Nucleation specific frequency

Evaluate the error obtained on and Bit. Give a value of A and t for each pressure. Calculate the growth reactivity and nucleation specific frequency at each pressure and their error. 

We note a very high error on nucleation specific frequency that is much more important than that on growth reactivity. 

Taking into account the relatively complex relations obtained for nucleation specific frequency, we will operate in a different way to compare the experiment and the models. We will determine the constants from one or two experimental values of the frequency and then calculate, with the help of the model, the other values starting from these constants. 

Table 19.27 displays the values of supersaturation, experimental average values of nucleation specific frequency, and the corresponding values of the coefficients a. and bj. Figure 19.16 gives the expected line. 

Table 19.27. Nucleation specific frequencies calculated according to the mode limited by the step of condensation leading to the maximum ofVolmer s curve...

We know that nucleation has always the form ( (see Chapter 8), thus, Yy(z)v the nucleation specific frequency (number of nucleus produced per second and per unit of volume) at time z, nucleation frequency takes the form ... 

To apply the preceding relations, it is necessary to be able to clarify the nucleation-specific frequency and the space function of growth. 

We will consider for the moment only the random nucleation, that is, formation of the nucleus at random (condition to apply equation [A.9.3] to express the rate). It is certain that the expression of nucleation-specific frequency can be formulated only starting from a mechanism describing the phenomenon. However, as we considered in section 8.5.1.2, the nucleus in volume can be created starting from two main categories of precursors ... 

From this classification, we could deduce two eategories of expressions for nucleation-specific frequency. 

In the first case, this is the model of nucleation, whieh we considered in section 8.5.1.2.1, We know whereas that if we work at constant temperature (which is the only run of creation of these precursors), we will be able to consider quasi-steady state modes (the space function is independent of time) for which nucleation frequency will not be time dependent (see sections 8.5.4.2 and 8.5.4.3). We will also be able to consider no quasi-steady state modes, leading to nucleation-specific frequency function of time, such as the models of section 8.5.4.4 and equation [8.55], which we again give in [A.9.9] ... 

To obtain the rate, it is enough to use equation [A.9.6] and to substitute into it the nucleation-specific frequency and the space function of selected growth. In the same way, we use [A.9.7] for kinetic law. We will consider three examples. 

If, in addition, we know molar volume and particle size of the initial solid, we can deduce the nucleation-specific frequency from this law and determine the growth, which was not possible by considering very large initial volume. 

We can develop two simple reaction pathways that represent each of the two types of condensations that we have just defined. These mechanisms enable us to illustrate the methods to calculate the reactivity and then the specific frequency of nucleation. 

However as the amounts of matter concerned in nucleation are often very low, we prefer to use, in the place of the rate, / the frequency of nucleation, that is, the number of nuclei created per unit of time and, in the place of reactivity g, y the specific frequency of nucleation (number of nuclei created per unit of time and per unit of area) that can replace the reactivity. If is the number of Avogadro, wq the amount of matter of reference, and n the size of the stable nucleus (in number of structure elements), we have at aity time... 

To calculate the reactivity and specific frequency of nucleation, it is enough to substitute this expression into equations [8.38] and [8.32]. 

This leads to a specific frequency of nucleation in the form... 

This expression gives obviously the reactivity of the formation of the nuclei and thus of the last step in consequence of the pseudo-steady state, it is also that of the first step that makes it possible to calculate xo(f) and that will thus lead to first-order kinetics with regard to the potential nuclei and consequently to a specific frequency of nucleation of the same form as equation [8.61],... 

Ultimately, if we stick to the simple laws of nucleation, we will be satisfied with the law at constant specific frequency, the exponential law of Avrami [8.64], or possibly with the law power [8.55] or [8.70]. We will study in Chapter 11 heterogenous reactions with surface nucleation using a constant specific frequency. In Appendix A. 9, we will use the various laws to discuss reactions with nucleation in the bulk. 

If y t) and S t) are the specific frequency of nucleation and its space function, respectively (free area or free volume following the case), nucleation always has the form ... 

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. 

Generally, we obtain expressions in the forms of integrals that we could calculate by numerical methods on the condition of knowing the rate of growth and the specific frequency of nucleation, which is not the case. We will avoid this difficulty using dimensionless variables. 

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)... 

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... 

Putting this value of reactivity, obtained from equation [11.15], in equation [10.31] and from the value of g, we calculate the specific frequency of nucleation 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. 

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. 

With respect to the specific frequency of nucleation, we do not have direct experimental method to obtain its variation with an intensive variable. We must apply the method using the model (section 11.7.1) by calculating yfor each value of the variables. 

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. 

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). 

In the following section, we will return for the effect on the curve-specific frequency of nucleation pressure. 

Several laws were proposed in the literature to describe the speed of formation of the nuclei during decomposition reactions they were listed and analyzed in particular by Galwey and Brouwn [GAL 99], These laws come either from simple condensations or from condensations with potential nuclei (see Chapter 8). In this last case, even with the pseudo-steady state modes we can lead to a specific frequency of nucleation function of time. 

The author examines the cases where each of the first three steps can be the ratedetermining step and calculates (Table 13.6) the specific frequency of nucleation, denoting the rate and the equilibrium constants of step i by and... 

Of course, the granular distribution is without an effect on the reactivity of growth and the specific frequency of nucleation, which are intrinsic properties of the reaction and which do not depend on granular distribution in pseudo-steady state modes (from which results the constancy of the energy of activation). 

We carry out pressure switches for a fractional extent a = 0.25, under an initial pressure Pq toward various pressures P. Calculate the value of the space function with a = 0.25 at the Ho pressure. Deduce the value of the reactivity of growth and the specific frequency of nucleation at each pressure Hi. 

At each pressure to calculate the reactivity of growth and the specific frequency of nucleation y, we determine the parameter of model A and the ratio 6 It of dimensionless time on real time, starting from the characteristics of the point of inflection given in Table 18.19. For that, we use Table A.7.2 (see Appendix 7) and carry out linear interpolations between two consecutive values of a. ... 

To calculate the reactivity of growth, if), and the specific frequency of nucleation, Y, corresponding to each quadruplet (ii, A, 6 ), we use the relation of definition (see section 10.5.5) of dimensionless time therefore. 

Table 18.21. Reactivities ofgmwth and specific frequencies of nucleation obtained from the high values of A...

Table 18.23. Specific frequencies of nucleation (nuclei m s ) at 353 K for various water pressures...

The reactivity of nucleation can be deduced from the specific frequency of nucleationby relation (see section 8.5.2) ... 

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