A note on representation-dependent Onsager closures for thermo-diffusive coupling

Non-isothermal diffusion can be formulated within different mixture-level approaches that account for diffusion-related transport either through an extended energy flux or through an extended entropy flux. This note compares two prominent approaches: the Gurtin approach, in which diffusion enters the energy balance through an additional diffusive energy flux, and the M & uuml;ller approach, in which the classical energy balance is retained while diffusion enters through an extended entropy flux. The comparison clarifies how the way diffusion-related transport is represented in each approach and the adopted thermodynamic force basis affect a subsequent Onsager-type closure. In the selected M & uuml;ller-type force basis, the diffusion-related thermodynamic force contains a temperature-gradient contribution. Consequently, a diagonal Onsager closure in this basis yields a diffusion flux with an apparent Soret-type contribution while the heat flux remains Fourier-like. In the selected Gurtin-type force basis, the corresponding diagonal closure yields uncoupled Fourier and Fick laws, whereas thermo-diffusive coupling appears through non-diagonal Onsager blocks. ... mehrThus, the apparent Soret/Dufour structure follows from how diffusion-related transport is represented in the mixture-level approach considered and from the basis of thermodynamic forces used for the Onsager closure. Although diffusion in solids provides the main motivation, the derivation does not rely on a specific constitutive law.