# Chapter 11 Species and charge transport

Understanding charge and species transport is critical to understanding how electric fields couple to fluid flow in dynamic systems. So far, the only species transport we have discussed is passive scalar diffusion in Chapter 4, and the only treatment of ion transport was the equilibrium distribution of ions specified by Boltzmann statistics in Chapter 9. Brief mention of the charge transport equation (albeit with diffusion ignored) was made in Section 5.2.1. Now, we describe a general framework for species and charge transport equations, which assists us in understanding electrophoretic separations (Chapter 12), dynamic modeling of electrical double layers (Chapter 16), and dielectrophoresis (Chapter 17), among other topics. In the following sections, we first describe the basic sources of species fluxes. These constitutive relations include the diffusivity (first discussed in the context of microfluidic mixing in Chapter 4), electrophoretic mobility, and viscous mobility. The species fluxes, when applied to a control volume, lead to the basic conservation equations for species, the Nernst-Planck equations. We then consider the sources of charge fluxes, which lead to constitutive relations for the charge fluxes and definitions of parameters such as the conductivity (first discussed in Chapter 3), as well as the molar conductivity. Since charge in an electrolyte solution is carried by ionic species (in contrast to electrons, as is the case for metal conductors), the charge transport and species transport equations are closely related—in fact, the charge transport equation is just a sum of species transport equations weighted by the ion valence and multiplied by the Faraday constant. We show in this chapter that the transport parameters D, μEP, μi, σ, and Λ are all closely related, and we write equations such as the Nernst-Einstein relation to link these parameters. These issues affect microfluidic devices because ion transport couples to and affects fluid flow in microfluidic systems. Further, many microfluidic systems are designed to manipulate and control the distribution of dissolved analytes for concentration, chemical separation, or other purposes.

### 11.1 Modes of species transport

•   11.1.1 Species diffusion
•   11.1.2 Convection
•   11.1.3 Relating diffusivity and electrophoretic mobility: the viscous mobility

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### 11.2 Conservation of species: Nernst-Planck equations

In this section, we describe the phenomena that lead to flux of species. These fluxes, when applied to a control volume, lead to the Nernst-Planck equations.

#### 11.2.1 Species fluxes and constitutive properties

In the absence of chemical reactions, the two mechanisms that lead to flow of species into or out of a control volume are diffusion and convection.

##### Diffusion

In the dilute solution limit with negligible thermodiffusion effects (which is applicable for most ionic species in the conditions used in microfluidics), Fick’s lawdefines a flux density of species proportional to the gradient of the species concentration and the diffusivity of the species in the solvent: where diff,i [mol∕s m2] is thediffusive species flux density (i.e., the amount of species i moving across a surface per unit area due to diffusion), Di is the diffusivity of species i in the solvent (usually water), and ci is the concentration of species i. Fick’s law is a macroscopic way of representing the summed effect of the random motion of species owing to thermal fluctuations. Fick’s law is analogous to theFourier law for thermal energy flux caused by a temperature gradient and the Newtonian model for momentum flux induced by a velocity gradient, and the species diffusivity Di is analogous to the thermal diffusivity α = k∕ρcp and the momentum diffusivity η∕ρ.

##### Convection

In addition to the random fluctuations of ions due to thermal motion, the deterministic motion of ions due to fluid flow and electric fields (i.e., convection) also leads to a species flux density: where conv,i [mol∕s m2]is the convectivespecies flux density (i.e., the amount of species i moving across a surface per unit area due to convection) and i is the velocity of species i. As described in Equation 11.4, the velocity of species i is given by the vector sum of (the velocity of the fluid) and EP,i (the electrophoretic velocity of the ion with respect to the fluid: (11.9)

Since we often use units of mol∕L for species concentration, the species concentrations must be converted to mol∕m3 if species fluxes are to be in SI units.

#### 11.2.2 Nernst-Planck equations

In general, the transport of a species i in the absence of chemical reactions can be described using theNernst-Planck equations: Here, Di is the diffusivity of species i and i is the velocity of species i. i is the vector sum of the fluid velocity and theelectrophoretic velocity of the species μEP,i . The first term in the brackets is thediffusive flux and the second is theconvective flux. Restated verbally, theNernst-Planck equations in this form say that the change in the concentration of a species is given by the divergence of the species flux density. This relation can be derived (Exercise 11.3) by drawing a control volume and evaluating the species fluxes. Such a control volume is shown (for a Cartesian system) in Figure 11.1. Figure 11.1: Species fluxes for a Cartesian control volume. ##### Comparing the Nernst-Planck and Navier-Stokes equations

Equation 11.10 can be reorganized into a form similar to the one we have used for the Navier-Stokes equations. For example, assuming the diffusivity is uniform and implementing the product rule ∇( ici) = i∇ci+ci∇⋅ i, we obtain (11.15)

As compared to the Navier-Stokes equations for momentum transport, the Nernst-Planck equations (shown here without chemical reaction) have no source term akin to a body force term, nor a pressure term. In place of these, the Nernst-Planck equations have a term (ci∇⋅ i) proportional to the divergence of the species velocity. For incompressible fluid flows, the divergence of the fluid velocity ∇⋅ is zero owing to conservation of mass; however, we cannot in general say that the divergence of the species velocity ∇⋅ i is zero, since systems with spatially-varying conductivity have finite divergence in the species velocity field.

### 11.3 Conservation of charge

•   11.3.1 Charge conservation equation
•   11.3.2 Diffusivity, electrophoretic mobility, and molar conductivity

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### 11.4 Logarithmic transform of the Nernst-Planck equations

The Nernst-Planck equations: (11.24)

can be difficult to solve numerically for microfluidic systems, since the variation in charged species in microfluidic systems (for example, near charged walls) is typically exponential and the derivatives are difficult to handle. Numerical simulations performed without extreme care often lead to non-physical solutions, for example negative concentrations, or numerical instability. One technique that greatly helps to address this is to make the substitution ci = exp(γi), leading to the equation This transformed equationhas two key advantages. First, it governs the transport of γi, not ci, and small errors no longer lead to nonphysical solutions such as negative concentrations. Second, the logarithm (γi) of the concentration is expected to vary linearly when the concentration ci varies exponentially, and thus γi is well suited for solving numerically on regular meshes.

### 11.5 Microfluidic application: scalar-imagevelocimetry

•   11.5.1 SIV using caged-dyeimaging
•   11.5.2 SIV using photobleaching

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### 11.6 Summary

In this chapter, we have described the basic sources of species fluxes, including constitutive relations such as the diffusivity, electrophoretic mobility, and viscous mobility. The species fluxes, when applied to a control volume, led to the basic conservation equations for species, the Nernst-Planck equations: This equation can be summed for all species and weighted by the species valence, leading to the charge conservation equation written here for negligible fluid flow and species with identical diffusivities: These equations lead to discussion of transport parameters D, μEP, μi, σ, and Λ, related by the Nernst-Einstein relation: and the definitions of conductivity and molar conductivity. These issues are central to nonequilibrium electrokinetic flow and microfluidic manipulation of chemical species.

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### 11.8 Exercises

1. Consider the distribution of an ion of valence z in a 1D potential field φ(y). Derive the Einstein relation by:

• writing c(φ),
• writing the 1D Nernst-Planck equations for ion transport in the y-direction, and
• showing that equilibrium requires that the Einstein relation holds.
2. Consider the 1D ion flux equation for a chemical species i: (11.29)

and show that the normalized flux ji∕ci is proportional to the spatial gradient of the electrochemical potential gi = gi+RT ln +ziFφ.

3. Most exercises are excluded from this web posting.
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