Violation of well-known reciprocity principle formulated by Lars Onsager in 1931.

It is shown here that for the cell model of the ion–exchange membrane, the Onsager principle of reciprocity is violated—the coupled cross kinetic coefficients are not equal. The violation is associated with the fact that the reciprocity principle takes place only for systems for which generalized fluxes are zero at thermodynamic forces other than zero within the framework of linear thermodynamics of irreversible processes.

The cell model of an ion exchange membrane assumes the replacement of a real system of randomly arranged ionite grains with a periodic lattice, in particular, identical porous charged balls enclosed in concentric spherical shells filled with electrolyte and forming a porous layer. The impact of neighboring particles in the cell method is considered by setting special boundary conditions on the surface of the liquid shell. It is assumed that the gradients of the external forces acting on the porous layer coincide with the local gradients on the unit cell.

The advantage of the described approach is that all the quantities included in the equations of electrolyte transfer through a porous layer: thermodynamic fluxes and forces, can be directly measured in experiments.

When calculating the kinetic coefficients L_ij of the Onsager matrix, as independent thermodynamic forces that are set during the experiment of transferring an electrolyte solution through a thin infinitely extended charged porous layer (membrane), we select gradients of pressure dp/dx, chemical dμ/dx and electrical dφ/dx potentials perpendicular to the surface of this layer.

As dependent thermodynamic parameters determined in the experiment, we take the flux densities: U—solvent (for example, water), I—mobile charges (electric current density), J—solute (density of the diffusion flux of the electrolyte).

It was shown based on experiments for heterogeneous MK-40 and homogeneous MF-4SK cation-exchange membranes that their cross coefficients differ little at small concentrations of aqueous solutions of NaCl electrolyte (up to 0.1 M).

At high concentrations, there is a significant quantitative and qualitative discrepancy between reverse osmotic L₁₃ and capillary osmotic L₃₁ coefficients, as well as for electrodiffusion coefficients L₂₃ and L₃₂. At the same time, the cross kinetic coefficients L₁₂ (regulates electroosmosis) and L₂₁ (regulates streaming current) practically coincide up to high values of the electrolyte concentration.

We have shown here that in the case of the cell membrane model, the Onsager reciprocity principle is violated—the matrix of kinetic coefficients is not symmetric. In this regard, it is necessary to be careful about determining the transport characteristics of membranes, which depend on the cross kinetic coefficients due to the asymmetry of the latter.

In present paper, it is shown that the coupled cross coefficients can differ not only quantitatively, but also qualitatively. It is crucial when considering the transport characteristics of charged membranes.

Anatoly N. Filippov. Asymmetry of cross kinetic coefficients in the cell model of a charged membrane. Asymmetry. 2025(1):0003, https://doi.org/10.55092/asymmetry20250003.