Contact Theorem

When the two walls approach each other, the density at the wall changes. The contact wall theorem states that the force per unit area is 

The two walls repel each other when the density of hquid at the walls increases upon approach. If the density decreases, the two surfaces attract each other. To obtain the force per unit area, we need to find out how the density at the walls changes when they come closer. 

Let us consider simple models for fluids and have a look at which structure they assume at an isolated waD. The simplest models for fluids are hard-sphere and Lennard-Jones fluids. In a hard-sphere fluid, each molecule is taken to be a sphere with a defined radius and volume. It does not interact at all with the other spheres or the wall, except when they get into contact. Then, it repels the other spheres or the walls with an infinitely steep potential so that no overlap occurs. No attraction is taken into account. 

Neglecting attraction between the molecules is a cmde assumption since, after all, the attraction leads to condensation and that is an essential feature of a liquid. Therefore, a better model for a liquid is the Lermard-Jones fluid. In the Lermard-Jones fluid, the potential energy between molecules is described by 

elj is the energy characterizing the interaction strength and a is the molecular radius. The first term describes a steep repulsion caused by overlapping electron orbitals. The second term accounts for the van der Waals attraction between molecules. 

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In [17] we have calculated (Ss(r)) and shown that the value of (,s(r)) on the wall verifies the contact theorem. The integral of (s(r)) gives the adsorption I. From the calculation of the free energy we have obtained the surface tension and shown that the Gibbs isotherm leads to the same F as the one calculated via the density profile. Thus, as in the case of the homogeneous Yukawa fluid we have a totally self consistent calculation. 

The density profile satisfies the so-called contact theorem [47], according to which,... 

The density profile pi z) for the uncharged surface has three different terms according to the contact theorem (73) and according to the expression (28) for the pressure. The first two of them describe the hard-sphere and ion-pairing contributions, respectively. They can be obtained in the framework of the associative version of the Henderson-Abraham-Barker (HAB) theory [53, 54], According to the obtained results [54],... 

According to the contact theorem, the expression (84) reproduces exactly the AMS A result (31) for the electrostatic part of the pressure,... 

Finally, the expression (78) for q(z) can be used to modify the nonlinear Poisson-Boltzmann theory in order to consider a highly charged surface [59, 60]. In this case, for the profile (h z) the new term appears which exactly reproduces the last electrostatic term in the contact theorem (73). 

Grahame equation and also as the contact theorem [6]. This, fundamentally, is a relationship between the surface charge density, (Tq (which is defined as o-q = — Jpedy, with a SI unit of C/m ), and the limiting value of the ionic density profile at the substrate-fluid interface. For a single fiat surface with an infinite extent of the adjacent liquid, an expression for co can be obtained from the Poisson-Boltzmann equation as... 

Putting it all together yields the general contact theorem for a planar on the average, but not necessarily smooth, surface... 

This theorem is a generalization of the previously derived contact theorems to the realistic case of non smooth electrode surfaces. It contains the previous results as particular cases. 

Replacing into the general contact theorem Eq.(1.29) gives [55]... 

There are several remarks about the Gouy-Chapman theory In spite of the apparent oversimplification the Poisson Boltzmann equation satisfies an overall dynamic equilibrium condition, that fixes the contact density at the electrode surface. This is the contact theorem... 

This contact theorem, as well as other sum rules that are valid for the charged interface will be discussed in the next section. The density profiles obtained from the Gouy-Chapman theory are monotonous, that is they show no oscillations. Since in this theory the contact theorem and the electroneutrality condition are satisfied, then, p,-(2) is pinned at the origin, and has a fixed integral, so that the density profile cannot deviate too much from the correct result. When the contact theorem is not... 

This equation is the plane electrode version of the Hypernetted Chain equation, called the HNCl [81]. It is completely defined in terms of short ranged quantities, which is not the case for the first form of the equation Eq.(1.76). The HNCl is the theory that has the closure with the largest number of graphs. It satisfies the electroneutrality relations and the Stillinger Lovett sum mles. One important observation about the HNCl is that it does not satisfy the contact theorem Eq.(1.29), but rather m ... 

We discuss the exact sum mles for the more complex, and realistic models, such as the contact theorem, which gives the amount of ions in the interface as a function of the excess charge, and the screening theorems, which are conditions on the charge distributions in an inhomogeneous system. 

To introduce solvation forces, we start with the contact theorem. Consider a fluid between two parallel planar walls as depicted in Figure 7.2. The contact theorem relates the local number density of molecules next to the walls, Q, to the pressure between the plates. This number density depends on the distance between the walls Qq(x). At infinite distance, the number density, written as Qq(cxd), is equal to that of an isolated wall. Qo(oo) is in general different from the number density in the bulk fluid. For example, recent experiments suggest that the density of water close to hydrophobic surface is depleted [1076-1078]. 

To calculate the force, we apply the contact theorem. Therefore, we first need to know the density at the walls. As a first approximation, we assume that the number density in the gap can be superimposed by the densities of the two walls according t0Q (5c,i) = e ( ) + e (x- )-e [I075,1096]. Here, Q (x, ) is the number density at position normal to a surface in a gap of width x. Q ( ) is the number density at position normal to an isolated surface, which is equal to that of a surface in an infinite gap (q ( ) = = oo, )). Q, is the bulk number density. Please note that... 

The same result is obtained if we calculate the density at the other wall at = Inserting this expression into the contact theorem (Eq. (10.1)) and considering that Qo(oo) = AQ + Qb leads to (see also Refs [31, 1097]) ... 

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