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: NernstPlanck 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 NernstPlanck 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 m^{2}] is thediffusive species flux density (i.e., the amount of species i moving across a surface per unit area due to diffusion), D_{i} is the diffusivity of species i in the solvent (usually water), and c_{i} 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 D_{i} is analogous to the thermal diffusivity α = k∕ρc_{p} 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 m^{2}]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∕m^{3} if species fluxes are to be in SI units.
11.2.2 NernstPlanck equations
In general, the transport of a species i in the absence of chemical reactions can be described using theNernstPlanck equations:
Here, D_{i} 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 μ_{EP,i}. The first term in the brackets is thediffusive flux and the second is theconvective flux. Restated verbally, theNernstPlanck 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 NernstPlanck and NavierStokes equations
Equation 11.10 can be reorganized into a form similar to the one we have used for the NavierStokes equations. For example, assuming the diffusivity is uniform and implementing the product rule ∇(_{i}c_{i}) = _{i}∇c_{i}+c_{i}∇⋅_{i}, we obtain
(11.15) 
As compared to the NavierStokes equations for momentum transport, the NernstPlanck 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 NernstPlanck equations have a term (c_{i}∇⋅_{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 spatiallyvarying 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 NernstPlanck equations
The NernstPlanck 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 nonphysical solutions, for example negative concentrations, or numerical instability. One technique that greatly helps to address this is to make the substitution c_{i} = exp(γ_{i}), leading to the equation
This transformed equationhas two key advantages. First, it governs the transport of γ_{i}, not c_{i}, and small errors no longer lead to nonphysical solutions such as negative concentrations. Second, the logarithm (γ_{i}) of the concentration is expected to vary linearly when the concentration c_{i} varies exponentially, and thus γ_{i} is well suited for solving numerically on regular meshes.
11.5 Microfluidic application: scalarimagevelocimetry
 11.5.1 SIV using cageddyeimaging
 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 NernstPlanck 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, μ_{EP}, μ_{i}, σ, and Λ, related by the NernstEinstein relation:
and the definitions of conductivity and molar conductivity. These issues are central to nonequilibrium electrokinetic flow and microfluidic manipulation of chemical species.
11.7 Supplementary reading
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11.8 Exercises

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 NernstPlanck equations for ion transport in the ydirection, and
 showing that equilibrium requires that the Einstein relation holds.

Consider the 1D ion flux equation for a chemical species i:
(11.29) and show that the normalized flux j_{i}∕c_{i} is proportional to the spatial gradient of the electrochemical potential g_{i} = g_{i}^{∘}+RT ln +z_{i}Fφ.
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