Thin film problems

If the plane source is on the surface of a semi-infinite medium, the problem is said to be a thin-film problem. The diffusion distance stays the same, but the same mass is distributed in half of the volume. Hence, the concentration must be twice that of Equation 3-45a ... 

The second basic class of thin-film problems involves the dynamics of films in which the upper surface is an interface (usually with air). In this case, the same basic scaling ideas are valid, but the objective is usually to determine the shape of the upper boundary (i.e., the geometry of the thin film), which is usually evolving in time. 

Although the solution (5 74) seems to be complete, the key fact is that the pressure gradient V.s//0) in the thin gap, and thus p(0 xs, 0, is unknown. In this sense, the solution (5-74) is fundamentally different from the unidirectional flows considered in Chap. 3, where p varied linearly with position along the flow direction and was thus known completely ifp was specified at the ends of the flow domain. The problem considered here is an example of the class of thin-film problems known as lubrication theory in which either h(xs) and us, or h(xs, 0) and uz are prescribed on the boundaries, and it is the pressure distribution in the thin-fluid layer that is the primary theoretical objective. The fact that the pressure remains unknown is, of course, not surprising as we have not yet made any use of the continuity equation (5-69) or of the boundary conditions at z = 0 and h for the normal velocity component ui° ... 

On the other hand, if a = 0, and the dynamics of the film is still dominated by body forces, then it appears from (6-3) that uc = eil1cpg/ii. In other cases, however, gravitational forces may play only a secondary role in the motion of the film, which is instead dominated by capillary forces. Then the appropriate choice for uc would involve the surface tension rather than either of the choices previously listed and the body-force terms in both (6-2) and (6-3) would be asymptotically small for the limit e -> 0. This then is a fundamental difference between this class of thin-film problems and the lubrication problems of the previous chapter. Here, the characteristic velocity will depend on the dominant physics, and if we want to derive general equations that can be used for more than one problem, we need to temporarily retain all of the terms that could be responsible for the film motion and only specify uc (and thus determine which terms are actually large or small) after we have decided which particular problem we wish to analyze. 

A generic problem, which is mathematically analogous to a number of thin-film problems for the shape function h, is the evolution of the radially symmetric concentration distribution that evolves at large times from a pulselike initial source. At very large times, the form of the concentration distribution becomes insensitive to its initial shape, i.e., it is independent of the details of the initial spatial distribution of c and depends on only the dimension of the distribution. In d dimensions, the form of the diffusion equation that describes the evolution of this radially symmetric concentration distribution is... 

Stone2 has summarized a generalization of the solution for this simple linear problem, due to Pattle3 and Pert4, which is extremely useftd in the analysis of thin-film problems. This is the development of similarity solutions for the (/-dimensional symmetric diffusion equation with a diffusivity that depends on the concentration,... 

It is also convenient to express (12-125) and (12-126) in dimensional form because the characteristic scales that were used in nondimensionalizing these equations are not the most appropriate choice for the thin-film problem (for example we used lc = R, whereas the gap width is a much more appropriate choice for a characteristic length scale in the narrow gap problem). Hence, reversing the nondimensionalization (12-119) and introducing the approximation (12-142), we have... 

The rate of thinning (drainage) of liquid films is drastically influenced by the rheological properties of the related adsorption layer. We will restrict ourselves to just a few examples. A detailed description of various sites of thin film problems is given for example by Ivanov (1988) and Hunter (1993). The immobilisation of a cylindrical plane film is a precondition for... 

From (2.18a), obviously, for the derivation of the full B-M model thin film problem, when the free surface deformation [5 = 0(1) in equation h = 1 + 5 Tj] plays an essential role, it is necessary to assume that the Froude number Fr is 0(1). As consequence we must consider the following incompressible limit process ... 

Metal Treatment. After rolling, the oxide scale on sheet steel is removed by acid treatment (pickling) (see Metal surface treatments). Phosphoric acid, a good pickling agent, leaves the steel coated with a thin film of iron phosphates. This process improves mst resistance but presents a problem if the steel is to be electroplated. 

Aquatic Toxicity. The standard tests to measure the effect of substances on the aquatic environment are designed to deal with those that are reasonably soluble ia water. Unfortunately this is a disadvantage for the primary phthalates because they have a very low water solubiUty (ca 50 p.g/L) and this can lead to erroneous test results. The most common problem is seen ia toxicity tests on daphnia where the poorly water-soluble substance forms a thin film on the water surface within which the daphnia become entrapped and die. These deaths are clearly not due to the toxicity of the substance but due to unsuitable test design. 

In most cases, CVD reactions are activated thermally, but in some cases, notably in exothermic chemical transport reactions, the substrate temperature is held below that of the feed material to obtain deposition. Other means of activation are available (7), eg, deposition at lower substrate temperatures is obtained by electric-discharge plasma activation. In some cases, unique materials are produced by plasma-assisted CVD (PACVD), such as amorphous siHcon from silane where 10—35 mol % hydrogen remains bonded in the soHd deposit. Except for the problem of large amounts of energy consumption in its formation, this material is of interest for thin-film solar cells. Passivating films of Si02 or Si02 Si N deposited by PACVD are of interest in the semiconductor industry (see Semiconductors). 

Another problem in the construction of tlrese devices, is that materials which do not play a direct part in the operation of the microchip must be introduced to ensure electrical contact between the elecuonic components, and to reduce the possibility of chemical interactions between the device components. The introduction of such materials usually requires an annealing phase in the construction of die device at a temperature as high as 600 K. As a result it is also most probable, especially in the case of the aluminium-silicon interface, that thin films of oxide exist between the various deposited films. Such a layer will act as a banier to inter-diffusion between the layers, and the transport of atoms from one layer to the next will be less than would be indicated by the chemical potential driving force. At pinholes in the AI2O3 layer, aluminium metal can reduce SiOa at isolated spots, and form the pits into the silicon which were observed in early devices. The introduction of a tlrin layer of platinum silicide between the silicon and aluminium layers reduces the pit formation. However, aluminium has a strong affinity for platinum, and so a layer of clrromium is placed between the silicide and aluminium to reduce the invasive interaction of aluminium. 

---