src/test/debye.c

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,

\displaystyle 0 = \nabla \cdot (e \omega_i Z_i c_i \nabla \phi) + \nabla \cdot (\omega_i k_B T \nabla c_i) \quad \mathrm{with} \quad \nabla \cdot (\epsilon \nabla \phi) = \sum_i e c_i

where \phi 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,

\displaystyle \hat{c}_+ = exp (-\hat{\phi}), \, \hat{c}_- = exp (\hat{\phi}) \quad \mathrm{with} \quad (\hat{\phi})_{xx} = 2 \sinh (\hat{\phi}).

#include "grid/multigrid1D.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, (|Z|=1).

scalar phi[];
scalar Cp[], Cm[];
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 \phi =1. The concentrations of the positive and negative ions depend exponentially on the voltage electrode.

phi[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

phi[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 \phi.

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

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 \phi 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 (phi, rhs);
}

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

Results

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

set xlabel 'x'
gamma = tanh(0.25)
fi(x) = 2*log((1+gamma*exp(-sqrt(2)*x))/(1-gamma*exp(-sqrt(2)*x)))
nplus(x) = exp(-fi(x))
nminus(x) = exp(fi(x))
plot 'log' u 1:2 notitle, fi(x) t '{/Symbol f}',\
     'log' u 1:3 notitle, nplus(x) t 'n+',\
     'log' u 1:4 notitle, nminus(x) t 'n-' lt 7

Profiles of electric potential and concentrations (script)

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