Boltzmann Statistics of Viscoelasticity in High-Strength and -Toughness Hydrogels

High strength and high toughness can be achieved in nanorestricted hydrogels through the movement of chain segments in highly entangled ionic clusters and polymer chains. Quantifying the relationship between the complex viscoelastic behavior and the mechanical properties generated by strong intermolecular interactions remains a challenge. In this work, inspired by the universal principle of the Boltzmann energy distribution and in combination with the upper convolution Maxwell (UCM) model, we established the physical constraint model controlled by the Boltzmann distribution (BDPhC). This model separately describes the viscous and elastic contributions of the constrained chain segments, yielding macroscopic mechanical properties that are in excellent agreement with the experimental results. Furthermore, the model reveals a scaling relationship between the Deborah number (De) and hysteresis in hydrogels and experimentally verifies that a near-perfect elasticity state would occur when De > 1. The model also effectively simulates the viscoelastic behavior of various reported high-strength hydrogels, demonstrating their broad applicability.