A Statistical Mechanics Framework for Polymer Chain Scission, Based on the Concepts of Distorted Bond Potential and Asymptotic Matching, with Implications to the Lake-Thomas Theory of Polymer Fracture

To design tough and resilient elastomers, the tools of statistical mechanics ought to be used to connect molecular constituents to macroscopic fracture toughness. Since polymer chain rupture is an enthalpically driven process, bond extensibility ought to be intrinsically incorporated in statistical mechanics-based extensible freely-jointed chain (FJC) models. This objective was rigorously satisfied only recently via the arbitrarily-extensible FJC (uFJC) model, formalized by Buche and colleagues. Despite this notable achievement, the uFJC model is unable to be fully cast in a numerically tractable fashion, limiting its applicability in an upscaled continuum setting. To rectify this, we develop the "composite" uFJC (cuFJC) model. Notably, the principles of asymptotic matching are utilized to derive a simple, quasipolynomial "composite" bond potential. This potential is then supplemented to a slightly-extended version of the uFJC model. The result is the cuFJC model, which is cast in an analytical closed form. Using the cuFJC model, a stochastic thermal fluctuation-driven chain rupture framework is developed based upon a tilted bond potential that accounts for distortional bond energy. Dissipated chain scission energy is then derived in a probabilistic sense, which is incorporated into the Lake-Thomas theory of polymer fracture. The sensitivity of the resulting richly-statistical Lake-Thomas fracture toughness with respect to loading rate and chain composition is presented.