Parallel slabs

A more detailed treatment has been given by Gurfein and his associates who chose as their pore model a cylinder with walls only one molecule thick. A few years later, Everett and Fowl extended the range of models to include not only a slit-shaped pore with walls one molecule thick, but also a cylinder tunnelled from an infinite slab of solid and a slit formed from parallel slabs of solid. 

Fig. 4.9 Enhancement of interaction potential in (i) a slit-shaped pore between parallel slabs of solid, (ii) a cylindrical pore in a block of solid. 0/0 is plotted against d/r (see text). (Reduced from a diagram of Everett...

Figure 2. (a) Density of water oxygens and hydrogens, (b) The water dipole density profile and the associated potential drop, (c) The total charge density profile and the potential drop, as a function of the distance between two parallel slabs of the Pt( 100) surface at T = 300 K. 

Consider a single-crystal its crystal structure belongs to one of the 32 point groups and prepare a thin foil for electron microscopy with the shape of a perfect parallel slab. Depending on the way this slab is cut from the single-crystal, 31 different types of specimen can be obtained if symmetry elements are taken into account. These 31 types of specimen will give 31 different types of diffraction pattern named diffraction groups. 

We shall explain this by means of the following model. Imagine a plane-parallel slab of semiconductor of thickness L, both surfaces of which contain chemisorbed particles. The energy band scheme of such a semiconductor in the case of negatively charged surfaces is shown on Fig. 26. Suppose first that Ly>l (Fig. 26a). Then the inner region of the semiconductor is electrically neutral, and the energy bands inside it are horizontal, as shown in Fig. 26a. From this condition for electrical neutrality, one determines the position of the Fermi level ,+ inside the crystal is thus insensitive... 

We now consider reflection and transmission of a wave Eiexp[iu(N2z/c — 0] normally incident on a plane-parallel slab of arbitrary material embedded in a nonabsorbing medium (Fig. 2.7). The reflected and transmitted waves are... 

Interaction between two parallel slabs, in a medium versus on a substrate, 67 Multiple layers, general scheme, 71 Smoothly varying s(z), 72... 

Interaction between two parallel slabs, in a medium versus on a substrate... 

If regions A and B have the same material properties as medium m, then the remaining interaction is that between two parallel slabs (see Fig. LI.27). 

PROBLEM Ll.ll Neglecting retardation, show how the interaction between two coated bodies, Eq. (LI.29) can be converted into the interaction between two parallel slabs, Eq. (LI.30). 

For the interaction of planar slabs, the Hamaker approach entails integration over finite ranges of zA or zB. For the interaction between a half-space A and a parallel slab of B of finite thickness b, this procedure is equivalent to subtracting from E(l) = — (AHam/12 /2) an amount — [AHam/12 (Z +b)2] (see Fig. L2.10). This subtraction yields a form equivalent to the equation of Table P.2.b.3 (see Fig. L2.ll) ... 

Microscopic approach of Hamaker between parallel slab surfaces... 

If we remember that Equation (538) is derived for a unit area = 1 of parallel slabs, it is clear from Equation (554) that the dispersion interaction potentials are the same if the effective interaction area in a sphere-surface interaction is equal to... 

Consider a composite wall made of three parallel slabs of cross-sectional area A (Fig. 2.3), The thickness and thermal conductivity of these slabs are 13 and k, respectively. Heat is transferred from a hot fluid at temperature 7 through this composite wall to a cold fluid at temperature To- Coefficients of heat transfer on the hot and cold sides are hi and ho, respectively. 

The last point which will be examined in this subsection is that of extracting the true transmission values from a DAC experiment, in order to calculate the absorption-coefficient spectrum. In the case of a plane-parallel slab of sample with a thickness d, a refractive index n, and an absorption coefficient a which is immersed in a transparent medium with an index n and negligible absorption, the theoretical transmission in the absence of interference fringes is... 

The spectral radiative heat flux in the direction normal to the parallel slab faces is found from the directional spectral intensity I (Q) by noting that 1 is per unit projected area (dA cos 0) and is in the 0 direction. Then the contribution from all directions to the normal heat flux is... 

FIGURE 34.8 Three models for the distribution of two phases in a material, (a) Parallel slabs, (b) continuous matrix phase, discontinuous particulate dispersion, and (c) large isolated grains separated by a continuous minor phase. 

For parallel slabs with heat flow parallel to the phase boundary ... 

Parallel slabs Continuous matrix Continuous minor... 

FIGURE 1 Maker Fringe Patterns from a parallel-slab cell (L = 375 pm) for an isotropic solution of PBT (cp = 0.021), with parallel polars for two incident wavelengths (a) 1542 nm and (b) 1907 nm. 

For the case of interaction between a molecule and two parallel slabs, the potential energy of interaction is calculated from the following equation, which is simply the sum of the two potential equations between a single atom or molecule and a single slab (from eq. 6.10-9). 

Figure 6.10-9 Plots of the reduced potential energy versus z/ai2 for two parallel slabs...

We have addressed the potential energy for a number of cases in the last sections (Sections 6.10-1 to 6.10-5). The analysis of the last two cases (i) a molecule and two parallel lattice planes and (ii) a molecule and two parallel slabs, are particularly useful for the study of adsorption of nonpolar molecules in slitshaped micropore solids, such as activated carbon. 

The integral in eq.(6.10-18) or (6.10-19) can only be evaluated if the relationship between the energy of interaction E and the pore half width r is known. This relationship is possible with the information learnt in Section 6.10-4 for two parallel lattice planes and Section 6.10-5 for two parallel slabs. The depth of the potential minimum is the interaction energy between the micropore and the adsorbate. 

Interaction Energy versus the Half Width for two Parallel Slabs... 

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