V. Electrostatics
Lecture 24: Diffuse Charge in Electrolytes
MIT Student
1. PoissonNernstPlanck Equations
The NernstPlanck Equation is a conservation of mass equation that describes the influence of an
ionic concentration gradient and that of an electric field on the flux of chemical species,
specifically ions. We can start with the general conservation of mass equation for an
incompressible fluid (
):
The background velocity in the convection term (
) is relatively easy to define in a dilute
solution, where it is just the (mass averaged) velocity of the solvent.
However, this velocity
becomes more difficult to define in concentrated solutions, since the distinction between the
“flux” of an ion and relative to the “flow” of the solvent becomes blurred. More general
prescriptions are available that treat all molecules (ions and solvent) on an equal footing, such as
the StefanMaxwell equations for coupled fluxes or the de GrootMazur equations of
nonequilibrium thermodynamics, but we neglect such complexities here, since most electrolytes
are dilute enough to be well described.
The flux density for the NernstPlanck Equation can be generally expressed as
Using the Einstein relation,
,
and the gradient of the chemical potential for a dilute solution,
we can rewrite the flux as
,
,
where the first term on the RHS is the flux due to diffusion and the second term on the RHS is
the flux due to electromigration (the nonlinear term).
We can now insert this expression for the
flux into the conservation of mass equations and we will obtain the NernstPlanck Equation for a
dilute solution:
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Lecture 24: Diffuse charge in electrolytes
10.626 (2011) Bazant
The NernstPlanck Equation gives us i equations with i+1 unknowns. Hence, in order to solve
the system of equations, we need to come up with one more equation. We can describe the
electrostatic potential by using the Poisson Equation (a mean field approach),
,
where
ρ
is the free charge density and
D
is the is the electric displacement field vector.
If we assume that we have a linear dielectric material, we can describe the electric displacement
field vector as
,
where
ε
is the permittivity of the material (mainly the solvent), and
E
is the electric field
generated by charges in the system. For electrostatics we also know that
Hence we can rewrite the expression for the electric displacement field vector as
We can now insert this expression into the Poisson Equation to arrive at our final form for this
equation:
Using a mean field approximation, we can get another equation for the free charge density,
defined in terms of the mean (volume averaged) ion concentrations,
This equation sums up all the charges of all the ions per unit volume.
Combining this equation
and the Poisson equation, we can get a new equation for the electrostatic potential to combine
with the Nernst Planck Equations:
Our full set of
PoissonNernstPlanck
(PNP) Equations is then:
The system is now fully specified with matching numbers of equations and variables.
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 Spring '03
 RogerD.Kamm
 Electrostatics, Electric charge, diffuse charge, Debye screening length

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