Parabolic defects

Figure 6.7 Parts of equally spaced parabolic cyclide surfaces are shown in the upper illustration. These were obtained from the parametrisation introduced at equations (6.114) to (6.117) with the mutually orthogonal parabolic defects (6.111) and (6.112) indicated by the two bold curves. The lower illustration shows a cross-section of the upper figure in the plane z = 0 the parabolic defect is evident and the corresponding smectic layers are arranged as shown with the other parabola being perpendicular to the page and passing through the point (0,0,0). By symmetry, there is a similar cross-section in the plane y — 0.

Salt solutions When a zinc sheet is immersed in a solution of a salt, such as potassium chloride or potassium sulphate, corrosion usually starts at a number of points on the surface of the metal, probably where there are defects or impurities present. From these it spreads downwards in streams, if the plate is vertical. Corrosion will start at a scratch or abrasion made on the surface but it is observed that it does not necessarily occur at all such places. In the case of potassium chloride (or sodium chloride) the corrosion spreads downwards and outwards to cover a parabolic area. Evans explains this in terms of the dissolution of the protective layer of zinc oxide by zinc chloride to form a basic zinc chloride which remains in solution. 

Current best estimates for natural plagioclase weathering rates are one to three orders of magnitude lower than laboratory rates. Surface characteristics which may play a role in determining rates and mechanisms of feldspar dissolution (including non-stoichiometric dissolution and parabolic kinetics) in the laboratory include adhered particles, strained surfaces, defect and dislocation outcrops, and surface layers. The narrow range of rates from experiments with and without pretreatments indicates that these surface characteristics alone cannot account for the disparity between artificial and natural rates. 

The Kf matrix is related to the conventional K matrix of MQDT by a frame transformation from parabolic to spherical co-ordinates the K matrix is then related by a further frame transformation to the quantum defects [43, 45], The first term in Eq. (4) gives a contribution to the photoionization intensity borrowed from the bound-state spectrum. The dlyom term represents direct photoionization, and the overall expression allows Fano-type interference between these terms. In Eq. (5) A is a phase shift in the parabolic rep-... 

These assumptions, however, oversimplify the problem. The parent (A,B)0 phase between the surface and the reaction front coexists with the precipitated (A, B)304 particles. These particles are thus located within the oxygen potential gradient. They vary in composition as a function of ( ) since they coexist with (A,B)0 (AT0<1 see Fig. 9-3). In the Af region, the point defect thermodynamics therefore become very complex [F. Schneider, H. Schmalzried (1990)]. Furthermore, Dv is not constant since it is the chemical diffusion coefficient and as such it contains the thermodynamic factor /v = (0/iV/01ncv). In most cases, one cannot quantify these considerations because the point defect thermodynamics are not available. A parabolic rate law for the internal oxidation processes of oxide solid solutions is expected, however, if the boundary conditions at the surface (reaction front ( F) become time-independent. This expectation is often verified by experimental observations [K. Ostyn, et al. (1984) H. Schmalzried, M. Backhaus-Ricoult (1993)]. 

Pulsed field ionization of an alkali atom differs from the description just given for H because of the finite sized ionic core, or equivalently, the nonzero quantum defects. There are three important effects. First, the zero field levels can only be spherical nim levels, not parabolic levels. Second, in the E > l/3n5 regime there are avoided crossings of states of different n. Third, ionization can occur at lower fields than in H. Specifically, in H blue states have higher ionization fields than red states, but in an alkali atom this is not the case due to nx changing ionization. 

By use of the proper experimental conditions and Ltting the four models described above, it may be possible to arrive at a reasonable mechanistic interpretation of the experimental data. As an example, the crystal growth kinetics of theophylline monohydrate was studied by Rodriguez-Hornedo and Wu (1991). Their conclusion was that the crystal growth of theophylline monohydrate is controlled by a surface reaction mechanism rather than by solute diffusion in the bulk. Further, they found that the data was described by the screw-dislocation model and by the parabolic law, and they concluded that a defect-mediated growth mechanism occurred rather than a surface nucleation mechanism. 

Fig. 16. Concentration profile and growth kinetics for the case of the diffusion of uncharged defects, (a) Linear concentration profile (b) parabolic growth kinetics.

FOLLOWING A SHORT introduction dealing with the relationship between diffusion process and field transport phenomena in tarnishing layers on metals and alloys, the mechanism of oxidation of iron is discussed. Epitaxy plays an important role on the gradient of the concentration of lattice defects and, therefore, on the validity of the parabolic rate law. Classical examples of metal oxidation with a parabolic rate law are presented and the various reasons for the deviation observed are elucidated on the systems Iron in CO/CO2 and CU2O in <>2. In addition, the oxidation of alloys with interrupted oxide-metal interfaces is treated. Finally, attention is focussed on the difficulties in explaining the low temperature-oxidation mechanism. 

The oxidation of iron at high temperatures, where several iron oxide phases form, obeys the parabolic rate law whereas in CO-CO2mixtures above 900°C, it obeys a linear rate law with the exclusive formation of an FeO layer (18). This result is understandable if one considers the high defect concentration in FeO of approximately 10%, which ensures high diffusion velocity. The linear rate constant - exhibits the following de-... 

A parabolic oxidation law is assumed so that simple relations can be derived. Jost (28) as well as Valensi (29) and Wagner (quoted in 23) have discussed more general assumptions. These relations, which have been derived elsewhere (1) in somewhat modified and generalized form, are based on the similarity of the defect and the distributions in each individual partial surface layer with variable value, and on the assumption that at the boundary of two partial layers a value of (jl and jug, respecitvely, prevails which is independent of the overall value, as, for example, in oxides near the equilib -... 

Figure 10.29 Response of an aligned smectic to layer dilation, (a) Initial equilibrium sample, (b) For a very small dilation Sh < Ink, the layer spacing simply increases, (c) A uniform rotation of the layers decreases the spacing toward that of equilibrium, but doesn t satisfy the boundary conditions, (d) Hence, the sample undergoes an mdulational instability, which also narrows the layer spacing while satisfying homeotropic boundary conditions, (e) For a large enough dilation, the undulation instability leads to formation of parabolic focal conic defects. (From Rosenblatt et al. 1977, with permission from EDP Sciences.)...

Parabolic focal conics are a special case of generic focal conic defects, which are composed of layers curved to form toroidal surfaces called Dupin cyclides (see Fig. 10-31). Each such structure contains a pair of disclination lines—one an ellipse and the other... 

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