Fundamental Theories for the Mechanics of Polymer Chains and Networks

205 pagesElastomeric materials contain many individual polymer chains crosslinked together, forming a polymer network. The rubber-like elasticity of these networks is primarily due to the entropic elasticity of the long polymer chains and weak intermolecular interactions above the glass transition temperature. New, important elastomeric materials are actively researched today, such as those that incorporate bond breaking to enhance mechanical properties. These emerging materials challenge us to develop physically-founded constitutive models to gain predictive power and a more fundamental understanding. This dissertation presents the development of such models, starting from the statistical mechanics of a single chain and ending with the mechanics of the polymer network. This dissertation begins with an additional contribution: a combined experimental and theoretical study of a metallopolymer and its mechanical properties. Neutral ligands are added to a polymer containing metal-coordination crosslinks, where the ligands bind at the metal centers, altering the crosslinking strength and allowing the macroscale mechanical properties to be tuned. Density functional theory is utilized to quantify the mechanics and thermodynamics of the crosslinks under this addition of ligands, in order to predict and understand observations from experimental mechanical tests. Accurate theoretical predictions are made when varying the number of added ligands, and qualitative conclusions are drawn in the case of varying the ligand type. Next, the methodical development of a constitutive model for polymer networks without bond breaking is presented. A careful statistical mechanical treatment highlights the important connection between the single-chain mechanical response and the equilibrium distribution of chains in the network, as well as the correspondence between different thermodynamic ensembles. Using an example single-chain model, these effects are directly studied. This statistical theory is then brought into the continuum scale, where the stress is obtained in terms of the applied deformation and molecular parameters, and the effects of statistical correspondences are again studied. The resulting framework serves as a directly physically-linked constitutive model for elastomers and hyperelastic materials in general. Following this, an asymptotic theory is presented for the statistical thermodynamics of classically-treated systems with strong interaction potentials. This development can be understood as the low-temperature analog of the well-established high-temperature perturbation theory. The asymptotic theory is applicable to approximating the mechanical response of single polymer chains, as well as to molecular modeling in general. The primary contribution in this dissertation focuses on another meticulous approach applied to a polymer network with bonds that break, whether reversibly or irreversibly. Beginning with the fundamentals of nonequilibrium statistical mechanics, classical transition state theory is extended to account for a continuous distribution of polymer chain extensions. In the process, an important connection is established between mechanically-sensitive reaction rates, mechanical response, and equilibrium distribution. Moving to the macroscale, the second law of thermodynamics is shown to be arbitrarily satisfied and the relation for the stress is obtained, once again in terms of the applied deformation and molecular parameters. With the general framework complete, a single-chain model is specified: the Morse-FJC model, a freely-jointed chain of Morse bonds, is introduced and asymptotically developed. The single-chain mechanical response and reaction rate functions are studied across a range in parameters. An original exact solution to the integro-partial differential equation governing the evolution of the network is presented. Two specialized versions of the general constitutive model – rate-independent irreversible breaking and transient breaking – are developed and applied to exemplary polymers from the literature. The general model is compared to these two specializations, a parameter study of the general model is performed across a wide range of molecular parameters, and finally the general model is applied to another polymer from the literature. The successes and shortcomings of the model are discussed considering all these results, and motivation for future theoretical development is provided