Interface advance rate

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]. 

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 (en) compound developed nuclei which advanced rapidly across all surfaces of the reactant crystals and thereafter penetrated the bulk more slowly. Kinetic data fitted the contracting volume equation [eqn. (7), n = 3] and values of E (67—84 kJ mole"1) varied somewhat with the particle size of the reactant and the prevailing atmosphere. Nucleus formation in the (pn) compound was largely confined to the (100) surfaces of reactant crystallites and interface advance proceeded as a contracting area process [eqn. (7), n = 2], It was concluded that layers of packed propene groups within the structure were not penetrated by water molecules and the overall reaction rate was controlled by the diffusion of H20 to (100) surfaces. 

Dehydration reactions. In early studies of dehydration reactions (e.g. of CuS04 5 H20 [400]), the surfaces of large crystals of reactant were activated through the incorporation of product into surfaces by abrasion with dehydrated material. An advantage of this pretreatment was the elimination of the problems of kinetic analysis of the then little understood relationship between a and time during the acceleratory process. Such surface modification resulted in the effective initiation of reaction at all boundary surfaces and rate studies were exclusively directed towards measurement of the rate of interface advance into the bulk. 

It is seen from these examples that, in appropriate systems, it is possible to introduce product into the reactant in such a manner that an effective reaction interface is established before the reactant has been heated to the decomposition temperature. Accordingly, the induction period is removed and the acceleratory process may be less pronounced or eliminated altogether. Artificial nucleation results in changes in reaction geometry with consequent variation in the a—time curve shape and the maximum value of da/dt but does not enhance the rate of interface advance. We have found no studies in which increases in reaction rates were quantitatively correlated with the increased interfacial area and/or density of nucleation which resulted from the reactant pretreatment. 

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