Interface advance

In addition to the above tenets, which are primarily concerned with the kinetics of interface advance within the reactant, a number of other factors may exert an appreciable control over kinetic characteristics. Some of the more important are mentioned in the following paragraphs. 

While the topochemistry of interface advance can be observed, the chemistry of reactions proceeding within that interface, localized inside the bulk of reactant particles, is less easily investigated. Intermediates cannot be isolated without destruction of the specialized environment in the strained region at the juxtaposition of two phases where these are generated. Identification of those species which are the necessary participants in a chemical transformation can be difficult since the total quantity... 

The systematic treatment of interface advance reactions given by Jacobs and Tompkins [28] remains a valuable survey of the kinetics of solid phase decompositions. A later account was given by Young [29]. Greater mathematical emphasis is to be found in the books by Delmon [30] and by Barret [31]. 

KINETIC EXPRESSIONS DERIVED FOR INTERFACE ADVANCE REACTIONS... 

Nucleation obeying a power law with constant rate of interface advance (normal growth)... 

This is often referred to as the contracting volume (cube or sphere) equation and is the simplest example of a more general family of expressions [28—31,432,453,458,459], which includes consideration of different rates of interface advance in different crystallographic directions and of variations in crystallite dimensions and shapes. The approach is readily extended, by use of solid geometry, to allow for angles between planar surfaces. Some examples of characteristic behaviour are conveniently discussed with reference to the expression... 

From the various possible geometric shapes of reactant crystallites, discussion here will be restricted to a consideration of reaction proceeding in rectangular plates knd in spheres [28]. A complication in the quantitative treatment of such rate processes is that reaction in those crystallites which were nucleated first may be completed before other particles have been nucleated. Due allowance for this termination of interface advance, resulting from the finite size of reactant fragments accompanied by slow nucleation, is incorporated into the geometric analysis below. 

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... 

Hulbert [77] discusses the consequences of the relatively large concentrations of lattice imperfections, including, perhaps, metastable phases and structural deformations, which may be present at the commencement of reaction but later diminish in concentration and importance. If it is assumed [475] that the rate of defect removal is inversely proportional to time (the Tammann treatment) and this effect is incorporated in the Valensi [470]—Carter [474] approach it is found that eqn. (12) is modified by replacement of t by In t. This equation is obeyed [77] by many spinel formation reactions. Zuravlev et al. [476] introduced the postulate that the rate of interface advance under diffusion control was also proportional to the amount of unreacted substance present and, assuming a contracting sphere (radius r) model... 

Kinetic expressions for appropriate models of nucleation and diffusion-controlled growth processes can be developed by the methods described in Sect. 3.1, with the necessary modification that, here, interface advance obeys the parabolic law [i.e. is proportional to (Dt),/2]. (This contrasts with the linear rate of interface advance characteristic of decomposition reactions.) Such an analysis has been provided by Hulbert [77], who considers the possibilities that nucleation is (i) instantaneous (0 = 0), (ii) constant (0 = 1) and (iii) deceleratory (0 < 0 < 1), for nuclei which grow in one, two or three dimensions (X = 1, 2 or 3, respectively). All expressions found are of the general form... 

Fig. 3. Reduced time plots, tr = (t/t0.9), for the contracting area and contracting volume equations [eqn. (7), n = 2 and 3], diffusion-controlled reactions proceedings in one [eqn. (10)], two [eqn. (13)] and three [eqn. (14)] dimensions, the Ginstling— Brounshtein equation [eqn. (11)] and first-, second- and third-order reactions [eqns. (15)—(17)]. Diffusion control is shown as a full line, interface advance as a broken line and reaction orders are dotted. Rate processes become more strongly deceleratory as the number of dimensions in which interface advance occurs is increased. The numbers on the curves indicate the equation numbers.

It is appropriate to emphasize again that mechanisms formulated on the basis of kinetic observations should, whenever possible, be supported by independent evidence, including, for example, (where appropriate) X-ray diffraction data (to recognize phases present and any topotactic relationships [1257]), reactivity studies of any possible (or postulated) intermediates, conductivity measurements (to determine the nature and mobilities of surface species and defects which may participate in reaction), influence on reaction rate of gaseous additives including products which may be adsorbed on active surfaces, microscopic examination (directions of interface advance, particle cracking, etc.), surface area determinations and any other relevant measurements. 

It is usually assumed in the derivation of isothermal rate equations based on geometric reaction models, that interface advance proceeds at constant rate (Chap. 3 Sects. 2 and 3). Much of the early experimental support for this important and widely accepted premise derives from measurements for dehydration reactions in which easily recognizable, large and well-defined nuclei permitted accurate measurement. This simple representation of constant rate of interface advance is, however, not universally applicable and may require modifications for use in the formulation of rate equations for quantitative kinetic analyses. Such modifications include due allowance for the following factors, (i) The rate of initial growth of small nuclei is often less than that ultimately achieved, (ii) Rates of interface advance may vary with crystallographic direction and reactant surface, (iii) The impedance to water vapour escape offered by... 

While there is agreement that the rates of clay dehydroxylations are predominantly deceleratory and sensitive to PH2G, there is uncertainty as to whether these reactions are better represented by the first-order or by the diffusion-control kinetic expressions. In the absence of direct observational evidence of interface advance phenomena, it must be concluded that the presently available kinetic analyses do not provide an unambiguous identification of the reaction mechanisms. The factors which control the rates of dehydroxylation of these structurally related minerals have not been identified. 

Microscopic examination has shown [102,922] that the compact nuclei, comprised of residual material [211], grow in three dimensions and that the rate of interface advance with time is constant [922]. These observations are important in interpreting the geometric significance of the obedience to the Avrami—Erofe ev equation [eqn. (6)] [59,923]. The rate of the low temperature decomposition of AP is influenced by the particle ageing [924] and irradiation [45], the presence of gaseous products [924], ammonia [120], perchloric acid [120] and additives [59]. 

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