Fluctuation theorems and work relations for single polymer rheology

Synthetic and biological polymeric materials are ubiquitous in nature and modern technology. The emergent properties afforded by these materials allows for wide a array of applications as found in adhesives, coatings, and synthetic polymers for plastics. Importantly, the molecular properties of polymeric systems ultimately determine their bulk macroscopic response and behavior in equilibrium and highly nonequilibrium conditions. As a result, the field of single polymer rheology can play a key role in establishing a molecular level understanding of polymeric systems by investigating the dynamics of single chains. Single polymer rheology is now a well-established approach to study polymer dynamics from experimental and computational perspectives. In general, this approach allows for the determination of molecular subpopulations, relaxation, and polymer chain dynamics in a wide variety of flows. Despite recent progress, current methods in single polymer rheology do not allow for the determination of equilibrium and nonequilibrium thermodynamic properties of polymeric systems. Moreover, it is challenging to connect backbone dynamics to key macroscopic rheological phenomena. In this context, the impact of single polymer rheology has remained limited for the past two decades. In this thesis, we address these challenges by developing and applying fluctuation theorems and nonequilibrium work relations to the field of single polymer rheology. The discovery of thermodynamic identities known as nonequilibrium work relations (NWRs) and fluctuation theorems (FTs) has catalyzed recent advances in statistical mechanics. In general, work relations provide an unprecedented route to extract fundamental materials properties of equilibrium and nonequilibrium systems. Furthermore, these identities have uncovered a broad range of unexpected and remarkable thermodynamic phenomena, including molecular level violations to the second law of thermodynamics. In the context of rheology and fluid mechanics, thermodynamics plays a key role in the understanding and design of a wide array of processes, including flow-induced phase separation and crystallization. As a result, there is a strong need for new methods to analyze the dynamics of complex fluids. In this thesis, we apply the Jarzynski/Hatano/Sasa equality and Crooks fluctuation theorem to determine equilibrium and nonequilibrium properties of polymer solutions in fluid flow. In particular, we use a combination of single molecule polymer experiments and computer simulations to probe the application of these NWRs to polymer dynamics in shear and extensional flows. Using this approach, we determine the equilibrium linear and nonlinear elasticity, the nonequilibrium free energies, and entropies of flowing polymer solutions. Interestingly, we also find that fundamental thermodynamic quantities are related to well known rheological functions such as the longest polymer relaxation time, viscosity, and stress. Overall, NWRs appear to provide a simple and distinct framework that connects thermodynamics and rheology, and this work opens new directions in an emerging field known as ``thermo-rheology"