Functional constraints on phenomenological coefficients

Thermodynamic fluxes (diffusion fluxes, heat flux, etc.) are often proportional to thermodynamic forces (gradients of chemical potentials, temperature, etc.) via the matrix of phenomenological coefficients. Onsager's relations imply that the matrix is symmetric, which reduces the number of unknown coefficients is reduced. In this article we demonstrate that for a class of nonequilibrium thermodynamic models in addition to Onsager's relations the phenomenological coefficients must share the same functional dependence on the local thermodynamic state variables. Thermodynamic models and experimental data should be validated through consistency with the functional constraint. We present examples of coupled heat and mass transport (thermodiffusion) and coupled charge and mass transport (electro-osmotic drag). Additionally, these newly identified constraints further reduce the number of experiments needed to describe the phenomenological coefficient.

Figure 1

Experimental verification of the proposed hypothesis for a Nafion membrane electrode assembly. Transport of water and protons are considered and the linear force-flux regime was identified to be for the range $\nabla \mathrm{\Phi }\in (0,0.2)$ of the applied electric potential $\nabla \mathrm{\Phi }$ across a membrane electrode assembly. Temperature and water activity at open circuit are as noted on the graph. The measured data suggest the same functional dependence of diagonal ($L_{pp}$) and cross- ($L_{pw}=L_{wp}$) coefficients on water concentration.

Figure 2

Experimental verification of the proposed hypothesis for a Nafion membrane electrode assembly. As data indicate, the functional dependency of proton conductivity $L_{pp}$ and water conductivity $L_{ww}$ on water concentration can be regarded equal (note that the shift in the logarithmic values corresponds to multiplication by a constant). These observations together with conclusions from Fig. 1 suggest that the proposed hypothesis holds true in the studied situation. Particularly, the whole matrix of phenomenological coefficients shares the same functional dependence on water concentration. The inset graph in the upper right shows the same data plotted as a function of water activity.