Excess surface chemical potential

The term (dFe/dni)T V is the chemical potential of the gas, //. Similarly, for the equilibrium state of the adsorbed phase, which is characterized by the variables A and T, we may define a surface excess chemical potential by the relation... 

By adopting the usual conventions of chemical thermodynamics, we are able to derive from the surface excess chemical potential pa a number of useful surface excess quantities. Our purpose here is to draw attention to the difference between the molar and the differential surface excess quantities. 

The surface excess chemical potential can be obtained by partial differentiation of G° (defined in Equation 2.28) with respect of the surface excess amount na, with the... 

We now return to the definition of the surface excess chemical potential fta given by Equation (2.19) where the partial differentiation of the surface excess Helmholtz energy, Fa, with respect to the surface excess amount, rf, is carried out so that the variables T and A remain constant. This partial derivative is generally referred to as a differential quantity (Hill, 1949 Everett, 1950). Also, for any surface excess thermodynamic quantity Xa, there is a corresponding differential surface excess quantity xa. (According to the mathematical convention, the upper point is used to indicate that we are taking the derivative.) So we may write ... 

We may derive the relation between these integral molar quantities of adsorption from Equations (2.20) using the expression of surface excess chemical potential p° given by Equation (2.41) and assuming the gas to be ideal ... 

If we want to define the surface excess chemical potential, fi% in terms of other surface excess parameters, we may write from Equations (201), (203) and (206)... 

The first point to be taken up in this section is the derivation, in terms of cell potential, of an alternative equation to 4.54. In order to do this let us consider a solution with only one electrolyte of univalent ions. Denoting the surface excess, chemical potential etc. of the cation by subscript + and the same of the anion by subscript —, the chemical potential of this electrolyte is... 

In metal deposition, the primary products form adsorbates on the electrode surface rather than a supersaturated solution. Their excess chemical potential is directly related to polarization and given by nFAE. The total excess surface energy = 2 S,o,. Otherwise, all the results described above remain valid. 

The second necessary ingredient in the primitive quasichemical formulation is the excess chemical potential of the metal-water clusters and of water by itself. These quantities p Wm — can typically be obtained from widely available computational packages for molecular simulation [52], In hydration problems where electrostatic interactions dominate, dielectric models of those hydration free energies are usually satisfactory. The combination /t xWm — m//, wx is typically insensitive to computational approximations because the water molecules coat the surface of the awm complex, and computational errors can compensate between the bound and free ligands. 

Periodic surface profiles on vicinal surfaces have received considerable attention in the past, both from a continuum as well as an atomistic point of view [8-18], Here we describe briefly some recent work for surfaces of miscut a (about 3-10°) based on continuum mechanics specifically designed to take the anisotropy of y(0) into account [18], The approach is based on eq. (1) and the excess chemical potential given by [2]... 

The classical theory of near-equilibrium surface relaxation was reviewed by Mullins (1963). In this theory, evolution of the surface profile h x) is driven by variations of an excess chemical potential... 

We take advantage of this presentation to make mention of the Scaled Particle Theory (SPT) [89,90] that depicts the mechanism of a test particle (or solute particle) insertion. Among the particle of diameter d, there is also a spherical cavity of radius Xd/2 that contains no particle centers. The local density of particles centers in contact with the surface of the cavity is denoted by pG(A, p), where p is the bulk density. G(X, p), whose precise form is not known, is the central function of the SPT. This function is related to the excess chemical potential by... 

Long ago, Langmuir suggested that the rate of deposition of particles on a surface is proportional to the density of particles in the vicinity of the surface and to the available area on the surface [1], However, the calculation of the available area is still an open problem. In a first approximation, one can assume that the available area is the total area of the surface minus the area already occupied by the adsorbed particles [1]. A better approximation can be obtained if the adsorbed particles, assumed to have the shape of a disk, are in thermal equilibrium on the surface, either because of surface diffusion and/or of adsorption/desorption kinetics. In this case, one can use one of the empirical equations available for the compressibility of a 2D gas of hard disks, calculate the chemical potential in excess to that of an ideal gas [2] and then use the Widom relation between the area available to one particle and its excess chemical potential on the surface (the particle insertion method) [3], The method is accurate at low densities of adsorbed particles, where the equations of state are accurate, but, in general, poor at high concentrations. The equations of state for hard disks are based on the virial expansion and only the first few coefficients of this... 

Figure 3.2 Variation of excess chemical potential of CHjF as a function of distance of the carbon atom from the liquid water-hexane interface at 310K (Pohorille and Wilson, 1996 Pratt and Pohorille, 2002). The hydrophobic contribution, obtained by eliminating electrostatic interactions, is the dot-dash curve and the electrostatic contribution is the dashed curve, lowest on the right. The water equimolar surface is at z = 0. The combination of these two contributions leads to interfacial activity for this simple solute.

It has been also indicated experimentally that ectoine enhances the thermodynamic stabilities of their folded (native) structures [24c]. This observation has been explained by the preferential exclusion model, which states that CS molecules are expelled from the protein surface [28,29] and the growth of the preferential exclusion corresponds with the increase of excess chemical potential of the protein [28,29]. In fact, onr previous MD simulation also indicated numerically that ectoine molecnles are preferentially excluded near the CI2 surface [39]. Thus, to understand how CS molecnles interact microscopically with proteins, and whether the addition of CS might indirectly stabilize them irrespective of their molecular properties, the hydration strnctnres have been studied not only for CI2 but also for a smaller... 

Because the volume of material associated with the surface is usually small compared with the bulk volume, it is convenient to work in the grand canonical ensemble, considering the bulk to be an infinite reservoir at a fixed chemical potential. Writing down the surface excess grand potential 3 for an area of surface A we have... 

If a drop is formed on the surface, the chemical potential of the system will be shifted away from the bulk coexistence value (Kelvin equation). Drops of a given size are only stable for a certain range of system sizes. I f the system size is too large, the system will rather dissolve the excess material homogeneously in the volume than pay the free energy cost of the liquid-vapor interface [121]. This is the analog of the droplet condensation-evaporation transition in the bulk [122]. 

The properties of the overlapping force fields may vary from one case to another depending on their origin. Consequently, the excess chemical potential, pex, can be influenced by forces underlying adsorption at the film surfaces, dispersion forces or electric forces acting between charged film surfaces. 

In general there are two factors capable of bringing about the reduction in chemical potential of the adsorbate, which is responsible for capillary condensation the proximity of the solid surface on the one hand (adsorption effect) and the curvature of the liquid meniscus on the other (Kelvin effect). From considerations advanced in Chapter 1 the adsorption effect should be limited to a distance of a few molecular diameters from the surface of the solid. Only at distances in excess of this would the film acquire the completely liquid-like properties which would enable its angle of contact with the bulk liquid to become zero thinner films would differ in structure from the bulk liquid and should therefore display a finite angle of contact with it. 

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