Gouy-Chapman Debye layer

The Debye layer is the ionic concentration and potential distribution structure that appears on the surface of a charged electrode in contact with solvents in which are dissolved ionic species. Louis Georges Gouy and David Leonard Chapman at the beginning of the XX century proposed a model of the Debye layer resulting from the combined effect of its thermal diffusion and its electrostatic attraction or repulsion. In effect, in a stationary situation and assuming fluid at rest, the Poisson-Nernst-Planck equations are,

$0=\nabla \cdot \left(e{\omega }_{i}{Z}_{i}{c}_{i}\nabla \varphi \right)+\nabla \cdot \left({\omega }_{i}{k}_{B}T\nabla {c}_{i}\right)\phantom{\rule{1em}{0ex}}\mathrm{\text{with}}\phantom{\rule{1em}{0ex}}\nabla \cdot \left(\epsilon \nabla \varphi \right)=\sum _{i}e{c}_{i}$

where $\varphi$ is the electric potential and ${c}_{i}$ is the number of $i$-ions per volume. ${\omega }_{i}$ and ${Z}_{i}$ are the $i$-ion mobility and valence. ${k}_{B}$ is the Boltzmann constant, $e$ is the electron charge, $\epsilon$ the electrical permittivity and $T$ the temperature.

The above equations, written in dimensionless form, reduces in the case of a fully dissolved binary system in a planar geometry to,

${\stackrel{^}{c}}_{+}=exp\left(-\stackrel{^}{\varphi }\right),\phantom{\rule{0.167em}{0ex}}{\stackrel{^}{c}}_{-}=exp\left(\stackrel{^}{\varphi }\right)\phantom{\rule{1em}{0ex}}\mathrm{\text{with}}\phantom{\rule{1em}{0ex}}\left(\stackrel{^}{\varphi }{\right)}_{xx}=2\mathrm{sinh}\left(\stackrel{^}{\varphi }\right).$

#include "grid/multigrid.h"
#include "diffusion.h"
#include "run.h"
#include "ehd/pnp.h"

#define Volt 1.0
#define DT 0.01

We assume a fully dissolved binary system labelling the positive ion as $Cp$ and the counterion as $Cm$. The valence is one, ($\mid Z\mid =1$).

scalar φ[];
scalar Cp[], Cm[];
double dt;
int Z[2] = {1,-1};
scalar * sp = {Cp, Cm};

Ions are repelled by the electrode due to its positive volume conductivity while counterions are attracted (negative conductivity).

#if 1
const face vector kp[] = {1., 1.};
const face vector km[] = {-1., -1.};
vector * K = {kp, km};
#endif

On the left the charged planar electrode is set to a constant potential $\varphi =1$. The concentrations of the positive and negative ions depend exponentially on the voltage electrode.

φ[left] = dirichlet(Volt);
Cp[left]  = dirichlet (exp(-Volt));
Cm[left]  = dirichlet (exp(Volt));

In the bulk of the liquid, on the right boundary, the electrical potential is zero and the ion concentrations match the bulk concentration i.e

φ[right] = dirichlet (0.);
Cp[right]  = dirichlet (1.);
Cm[right]  = dirichlet (1.);

Initially, we set the ion concentration to their bulk values together with a linear decay of the electric potential $\varphi$.

event init (i = 0)
{
foreach() {
φ[] = Volt*(1.-x/5.);
Cp[] = 1.0;
Cm[] = 1.0;
}
boundary ({φ, Cp, Cm});
}

event integration (i++) {
dt = dtnext (DT);

At each instant, the concentration of each species is updated taking into account the ohmic transport.

#if 1
ohmic_flux (sp, Z, dt, K);
#else
ohmic_flux (sp, Z, dt); // fixme: this does not work yet
#endif

Then, the thermal diffusion is taken into account.

for (scalar s in sp)
diffusion (s, dt);

The electric potential $\varphi$ has to be re-calculated since the net bulk charge has changed.

scalar rhs[];
foreach() {
int i = 0;
rhs[] = 0.;
for (scalar s in sp)
rhs[] -= Z[i++]*s[];
}
poisson (φ, rhs);
}

event result (t = 3.5) {
foreach()
fprintf (stderr, "%g %g %g %g \n", x, φ[], Cp[], Cm[]);
}

Results

We compare the numerical results (symbols) with the analytical solution (lines).

int main() {
N = 32;
L0 = 5;
TOLERANCE = 1e-4;
run();
}