Statistical Field Theory Model for Elastomers

The macroscopic properties of elastomers depend upon the chemical nature of their polymer segments, their chain architecture, and polymer inter-segment interaction. Existing elastic models for rubbery polymers can be broadly classified into phenomenological and micromechanical. Phenomenological models do not consider polymer chain molecular structure, whereas micromechanical models typically do not account for polymer inter-segment interaction. To address these limitations and understand the role of inter-segment interaction, we have developed a statistical mechanics-based field theoretic model for elastomers.

We start with a continuous Gaussian chain model which allows us to account for short-ranged interactions associated with the bonding constraint between adjacently connected polymer segments. Building on this, in an elastomer network with many cross-linked polymer chains, the long-ranged inter-segment interactions between polymer segments of different chains that are close to each other in space are accounted through statistical field theory. We model this long-ranged interaction through an excluded volume approach. The contact of each chain with many other chains in its vicinity produces a damping effect on the molecular correlations. This leads to mean field theory to be effective for modeling elastomers. The complete partition function for the elastomer network is evaluated by using the mean field assumption; the mean field is obtained self consistently by solving the stationary conditions for total free energy.

Motivated by the Arruda-Boyce 8-chain network, we consider chain networks that are subject to affine deformations in the principal frame of the deformation gradient to perform the chain orientational averaging. The total free energy and equilibrium elastomer density are obtained as a function of applied deformation gradient using the Finite Element Method. We find that in the absence of long-ranged interaction, the elastic response of the polymer chain matches with the classical rubber elasticity, whereas a sufficiently strong long-ranged inter-segment interaction leads to unexpected instabilities in the structure and response of the polymer network.

We extend our statistical mechanics-based field theoretic elastomer model for Liquid Crystal Elastomers (LCEs) in which relatively stiff liquid crystal molecules are connected by flexible polymeric chains. LCEs show reversible shape change with temperature driven by the nematic-isotropic phase transition of the liquid crystals. The polymer chain elasticity is entropic while the liquid crystalline free energy is based on the Maier-Saupe mean field theory for liquid crystals. The existing models for LCE are typically based on continuum mesoscale approaches such as phase field method. While these models provide macroscopic predictive capability once material constants are calibrated, they are unable to predict mesoscale structure and response. We probe these limitations using our model, and further examine the effects of temperature, geometry and loading conditions on LCE response.