# Ions near charged surfaces

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## Poisson-Boltzmann Model

 Adsorbed ions determine the surface potential, $\Phi _{0}$. The counterions (valence $z_{i}$) form a diffuse cloud near the surface. The Guoy-Chapman model assumes the surface potential is uniform and the counterions are point charges. The concentration of ions, $n_{i}$, depends on concentration of ions far from the surface, $n_{i0}$, and the local electric potential,$\Phi (x)$: $n_{i}=n_{i0}\exp \left( -\frac{z_{i}e\Phi (x)}{kT} \right)$ The local charge density, $\rho$, depends on the charges and the concentrations of ions. $\rho =\sum{z_{i}}en_{i}=\sum{z_{i}en_{i0}\exp \left( -\frac{z_{i}e\Phi }{kT} \right)}$ The Poisson equation relates the local charge density to the Laplacean of the electric potential. $\nabla ^{2}\Phi =-\frac{\rho }{D\varepsilon _{0}}$ Hence the Poisson-Boltzmann equation: $\nabla ^{2}\Phi =-\frac{1}{D\varepsilon _{0}}\sum{z_{i}en_{i0}\exp \left( -\frac{z_{i}e\Phi }{kT} \right)}$ No closed solution has been found for the P-B equation. Approximate solutions abound. For small potentials: Expand the P-B equation in potential; the first term is zero, the second term gives: $\nabla ^{2}\Phi =\kappa ^{2}\Phi$ ${1}/{\kappa }\;$ has units of length and is called the Debye length. $\kappa ^{2}=\frac{e^{2}}{D\varepsilon _{0}kT}\sum{n_{i0}z_{i}^{2}}$ For water at 25, $n_{i0}$ in mol/L: $\kappa =2.328\left( \sum{n_{i0}z_{i}^{2}} \right)^{1/2}nm^{-1}$ For the planar interface: $\Phi =\Phi _{0}\exp \left( -\kappa x \right)$ For a sphere, radius a: $\Phi =\frac{a\Phi _{0}}{r}\exp \left[ -\kappa (r-a) \right]$

## Stern's model for a charged surface

Stern presents a modified version of the Gouy-Chapman model to describe ions near charged surfaces. Stern's model assumes a uniformly-charged surface layer, likely formed by adsorbed ionic surfactant, surrounded by a diffuse double layer. The "Stern layer", which contains all if the charges is separated from the surface by a small region where no surface charges are present. In the figure below, the stern layer is located approximately where the vertical red line is. This smaller layer permits one to approximate ions in the diffuse region as point charges. The gap between the diffuse and stern layer accounts for the closest approach that an ion in solution may take to the charged surface. The stern layer serves to balance the charges in the diffuse region with the charge of the surface and is in equilibrium with the surface. The potential at the surface is taken to be $\zeta$ (usually called the zeta-potential); the potential decreases with distance from the surface roughly exponentially:

$\text{Potential }=\zeta \exp (-\kappa x)\,\!$

 The decay constant is a function of the ionic concentration in the bulk of the liquid: $\kappa =\sqrt{\frac{e^{2}\sum\limits_{i}{c_{i}z_{i}^{2}}}{D\varepsilon _{0}kT}}\,\!$

The interesting term in the expression is familiar in physical chemistry:

$I=\frac{1}{2}\sum\limits_{i}{c_{i}z_{i}^{2}}\,\!$

and is called the ionic strength. Note that the ionic strength varies as the square of the charge on the ions. Addition of electrolyte to the solution results in a increases not only the concentration of ions in the diffuse region but also transfers more ions to the stern layer as a new equilibrium is achieved. This ultimately results in a decrease int he stern potential.

The Stern model has been tested experimental for a range of tropical soil samples. These studies compared the measured net surface charge to theoretically predicted values for different concentrations of electrolytic solutions. The Stern model was more accurate than its predecessor, yet significant discrepancies appeared suggesting that further explanation was necessary for these soils. Indeed a more complicated Surface Complexation model has been developed to describe properties of colloidal soil samples.

(Reference: Environmental Soil Chemistry by Donald L. Sparks)

## Limitations of the Poisson-Boltzmann equation

Since the Poisson-Boltzmann equation is a continuum theory, it is bound to fail at small distances where it no longer faithfully describes the ionic distribution and forces between two charged surfaces. More specifically, the following interactions come into play at short separation:

1) Ion-correlation effects: the mobile counterions in the diffuse double-layer constitute a highly polarizable layer at each interface. Consequently, these two conducting layers experience an attractive van der Waals force which is not included in the Poisson-Boltzmann (nor the Lifshitz theory for that matter). This attraction becomes significant at small distances and it increases with the surface charge density $\sigma$ and valency z of the counterions.

2) Finite ion size effects: which enhances the repulsion between two surfaces. The effect is due to the excluded volume of the counterions adsorbed on surfaces. These create a steric repulsion between the two Stern layers, when they are sufficiently close to each other.

3) Discreteness of surface charges: surface charges are discrete and not continuous, as implied by the Poisson-Boltzmann theory. Discrete ions generally contribute an attractive force.

4) Solvation forces: have to do with the solvent rather than the counterions. These short-range solvation (or hydration) forces can be attractive, repulsive or oscillatory.

(Reference: J. Israelaschvili, Intermolecular & Surface Forces, Academic Press, 1985)