Rate determining step internal interface

For a mode with the internal interface reaction as the rate determining step ( X interface), we will have (9 = and Q = = C° and thus ... 

Outward development with the rate-determining step at the internal interface (A/B)... 

We now consider the two possible directions of development (inward and outward) and for each of them, the rate-determining step can be a reaction step at the internal interface or the external interface or diffusion through the layer of the formed solid. Therefore, we have six possibilities. 

For example, for a spherical grain with inward development and a reaction located at the internal interface as the rate-determining step, comparing equations... 

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. 

Ug is the initial amount of A in an i grain Sp indicates at the i time for a nucleus i, bom at the initial time (i.e. in our case for an A grain), the area of the active surface, that is, the area of the internal interface if the growth has internal interface reactions or diffusion as rate-determining steps and the area of the external interface if the growth has external interface reactions (or gas adsorption or desorption) as rate-determining steps. 

Example of a spherical sample with a reaction growth with inward development and an internal interface rate-determining step... 

We have seen that the diffusion eould not be a rate-determining step in an isotropic growth we will thus have four possible modes of growth, inward development with internal or external interface step as rate determining and outward development with internal or internal interfaee step as rate determining. 

As an example, consider the case of spherical grains all identical with a reaction with inward development and an internal interface reaction as the rate-determining step. Table A.3.1 of Appendix 3 (second column, fourth line) gives rate in a form independent of the variations of with temperature... 

If we compare, for example, for the model of attack of spherical grains with instantaneous nucleation, inward development in a mode with an internal interface reaction as the rate-determining step, the ciuves obtained for various values of the average radius, we observe that the distribution has especially an influence on the end of the curves (Figure 13.14). 

Thus, we consider the case of a spherical grain A with radius rA(o> surrounded by two spherical grains C. We assume that an internal interface reaction on A is the rate determining step and an inward development into A. is the reactivity and F a is the molar volume of A. The ratio of the initial radii of A and C is such that the composition is stoichiometric. We assume that two nuclei occur at contact points Oi and O2 (Figure 14.17) at times and At any time t, the radii of the interfaces are ri = OiM and 2 = O2M, such that ... 

We can now apply the method of resolution of the pure kinetics, with one of the preceding reactions as the rate determining step in pseudo-steady state mode. For each type of solid MG, we will obtain three solutions according to whether the determining step is adsorption, the reaction at internal interface i or at the external interface e (we exclude the modes limited by the heart reaction that we have never encountered). Table 15.4 provides the results obtained in conditions far from equilibrium. To obtain the expression of the reactivity in the opposite case, closed to equilibrium conditions, we have to multiply the preceding relations by the term ... 

Within the fiamewoik of the formation of a sohd MG starting from gas G2, the multiplying coefficients of the steps of adsorption, external interface, and internal interface are I/2, 1, and 1, respectively, from which the reactivities in pure modes according to the reactivities of the rate determining step are ... 

We note that the pressure law when the internal interface is the rate determining step is different from that given in Table 15.4, whereas it remains the same when the external interface reaction is the rate determining step. In both cases, the kinetic law remains linear. 

We will develop two cases as examples a mixed mode with internal and external interface reactions as rate determining steps and a mixed mode with external interface reaction and diffusion through the formed layer as rate determining steps. 

Mixed modes with internal and external interface reactions as rate determining steps... 

The two rate determining steps are the internal and external interface reactions. We will consider again the reasoning of section 7.8. 

When F diffusion is the rate determining step, F concentration is fixed at the external interface by the other steps which are constantly at equilibrium but the F concentration in any point of the solid always decreases and there is no chemical equilibrium in the bulk of the solid which would impose a concentration at the internal limit of the way of diffusion it is thus impossible to retain a pseudo-steady state as it is shown in the evolution of concentration profiles with time (see Figure... 

We can thus consider two categories of pure kinetics according to an inward or outward development. In each one of them, the rate-determining step conld be a reaction at the internal interface, a reaction at the external interface, or a step of diffusion through the layer of formed B. 

We must now study the case of the outward development it is easy to realize that the area of the external interface grows, whereas that of the internal interface remains constant. Thus, the rate-determining steps take place at the internal interface and the layer has an outward development. 

The fact of observing small islands of B on A suggests a phenomenon of slow nucleation and isotropic growth on spherical particles, which excludes the diffusion as the rate-determining step, and thus, we adopt a mode of growth limited by a step taking place at the internal interface, with inward development. We will thus use the Mampel model for spheres. 

Table 19.5 and curves in Figures 19.4 and 19.5, which are hnked, give the obtained results for an internal interface rate-determining step, an external interface rate-determining step, and diffusion as rate-determining step, respectively. 

Figure 19.4. Interface steps as rate-determining steps (b) internal and (a) external...

Examine where this determining step can be located. It may be either at the internal interface or at the external one, but in these two cases, the space function is independent of time. It remains only the possibihty of difihsion as rate-determining step. This can be either a diffusion of point defects through the layer of formed sulfide or a diffusion of vacancies in the metal. It is indeed known that, for plates, a mode limited by a diffusion through thick layers leads to the parabolic law. 

Table A.5.2. Spherical grains, inward development with external reaction or outward development with internal interface as rate-determining step (z=l)...

These tables are drawn up for two shapes of powder samples that have an internal interface reaction as a rate-determining step and inward development. They give, for various values of the model parameter, the remarkable properties of the kinetic curves fractional extents and dimensionless times at the points of iirflection (if it exists) and the dimensionless time at the fractional extent 0.5 in two-process models with isotropic growth. 

We will only use pure mode with internal interface as rate-determining step and inward development. These modes are most frequent because the concerned reactions do not often use matter transport for growth. 

If the rate-determining step is the diffusion of species through layer B, the spatial function will depend on the shape of the diffusion. If Sj is the area of the internal interface, the spatial function is given by equation [14.6], where G is the geometric factor given by the second colnttm in Table 14.1 ... 

We will schematize a transformation that will show us that under identical conditions, it is possible that speeds are different. To illustrate it, assume a pseudosteady state mode with an internal interface elementary step as rate determining with inward development. 

In what follows we will stiek to the first ease of inward development with internal interface step as rate determining (the laws that correspond to the other cases were not established). Indeed, the laws that correspond to the isotropic growth are primarily known for reactions in which the initial solid A is the single reactant, that is, for the thermal decompositions and the polymorphic transformations. It is known that in these cases, the essence of chemistry proceeds at the internal interface, and as there is no other reactant, the outward development is not conceivable. In the other cases, if the initial solid is not a single reactant, the experiment shows that we have, in the large majority of the cases, either a one-process model with instantaneous nucleation or a two-process model with surface nucleation and anisotropic radial growth developed earlier. 

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