Thermodynamically consistent multiscale asymptotics: generalisation of Donnan theory to heterogeneous media
We shall consider a two-phasic mixture consisting of fluid permeating through solid. The fluid phase is a carrier of charge and there is also a fixed charge at the fluid-solid interface. The microstructure is such that it is mildly varying in space and hence not strictly periodic. One of the main goals is to assess the effect of this spatial variation on the microscale in the upscaled version of the PNP equations. The source of this spatial variation can be of two origins - either geometrical (the details of the interface change on the macroscale) or the concentration of the fixed charge can vary.
After restating the classical Poisson–Nernst–Planck equations, the multiple scales approach is employed to upscale the microscale description for a system with weak non-periodicity. We highlight the key steps including the appropriate scaling of the density of fixed charge located at the fluid-solid interface, the use of the level set approach for appropriately expanding the boundary and interface conditions, or the choice of a correct Ansatz with an insight from non-equilibrium thermodynamics.
To our best knowledge, the PNP equations with a mild variation in space have not been upscaled and, in addition, we keep in mind the thermodynamic consistency of the upscaled variant, namely the force-flux relations and the fluctuation dissipation theorem. As a result, we retained the gradient-flow structure from the microscale formulation with the same definition of thermodynamic driving force, only the phenomenological coefficient contains a finer (geometrical) detail of the microscale.
Donnan’s theory is the most widely applied theory to represent the osmotic pressure in porous media like cartilage, although it is clear from its derivation (osmotic pressure as a result of a semipermeable membrane) that the theory is not appropriate for such an application. Here we shall argue that the upscaled version of Poisson-Nernst-Planck equation in heterogeneous medium can be used as a generalisation of Donnan theory. The effect of spatial variation of material properties is significant as follows from the expression for the electric potential $\psi$ (being valid in a 1D simplification) \(-\varepsilon L \psi'' = 2 \varphi \sinh\left(-\psi/L\right) 2 \rho_Q(x) \sqrt{\pi (1-\varphi)} ,\) where $\varepsilon$ is a small parameter (related to Debye lenght to macroscale length ratio), $L$ is a dimensionless constant, $\varphi$ is the fluid volume fraction, and $\rho_Q$ is the surface charge. Hence, one can immediately observe that the osmotic pressure constitutes of two distinct parts: i) the boundary layer solution that enables satisfying the boundary conditions via a rapid change in a small transition region, and ii) the bulk solution that emerges from the dominant balance between the fixed charge density $\rho_Q$ and ions represented by $\sinh$ term. The former can be perceived as a form of Donnan theory while the latter generates a non-zero contribution to the osmotic pressure only if there is a spatial variation in the fixed charge density. In cartilage tissue, preliminary data suggest that the above correction has the potential to play a significant role in tissue response.