On the statistical physics and thermodynamics of polymer networks: An Eulerian theory for entropic elasticity

Abstract

This study presents an Eulerian theory to elucidate the molecular kinematics in polymer networks and their connection to continuum deformation, grounded in fundamental statistical physics and thermodynamics, and free from phenomenological assumptions and additional parameters. Three key innovations are incorporated:

1. The network behavior is described through a global thermodynamic equilibrium condition that maximizes the number of accessible microstates for all segments, instead of directly dealing with the well-established single-chain models commonly adopted in traditional approaches. A variational problem is then posed in the Eulerian framework to identify this equilibrium state under geometric fluctuation constraints. Its solution recaptures the classical single-chain model and reveals the dependence of chain kinematics on continuum deformation.

2. The chain stretch and orientation probability are found to be explicitly specified through the Eulerian logarithmic strain and spatial direction. The resulting hyperelastic model, with a minimal number of physical parameters, outperforms the extant models with same number of parameters. It further provides a physical justification for prior models exhibiting superior predictive capabilities: the model becomes equivalent to the Biot-chain model (Zhan et al. 2023b) at moderate deformations, while converging to the classical Hencky strain energy in the small strain limit.

3. A novel biaxial instability emerges as a phase transition in chain orientation. At sufficiently large deformation, chains increasingly align with the primary stretched direction, depleting their density in other directions. Consequently, the stresses in non-primary stretched directions would decrease as the loss in chain density outweighs the gain in chain force. For equal biaxial tension, instability is therefore triggered because perfect equality of the two principal stretches without any perturbation is practically unachievable.

The theory establishes a physical picture of the network response under thermodynamic equilibrium: chain constrains but segment fluctuates. It is the statistical behavior of the latter, under the structural influence of the former, that governs the continuum response. Specifically, the macroscopic response of the network may not emerge from simplistic extensions of the chain-level models, but arises as a natural consequence of the underlying segment-level statistics. The theory also necessitates an Eulerian statistical perspective, as random thermal fluctuations prevent continuous tracking of any specified chains (material description) during deformation. Consequently, the Lagrangian framework may not be well suited in this context and chain stretch and orientation probability need be treated as Eulerian/spatial field variables. These perspectives not only advance the theoretical foundations of constitutive modeling for soft polymer networks, but also offer new insights into the microstructural origin of their macroscopic behavior.