Topic D — Nernst–Planck Electrochemistry with Migration

Binary electrolyte · electroneutrality · current conservation · Butler–Volmer with a solved potential · tier: advanced

Motivation

With excess supporting electrolyte, ion transport is plain convection–diffusion and the electrode overpotential can be prescribed (topic A). Without it, ions also migrate in the electric field, and the overpotential depends on the solution potential, which is itself an unknown. For a cation reduced at a cathode, migration and diffusion cooperate: the limiting current exceeds the diffusion-only value — for a 1:1 electrolyte with equal diffusivities by a factor of exactly 2. Your model has to reproduce that number.

Course weeks: 8–9 (Newton, continuation), 11 (null spaces, pinning a potential), 13 (advection–diffusion). New: migration, a solution-dependent conductivity, a boundary condition coupling two fields.

Model

Channel with the cathode on the lower wall and the bulk boundary on the upper wall; the 1D layer solution (schematic).

Binary 1:1 electrolyte M^+X^-, cation reduced at the electrode, anion inert. Fluxes (\phi scaled by RT/F): \mathbf N_\pm=-D_\pm\nabla c_\pm\mp D_\pm c_\pm\nabla\phi+c_\pm\mathbf u,\qquad\nabla\cdot\mathbf N_\pm=0 . Electroneutrality c_+=c_-=:c replaces the Poisson equation outside the (unresolved) double layer. Adding and subtracting the two species equations gives one equation for c and current conservation \nabla\cdot\mathbf i=0, \mathbf i\propto\mathbf N_+-\mathbf N_-, for \phi — derive both; the conductivity \propto(D_++D_-)\,c depends on the solution. \phi is determined up to a constant: pin it (where? justify) or use a null space.

Electrode (e.g. part of the lower wall of a channel): \mathbf N_-\cdot\mathbf n=0 and \mathbf N_+\cdot\mathbf n=-\frac{i_n}{F},\qquad i_n=i_0\,\frac{c}{c_\text{ref}}\Big[e^{\alpha_a\eta}-e^{-\alpha_c\eta}\Big],\qquad \eta=\phi_\text{metal}-\phi|_\text{electrode}-E_\text{eq}, nonlinear in c and \phi. \phi_\text{metal} is the control parameter (potentiostatic).

Design output: current against \phi_\text{metal}, with migration and with migration switched off (supporting-electrolyte model), and their ratio.

Note

Analysis questions (required). (i) Derive the c and \phi equations from the two Nernst–Planck equations under electroneutrality; which terms cancel in \mathbf i when D_+=D_-? (ii) Is the \phi problem coercive? What is its null space, and how does your boundary data remove it? (iii) What happens to the \phi equation as c\to0 at the limiting current?

Required tasks

  1. Nernst–Planck for both ions with a prescribed \phi (check the migration term alone).
  2. The coupled (c,\phi) problem with a prescribed \eta on the electrode.
  3. The full electrode condition with \eta from the solved \phi; Newton with continuation in \phi_\text{metal}.
  4. Polarization curves with and without migration; ratio of the limiting currents.

Verification specific to this topic

  • 1D exact solution (required, do this first): stagnant layer 0<y<\delta, bulk c=c_b, \phi=0 at y=\delta, electrode at y=0. With D_+=D_-=D the limiting state is c(y)=c_b\,y/\delta, \phi=\ln(c/c_b), cation flux 2Dc_b/\delta: twice the supporting-electrolyte value Dc_b/\delta. Below the limit the profile is still linear in c and logarithmic in \phi — an exact solution for an MMS-free convergence study.
  • Enhancement factor I_\text{lim}/I_\text{lim}^\text{no migr}\to1/(1-t_+)=2 in the 2D model as the cathodic polarization increases — exactly, in any geometry, provided no boundary exchanges the inert anion (add the two Nernst–Planck equations to see why). At small polarization the ratio may well be below 1 (the migration model also carries the ohmic drop); it is the plateau that must reach 2. A plateau below 1 means a sign error: check the sign of the electrode term in the \phi equation (it must make \mathbf N_-\cdot\mathbf n=0) and the sign of \phi in \eta.
  • Anion impermeability: compute the electrode flux of the anion (from the weak-form residual, not from boundary gradients) — it must be \approx0.
  • Where the current returns: the \phi Dirichlet boundary is the counter electrode. Make it a bulk boundary (c and \phi prescribed) or the inlet; a \phi-only condition on a no-flux wall lets that wall exchange anions, and the plateau stalls below 2.
  • Supporting-electrolyte limit (stretch 1) reproduces the migration-free model.
  • MMS for the coupled (c,\phi) system in 2D.

Stretch goals

  1. A third, inert ion at increasing concentration: the limiting current moves from the migration value to the supporting-electrolyte value.
  2. D_+\ne D_-: test 1/(1-t_+) in general.
  3. Galvanostatic control: prescribe the total current, solve for \phi_\text{metal}.
  4. Resolve the double layer in a thin sub-region with the Poisson equation.

Milestones

M1 1D stagnant layer (exact solution, factor 2) · M2 2D with prescribed \phi · M3 2D with prescribed \eta · M4 coupled electrode condition and continuation · M5 the two polarization curves.

Pitfalls

  • Concentration can go negative during Newton near the limiting current: variational-inequality Newton (vinewtonrsls) with c\ge0, continuation in \phi_\text{metal}, and line search.
  • ufl.exp in a boundary integral defeats the automatic quadrature degree estimate: set metadata={"quadrature_degree": …} on the electrode ds and check that the current does not change when you raise it.

References

  • J. Newman, K. E. Thomas-Alyea, Electrochemical Systems, 3rd ed., Wiley 2004 — dilute-solution theory, migration, limiting currents.
  • A. J. Bard, L. R. Faulkner, Electrochemical Methods, 2nd ed., Wiley 2001.