Reactivity of growth

The modeling of the space function of growth can develop in a generic way by considering the reactivity of growth, specific to a particular reaction under well-defined conditions, such as a parameter likely to vary with the intensive constraints (partial pressures, concentrations, temperature) and independent of time insofar as these constraints are maintained constant. 

To model the reactivity of growth, it is necessary to establish a reaction mechanism that is solved by the methods developed in Chapter 7. The reaction mechanism is specific to each reaction and exists, however, as strong analogies for the reachons of the same family. We will approach examples belonging to various families of reachons in Chapters 12 to 16. 

To derive the space function, we assume isothermal and isobaric conditions so that the reactivity of growth remains constant. We will successively study both isotropic and radial anisotropic models of growth. 

We will indicate by the reactivity of growth and by the space function for a grain /, with... 

Important remark.- In all the cases (see tables of Appendix 3) of this kind of relationship between the rate and the fractional extent, we note that the space function depends only on the fractional extent and thus does not depend on the reactivity of growth, whatever is the past of the sample, and the space function is completely determined by the fractional extent. The past of the sample has no influence on speed. Consequently, the variations of the rate with an intensive variable will give (with a coefficient which will be the space function) those of the rate of growth on the condition of studying these variations with constant space function, that is, with constant fractional extent. 

For the total rate, we consider Nq grains with the same reactivity of growth ... 

In addition, if growth is with separable rate and if we indicate by Sp the area of the active surface for growth, by G the dimensional factor of flux of diffusion, these two properties being the functions of time t and age of birth of nucleus r, and by (j) its reactivity of growth, which will be function of only time t (if the system is not at constant temperature and partial pressures), the speed of growth of the nucleus is given as ... 

We assume that the reactivity of growth and the specific frequency of nucleation are independent of time (pseudo-steady state modes at constant tenperature and partial pressures). We will thus refer to relations [10.16] and [10.18], but in this case, a nucleus corresponds to a grain we can thus reveal in these expressions the space function of growth of a grain. 

In this expression, and xq, respectively, indicate the area of surface and the value of characteristic size (Table A. 1.1 of Appendix 1) of a grain, /q has the usual values (1 with diffusion as the rate-determining step for growth and Xq in the case of the interface reaction). This parameter is expressed in the number of nuclei per mole of A solid involved and measures finally the relative values of the specific frequency of nucleation and the reactivity of growth. Substituting into [10.30], we obtain... 

Actually, if the trarrsformations are non-isothermal and/or tsobaric orres, it means that the reactivity of growth and the specific frequency of nucleation are functions of time (even in pseudo-steady state mode) through the fimctions and y(T,Pi)... 

It is noted that at temperature Ti the curves merged and thus, we show in this example that the rate for a given fractional extent does not depend on the previous values of the reactivity of growth. 

As an example, we will consider again the case of temperature break (Figure 10.12) and we note now that for a given fractional extent, the rate depends on the former values of the reactivity of growth and the frequency of nucleation, that is, depends on the past of the sample. 

We thus determined a certain number of models that allow the description of the kinetic and rate curves according to time and the fractional extent. It is certain that the development of numerical calculation by data processing made it possible to undertake modeling, which was not conceivable previously. Such modeling uses the reactivities of growth and the specific frequencies of nucleatiom We will note in Chapter 11 that the models, by confrontation with the experiment, enable us to determine these properties. The purpose of Chapters 12 to 16 is to connect these properties with the mechanisms that require the approach by reactional families. 

Remark.- This chapter is based on the assumptions of pseudo-steady state and separable rate. If these assumptions are not checked, in particular the second one for the growth, we no longer define the reactivity of growth and the space function, chemistry and morphology are narrowly frays, and some solutions are discussed only in some simple cases of massive plate samples, and for pseudo-steady state mixed modes. We will discuss such an approach in Chapter 15. 

In this case, we will test the various laws adjusted to the shape of the grains. These laws include only one parameter, which is the reactivity of growth or the specific frequency of nucleation (see tables of Appendix A.3). To cany out these tests, we may find it beneficial to consider the laws giving the rate according to the fractional extent. While tracing SR versus E oc), the correct law gives a line with a slope (j) or y. 

Calculations of the reactivity of growth and the specific frequency of nucleation... 

In the one-process models, the reactivity of growth (or the frequency of nucleation) is given directly as a parameter of the model (see section 11.5.2.1). 

