Statistical field theory of polarizable polymer chains with nonlocal dipolar interactions

The electromechanical response of polymeric soft matter to applied electric fields is of fundamental scientific interest as well as relevant to technologies for sensing and actuation. Several existing theoretical and numerical approaches for polarizable polymers subject to a combined applied electric field and stretch are based on discrete monomer models. In these models, accounting for the interactions between the induced dipoles on monomers is challenging due to the nonlocality of these interactions. On the other hand, the framework of statistical field theory provides a continuous description of polymer chains that potentially enables a tractable way to account for these interactions. However, prior formulations using this framework have been restricted to the case of weak anisotropy of the monomer polarizability. This paper formulates a general approach based in the framework of statistical field theory to account for the nonlocal nature of the dipolar interactions without any restrictions on the anisotropy or nonlinearity of the polarizability of the monomer. The approach is based on three key elements: (1) the statistical field theory framework, in which the discrete monomers are regularized to a continuous dipole distribution, (2) a replacement of the nonlocal dipole-dipole interactions by the local electrostatics partial differential equation with the continuous dipole distribution as the forcing, and (3) the use of a completely general relation between the polarization and the local electric field. Rather than treat the dipole-dipole interactions directly, the continuous description in the field theory enables the computationally tractable nonlocal-to-local transformation. Further, it enables the use of a realistic statistical-mechanical ensemble wherein the average far-field applied electric field is prescribed, rather than prescribing the applied field at every point in the polymer domain. The model is applied, using the finite element method, to study the electromechanical response of a polymer chain in the ensemble with fixed far-field applied electric field and fixed chain stretch. The nonlocal dipolar interactions are found to increase, over the case where dipole-dipole interactions are neglected, the magnitudes of the polarization and electric field by orders of magnitude as well as significantly change their spatial distributions. Next, the effect of the relative orientation between the applied field and the chain on the local electric field and polarization is studied. The model predicts that the elastic response of the polymer chain is linear, consistent with the Gaussian approximation, and largely unchanged by the orientation of the applied electric field, though the polarization and local electric field distributions are significantly impacted.