11.1 Modes of species transport [species/charge transport top]

Chemical species are transported through fluid systems owing to both diffusion and convection, as described below. We omit consideration of chemical reactions for the purposes of this discussion.

11.1.1 Species diffusion

Species diffusion refers to the net migration of species owing to Brownian motion in the system. The thermal energy in the system leads to species moving in random directions with temporally varying velocities whose magnitude are proportional to the energy of the system (and therefore proportional to RT ). The net effect is a net migration of species away from high-concentration regions and toward low-concentration regions.

11.1.2 Convection

In addition to the random fluctuations of chemical species due to thermal motion, the deterministic motion of chemical species due to fluid flow and electric fields (i.e., convection) also leads to a species flux, as the species are carried along when the fluid moves. Considering chemical species as passive scalars (i.e., parameters described with scalar variables that are simply carried along with the fluid) leads to the passive scalar diffusion equation (Equation 4.6), which describes chemical species diffusion in the absence of electric fields.

In the presence of electric fields, though, charged ions move in response to the Coulomb force they feel in that electric field— a process termed electrophoresis. The forceexerted by an electric field on an ion is (cf. Section 5.1.10)

(11.1)

where e is the magnitude of the charge of an electron(e = 1.6×10^-19C) and z is the charge number or valenceof the ion (for example, z = 1 for Na⁺ and z = -2 for SO₄^-2). The steady-state response of the ion occurs at equilibrium between two equal and opposite forces: the Coulomb force from the electric field and the “drag” force caused by the solvent molecules. This drag force cannot be predicted precisely without detailed molecular dynamics calculations, but one achieves results within an order of magnitude for most ions if one simply models the ion as a sphere in Stokes flow with a radius commensurate with its size:

(11.2)

where _i is the velocity of the ion i and r_hydrated is the hydrated radius of the ion, i.e., the radius of the ion and any water molecules that are bound to it at equilibrium. Clearly, the Stokes flow equations do not apply at the length scale of an ion; however, the effects have the same scaling, and the magnitude of the “drag” force is surprisingly close, despite the incorrectness of the model.

At equilibrium, which for an ion is essentially instantaneous, the electromigratory velocity of species i caused by the electric field is written as

where μ_EP,i [m²∕Vs] is termed the electrophoretic mobility. The motion of an ion (or a charged particle) in an electric field is termedelectrophoresis. Example electrophoretic mobilities are listed in Table 11.1.

Given the motion of the ion with respect to the solvent, the total velocity of the ion _i is given by

where  refers strictly to the velocity of the fluid and _i refers to the total velocity of the ion.

11.1.3 Relating diffusivity and electrophoretic mobility: the viscous mobility

Though they find application in different flux terms, species diffusivity and electrophoretic mobility are closely related phenomena, and can be calculated from one another with simple equations. Diffusivity is a measure of a species’ ability to move randomly because of random thermal molecular motion, and is used to describe diffusive fluxes of species. Electrophoretic mobility is a measure of the species’ ability to move in response to an electric field, and is a component of convective fluxes of species. The molecular collisions that tend to limit both of these motions are the same. The difference is the driving force—thermal motion (proportional to the thermal energy of a mole of ions, and therefore proportional to RT ) vs Coulomb force (proportional to the charge of a mole of ions zF ). Consistent with this, thediffusivity of a species and itselectrophoretic mobility are related to a single property and to each other by theNernst-Einstein relation:

(11.5)

On a per mole basis, we can equivalently write

We call μ_i [m∕N s] theviscous mobility of species i. The viscous mobility measures the extent to which a species can move in the solvent in response to the force—it is a viscous property of the ion and the solvent, and 1∕μ_i [N s∕m] is a molecular scale analog of the drag coefficient that might be defined for a macroscopic object. In fact, a viscous mobility can be defined for a macroscopic object, for example, for a sphere of radius a in Stokes flow (see Section 8.3.1), the viscous mobility is given by μ = .

The Nernst-Einstein relation applies rigorously for species that can be modeled as being a point charge. The point charge assumption is correct for small ions and is approximately correct for proteins (even though the charge on a protein in general is distributed). However, it fails when the hydrodynamic effects of diffusion and the electrostatic effects of electrophoresis behave differently owing to the structure of the molecule. It cannot, for example, be used to relate the diffusivity of charged macromolecules like DNA to their electrophoretic mobility (see Chapter 14) owing to the difference in the hydrodynamic and electrostatic interactions of the components of the polyelectrolyte.

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