Publication 21-CNA-015
Statistical mechanics of a dielectric polymer chain in the force ensemble
Matthew Grasinger
Materials and Manufacturing Directorate, Air Force Research Laboratory
UES, Inc.
matthew.grasinger.ctr@afresearchlab.com
Kaushik Dayal
Center for Nonlinear Analysis
Department of Civil and Environmental Engineering
Department of Materials Science and Engineering
Carnegie Mellon University
Pittsburgh, PA 15213
Kaushik.Dayal@cmu.edu
Gal deBotton
Department of Mechanical Engineering
Department of Biomedical Engineering
Ben-Gurion University
Prashant K. Purohit
Department of Mechanical Engineering and Applied Mechanics
University of Pennsylvania
purohit@seas.upenn.edu
Abstract: Constitutive modeling of dielectric elastomers has been of long standing interest in mechanics. Over the last two decades rigorous constitutive models have been developed that couple the electrical response of these polymers with large deformations characteristic of soft solids. A drawback of these models is that unlike classic models of rubber elasticity they do not consider the coupled electromechanical response of single polymer chains which must be treated using statistical mechanics. The objective of this paper is to compute the stretch and polarization of single polymer chains subject to a fixed force and fixed electric field using statistical mechanics. We assume that the dipoles induced by the applied electric field at each link do not interact with each other and compute the partition function using standard techniques. We then calculate the stretch and polarization by taking appropriate derivatives of the partition function and obtain analytical results in various limits. We also perform Markov chain Monte Carlo simulations using the Metropolis and umbrella sampling methods, as well as develop a new sampling method which improves convergence by exploiting a symmetry inherent in dielectric polymer chains. The analytical expressions are shown to agree with the Monte Carlo results over a range of forces and electric fields. Our results complement recent work on the statistical mechanics of electro-responsive chains which obtains analytical expressions in a different ensemble.
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