Thin-film approximation formulas

Let us compare the thin-film approximation formulas for (1) the transmissivity (1.98) (2) the reflectivities for the external reflection from this film deposited onto a metallic substrate (1.82) (3) the internal reflection at (p > (pc (1.84) and (4) the external reflection from this film deposited on a transparent substrate (dielectric or semiconducting) (1.81) (Table 1.2). In all cases s-polarized radiation is absorbed at the frequencies of the maxima of Im( 2), vroi (1.1.18°), whereas the jo-polarized external reflection spectrum of a layer on a metallic substrate is influenced only by the LO energy loss function Im(l/ 2) (1.1.19°). The /7-polarized internal and external reflection spectra of a layer on a transparent substrate has maxima at vro as well as at vlq. Such a polarization-dependent behavior of an IR spectrum of a thin film is manifestation of the optical effect (Section 3.1). 

A comparison between band intensities obtained with the exact formulas and those obtained by the thin-film approximation (Fig. 3.21c, d) reveals that the highest deviation from linearity is observed for the modes polarized in the film plane. 

A general formula for calculation of the dispersion molecular interactions in any type of condensed phases has been proposed in [148], The attraction between bodies results from the existence of fluctuational electromagnetic field of the substance. If this field is known in a thin film, then it is possible to determine the disjoining pressure in it. The more strict macroscopic theory avoids the approximations assumed in the microscopic theory, i.e. additivity of forces integration extrapolation of interactions of individual molecules in the gas to interactions in condensed phase. The following function for IIvw was derived in [148] for thick free films... 

It was already mentioned that thin film chromatography is used firstly as a qualitative analysis method. There is no accurate quantitative analysis that is possible by thin film chromatography. However, there is an empirical formula, equation (2.3), which allows the approximate calculation of the weight of the substance (W) from the spot area (A). This method is very inaccurate and can be used only for gross estimation. 

In this section, we list the exact formulas of Hansen [99] and Abeles [125] for isotropic films and the thin-fihn approximation of Yamamoto and Ishida [126] for the calculation of dansmission spectra of anisotropic ultrathin films on planar isotropic supports. The spectral features and dependences predicted by these formulas are discussed in Sections 2.1 and 3.3.4. 

Figure 3.21 allows one to deduce the applicability of the thin-fihn approximation to IRRAS of an anisotropic inorganic layer at the ZnSe and Ge substrates. The deviation from the results of the exact formulas is greatest (up to 50% for a 200-nm-thick film) for the vlo band and the Ge substrate. 

On the upper and lower thin film surfaces there arises an inherent liquid phase that wets well and partially (not always completely) dissolves crystal grains of the thin film on the boundaries (Fig. 15 (a)). According to the experimental results obtained by AFM a thin film, e.g. 50 nm thick, consists of no more than two layers of crystalline grains. Liquid layer thickness depends on temperature. For example, dispersion of a gold film 50 nm thick was observed within the temperature range 903-1003 K. Formula (10) indicates that both at 903 K and at 1003 K on the upper and lower surfaces of a thin gold film there has to exist a liquid layer of more or less uniform thickness approximately 12 and 16 nm, respectively. 

Equation 5.2 results from the stationary, linearized Navier-Stokes equation in the lubrication approximation with a no-slip condition on the solid substrate, complemented by a disjoining pressure term in the mesoscopic range. The occurrence of the disjoining pressure characterizes a "thin" film the chemical potential of the molecules depends on the film thickness f, which means that the film is neither a three-dimensional (3D) nor a two-dimensional (2D) phase. As mentioned before, thin films are mesoscopic if t is large compared to molecular sizes, they are treated as continuous media, and they obey general formulae. Molecular films are obviously quite specific. 

The gap between two colliding particles (bubbles, droplets, solid particles, surfactant micelles) in a colloidal dispersion can be treated as a film of uneven thickness. Then, it is possible to utilize the theory of thin films to calculate the energy of interaction between two colloidal particles. Deijaguin [276] has derived an approximate formula which expresses the energy of interaction between two spherical particles of radii and i 2 through integral of the excess surface free energy per unit area, f h), of a plane-parallel film of thickness h [see Eq. (161)] ... 

Schmidt et al. [136] also reported room-temperature spectroscopic ellipsometry results on pulsed laser deposition-grown wurtzite MgxZni xO (0thin films. The refractive index data were fit to a three-term Cauchy approximation type formula (Equation 3.105), and the anisotropic Cauchy model parameters A, B, and C for ZnO were obtained as 1.916,1.76, and 3.9 for E J c and 1.844,1.81, and 3.6 for... 

Up to now we have discussed two extreme limits, the band picture on the one hand, and strong localization associated with interruptions in the metallic chains on the other. In fact, from work on thin metallic films and metallic glasses it is known that there is an intermediate region, that of weak localization. This occurs when the mean free path for elastic scattering (Lel) is only somewhat larger than, or comparable with, that for inelastic processes (Lin). In the first approximation there are corrections to the Boltzmann transport formula which depend on the ratio Lin/Lel in different ways for one-, two-, and three-dimensional materials. Weak localization... 

In a first approximation, one can assume that the viscous dissipation of kinetic energy happens mostly in the thin liquid film intervening between two drops. (In reality, some energy dissipation happens also in the transition zone between the film and the bulk continuous phase.) If the drop interfaces are tangentially immobile (owing to adsorbed surfactant), then the velocity of approach of the two drops can be estimated by meanss of the Reynolds formula for the velocity of approach of two parallel solid disks of radius R, equal to the film radius (142) ... 

The investigation of the influence of the interfacial viscosity on the rate of film thinning and the shape of the film surfaces is a computationally difficult task, which could be solved only numerically. The results for a symmetrical plane-parallel foam and emulsion films, obeying the classical Boussinesq-Scriven constitutive law (see Sec. III.F) are presented in Refs. 5, 58, 267, 479, and 480. Ivanov and Dimitrov [5,481] showed that to solve this problem, it is necessary to use the boundary conditions on the film ring (at r = jR) for that reason, the calculations given in Refs. 479 and 480 may not be realistic. The only correct way to solve the boundary problem is to include the influence of the Plateau border in the boundary conditions however, this makes the explicit solution much more difficult. As a first approximation, in Ref. 267 an appropriate asymptotic procedure was applied to foam films and the following formula for the velocity of thinning was obtained ... 

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