Planar infinite surfaces

Finally, the term A/to can be expressed as a function of the molar fractions C and Coo (he. concentration, or activity, or partial pressure, depending on the system), the concentrations near the growing surface, and around an infinite planar surface of the sohd formed, respectively. The supersaturation parameter S is generally defined as the ratio C/Coo, thus ... 

Clearly, an infinite planar surfaces is a model of a macroscopically large, although not infinite, surface... 

In equation (4), A is the number density of atoms per unit surface area A is the dispersion constant the subscripts 5 and / refer to the adsorbent and adsorbate, respectively and do = 0.S5 asf is the z-coordinate at which the 10-4 potential for a single planar surface passes through its zero-point value. The 10-4 potential is obtained by integration of the Lennard-Jones 12-6 potential over an infinite planar surface. The dispersion constants A and Aff represent the adsorbate-adsorbent and adsorbate-adsorbate interactions, respectively these coefficients are calculated from the Kirkwood-Muller equations in the original HK paper [6], Combining equations (2-4) yields an equation that relates filling pressure to pore width ... 

Consider now such a semi-infinite system, confined on one side by an infinite planar surface, and assume that a given potential I .s is imposed on this surface. The interior bulk potential is denoted g. Having h.s implies that the mobile... 

M. Verbrugge, Galvanic corrosion over a semi-infinite, planar surface, Corros. Sci. 48 (2006) 3489-3512. 

The surface is an energetically homogeneous, infinite planar surface. 

Consider an electrode reaction occurring on an infinite planar surface. Assume electrons or molecules of the electrochemically active compound are transported into the reaction zone (i.e. the electrode surface) only by means of molecular diffusion. The compound is transferred to the surface only tangentially, in other words - along only one axis. In this case, we have a nonlinear diffusion. It is also nonstationary, dCox x, t)/dt 0. Let s find the function of concentration distribution as well as the current-time relationship. 

The surface charges turn out to be very close to the bulk charges. The rough balance between electrostatic and covalent effects invoked for semi-infinite planar surfaces takes place in most cases, except for the smallest thicknesses in the less-dense orientations, when the coordination number is smaller than or equal to 2. 

The solid angle under which the plane is seen from the front and back sides does not depend on the position of the point p and is equal to + 2n, respectively. In other words, planar surface masses with infinite extension and constant density create a uniform field in each half space ... 

In the sections above, only infinite planar interfaces between air and an aqueous phase or two immiscible liquids like water and DCE were considered. Reducing the question to this class of surfaces only would be a severe limitation in the scope of the review as more reports appear in the literature debating on the SH response from small centro-symmetrical particles [107-110]. It is the purpose of this section to discuss the SHG response from interfaces having a radius of curvature of the order of the wavelength of light. 

In the theoretical section above, the nonlinear polarization induced by the fundamental wave incident on a planar interface for a system made of two centrosymmetrical materials in contact was described. However, if one considers small spheres of a centrosymmetrical material embedded in another centrosymmetrical material, like bubbles of a liquid in another liquid, the nonlinear polarization at the interface of a single sphere is a spherical sheet instead of the planar one obtained at planar surfaces. When the radius of curvature is much smaller than the wavelength of light, the electric field amplitude of the fundamental electromagnetic wave can be taken as constant over the whole sphere (see Fig. 7). Hence, one can always find for any infinitely small surface element of the surface... 

Thus the time during which the transport process attains the steady state depends strongly on the radius of the sphere r0. The steady state is connected with the dimensions of the surface to which diffusion transport takes place and does, in fact, not depend much on its shape. Diffusion to a semispherical surface located on an impermeable planar surface occurs in the same way as to a spherical surface in infinite space. The properties of diffusion to a disk-shaped surface located in an impermeable plane are not very different. The material flux is inversely proportional to the radius of the surface and the time during which stationary concentration distribution is attained decreases with the square of the disk radius. This is especially important for application of microelectrodes (see page 292). 

For the simple case of a planar, infinitely extended planar surface, the potential cannot change in the y and z direction because of the symmetry and so the differential coefficients with respect to y and z must be zero. We are left with the Poisson-Boltzmann equation which contains only the coordinate normal to the plane x ... 

In order to determine the interaction between macroscopic solids, in the first step we calculate the van der Waals energy between a molecule A and an infinitely extended body with a planar surface made of molecules B. This is also of direct relevance in understanding the adsorption of gas molecules to surfaces. We sum up the van der Waals energy between molecule A and all molecules in the solid B. Practically this is done via an integration of the molecular density pB over the entire volume of the solid ... 

By decoration of these various infinite two-dimensional manifolds (just as the sphere has been decorated with closed networks) several related structures have been proposed for graphite nets. These are mostly based on the P, D and G surfaces (the first two due to Schwarz (1890) and the last, the gyroid, discovered by Schoen (1970). However, many other surfaces (perhaps 50) are available for consideration. Some fit naturally with hexagonal sheets and others with sheets of square or lower symmetry. In general, the P, D and G surfaces are the least curved from planarity. Surfaces parallel to the surfaces of zero mean curvature have lower symmetry than those with H = 0. When decorated with graphite nets the symmetry may be further lowered to that of a sub-group of the symmetry group of the surface itself. 

In the limit of an infinite micellar radius, i.e. a charged planar surface, the salt dependence of Ge is solely due to the entropy factor. A difficult question when applying Eq. (6.13) to the salt dependence of the CMC is if Debye-Hiickel correction factors should be included in the monomer activity. When Ge is obtained from a solution of the Poisson-Boltzmann equation in which the correlations between the mobile ions are neglected, it might be that the use of Debye-Hiickel activity factors give an unbalanced treatment. If the correlations between the mobile ions are not considered in the ionic atmosphere of the micelle they should not be included for the free ions in solution. 

Small sphere, radius R, far from planar surface of an infinitely thick wall... 

For the moment let us follow closely the analysis of Richmond [68,69] who considered two planar surfaces, one at z = 0 and one at z = h, with arbitrary nonuniform distributions of charge or potential y s) and y2(s). The boundaries mark the ends of two infinite planar half spaces with dielectric permittivities, g] and S3, separated by a third, intermediate dielectric continuum of width h and permittivity, e2. This central medium contains a simple electrolyte solution. The generic function, y(s), we use to represent either a surface potential, P(s), or a surface charge, cr(s). s = (x,.y) is again the position vector in the plane of a surface. An arbitrary source distribution can be represented by the Fourier integral,... 

At a relative humidity of 40%, p = 0.4(23,8 = 9.52 mm, By linear interpolation in the table, this corresponds to a 47.9% solution of sulfuric acid with infinite radius of curvature (planar surface). By (9.17)... 

In the planar parallel system consisting of two infinitely large surfaces, the force is perpendicular to the surfaces and, consequently, F = —(dF/dd), where d is the separation between the two objects F > 0 corresponds to a repulsive and F < 0 to an attractive force. Let us comment here the Derjaguin procedure [58] which applies the forces in the planar parallel geometry to the curved systems. This procedure is justified when the curvatm-es are small and when the interactions are isotropic. In particular, the latter is not the case when discussing liquid crystals, therefore one should be veiy careful when applying the Derjaguin approximation. 

From these expressions it can be deduced that, in contrast to cyclic voltammet-ric responses for solution systems with semi-infinite planar diffusion, for redox processes for confined or thin layer systems close to the electrode or adsorbed molecules at the electrode surface, the peak current expression becomes linear with respect to the scan rate (Eqs. II.1.13a and II.1.13b). 

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