously treats bulk and frontal polymerization under strong thermo-chemo-mechanical coupling. It integrates stress-influenced reaction kinetics and the evolution of the stress-free configuration via a multiplicative decomposition and a flow-rule–like evolution for transformation deformation, enabling analytical treatment in a one-dimensional, narrow-front limit. A generalized Zeldovich-number–type stability criterion is derived, incorporating heat loss and mechanical loading, along with closed-form expressions for front velocity and temperature, and a phase diagram distinguishing stable, unstable, and quenched fronts. The framework further yields an analytical force response in a uniaxial FP setup and demonstrates qualitative agreement with experiments, offering a mechanism for residual-stress prediction and mitigation in FP-based manufacturing. Overall, it provides a rigorous foundation for predicting front dynamics and mechanical outcomes in polymerization, with pathways to three-dimensional simulations and broader loading conditions.

Abstract

Polymerization is a fundamental chemical process enabling large-scale production of material components across modern industries. By transforming a monomer mixture into a cross-linked polymer network, polymerization induces changes in temperature and material properties such as density and stiffness, which can generate residual stress and warping through coupled mechanisms that remain incompletely understood. Depending on processing conditions, polymerization may occur either in the bulk, sustained by continuous external energy input, or as a self-sustaining exothermic reaction front, commonly referred to as frontal polymerization. While frontal polymerization offers rapid and energy-efficient curing, its localized reaction zone produces sharp spatial gradients that amplify thermo-chemo-mechanical coupling effects. In this work, we develop a thermodynamically consistent framework that captures both bulk and frontal polymerization, incorporating stress-dependent reaction kinetics and the evolution of the stress-free configuration during curing. Using a narrow reaction-zone approximation in a uniaxial setting, we derive analytical predictions for propagation velocity, residual stress development, and stability. A perturbation analysis yields a stability criterion that generalizes the classical Zeldovich number by accounting for heat loss and mechanical loading, and enables construction of a phase diagram distinguishing stable, unstable, and quenched propagation regimes.

Figures (4)

• Figure 1: Illustration of assumed kinematics via multiplicative decomposition of the deformation gradient $\bm{F} = \bm{F}_e\bm{F}_i$ where $\bm{F}_e$ is the elastic deformation gradient and $\bm{F}_i$ is the inelastic deformation gradient. The inelastic deformation gradient $F_i$ can be further decomposed as $\bm{F}_i = \bm{F}_*\bm{F}_v$, where $\bm{F}_v$ is a spherical tensor that represents the volume change, and $\bm{F}_*$ is the transformation deformation gradient.
• Figure 2: Comparison between bulk and frontal polymerization regimes obtained from numerical simulations of the reduced one-dimensional governing equations \ref{['eq:reac_diffusion']}. The top panels show the temporal evolution of the temperature profiles, while the bottom panels present the corresponding reaction-rate distributions $\dot{\alpha}(x,t)$, indicating the reaction zones where polymerization occurs. (a) Bulk polymerization, where continuous heating is applied at $x = 0$; (b) Frontal polymerization.
• Figure 3: Investigation of the influence of thermal dissipation and mechanical loading on propagation dynamics: (a) Variation of normalized velocity $\bar{v}$ with the elastic stretch ${\lambda}_e$, evaluated with moderate dissipation $\bar{h} = 0.015$. The shaded regions indicate distinct propagation regimes, corresponding to the phase diagram in (c). (b) Time evolution of the normalized front velocity $\bar{v}(\bar{t})$ from numerical simulations for different values of the normalized parameter $\bar{Y}/Y_c$. (c) Phase diagram showing the influence of the normalized dissipation coefficient $\bar{h}$ and elastic stretch $\lambda_e$ on the propagation regime.
• Figure 4: (a) Illustration of the experimental set up. A soft polymer sample is mounted on the mechanical testing machine that controls its length and measures the applied force; (b) Simplified schematic of the uniaxial setup, the front propagation behavior is simplified as a shock-like wave leading to discontinuities in distribution of material properties; (c) Experimental measurement of the uniaxial stress as a function of time, here for the convenience of comparison, the time is normalized as $\bar{t} =t/t_0$, where $t_0 = 1300s$ denotes the time at which the sample is fully cured. Thus $\bar{t}=1$ represents the instant when the front reach the end of the sample. (d) Theoretical prediction of the stress response during FP process.