Chapter 4
Charge transport
Context: In this learning module, we will introduce the specific equations that describe charge transport. Similar equations can be found for charge transport in semiconductors and in electrolytes. In particular, similar equations should have already appeared in the lecture “Semiconductor Physics”. We will develop the equations here in the context of electrochemistry. The aim of the chapter is to introduce the Poisson-Nernst-Planck equation, which we will solve numerically in the rest of the course.
4.1 Electrostatics
We will repeat the basics of electrostatics here. A point charge \(q\) at the position \(\v {r}_0\) generates an electrostatic potential of the form \begin {equation} \Phi (\v {r}) = \frac {1}{4\pi \varepsilon } \frac {q}{|\v {r}-\v {r}_0|}, \label {eq:pointcharge} \end {equation} where \(\varepsilon =\varepsilon _0 \varepsilon _r\) is the permittivity. In vacuum, \(\varepsilon _r=1\). We will only discuss (aqueous) electrolytes here, i.e. ions dissolved in water. For water, \(\varepsilon _r\approx 80\). Equation \eqref{eq:pointcharge} is the specific solution of the Poisson equation, \begin {equation} \nabla ^2 \Phi (\v {r}) = -\frac {\rho (\v {r})}{\varepsilon } \end {equation} for a point charge \(\rho (\v {r})=q\delta (\v {r}-\v {r}_0)\).
The Poisson equation has the same form as the (stationary) diffusion equation from chapter 3. We can also split this into two equations. First, the electric field \(\v {E}\) is given by \begin {equation} \v {E} = -\nabla \Phi , \end {equation} the (negative) gradient of the potential. (In the sense of the analogy to the diffusion equation, the field is a kind of current density.) The “continuity equation” for the field is given by \begin {equation} \nabla \cdot \v {E} = \frac {\rho }{\varepsilon }. \end {equation} Together, these equations yield the Poisson equation.
We will need the Poisson equation to calculate the electrostatic potential (and thus the electric field) within an electrolyte. Within the electrolyte, we usually have a positively and a negatively charged species, with corresponding concentrations \(c_+(\v {r})\) and \(c_-(\v {r})\). The corresponding charge density is then proportional to the difference of these concentrations, \(\rho (\v {r})=|e|(c_+(\v {r})-c_-(\v {r}))\).
4.2 Drift in an electric field
The charges in our electrolyte not only generate an electric field, they also react to it. The force \(\v {f}\) acting on a particle with charge \(q\) is given by \begin {equation} \v {f}_{\text {E}} = q \v {E}. \end {equation} Positively charged particles move in the direction of the electric field, negatively charged particles against it.
So there is a force acting on our ions due to the electric field. This force alone would lead to an acceleration of the ions, i.e. a continuous increase in speed. Since the ion moves in a medium (solvent, e.g. water), it experiences a flow resistance (see Fig. In the case of laminar flow around a spherical particle with radius \(R\), this is caused only by internal friction within the fluid. The resulting force acts against the direction of motion and is described by Stokes’ law, \begin {equation} \v {f}_{\text {Stokes}} = -6\pi \eta R \v {v} = -\v {v}/\Lambda , \end {equation} with \(\Lambda =(6\pi \eta R)^{-1}\) is described. Here, \(\eta \) is the viscosity of the liquid. The quantity \(\Lambda \) is called mobility. In equilibrium \(\v {f}_{\text {E}} + \v {f}_{\text {Stokes}}=0\), the drift velocity is obtained \begin {equation} \v {v} = q \Lambda \v {E}. \end {equation} This drift velocity, together with \(\v {j} = c \v {v}\), gives the drift current caused by the electric field, \begin {equation} \v {j} = q\Lambda c \v {E} = \sigma \v {E} \end {equation} with \(\sigma = q\Lambda c\). The quantity \(\sigma \) is also called conductivity. An equivalent law applies, for example, to electron conduction in metals.
4.3 Nernst-Planck equation
A diffusion current in combination with drift in the electric field yields the Nernst-Planck equation. The current density for ionic species \(\alpha \) is given by \begin {equation} \v {j}_\alpha = - D_\alpha \nabla c_\alpha + q_\alpha \Lambda _\alpha c_\alpha \v {E}, \end {equation} where we have explicitly indicated by the index \(\alpha \) that the transport parameters (\(D\), \(\Lambda \)), the charge \(q\) and the concentration \(c\) depend on the ionic species. Using the Einstein-Smoluchowski relationship, \(D = \Lambda k_B T\), the mobility \(\Lambda \) can be expressed in terms of the diffusion constant \(D\). This leads to the usual form of the Nernst-Planck equation, \begin {equation} \v {j}_\alpha = - D_\alpha \left (\nabla c_\alpha + \frac {q_\alpha }{k_B T} c_\alpha \nabla \Phi \right ), \label {eq:NPcurrent} \end {equation} in which we have expressed the electric field as \(\v {E}=-\nabla \Phi \).
