Nonequilibrium thermodynamics for soft matter made easy(er)

Abstract

We use straightforward energy and entropy balances to test the thermodynamic consistency of microstructural rheological models. The method utilizes the same mathematical methods as classical transport phenomena, so it is much simpler to use than the much more rigorous GENERIC formalism. The cost of this simplicity is that fewer restrictions are actually tested than those in either the single-generator or the two-generator formalisms. The proposed test does provide, however, a separation of energy and entropy, leading to an interesting internal energy balance. More importantly, it leads to two requirements for non-negative entropy production: one closely related to a virtual work argument, important during flow, and a second that guarantees adherence to the second law of thermodynamics during microstructural relaxation. These criteria do not appear to be in conflict with the requirements of the more rigorous formulations and are much simpler to implement. Several illustrative examples are given with models using the conformation tensor level of description. As expected, the models that use a relaxation function that is proportional to the free energy gradient are straightforward to check. These include the Hookean dumbbell, the FENE-P, and the Giesekus models, which are shown to satisfy the first and second laws. With a little more work, models with relaxation functions not driven by the free energy gradient only can also be checked for thermodynamic compliance with the proposed formalism. Examples of these are the GLaMM and Rolie–Poly models; they violate the first and second laws of thermodynamics, respectively. A recently proposed FENE-mode model is also checked, which superficially satisfies both laws, but fails to have an analytic free energy. Finally, the application of the formalism to a nonlinear dumbbell model that uses the probability density for chain conformations is also illustrated. In that case, satisfaction of the fluctuation-dissipation theorem and a positive mobility guarantee compliance with the first and second laws.