Dissipation Potentials for Reaction-Diffusion Systems

Abstract

This work considers strongly dissipative reaction–diffusion systems with constitutive equations given by Edelen’s dissipation potentials, whose existence is tantamount to nonlinear Onsager symmetry. As the main goal of this work, it is shown that the associated extremum principle yields the relevant steady-state species balances and concentration fields for well-mixed reactors as well as for spatially inhomogeneous reactions accompanied by diffusion. The above principle is contrasted briefly with the popular notion of maximum entropy production. Potential applications of the theory to a wide range of reaction–diffusion systems are discussed, and questions are raised as to the possible breakdown of strong dissipation and nonlinear Onsager symmetry arising from reversible chemical or chemomechanical couplings.

A multi-field model for charging and discharging of lithium-ion battery electrodes

“…, r }. The driving forces of the process, i. e., the reaction affinities A j , are assumed to derive from a variational principle where the chemical species are subject to constant supply rates [18]. The chemical affinity A j of the j-th reaction is then defined as…”

“…To express the dissipation inequality (19) in terms of process energy contributions, we make use of an alternative thermodynamic potential, namely the Helmholtz free energy density Ψ = U − T S, obtained from (18) through a Legendre transform. Inserting its rate and (18) into (19) gives the dissipation inequality…”

“…The chemical affinities A j can also be derived from dissipation potentials subjected to the stoichiometric constraints, i.e., they are work conjugate to the reaction-path rates J j , cf. [18]. Additionally, we split the mechanical stress into an equilibrium part P eq , depending on the variables of state only, and a rate-dependent dissipative contribution P v , also depending on the rate of deformationḞ,…”

Dissipation potentials from elastic collapse

“…Dissipative systems of this type have received attention in several past and recent studies, e.g. [3,9,12]. We distinguish at the outset our macroscopic description of applied kinematics or forces from a detailed microscopic description of dynamics based on specific micromechanical models such as those considered in previous models of plasticity, fracture and viscous flow, e.g.…”

“…We first consider a discrete system and then extend the results to continuum mechanics. While our treatment can be viewed as mechanical in nature, it can with slight modification be extended to other dissipative processes such as networks of chemical reactions accompanied by diffusive transport of mass and heat [9].…”