4.4 Poisson-Nernst-Planck equations
We now combine the Nernst-Planck transport problem with the solution of the Poisson equation to determine the electrostatic potential \(\Phi \). For this purpose, we have to consider two ion species here, one positively (charge \(q_+\)) and one negatively (charge \(q_-\)) charged. Thus, in addition to the potential \(\Phi \), we have to determine two concentrations \(c_+\) and \(c_-\).
The coupled system of equations that describes the transport processes in our electrolyte solution therefore looks like this: \begin {align} \frac {\partial }{\partial t} c_+ + \nabla \cdot \v {j}_+ = 0 & \quad \quad \text {(conservation of positive species)} \label {eq:continuityplus} \\ \v {j}_+ = - D_+ \left (\nabla c_+ + \frac {q_+}{k_B T} c_+ \nabla \Phi \right ) & \quad \quad \text {(transport of the positive species)} \label {eq:currentplus} \\ \frac {\partial }{\partial t} c_- + \nabla \cdot \v {j}_- = 0 & \quad \quad \text {(Conservation of the negative species)} \label {eq:continuityminus} \\ \v {j}_- = - D_- \left (\nabla c_- + \frac {q_-}{k_B T} c_- \nabla \Phi \right ) & \quad \quad \text {(Transport of the negative species)} \label {eq:currentminus} \\ \nabla ^2 \Phi = -\frac {q_+ c_+ + q_- c_-}{\varepsilon } & \quad \quad \text {(Electrostatic potential)} \label {eq:poissonfinal} \end {align}
We will solve this coupled system of differential equations using the finite element method as part of this course. These equations are called the Poisson-Nernst-Planck equations.
Note: A set of equations identical to Eq. \eqref{eq:continuityplus} to \eqref{eq:poissonfinal} describes the transport of charge carriers in semiconductors. The positive charge carriers are holes and the negative ones are electrons. What is referred to here as the “chemical potential” is sometimes called the quasi-potential level there. This type of charge carrier transport has already been discussed in the lecture “Semiconductor Physics”.
In addition to the transient solution of the problem, i.e. the time propagation of the two concentrations \(c_{+}\) and \(c_{-}\), the stationary solution is also interesting. For the stationary solution, the time dependence, i.e. \(\partial c_{+/-}/\partial t = 0\), disappears in these equations. Here, we will consider both the transient and the stationary solution of this and similar systems of equations.
4.5 Poisson-Boltzmann equation
The Nernst-Planck equation can be simplified by introducing a chemical potential. The chemical potential integrates the effect of diffusion into an effective potential \begin {equation} \mu _\alpha (\v {r}) = q_\alpha \Phi (\v {r}) + k_B T \ln c_\alpha (\v {r}). \label {eq:chempot} \end {equation} The term \(q_\alpha \Phi \) is the potential energy of an ion with charge \(q_\alpha \) in an electric field. The term \(k_B T \ln c_\alpha \) is the free energy of an ideal gas with density \(c_\alpha \). We can describe the ions as an ideal gas here because (in our model) they only interact via the electrostatic potential. The current density then becomes proportional to the gradient of the chemical potential \(\mu \), \begin {equation} \v {j}_\alpha = -\frac {D_\alpha }{k_B T} c_\alpha \nabla \mu = -\Lambda _\alpha c_\alpha \nabla \mu . \label {eq:NPcompact} \end {equation} Inserting eq. \eqref{eq:chempot} into eq. \eqref{eq:NPcompact} gives eq. \eqref{eq:NPcurrent}.
Equation \eqref{eq:NPcompact} tells us that no current flows if the chemical potential \(\mu \) is spatially constant. This is exactly the case if \begin {equation} c_\alpha (\v {r}) = c_0 \exp \left (-\frac {q_\alpha \Phi (\v {r})}{k_B T}\right ) \end {equation} with a constant \(c_0\). This equation in conjunction with the Poisson equation to determine \(\Phi \) is also called the Poisson-Boltzmann equation.
4.6 Example: Supercapacitor
In the following video, we discuss the application of the Poisson-Nernst-Planck equation for modeling charge transport in supercapacitors with porous electrodes.