Statistical Field Theory for Polymer Networks and Electro-mechanical Polymers: Accounting for Nonlocal Interactions

Polymer networks formed by cross-linking flexible polymer chains are ubiquitous in many natural and synthetic soft matter systems. Knowing the physics of polymer networks is essential to tailor their elastic response for modern soft material-based technologies. Existing phenomenological and micro-mechanical elastic models for polymer networks as well as constraint-based molecular theories for polymers are unable to account for both polymer molecular structure and inter-segment interactions in the network. In the first part of this dissertation, we present a statistical mechanical model framework for polymer networks. The framework is based on three main pillars: statistical mechanics of single polymer chain allowing to consider polymer molecular structure, statistical field theory enabling to account for inter-segment interactions, and non-linear elasticity used to obtain mesoscale elastic response of the polymer network. For any pairwise and regularized inter-segment interaction potential, we derived expressions for the partition function and average segment density of polymer network as a function of an applied deformation gradient. We use an excluded volume approach for modeling the inter-segment interactions in the polymer network. The model is solved numerically in 3D with a finite element approach to obtain average segment density and strain energy density of the polymer network. We find that in the absence of inter-segment interactions, the elastic response of a polymer chain matches with the classical rubber elasticity. The model demonstrates swelling effect observed in real polymer networks. By assuming polymer network system as hyper-elastic solid, we evaluate its mesoscale elastic response and elastic moduli and show behavior that is consistent with elastomer materials. This developed model framework has the potential to allow engineers to tailor the microscopic as well as bulk elastic response of polymer network-based soft material systems with inputs such as polymer chain parameters, chain connectivity in the network, and choice of inter-segment interactions.

Polarization of polymeric soft matter under the external electric or magnetic field provides sensing and actuation, a highly sought feature in many engineering and scientific applications. To advance and tailor the functional response of such material systems, it is important to know the detailed physics of polarizable polymeric soft matter under external fields. Existing theoretical approaches for polarizable polymers subject to a combined applied electric field and stretch are based on discrete monomer models. In this framework, it is challenging to account for the nonlocal dipole-dipole interactions within polymer segments. The prior works typically consider only the interaction between the applied field and the dipoles or use assumptions such as weak anisotropy of polymer polarizability. In the later part of this dissertation, we present a statistical field theoretic framework for polarizable polymer that is based on three main pillars: electrostatics, continuous description of polymer chain using statistical mechanics that allows to consider polymer molecular structure, and newly introduced self-consistent field theory formulation in terms of polymer chain density fields that enables to account the nonlocal dipole-dipole interactions. To address the setting of constant applied electric field ensembles, the proposed self-consistent formulation allows us to transforms the nonlocal interactions into a PDE constraint corresponding to the Gauss’ equation in electrostatics. We implement the model in a finite element method to numerically compute the average segment density, average polarization, and the free energy of polymer chain at thermodynamic equilibrium. We find that the presence of dipole-dipole interactions leads to qualitative changes in the dipole distributions, total polarization, and equilibrium electric field in the domain. We also investigate the effect of orientation and strength of the applied electric field on the equilibrium properties of polymer chain. We notice a sharp instability leading to a collapse of the chain under the strong electric fields as a consequence of dipole-dipole interactions. The insights gained from this developed model would allow designing functional response of polarizable polymeric soft matter using inputs such as polymer chain parameters and externally applied fields. 

In general, the statistical field theory for the polymer networks and electro-mechanical polymers developed in this dissertation provides a framework for the design and discovery of polymeric soft matter having tailored elastic, functional properties, and embedded physical intelligence.