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;
      }
      boundary ({phi, 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 \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)

    Profiles of electric potential and concentrations (script)

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