Space function 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. 

Ultimately, it is noted that in all the cases, except that of the isotropic growth on a sphere (also true for a cylinder) with inward development and the rate-determining step located at the internal interface, the space function of growth of a nucleus is a monotonous function of time, either constantly increasing or constantly decreasing, or constant in time if the active surface is of invariable area. 

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. 

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

Tables of Appendix 2 give the various expressions of the space function of radial anisotropic growth for various shapes of grains. Examining all these expressions for the space function according to time, we can write this function in a general form, using the initial volume vq and area of grain...

We see that if the growth is with separable rate, the total reaction is also separable and the space function of the growth is worth in the general case ... 

If we indicate by e t, t) the space function of anisotropic growth of a grain (which is brought back to the amormt of initial matter of a grain) and by No the nirmber of grains of the powder, eiqiressions [10.16] and [10.18] become ... 

We will not reconsider the modeling of the space function for growth already abimdantly covered in Chapter 10 but will examine some chemical models for growth and nucleation. 

Tables A.2.1 to A.2.3 give the space functions of anisotropic growths for a grain and for the various types of samples in different modes, having inward and outward...

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. 

Moreover, if growth is given by [14.3] and is the growth space function of the spot, t the time function, t the age of the spot, and ( ) its reactivity that will only be a function of the intensive physico-chemical parameters if the system evolves at constant temperature and partial pressures the growth rate is ... 

The generalized Fisher theorems derived in this section are statements about the space variation of the vectors of the relative and absolute space-specific rates of growth. These vectors have a simple natural (biological, chemical, physical) interpretation They express the capacity of a species of type u to fill out space in genetic language, they are space-specific fitness functions. In addition, the covariance matrix of the vector of the relative space-specific rates of growth, gap, [Eq. (25)] is a Riemannian metric tensor that enters the expression of a Fisher information metric [Eqs. (24) and (26)]. These results may serve as a basis for solving inverse problems for reaction transport systems. 

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