Surface integral equation

T v exp +Eq / kT) is the average time of contact of adsorption particle with the surface. Integrating equation (1.10) one gets... 

Glisson, A.W., Kajfez, D., and James, J., 1983, Evaluation of modes in dielectric resonators using a surface integral-equation formulation, IEEE Trans. Microwave Theory Tech. 31(12) 1023-1029. 

Liu,Y., Safavi-Naeini, S., Chaudhuri, S.K., and Sabry, R., 2004, On the determination of resonant modes of dielectric objects using surface integral equations, IEEE Trans. Antennas Propagat. 52(4) 1062-1069. 

We note that for a nontrivial magnetic permeability of the particle, a volume-surface integral equation has been derived by Volakis [245], and a pure volume-integral equation has been given by Volakis et al. [246]. 

The null-field method leads to a nonsingular integral equation of the first kind. However, in the framework of the surface integral equation method, the transmission boundary-value problem can be reduced to a pair of singular integral equations of the second kind [97]. These equations are formulated in terms of two surface fields which are treated as independent unknowns. In order to elucidate the difference between the null-field method and the surface integral equation method we follow the analysis of Martin and Ola [155] and review the basic boundary integral equations for the transmission boundary-value problem. We consider the vector potential Aa with density a... 

The first integral on the right-hand side is zero it becomes a surface integral over the boundary where (W - ) = 0. Using the result in the previous equation, one obtains... 

Fig. 21. Control volumes for appHcation of the integral equations of motion where 1, 2, and 3 are the location of control surfaces and Sy (a) general,...

In this book, the theory of gases and liquids and its application to surface and interfacial problems is considered. In this chapter, theories based on integral equations are reviewed. Since these theories were developed originally for bulk fluids, we will consider this case first and indicate the extension of the theories to interfaces. 

Integral equations have been developed for inhomogeneous fluids. One such integral equation is that of Henderson, Abraham and Barker (HAB) [88] who assumed the OZ equation for a mixture and regarded the surface as a giant particle. For planar geometry they obtained... 

The present chapter is organized as follows. We focus first on a simple model of a nonuniform associating fluid with spherically symmetric associative forces between species. This model serves us to demonstrate the application of so-called first-order (singlet) and second-order (pair) integral equations for the density profile. Some examples of the solution of these equations for associating fluids in contact with structureless and crystalline solid surfaces are presented. Then we discuss one version of the density functional theory for a model of associating hard spheres. All aforementioned issues are discussed in Sec. II. 

Second-Order Integral Equations for Associating Fluids As mentioned above in Sec. II A, the second-order theory consists of simultaneous evaluation of the one-particle (density profile) and two-particle distribution functions. Consequently, the theory yields a much more detailed description of the interfacial phenomena. In the case of confined simple fluids, the PY2 and HNC2 approaches are able to describe surface phase transitions, such as wetting and layering transitions, in particular see, e.g.. Ref. 84. 

This is the fundamental differential equation. The reader who is acquainted with the rules for transforming the variables in a surface integral will observe that it has the geometrical interpretation that corresponding elements of area on the (v, p) and (s, T) diagrams are equal (cf. 43). 

The role of electrolyte is critical in these nanoscopic interfaces, but is difficult to predict and quantify. For sufficiently large rigid interfacial structures, one can apply the model of electrolyte interaction with a single charged surface in Figure 1(a). The double-layer theories or the recent integral-equation theories have been applied. Reviews of this subject are available in the literature [4,5]. For electrolytes in a nanostructure, the double layers from two surfaces overlap and behave differently from the case of a single surface. Ad-... 

The second fundamental feature of the field g is described by Equation (1.26), and it states that the flux of this field through any closed surface characterizes amount of mass in the volume V enclosed by this surface. Both Equations (1.26 and 1.29) represent the system of equations of the attraction field in the integral form ... 

Now we are prepared to formulate boundary conditions for the potential of the attraction field, which uniquely define this field inside the volume V. With this purpose in mind suppose that the surface integral on the right hand side of Equation (1.78) equals zero. Then... 

In other words, if the surface integral in Equation (1.78) vanishes, these solutions can differ by a constant only ... 

Indeed, it satisfies Laplace s equation everywhere except at the point p, since it describes up to a constant the potential of a point mass located at the point p. Also, it has a singularity at this point and provides a zero value of the surface integral over the hemisphere when its radius r tends to infinity. Correspondingly, we can write... 

By definition, any plane 0 — constant is a plane of symmetry. In other words, there are always two elementary masses, which are equal to each other, and located at opposite sides of this plane but at the same distance. As is seen from Fig. 1.5d, the sum of 0-components, caused by both masses is equal to zero. Representing the total mass as a sum of such pairs we conclude that the 0-component, gg, due to the spherical mass is absent at every point outside and inside the body. In the same manner we can prove that — 0. Of course, volume integration, Equation (1.6), can prove this fact, but this procedure is much more complicated. Thus, the attraction field has only a radial component, g, and the field is directed toward the origin, 0. In order to determine this component we will proceed from Equation (1.26) and consider a spherical surface with radius R, as is shown in Fig. 1.5c. Inasmuch as dS — dSiR and the scalar component g is constant at points of the spherical surface, we have for the flux ... 

One can say that we have expressed the disturbing potential in terms of an unknown density. Now we demonstrate that this transition is justified because it is possible to obtain the integral equation with respect to a. In Chapter 1, it was shown that the discontinuity of the normal components of the field at both sides of the surface masses is equal to —2nka. Correspondingly, we have... 

Thus, we have derived the integral equation with respect to the function a. As in the case of Stokes problem it is possible to apply the spherical approximation, that is, the magnitude of the normal field at points of the surface S is... 

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