ArBs layer

Note that no reactions proceed within the bulks of both layers ApBq and ArBs. Layer bulks are no more than a transport medium for diffusing atoms. Chemical transformations take place only at the phase interfaces which are regarded as transition regions between the interacting phases, whose widths are not very different from the lattice spacings of those phases. 

To establish differential equations relating At to the increases, AxBi, AxA2, AyB2 and Ay, 13, in thicknesses of the ApBq and ArBs layers, use is again made of the postulate about the summation of the time of diffusion of the A or B atoms and the time of subsequent chemical transformations for each of four partial chemical reactions taking place at phase interfaces 1, 2 and 3. This yields... 

The system of equations (2.27) is seen to be rather complicated. Its solution, if obtainable at all in quadratures, must probably be even more complicated. However, in experiments certain conditions which enable the initial equations to be simplified are usually fulfilled. Consider limiting cases of particular interest from both theoretical and practical viewpoints.134,136,139,140 The process of growth of the ApBq and ArBs layers will be analysed in its development with time from the start of the interaction of initial substances A and B up to the establishment of equilibrium at which, according to the Gibbs phase rule (see Refs 126-128), no more than two phases should remain in any two-component system at constant temperature and pressure. 

Initial linear growth of the ApBq 2Liu ArBs layers... 

Evidently, in an initial period of interaction of substances A and B when the thicknesses of the ApBq and ArBs layers are relatively small the conditions k0Bi kim/x KA2 k iA2lx> k0B2 kw2 y and k0A3 klA3/y are satisfied. Hence, at low t the terms of the type kQxlkx and k0y kx can be neglected in comparison with unity. Therefore, the system of equations (2.27) is simplified to... 

If kom + k 0A2 = (rg/ p)k QB2, then dx/dt = 0. This corresponds to the stationary state where the rate of growth of the ApBq layer due to partial chemical reactions (2.11) and (2.12) is equal to the rate of its consumption in the course of formation of the ArBs layer by reaction (2.2 ). If the ApBq layer were in the initial specimen A B, then its thickness would remain constant (Fig. 2.3b). At the same time, the ArBs layer continues to grow linearly. 

From the system of equations (2.30), it follows that the thickness of the ArBs layer remains constant or equal to zero if k 0B2 + k0A3 = (q/sg)k 0A2. If the inequality... 

It should be noted that kinetic data must necessarily be supplemented by measurements of the displacement of phase interfaces relative to inert markers located within the bulks of growing ApBq and ArBs layers (Fig. 2.4). Otherwise, it is impossible to find all four chemical constants, kom, k 0A2, k 0B2 and k0A3, from the system (2.30) containing only two equations. 

After measuring the thicknesses of the ApBq and ArBs layers grown in time tu the values of k QB2 and k 0A2 are calculated from the system of two algebraic equations with two unknown quantities ... 

Minimal thickness of the ArBs layer necessary for the ApBq layer to occur... 

The absence of the ApBq compound layer from the A-B reaction couple during some period of time in the initial stage of interaction of substances A and B does not clearly mean that this compound cannot then occur and grow between A and ArBs. Indeed, equation (2.271) shows that the rate of consumption of the ApBq layer gradually decreases with increasing thickness of the ArBs layer. 

Hence, a certain minimally necessary value, ymtn, of the thickness of the ArBs layer is always reached at which the condition... 

The delay with the formation of the ApBq layer is thus due to purely kinetic reasons and is not associated with either the thermodynamic stability of the ApBq compound or its nucleation rate. It should be noted that after the appearance of the ApBq layer the rate of growth of the ArBs layer appreciably decreases because it starts to be consumed in the process of formation of the ApBq compound. 

If the ArBs layer was initially missing from the A-B reaction couple, it occurs between ApBq and B after the ApBq layer has reached a minimally necessary value xmm to satisfy the following equation... 

Fig. 2.5. Schematic illustration of the concept of a minimal thickness of the ArBs layer necessary for the ApBq layer to occur and grow in the A-B reaction couple.

The latter equation of this system indicates that the ArBs layer linearly grows with passing time. The x-t dependence is more complicated. Let us briefly analyse its possible variants. 

Similar considerations clearly apply to the ArBs layer if the regime of its growth is diffusion controlled with regard to component A, so that... 

Effect of the critical thickness of the ApBq layer with regard to components on the process of growth of the ArBs layer With further thickening the ApBq layer its growth regime becomes diffusion controlled (in the theoretical definition) not only with regard to component... 

Fig. 2.7. The non-linear-linear stage of simultaneous growth of the ApBq (line l)and ArBs (line 2) layers between substances and B. Growth of the ArB, layer is linear. Growth of the ApBq layer is (a) parabolic, (b) asymptotic or (c) close to parabolic in the t - t2 range and almost linear at t > t2.

Fig. 2.8. Decreasing the growth rate of the ArBs layer after the ApBq layer has reached its critical thickness, x f2, with regard to component. 1, thickness of theApBq layer 2, thickness of the T/f layer.

Indeed, at x < x f2 the ArBs layer grows at the expense of diffusion of both components A and B followed by partial chemical reactions (2.21) and (2.22), while at x>x fl it only grows at the expense of diffusion of component B and subsequent partial chemical reaction (2.21). Therefore, the y t curve will exhibit a deviation from its former course reflecting a... 

Decreasing the growth rate of the ArBs layer is clearly smooth, not abrupt, since a relative deficit of the diffusing A atoms in comparison with the reactivity of the surface of substance B towards these atoms becomes perceptible somewhat earlier than the thickness of the ApBq layer reaches... 

If the ApBq layer grows in the diffusion controlled regime with regard to components A and B, while the ArBs layer grows in the reaction controlled... 

Diffusion controlled growth of the ApBq and ArBs layers Increasing the thickness of the ArBs layer will inevitably result in a change of its growth regime from reaction to diffusion controlled with regard to... 

Thus, at x > x fl and y > the ApBq layer grows at the expense of diffusing A atoms and subsequent partial chemical reaction (2.12), while the ArBs layer grows at the expense of diffusing B atoms and subsequent partial chemical reaction (2.2]). Both reactions take place at their common interface 2 (Fig. 2.12). 

When the ArBs layer reaches the thickness y, a deviation appears on the dependence, x-t, of the thickness of the ApBq layer upon time. Note that in the case under consideration the deviations on the kinetic dependences are caused solely by quantitative changes in the thicknesses of growing compound layers. Neither their number, nor phase composition, nor structure change. This is an instructive example of the transformation of quantity into quality. 

The late diffusional stage of growth of the ApBq and ArBs layers is the one where the conditions x x fl and y y f2 are satisfied. This means that Ka2 x and k )B2 Kb2 y Therefore, by omitting, as physically meaningless, both the first term of the right-hand side of equation (2.271) and the second term of the right-hand side of equation (2.272) and neglecting unity in the denominators of the other terms, one obtains... 

Positive values of the derivatives dx/dt and dy/dt are a necessary condition for the simultaneous growth of the ApBq and ArBs layers. Therefore, instead of the system of equations (2.46), the following system of in-... 

These inequalities must be satisfied simultaneously, within which the ratio of the thicknesses of the ApBq vary are as follows Hence, the limits and ArBs layers can... 

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