Self profile

The above recipe can be repeated indefinitely, and the mathematical result would be what is called a self-similar profile. That is, successive magnifications of a section (in this case by factors of 3) would give magnified jagged lines which could be superimposed exactly on the original one. In this limit, one finds for this case that... 

Theoretical studies of diffusion aim to predict the distribution profile of an exposed substrate given the known process parameters of concentration, temperature, crystal orientation, dopant properties, etc. On an atomic level, diffusion of a dopant in a siUcon crystal is caused by the movement of the introduced element that is allowed by the available vacancies or defects in the crystal. Both host atoms and impurity atoms can enter vacancies. Movement of a host atom from one lattice site to a vacancy is called self-diffusion. The same movement by a dopant is called impurity diffusion. If an atom does not form a covalent bond with siUcon, the atom can occupy in interstitial site and then subsequently displace a lattice-site atom. This latter movement is beheved to be the dominant mechanism for diffusion of the common dopant atoms, P, B, As, and Sb (26). 

To achieve the lowest possible delay a bipolar switching transistor developed by IBM minimizes parasitic resistances and capacitances. It consists of self-aligned emitter and base contacts, a thin intrinsic base with an optimized collector doping profile, and deep-trench isolation (36). Devices must be isolated from each other to prevent unwanted interactions in integrated circuits. While p—n junctions can be used for isolation, IBM s approach etches deep trenches in the siUcon wafer which are filled with Si02 to provide electrical insulation. 

Similarity Variables The physical meaning of the term similarity relates to internal similitude, or self-similitude. Thus, similar solutions in boundaiy-layer flow over a horizontal flat plate are those for which the horizontal component of velocity u has the property that two velocity profiles located at different coordinates x differ only by a scale factor. The mathematical interpretation of the term similarity is a transformation of variables carried out so that a reduction in the number of independent variables is achieved. There are essentially two methods for finding similarity variables, separation of variables (not the classical concept) and the use of continuous transformation groups. The basic theoiy is available in Ames (see the references). 

Region of established flow—the principal region of the jet. In this region, the velocity profile transverse to the jet is self-preserving when normahzed by the centerline velocity. 

FIG. 16 36 Dimensionless time-distance plot for the displacement chromatography of a binary mixture. The darker lines indicate self-sharpening boundaries and the thinner lines diffuse boundaries. Circled numerals indicate the root number. Concentration profiles are shown at intermediate dimensionless column lengths = 0.43 and = 0.765. The profiles remain unchanged for longer column lengths. 

Wool [32] has considered the fractal nature of polymer-metal and of polymer-polymer surfaces. He argues that diffusion processes often lead to fractal interfaces. Although the concentration profile varies smoothly with the dimension of depth, the interface, considered in two or three dimensions is extremely rough [72]. Theoretical predictions, supported by practical measurements, suggest that the two-dimensional profile through such a surface is a self-similar fractal, that is one which appears similar at all scales of magnification. Interfaces of this kind can occur in polymer-polymer and in polymer-metal systems. 

THEORY OF SURFACE SEGREGATION SELF-CONSISTENT DETERMINATION OF THE CONCENTRATION PROFILE... 

Repairs to the coating are made with either hot-melt polyethylene sticks or polyethylene sheet patches with mastic profiling compounds for small damaged areas. Large repair areas are best treated as for field weld joint coating, where either heat-shrink crosslinked polyethylene sleeves or cold-applied self-adhesive laminate tapes are employed. 

A. Hydration energy profile, using the Bom formalism (Eqn. 1), shows the drop of ion self energy as a function of the radius of a hydration sphere. Note that even with a hydration shell of 10 A radius not all of the hydration energy is obtained. 

In this study the water temperature is changed to other values using the Pb(WW) and J(WW) parameters shown in Table 3.2. A profile of self-diffusion as a function of temperature can be derived from these results. 

At large distances from the nozzle, the axial velocity exhibits self-preserving similarity in that, when plotted in the above dimensionless form, velocity profiles at all cross-sections downstream collapse onto a single curve. It is probable that strict similarity is preserved only at axial distances in excess of 30-40 diameters. However, the above formulation is frequently used to describe the velocity profile at all points downstream of the flow development region. For work performed with M. citrifolia, the numerical values recommended by Panchapakesan and Lumley [131] for the constants K and C in Eqs. (12) and (13), i.e. 75.2 and 6.06, respectively, are employed. 

Optimization sequence (experimental data, arbitrary units) Runs 1 and 2 are initial experiments. From run 3 to run 6 the amounts of A, B, G, and feed rate of G are fixed. These constraints are relaxed for runs 7 and 8. (Reprinted from Marchal-Brassely et al. (1992), Optimal operation of a semi-batch reactor by self-adaptive models for temperature and feed profiles . Copyright (1992), with permission from Elsevier Science). 

From the asymmetrical concentration profile with front tailing (see Figure 2.4b), it can correctly be deduced that (1) the adsorbent layer is already overloaded by the analyte (i.e., the analysis is being run in the nonlinear range of the adsorption isotherm) and (2) the lateral interactions (i.e., those of the self-associative type) among the analyte molecules take place. The easiest way to approximate this type of concentration profile is by using the anti-Langmuir isotherm (which has no physicochemical explanation yet models the cases with lateral interactions in a fairly accurate manner). 

---