Geometry of interfaces

Measurements of overall reaction rates (of product formation or of reactant consumption) do not necessarily provide sufficient information to describe completely and unambiguously the kinetics of the constituent steps of a composite rate process. A nucleation and growth reaction, for example, is composed of the interlinked but distinct and different changes which lead to the initial generation and to the subsequent advance of the reaction interface. Quantitative kinetic analysis of yield—time data does not always lead to a unique reaction model but, in favourable systems, the rate parameters, considered with reference to quantitative microscopic measurements, can be identified with specific nucleation and growth steps. Microscopic examinations provide positive evidence for interpretation of shapes of fractional decomposition (a)—time curves. In reactions of solids, it is often convenient to consider separately the geometry of interface development and the chemical changes which occur within that zone of locally enhanced reactivity. 

If the phases present can be unambiguously identified, microscopy can be used to determine the geometry of interface initiation and advance, and to provide information about particle sizes of components of mixed reactants in a powder. Problems of interpretation arise where materials are poorly crystallized and where crystallites are small, opaque, porous or form solid solutions. With the hot-stage microscope, the progress of reactions can be followed in some instances and the occurrence of sintering and/or melting detected. 

Substitution of appropriate functions for nucleation and growth rates into eqn. (1) and integration yields the f(a)—time relation corresponding to a particular geometry of interface advance. In real systems, the reactant... 

Although some progress has been made in determining the geometry of interface advance through interpretation of observed f(a)—time relationships for individual salts, the reasons for differences between related substances have not always been established. Nickel carboxylates, for which the most extensive sequence of comparative rate studies has been made [40,88,375,502,1106,1107,1109], show a wide variety of kinetic characteristics, but the controlling factors have not yet been satisfactorily determined. Separate measurements of the rates of nucleation and of growth are not usually practicable. 

The geometry of interfaces — including the binding of substrates, effectors and drugs. What is the nature of the packing at an interaface Is there room for another methyl group ... 

Reactant particle size distributions Kinetic characteristics of some reactions of solids depend sensitively on reactant particle sizes (29). Ideally, reactants to be used in kinetic studies should be composed of crystallites of identical (known) sizes and shapes, to which the geometry of interface advance can be related quantitatively. This is not, however, always (or easily) achieved in experimental studies, and most powder samples contain particles of disparate sizes for example, sometimes crystals are mixed with fine powder. The kinetic model giving the best apparent fit to the data then may not accurately represent the reaction. Dependencies of rate on particle size are only rarely investigated. The state of subdivision of a solid reactant is most frequently described in literature reports only by qualitative terms, such as single crystals or crushed powder. 

The aforementioned behavior patterns and controls (rates of nucleation followed by growth) may be combined (71) to formulate rate expressions that represent the developing geometry of interface reactions proceeding in solid reactant particles. Through this approach, the following groups of rate equations have been... 

The Champ-Sons model is a most effieient tool allowing quantitative predictions of the field radiated by arbitrary transducers and possibly complex interfaces. It allows one to easily define the complete set of transducer characteristics (shape of the piezoelectric element, planar or focused lens, contact or immersion, single or multi-element), the excitation pulse (possibly an experimentally measured signal), to define the characteristics of the testing configuration (geometry of the piece, transducer position relatively to the piece, characteristics of both the coupling medium and the piece), and finally to define the calculation to run (field-points position, acoustical quantity considered). 

Another way to define the geometry of a molecule is as a set of Cartesian coordinates for each atom as shown in Figure 8.2. Graphic interface programs often generate Cartesian coordinates since this is the most convenient way to write those programs. 

The geometry of Fig. 10.3 leads to a result known as Snell s law, which relates the refractive index of the medium to the angles formed by two wave fronts with the interface. Defining 6q and 6, respectively, as the angles between the phase boundary and the wave front under vacuum and in the medium of refractive index n, show that Snell s law requires n = sin Oo/sind. 

The proteins thus adapt to mutations of buried residues by changing their overall structure, which in the globins involves movements of entire a helices relative to each other. The structure of loop regions changes so that the movement of one a helix is not transmitted to the rest of the structure. Only movements that preserve the geometry of the heme pocket are accepted. Mutations that cause such structural shifts are tolerated because many different combinations of side chains can produce well-packed helix-helix interfaces of similar but not identical geometry and because the shifts are coupled so that the geometry of the active site is retained. 

Figure 3.13. Simple relationships between properties and microstriictural geometry (a) hardness of some metals as a function of grain-boundary density (b) coercivity of the cobalt phase in tungsten earbide/coball hard metals as a function of interface density (after Exner 1996).

Tenets (i) and (ii). These are applicable only where the reactant undergoes no melting and no systematic change of composition (e.g. by the diffusive removal of a constituent) and any residual solid product phase offers no significant barrier to contact between reactants or the escape of volatile products [33,34]. When all these conditions are obeyed, the shape of the fraction decomposed (a) against time (f) curve for an isothermal reaction can, in principle, be related to the geometry of formation and advance of the reaction interface. The general solution of this problem involves intractable mathematical difficulties but simplifications have been made for many specific applications [1,28—31,35]. 

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