# What equation describes the electrostatic potential in these circumstances?

I have a solver for Poisson's equation and it works nicely. It uses finite differences. It works in the presence of multiple dielectrics.

It also solves the Poisson Boltzmann equation. That is, fixed charges with free moving charges, as in a molecule immersed in a solution with salt assuming that the molecule and liquid can be approximated as a continuum medium.

Now, what happens if there are currents? this violates the assumption of equilibrium required for Poisson Boltzmann. I'm looking for the equation that describes this situation. I guess it should have the form

$$\nabla\cdot(\epsilon\nabla \phi) = -\rho_\text{fixed} + \text{<ions effect>} + \text{<current effect>}$$

I'm pretty sure this has already been studied. Can anyone direct me towards where to look for more detail? is there an equation with a name (like Poisson Boltzmann) for this?

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Erm.. Maxwell's laws? – Manishearth Mar 9 '12 at 13:41
of course Maxwell describes the phenomena, but to go from there to salt concentration using only Maxwell it's computationally impossible I think. I guess you mean doing a full molecular dynamics simulation. I don't think simulations that long have been done. Maybe that's the only way though. – jbcolmenares Mar 9 '12 at 13:55
Naah, there probably is something.. I'm not too familiar with this particular field though.. – Manishearth Mar 9 '12 at 14:01
$j = \sigma E$ ? – Physiks lover Nov 18 '12 at 19:47

In classical electrostatics, Gauss' Law can be used to derive the relationship between the electric potential, $\varphi$ for a homogeneous medium (constant permittivity, $\epsilon$) and volume charge density, $\rho_{V}$ in the form of the Poisson equation, which, in cartestian co-ordinates, is given by:

$$\nabla^{2}\varphi=\frac{\partial^{2} \varphi}{\partial x^{2}}+\frac{\partial^{2} \varphi}{\partial y^{2}}+\frac{\partial^{2} \varphi}{\partial z^{2}}=-\frac{\rho_{V}}{{\epsilon}_{r}{\epsilon}_{0}},$$

where ${\epsilon}_{r}$ is the relative permittivity (dielectric constant) for the homogenous medium.

Now, for an ionic solution at thermal equilibrium and at temperature $T$, the charges are uniformly distributed.

Under the effect of an electrostatic field, the positive ions are attracted towards the negative electrode and negative ions attracted towards the positive electrode. Furthermore, positive ions become repelled by other positive ions and similarly negative ions become repelled by other negative ions, until a new equilibrium is reached. At equilibrium, the ions are distributed with various energies, $E$ given by the Maxwell-Boltzmann distribution law, in which the probability of a particle having energy $E$ is proportional to $\exp(-\frac{E}{kT})$ where $k$ is Boltzmann's constant and $T$ is the (absolute) temperature [in Kelvins].

If $z_{i}$ is the number of charges of the $i^{th}$ ionic species, then its electric potential energy is $z_{i}e\phi$ where $e$ is the elementary electric charge ($e = 1.602\times10^{-10}$ Coulombs). The concentration (number density) of the $i^{th}$ ionic species at position $\textbf{r}$ is then given by:

$$n_{i}(\textbf{r})=n_{i}^{\infty}\exp\left(-\frac{z_{i}e\phi(\textbf{r})}{kT}\right),$$

where $n_{i}^{\infty}$ is the number concentration of the $i^{th}$ ionic species in the bulk solution. The volume charge density is therefore:

$$\rho_{V}(\textbf{r})=\sum\limits_{i=1}^Nz_{i}en_{i}(\textbf{r})=\sum\limits_{i=1}^Nz_{i}en_{i}^{\infty}\exp\left(-\frac{z_{i}e\phi(\textbf{r})}{kT}\right).$$

Which, when substituted into the Poisson equation, gives the Poisson-Boltzmann equation for potential of an ionic solution:

$$\nabla^{2}\varphi=-\frac{1}{{\epsilon}_{r}{\epsilon}_{0}}\sum\limits_{i=1}^Nz_{i}en_{i}^{\infty}\exp\left(-\frac{z_{i}e\phi(\textbf{r})}{kT}\right).$$

This is a non-linear partial differential equation, the solution of which is dependent upon the specific geometry and properties of the electrolyte. For the case of an infinite sheet (plate) electrode located in the y-z plane at the origin, with the potential of the plate $\phi=\phi_{0}$ at $x=0$ and the potential in the solution $\phi\rightarrow0$, $\frac{d \phi}{dx}\rightarrow0$ as $x\rightarrow\infty$, the solution, for low potential, $\phi$, that is, $\lvert\frac{z_{i}e\phi}{kT}\rvert\ll 1$, the Poisson-Boltzmann equation becomes linearized to $\nabla^{2}\varphi=\kappa^{2}\phi$ (the Debye-Hueckel equation) which has the solution:

$$\varphi(x)=\varphi_{0}\exp(-\kappa x)$$

where $\kappa=\sqrt{\frac{2z^{2}e^{2}n}{\epsilon_{r}\epsilon_{0}kT}}$.

For the non-electrostatic case (ie: current flow and/or ion flow), the ions in the solution will undergo transport under the effect of the applied electric field. These ions will initially accelerate up to a speed, $s$, limited by the hydrodynamic properties of the solute (Stokes law).

We can calculate the hydrodynamic force, $F_{H}$ exerted on an ion of radius $r$, travelling at speed, $s$ through the solute of density, $\rho$ and viscosity, $\eta$ to get $F_{H}=6\pi\eta rs$ from which the Stokes-Einstein equation for the diffusion coefficient is derived:

$D=\frac{kT}{6\pi \eta r}$

From the Nernst equation we can calculate the electrochemical potential of an ionic species from ionic concentration (activity) gradients present in solution. When coupled with the conservation of mass, we get the Nernst-Planck equation:

$$\frac{\partial n_{i}}{\partial t}= \nabla \cdotp \lgroup D_{i}(\nabla n_{i}+\frac{q_{i}n_{i}}{kT}\nabla \phi)\rgroup.$$

Of course, these models include assumptions which may not always be valid, such as the use of Stokes law or the assumption of non-interaction of ions. More accurate numerical models may require the use of empirical data to account for these and other effects, including chemical reaction kinetics.

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