COMSOL 6.4 - Diffuse Double Layer

Close to an electrode surface, ions in the electrolyte are attracted and repelled by unscreened excess charge on the electrode. This region is called the diffuse double layer. Its size is normally a few nanometers away from the electrode surface. The electrical interactions here mean that charge separation can occur, and the assumption of local electroneutrality is not valid. The study of the diffuse double layer is important to applications that consider very thin layers of electrolyte, such as electrochemical capacitors, atmospheric corrosion problems, ion-selective field effect transistors (ISFETs), and nanoelectrochemistry.

This example shows how to define a double layer model by combining the Transport of Diluted Species and Electrostatics physics interfaces to account for mass transfer and charge transfer, respectively. The model contains one charged electrode adjacent to bulk solution.

One of the simplest physical models of the double layer is the Gouy–Chapman–Stern (GCS) model. In Gouy–Chapman theory (Ref. 1, Ref. 2), the diffuse double layer is treated as a multiphysics coupling of the Nernst–Planck equations for mass transport of all of the ions, with Poisson’s equation (Gauss’s law) for the charge density and electric field. The combination of these equations is frequently referred to as the Poisson–Nernst–Planck (PNP) equations or Nernst–Planck–Poisson (NPP) equations.

Gouy–Chapman theory predicts the spatial extent of the diffuse double layer to be of the same order as the Debye length, xD, for the solution, defined for a monovalent binary electrolyte with concentration cbulk and solvent relative permittivity εr,D:

(1)

The Stern modification (Ref. 3) to the Gouy–Chapman theory additionally considers the electric field inside a plane called the outer Helmholtz plane (OHP). This is the plane of closest approach to the electrode for dissolved ions in solution, due to the finite size of the ions and surrounding solvent molecules. In the distance of a fraction of a nanometer between this plane and the electrode surface, Stern’s theory describes the electrolyte as a dielectric of constant permittivity. This uncharged region of the double layer is sometimes called the compact double layer.

When there is a low applied potential (< RT/F ~ 25 mV), the Gouy–Chapman–Stern theory has an analytical solution. The capacitance, Cdl, is predicted to be the same as for a parallel plate capacitor whose electrode separation is equal to the sum of the Stern layer size and the Debye length:

(2)

In the diffuse layer, the relative permittivity for bulk water is used, whereas within the Stern layer, due the strong electric field, the permittivity is lowered to εr,S = 10 (Ref. 5).

Model Definition

The model geometry is in 1D (a single interval between 0 and L) and consists of a single domain, representing the electrolyte phase from the electrode through the diffuse double layer, as far as the electroneutral bulk solution. The compact component of the double layer is handled using a boundary condition set at x = 0.

Domain equations

The concentrations, ci (SI unit: mol/m3, i = +, −), of two ions of opposite charge (+1/−1) are solved for in the electrolyte phase. The fluxes (Ji, SI unit: mol/(m2·s)) of the ions are described by the Nernst–Planck equation

with Di (SI unit: m2/s) being the diffusion coefficient, um,i (SI unit: s·mol/kg) the mobility, zi the charge, F (SI unit: C/mol) the Faraday constant, and ϕ (SI unit: V) the electric potential in the electrolyte phase.

Assume that there are no homogeneous reactions of the ions in the solution. Then, conservation of mass requires that, for both species:

For the potential, the Poisson equation (Gauss’s law) states

where ε is the permittivity (SI unit: F/m) and ρ the charge density (SI unit: C/m3). The charge density depends on the ion concentrations according to:

Boundary Conditions

Choose the bulk solution as the ground condition for the electrolyte potential:

which is set at the outer boundary.

The ion concentrations are set to their bulk values at the outer boundary. Because bulk solution is electroneutral, the positive and negative ions have equal concentrations here.

From Gauss’s law, the electric field inside the compact double layer is constant because this layer is uncharged and has a uniform permittivity. Hence, according to the Stern theory for a compact double layer of a constant thickness, λS (SI unit: m), the following Robin-type boundary condition applies for the electrolyte potential:

where ϕM (SI unit: V) is the applied potential of the electrode, as measured against bulk solution, and n is the boundary normal vector, pointing outward from the electrolyte domain.

For the case of a nonzero Stern layer thickness, the condition can be reformulated as a surface charge condition that depends on the potential difference, ϕΔ (SI unit: V), between the electrode potential and the electrolyte potential at the outer Helmholtz plane (x = 0):

where

The condition in is used to define the above condition. The problem is solved for a sweep of values of ϕM from 1 mV to 10 mV.

If the electrode is inert to electrolysis of the ions at the electrolyte–electrode interface, no flux of the ions into the electrode surface can occur. Hence, conditions are used for both ion concentrations at the electrode boundary.

Results and Discussion

Figure 1 shows the potential profile in the electrolyte as a line graph and the potential at the electrode as a point.

Figure 1: Electrolyte potential profile (blue line) and 10 mV applied electrode potential versus bulk solution (red dot).

The potential difference between the red dot (electrode surface) and the value of the blue line at x = 0 represents the potential difference over the compact double layer. The electrolyte potential falls off exponentially on the Debye length scale (Equation 1), as predicted by Gouy–Chapman theory.

Figure 2 shows the concentrations of the two ionic species. Because the electrode is polarized positively against bulk solution, it is positively charged and attracts anions while repelling cations. The concentration profiles confirm that the anion in the electrolyte is accumulated at the surface, while the cation is depleted.

Figure 2: Concentration profile for the two ions, showing cation depletion (blue line) and anion accumulation (green line).

Figure 3 shows the surface charge dependence on the applied potential. This agrees precisely with the low potential analytical expression (Equation 2) for the smaller potentials. At 10 mV applied potential, a small deviation is observed as the low-potential approximation becomes less precise.