Chemical potential curvature effect

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. 

The Saam-Cole approach has several approximations, among which are the neglect of the sohd-adsorbate interaction and curvature effects on the adsorbate chemical potential, and curvature effects on surface tension in symmetrical and asymmetrical states, while modeling the multilayer region. Here, a more accurate version of the above approach has been introduced and tested for explaining the reversibihty of adsorption in MCM-41. For fluid molecules inside a cyhndrical pore of radius R, the incremental potential function has been expressed as [4,6,7]... 

Full density is thus obtained only when the atomic processes associated with coarsening are suppressed, while those associated with densification are enhanced. It follows that in order to understand and be able to control what occurs during sintering the various atomic processes responsible for each of the aforementioned outcomes are identified and described. Before that, however, it is imperative to understand the effect of curvature on the chemical potential of the ions or atoms in a solid. 

This equation is also called Gibbs-Thompson equation and the effect (surface curvature, vapor pressure and chemical potential) is also called the Gibbs-Kelvin effect or Kelvin effect. 

Capillarity effects arise in physical situations where significant geometrical curvatures are present. Because of the existence of surface energy, the local chemical potential becomes a function of the local radius of curvature K according to the Gibbs-Thompson equation [93Bonl] ... 

The last term in parentheses in Equation 6.66 represents the effect of interfacial tension and curvature on the equilibrium freezing temperature. The basic idea is that curvature produces a difference in pressure between solid and melt. Since the chemical potentials in the two phases are fimctions of pressure, and since they must be equal at equilibrium, the equilibrium freezing temperature depends on curvature (see Problem 6.7). 

In general, the chemical potential is measured relative to some refo-ence value p,vo. and as found earlier, p. contains a vacancy concentration tom. An expression for pv that incorporates not only curvature but also pressure and concentration effects can therefore be written as... 

The dependence of the capillary pressure on the interfacial curvature leads to a difference between the chemical potentials of the components in small droplets (or bubbles) and in the large bulk phase. This effect is the driving force for phenomena like nucleation [224,225] and Ostwald ripening (see Section 4.3.1.4). Let us consider the general case of a multicomponent two-phase system we denote the two phases by a and p. Let phase a be a liquid droplet of radius R. The two phases are supposed to coexist at equilibrium. Then we can derive [4,5,226,227]... 

The effect of curvature on the chemical potential - the Gibbs-Thomson equation. 

This equation relates the droplet capillary pressure (left-hand side where K (x) is the local curvature of the liquid surface at a specific coordinate x normal to the striped pattern) to the chemical potential difference A p(a A T) between the substrate and the liquid reservoir (at T = To) and the partial derivative of the effective interface potential co with respect to the liquid thickness /(x) (often referred to as the disjoining pressure, see also chapter 3). 

Using again the relationship between chemical potential and surface curvature, given by Eq. (11.26), we obtain for the effect of diffusion... 

---