In the cases of two-process models, it is initially advisable to determine the relationship between real time t and dimensionless time 9. For this, we determine the correspondence between t and the fractional extent on the ejqierimental eurve, on the one hand, and the correspondence between the fractional extent and dimensionless time given by the model, on the other hand. We also plot dimensionless time versus real time (for the various fractional extents). In accordance with relation [10.2], we must obtain a line whose slope allows us to calculate the reactivity of growth (/o = 1 for diffusion as the rate determining step and xb for interface reaction as the rate determining step) as follows ... 

There are two methods to determine the variations of the reactivity of growth with the intensive variables (temperature, partial pressures, etc.). The first method uses the morphological model, and the second method is given directly by the experiment. With regard to the specific frequency of nucleation, only the first method is applicable. 

Determination of the variation in the reactivity of growth Parting from the morphological model... 

Repeating several experiments for various values of an intensive variable (temperature, partial pressures) and determining each time, y and (, as previously (section 11.5), we obtain the variations of the specific frequency of nucleation and the reactivity of growth with this intensive parameter. These variations are thus obtained starting from the morphological model. 

Insofar as the rate is separable, we can determine, directly by the experiment, with an unknown constant multiplicative factor E, the variations of the reactivity of growth with an intensive variable. For this, it is sufficient to measure the variations of the rate with this variable at a constant space function. Two methods are... 

Figure 11.8. Determination of the variation of the reactivity of growth with the temperature (a) for a one-process model (b) in the general case...

The space function in all the cases has the same value E, that is, the one reached at point I imder the same conditions. We thus obtain the variations of the reactivity of growth with the variable (here temperature) with an imknown coefficient Ei. 

Figure 11.9. Comparison of the reactivity of growth resultingfrom the model with the reactance after switch for various values of the variable...

The ehoiee of the kinetie modes is carried out by solving the mechanisms and by comparing the expressions obtained for the reactivity of growth and the speeifie frequeney of nueleation with the experimental variations of these properties with the intensive variables (tenperature, partial pressures, eoneentrations, ete.). 

Remark.- There is very httle probability that speed obeys the Arrhenius law, with energy of activation having a physical meaning. For this, in the one-process model, reactivity of growth must obey the Arrhenius law and in the two-process models this opportunity is very rare. 

For this research, we can use the flow chart of Figure 11.11. We determine initially the category of model (one- or two-process model). It will be noted that if the kinetic curve presents a point of inflection, we can move directly toward the cases of two-process models. From this, we can proceed to the identification of the morphological model and the determination of the reactivity of growth and/or the specific frequency of nucleation starting from the experimental kinetic curve. 

Then, we can repeat experiments while vaiying one or the other variables (constant temperature and various partial pressures, and then with constant partial pressures and variation of temperature). By calculating in each case (j) and y (section 11.7.1), we obtain the variations of these kinetic properties with the variables. We verify the variations in the reactivity of growth by the direct experiment (see sections 11.7.2 and 11.7.3). 

We can set up the sequence of elementaiy steps for growth, and if it is necessaiy for the nucleation, we can solve the various modes. Starting from the possible modes identified in morphological modeling (direction of development, place of the rate determining step), we can choose the suitable mode by comparison of the expressions of the reactivity of growth (or the frequency of nucleation) according to the partial pressures with these ejqterimental results obtained. 

Remark.- if the assumptions of the pseudo-steady state mode and of separable rate are not verified, in particular the second for the growth, we cannot define a reactivity of growth and a space function anymore chemical and morphological aspects are narrowly merged. Solutions are approached only in simple cases of massive plate samples in mixed modes. We will consider such an approach in Chapter 15. 

As we can note in section 13.4.4, this effect on the motion velocity is reflected on the reactivity of growth and frequency of nucleation. 

Table 13.2. Simplified expressions of the reactivity of growth in the model with...

Table 13.5. Reactivity of growth from the mechanism with diffusion of defects in the formed solid...

Bouineau [BOU 98], by a systematic analysis of the variations of the reactivity of growth and specific frequency of nucleation, found that the curve of the reactivity versus the water pressure (Figure 13.7a) and the one of the frequency of nucleation versus the water pressure also presented two extremums in the same range of pressure (Figure 13.7b). 

Figure 13.8. Curve reactivity of growth-water pressure in the case of two successive reactions (CUSO4, 5H2O)...

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