Thin-film model layer approximation

We use a thin-film model where the film thickness is assumed to be much smaller than the characteristic wavelength of the undulations in the lateral plane. Under such a long-wave approximation the Navier-Stokes equations lead to the following boundary-layer equations [31-33] ... 

The effects of mercury film electrode morphology in the anodic stripping SWV of electrochemically reversible and quasi-reversible processes were investigated experimentally [47-51], Mercury electroplated onto solid electrodes can take the form of either a uniform thin film or an assembly of microdroplets, which depends on the substrate [51 ]. At low sqtrare-wave frequencies the relationship between the net peak crrrrent and the frequency can be described by the theory developed for the thin-film electrode because the diffusion layers at the snrface of microdroplets are overlapped and the mass transfer can be approximated by the planar diffusion model [47,48],... 

In a well-fluidized gas-solid system, the bulk of the bed can be approximated to be isothermal and hence to have negligible thermal resistance. This approximation indicates that the thermal resistance limiting the rate of heat transfer between the bed and the heating surface lies within a narrow gas layer at the heating surface. The film model for the fluidized bed heat transfer assumes that the heat is transferred only by conduction through the thin gas film or gas boundary layer adjacent to the heating surface. The effect of particles is to erode the film and reduce its resistive effect, as shown by Fig. 12.3. The heat transfer coefficient in the film model can be expressed as... 

The Nernst film model is used to quantify diffusional transport through the static boundary layer. This model approximates the low velocity boundary layer as a thin, static film between the surface and the free fiowing solution. Pick s first law gives the diffusional flux (/, mol/m sec) through this film (Figure 7.6). 

It is claimed to be able to measure thicknesses with an accuracy of about 0.3 A, approximately the thickness of an atomic layer. Because the method relies on polarized fight, ellipsometry is a nondestructive technique, which makes it suitable for in situ measurements in some cases. One disadvantage is that the substrate must be reflective, so gold or sUicon wafer is often used. However, a major disadvantage with the technique is interpretinging the data, which is not trivial models of the air—thin film-substrate must be used. The dielectric and optical properties of the thin film must be known accurately to calculate the thickness of the film, and even then the modelling usually assumes a homogeneous layer which may not always be justified. 

The full B-M model problem, [(4.3a,b,c), (4.5a,b,c,d), (4.6), (4.7)], is a very complicated problem, but for a thin film flow we can apply the long wave approximation. In this case, we derive (as in next Section 4.2) a simplified Benard-Marangoni Boundary Layer (B-Mbl) model problem for the high Reynolds numbers. 

Constructing an optical model. In the data analysis procedure in ellipsometry, an optical model corresponding to the investigated sample structures must be constructed firstly. An optical model is represented by the complex refractive index and layer thickness of each layer, normally, it consists of an air/thin film/ substrate structure. It should be decided if any layer is anisotropac at this stage, and whether or not interface layers are to be modeled as a single effective medium approximation, or is a more complicated graded interface to be used for the sample. 

The above equation indicates that a thick layer thins out more rapidly than a thin one. This in turn indicates that a nonuniform layer should become increasingly more uniform as centrifugation continues. The following empirical relation approximately models the height of the thin film, h, for a constant evaporation assumption ... 

Several advantages of the inlaid disk-shaped tips (e.g., well-defined thin-layer geometry and high feedback at short tip/substrate distances) make them most useful for SECM measurements. However, the preparation of submicrometer-sized disk-shaped tips is difficult, and some applications may require nondisk microprobes [e.g., conical tips are useful for penetrating thin polymer films (18)]. Two aspects of the related theory are the calculation of the current-distance curves for a specific tip geometry and the evaluation of the UME shape. Approximate expressions were obtained for the steady-state current in a thin-layer cell formed by two electrodes, for example, one a plane and the second a cone or hemisphere (19). It was shown that the normalized steady-state, diffusion-limited current, as a function of the normalized separation for thin-layer electrochemical cells, is fairly sensitive to the geometry of the electrodes. However, the thin-layer theory does not describe accurately the steady-state current between a small disk tip and a planar substrate because the tip steady-state current iT,co was not included in the approximate model (19). 

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