An exact sum rule, due to Henderson, Blum, and Lebowitz, for the contact value of the density profile of ions in a primitive model electrolyte next to a planar, nonpolarizable charged hard wall, has been known for some years. This result has a pleasing physical interpretation and is local. It has been useful in assessing the accuracy of theoretical approximations. However, a sum rule for the contact value of the charge profile for the same system has, until recently, not been known. A few years ago, Boda and Henderson proposed what they thought might be a useful, but approximate, local expression for the contact value of the charge profile at a weakly charged electrode. Very recent computer simulations indicate that this expression may well be exact at low electrode charge. Recently, Holovko, Badiali, and di Caprio have obtained a more general, but nonlocal, sum rule for the contact value of the charge profile that is valid for all electrode charge. In this paper, we develop an alternative, nonlocal, but nonrigorous expression for this quantity. Both the expression of Holovko et al. and our new expression are examined by means of computer simulations. The Holovko et al. expression is exact and, within numerical uncertainties, seems supported by our simulations. Although admittedly nonrigorous, our simpler expression is in seemingly reasonable agreement with simulation and thus appears to be useful. The relation between the two expressions has not been established.
Self-assembly and crystallisation of indented colloids at a planar wall
