6.2 Integral analysis of Coulomb forces on the electrical double layer [electroosmosis top]

A variety of processes (for example, ion adsorption and acid-base reactions) lead solid surfaces to acquire a nonzero surface charge density in an aqueous solution, and this charge makes the electrical potential at the wall different from that in the bulk. Near the surface, the charge of the wall is counteracted by a thin cloud of oppositely charged ions, which is contained in what we call theelectrical double layer. Unlike the bulk solution, which is electroneutral, meaning that its net charge density ρ_E is zero, the solution in the electrical double layer has a nonzero net charge density. This electrical double layer has a finite thickness, which we denote approximately as λ_D. Electroosmosis stems from the net Coulomb force felt by the fluid near the wall because of the cloud of ions.

To describe this phenomenon, we start by defining adouble layer potential φ = ϕ-ϕ_bulk so that φ is, by definition, zero in the bulk. The value of φ thus specifies how a potential differs from that in the bulk far from walls. The potential difference between the wall and the bulk solution is denotedby φ₀. If an extrinsic electricfield is applied, the force that this field applies to the ions near the wall induces fluid flow in the system. In this section, we show that all we need to predict the bulk fluid flow is the potential change between the bulk fluid and the wall.

We can describe the fluid flow induced by an extrinsic electric field using an integral treatment of theNavier-Stokes equations with an electrostatic source term ρ_Eext. The domain we consider is shown in Figure 6.3—the limits of integration are the wall and a point in the fluid at which the potential can be approximated by φ = 0, typically a point that is much more than λ_D away from the wall. We assume that three quantities are known: φ at the top (φ = 0) and bottom (φ = φ₀) surfaces, as well as the velocity at the bottom surface (u = 0). We also assume that this domain is small enough that we can assume the extrinsic electric field _ext is uniform. The integral analysis then determines the velocity at the top surface.

We write the uniform-viscosity Navier-Stokes equations for a thin region near the wall, and include the electrostatic body force term ρ_Eext:

Here, we assume that _ext is caused by an external power supply and is uniform within the electrical double layer, with a value equal to _ext,wall. The intrinsic electric field - is caused by the chemistry at the surface and is spatially nonuniform.

If we consider steady isobaric flow strictly along a wall aligned in the x-direction, with velocity and potential gradients only in the y-direction, Equation 6.1 reduces to a simple conservation of x-momentum equation:

(6.2)

In this simplified geometry, _ext,wall is tangent to the wall. Recall next the uniform-permittivity Poisson equation (Equation 5.20):

(6.3)

Using Equation 6.3 to substitute for ρ_E, and retaining only y-gradients, we find

(6.4)

The intrinsic field impacts Equation 10.28 through the y-gradients in velocity and electrical potential, while the external field impacts Equation 10.28 explicitly. Rearranging, we obtain

(6.5)

Now we integrate from the wall (y = 0) to a point outside the double layer (y ≫ λ_D, at which φ = 0 by definition). We find from this integral that

(6.6)

Applying theno-slip boundary condition, and forcing the velocity to be bounded at y = ∞, we obtain

(6.7)

where φ₀ is defined as the value of φ at the wall, i.e., at y = 0.

Inner solution. Equation 6.7 is the inner solution for electroosmotic flow, and we can write this solution with a subscript for clarity:

The inner solution is valid only near the wall, where it is correct to assume that E_ext,wall is uniform.

The velocity and the intrinsic potential are similar (i.e., proportional to each other) inside the electrical double layer. Thus the spatial variation in the electrical potential is the same as the spatial variation in the velocity. The spatial distribution of φ is as yet underived, and thus Equation 6.8 only describes one unknown (the velocity) in terms of another unknown (the electrical potential). However, qualitative schematics of the velocity distribution and associated potential distribution, shear, and Coulomb forces predicted for electrical double layers are shown in Figure 6.4. This equilibrium velocity distribution involves counterbalanced (a) viscous shear forces and (b) Coulomb forces near the wall. The viscous shear forces are related to velocity gradients, which decrease as the distance from the wall increases, while the net Coulomb forces are induced by large net charge densities, which also decrease as the distance from the wall increases.

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