Interfaces integral equation formalism

How to represent an interface between an electrolyte and a solid or fluid medium (of nanometric radius of curvature or of planar geometry) within integral eqnations The simplest approach is to consider the solid-air object as a new species at infinite dilntion, labelled 0 in the following, say a sphere of radius R which adds to the previous ion components, and to repeat the integral equation formalism for the new mixture. Since the new object is at zero density, the ion-ion correlations are not perturbed... 

Density profiles are the central quantity of interest in computer simulation studies of interfacial systems. They describe the correlation between atom positions in the liquid and the interface or surface . Density profiles play a similarly important role in the characterization of interfaces as the radial distribution functions do in bulk liquids. In integral equation theories this analogy becomes apparent when formalisms that have been established for liquid mixtures are employed. Results for interfacial properties are obtained in the simultaneous limit of infinitesimally small particle concentration and infinite radius for one species, the wall particle (e.g., Ref. 125-129). Of course, this limit can only be taken for a smooth surface that does not contain any lateral structure. Among others, this is one reason why, up to now, integral equation theories have not been able to move successfully towards realistic models of the double layer. 

The derivation of the given whole field formulation, introducing the Dirac delta function (d/) into the surface tension force relation to maintain the discontinuous (singular) nature of this term, is to a certain extent based on physical intuition rather than first principles (i.e., in mathematical terms this approach is strictly not characterized as a continuum formulation on the differential form). Chandrasekhar [31] (pp 430-433) derived a similar model formulation and argued that to some extent the whole field momentum equation can be obtained by a formal mathematical procedure. However, the fact that the equation involves /-functions means that to interpret the equation correctly at a point of discontinuity, we must integrate the equation, across the interface, over an infinitesimal volume element including the discontinuity. 

Although this expression for uf is formally complete, it contains the unknown shape function h(xs,t). The pressure gradient, on the other hand, is determined from the conditions on pressure at the interface, z = h, by means of Eq. (6-23). To obtain a governing equation from which we can determine the unknown shape function, we can follow either of the two paths outlined in the preceding chapter, namely, either integrate the continuity equation, (6-1), to obtain an expression for uf and apply boundary conditions (6-5) and (6-19), or integrate the continuity equation first to obtain (5-75), and then apply the boundary conditions to evaluate this integral constraint. We follow the latter route. 

We now will outline the formalism which we used to derive the renormalization group flow equations. First we reformulate (1) by using path integrals. To do this we rewrite (1) as a Fokker-Planck equation for the probabihty density P(z(x),t) for the interface profile z(x) at time t The conditional probabihty P z x)yt] z x),t ) to have the profile z x) at time t having the profile (x) at time averaged over the disorder can be represented by a path integral as... 

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