LiquidElectrolytes.jl
LiquidElectrolytes — Module
LiquidElectrolytes.jl
This package is a solver for generalized Poisson-Nernst-Planck models taking into account finite ion size, solvation and electroosmotic pressure based on formulations derived from first principles of nonequilibrium thermodymamics. It utilizes a thermodynamically consistent finite volume space discretization approach which uses the sum of the electrotstatic potential and the excess chemical potential as convective terms. It is realized on top of the solver kernel VoronoiFVM.jl for coupled nonlinear PDE systems which takes advantage of ForwardDiff.jl to generate the full linearization of the coupled nonlinear system as a basis for a robust Newton solver for the discrete nonlinear systems.
Parts of the package are still work in progress.
Poisson-Nernst-Planck (PNP) system
The package is devoted to the numerical solution of the generalized Poisson-Nernst-Planck (PNP) system in the following formulation:
Poisson equation:
\[-\nabla \varepsilon\varepsilon_0 \nabla \phi = q = F\sum_{i=1}^N z_i c_i\]
Momentum balance taking derived from the Navier-Stokes equation in mechanical equilibrium:
\[ -\Delta p + q\nabla \phi = 0\]
Continuity equation:
\[\partial_t c_i + \nabla \cdot \mathbf N_i = 0 \qquad (i=1\dots N)\]
Generalized Nernst-Planck flux:
\[ N_i=-D_i c_i \left( \nabla\log \left(\gamma_i\frac{c_i}{c^\circ}\right) + z_i\frac{F}{RT}\nabla\phi\right) \qquad (i=1\dots N)\]
The activity coefficients $\gamma_i = \gamma_i(c_1\dots c_N, p)$ encode a particular electrolyte model. In general, the fluxes $N_i$ are coupled due to the dependency of $\gamma_i$ on all concentrations and the pressure. E.g. $\gamma_i=1$ describes the classical Guoy-Chapman model. The default implements the model of Dreyer, Guhlke, Müller, Landstorfer. See the section on Activity coefficients for more.
Poisson-Boltzmann (PB) system
In thermodynamic equilibrium, the fluxes $N_i$ of the PNP system are zero, which allows to replace the continuity and flux equations by the algebraic condition
\[c_i=c_i^b \frac{\gamma_i(c_1^b\dots c_N^b, p^b)}{\gamma_i(c_1\dots c_N,p)}\exp\left(z_i\frac{F}{RT}(\phi^b-\phi)\right), \]
where $\phi^b, p^b, c_1^b\dots c_N^b$ are reference values usually imposed as boundary conditions at the interface to the bulk of the electrolyte. Poisson equation and momentum balance are kept. In general, this algebraic condition for given $\phi, p$ comprises a nonlinear system of equations due to the dependency of $\gamma_i$ on all concentrations and the pressure.