# Plate-Plate interactions

Problems with the Poisson-Boltzmann equation
The Poisson-Boltzmann (PB) equation is usually used to calculate the potential and force between two interfaces in the form of plate-plate geometry and is a critical component in the primitive classical DLVO theory of colloidal stability. In the post-DLVO era of highly-ionic solutions which are biological in nature and occurs in numerous engineering contexts, we must examine its assumptions in order to understand our limitations:
1. Linearisation. This mathematical approximation in the form of Debye-Huckel linearisation is assumed correct for weak potentials which is on the edge of validity for air-water interfaces. Beyond this, numerical methods must be applied, but, with the increased sophistication of potentials this is an entirely non-trivial task and sometimes with not too much gain for the induced MATLAB stress.
2. Finite size effects (Steric). The PB does not take into account the finite size of the ions (to the order of Angstroms) which usually leads to an infinite potential at short distances as seen in foam literatures. This is of course, a completely whacky assumption, as it assumes an infinite surface for ions to adsorb onto. We usually remedy this by neglecting this very short-ranged interaction although we perhaps shouldn’t if we are being sophisticated.
3. Dispersion forces. The PB does not take into account the ion-interface interactions, which also take control of ion-specific effects. (To those of you that are more chemistry-inclined, this provides a partial model to the Hopfmeister effect of ions.) To bubbleologists, this is utterly critical to the coalescence process of foaming and cannot be neglected in our calculations. This is also partly why the DLVO theory fails so completely at higher concentrations of ionic electrolytes and the incorrect calculation of disjoining pressures between foam films.
4. Solvation forces. The PB neglects the behaviour of ions in the interfacial regions and assumes the bulk behaviour. Recent simulations estimates the change in the self-energy of ions as they approach the interface and can potentially be dominant over the dispersion forces if we’re interested in a precise calculation of the surface tension.
5. Image repulsion. This goes back to the Onsager-Samaris theory of surface tension and the repulsive image forces between ions. Onsager-Samaris assumed no separation of charge normal to the surface and so the electrostatic potential is zero. This is a dramatic simplification which did not correspond to experiments except in specific cases. For example, the inclusion of dispersion forces results in charge separations and electrostatically-significant potentials. Nonetheless, image repulsions do exist, just not the sole interaction here.
6. Discrete charge. The final approximation is perhaps a reluctant but necessary one, the PB assumes a smooth and smeared our surface and counter-ion charge, whereas they tend to be discrete in real systems. However, there does not seem to be a simple model of discrete charge that is easy to use (a very unfortunate fact of science that complicated but correct expressions are used by virtually nobody whilst simple, beautifully flawed approximations are universally adored)

If we assume 6, not assume 1 and take into account 2,3,4,5, then we’ll have the bombastically non-linear Poisson-Fermi (because of 2) equation [assuming 1:1 electrolyte]

$\displaystyle \phi''=\frac{\sinh(\phi+U)}{1-\phi_{0}+\phi_{0}\cosh(\phi+U)}$

where $\displaystyle U=\frac{Ae^{-x}}{x}+\frac{B}{x^{3}}+Ce^{-x}$ is the potential responsible for image, dispersion and solvation force, respectively. Here $\phi_0=2a^3c_0$ with $a$ being ion-size and $c_0$ initial ionic concentration.

We can reduce this imposing Poisson-Fermi equation to the usual linearised PB by letting $\phi_0\rightarrow 0$ (neglecting finite size effects) and let $\sinh x\approx x$. We then still need to solve the resulting equation, slightly easier than the Poisson-Fermi but still quite formidable.

To illustrate plate-plate interactions, here’s one from London Bridge Tube station that I took last week.

[Onboard the Death Star, London Bridge, 11/3/15]

In a related me-with-a-camera update, I’m now on 500px as well as Flickr for no good reasons at all!