ApBq and ArBs layers

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 

It does not mean, however, that the ApBq and ArBs compound layers must always occur at the A-B interface simultaneously. Clearly, their simultaneous occurrence is only possible if the derivatives dx/dt and dy/dt in equations (2.30) are positive (dx/dt 0 and dy/dt 0), and consequently the inequalities k0m+k 0A2 (rg/p)k 0B2 and k 0B2+k0A3 (q/sg)k 0A2 are satisfied. In such a case, both layers can grow simultaneously from the very start of interaction of phases A and B according to the linear law (2.31), as illustrated in Fig. 2.3a. Note that they will grow at the highest rates possible under given (constant) temperature-pressure conditions. 

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. 

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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... 

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 ... 

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... 

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-... 

It is seen, firstly, that the ratio of the thicknesses of the ApBq and ArBs layers depends upon (/) the values of the physical (diffusional) constants, (ii) the ratio of the molar volumes of the ApBq and ArBs compounds, (///) the stoichiometry of these compounds. ... 

During the natural course of the process of formation of the ApBq and ArBs layers between elementary substances A and B when an A B specimen is given to itself at constant temperature and pressure, a correct ratio of their thicknesses is established automatically. However, if an A-ApBq-ArBs B specimen was prepared artificially, this ratio can hardly be expected to be correct. Therefore, during subsequent isothermal annealing of the specimen, one of the layers will shrink and can even disappear as occurred before its turn if, of course, by that time the other layer has not reached a minimal thickness required for the former to occur. Such a phenomenon was observed, for example, by G. Ottaviani and M. Costato74 with the PtSi layer in Pt-Pt2Si-PtSi-Si specimens and by K.N. Tu et alm with the CoSi layer in Co-Co2Si-CoSi-Si specimens. 

Therefore, an attempt can be undertaken to describe the experimental dependence of the thickness of the layers upon time in terms of the dif-fusional constants k[A2 and k[B2. If these prove to be indeed constant at all experimental values of the thicknesses of the ApBq and ArBs layers, then the system of equations (2.46) properly describes the process of growth of two chemical compound layers. Otherwise, this portion of the layer thicknesstime dependence appears to be not purely diffusional, and use must be made of other mathematical equations giving a more adequate fit to the experimental data. 

In fact, chemical transformations taking place at the interfaces between reacting phases are responsible for the occurrence of the barriers to diffusing atoms at the critical values of the thickness of the ApBq and ArBs layers. Their rate is also decisive in determining the sequence of formation of the layers of those compounds in the A-B reaction couple. 

A schematic diagram illustrating the growth process of the layers of two chemical compounds ApBq and ArBs, with p, q, r and s being positive numbers, at the A B interface is shown in Fig. 2.1. Note that the lines showing the distribution of the concentration of components A and B in the phases involved in the interaction are parallel to the distance axis since (/) the formation of the layers of chemical compounds which have narrow, if any, ranges of homogeneity is considered and (ii) initial substances are assumed to be mutually insoluble. 

Fig. 2.1. Schematic diagram to illustrate the growth process of the layers of two chemical compounds ApBq and ArBs at the interface between mutually insoluble elementary substances andB.

Direct chemical reaction between elementary substances A and B clearly ceases after the formation of compound layers ApBq and ArBs, a few crystal-lattice units thick, which separate the reacting phases from each other. Subsequently, four partial chemical reactions take place at the layer interfaces. These are as follows ... 

Let us now return to analysing the process of growth of the ApBq and ArBs compound layers at the interface between elementary substances A and B. From equation (2.21), it follows that the ratio of the mass of the ApBq compound entering into partial chemical reaction (2.21) to the mass of the ArBs compound formed as a result of this reaction is equal to the ratio of the molecular masses of the compounds ApBq and ArBs with the factors r and p, respectively ... 

Substituting into these equations the expressions (2.8 i)-(2.92) for the increases of the thicknesses of the ApBq and ArBs compound layers, one obtains the required general system of two differential equations describing their growth rates at the A B interface ... 

The system of equations (2.27) belongs to the so-called autonomous systems which are analysed in detail, for example, in Refs 203 and 204. Note, however, that, to avoid misleading conclusions concerning layer-growth kinetics, an approach to its solution should by no means be formally mathematical. Namely, besides the initial conditions x = 0 and y = 0 at t = 0, the existence of the critical values of the thicknesses of the ApBq and ArBs compound layers must necessarily be taken into account. These are as follows (see equations (1.17) and (1.22) in Chapter 1) ... 

Clearly, in such a case x and y correspond to experimentally measured values of the thicknesses of the ApBq and ArBs compound layers. This form of writing mathematical equations also makes it possible to easily take account of a change in volume of the system resulting from the formation of chemical compounds, which in many cases is too considerable to be neglected without a noticeable error. 

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... 

From the system of equations (2.46), it follows that the layers of the ApBq and ArBs compounds already present in an A-ApBq-ArBs B specimen should not necessarily simultaneously grow during its further isothermal annealing. If their initial thicknesses, x0 and y0, are such that, for example, the derivative 6x/ t)t=h is negative and the derivative (dy/ dt)l=l is positive, then the thickness of the ApBq layer will decrease, while the thickness of the ArBs layer will increase until the x/y ratio falls into the range defined by inequality (2.48). Subsequently, both layers will grow simultaneously. 

In the reaction controlled regime the layer of each of two chemical compounds ApBq and ArBs grows at the expense of two partial chemical reactions taking place at its interfaces with adjacent phases. 

In the diffusion controlled regime the growth of each of two compound layers is due to one partial chemical reaction taking place at its common interface with another growing layer. In this case, only the A atoms diffuse across the ApBq layer adjacent to initial phase A, while only the B atoms diffuse across the ArBs layer adjacent to initial phase B. No partial chemical reactions proceed at the A ApBq and ArBs-B interfaces in view of the lack of appropriate diffusing atoms. 

If dx d/ < 0 and dz/dt < 0, then the ApBq and AiBn layers cannot occur at all and therefore only the ArBs layer will grow at the interface between substances A and B according to a linear law (Fig. 3.2d). If the layers of all three compounds were present in initial specimens, the thicknesses of the ApBq and A/Btl layers should decrease and they can and will disappear completely after some time, if of course the thickness of the ArBs layer does not exceed the minimal values necessary for the growth of these layers to start. 

It should be noted that the disappearance of each of the ApBq and AiBn layers considerably reduces the rate of linear growth of the ArBs layer, as shown schematically in Fig. 3.2d. This is due to the fact that the closer the compositions of adjacent non-growing phases to the composition of any growing compound layer, the greater is the growth rate of that layer. This question will be considered in more detail in Chapter 4. 

The ApBq and A Bn layers are seen to grow parabolically, whereas the thickness of the ArBs layer will gradually decrease with passing time. Eventually, this layer will disappear. It is easy to notice that in this case the values of the diffusional constants k[A2 and kim can readily be determined from the experimental dependences x2- t and z2- t, respectively, using an artificially prepared specimen A-ApBq-ArBs-A iB,-B or A-A,B-B. It is essential to mention that both the ApBq and AtBn layers must be the first to occur at the A-B interface. The diffusional constant k[A2 thus obtained is the reaction-diffusion coefficient of the A atoms in the ApBq lattice, while the diffusional constant klB3 is the reaction-diffusion coefficient of the B atoms in the Afin lattice, to be compared with respective self-diffusion coefficients determined using radioactive tracers. 

During further isothermal holding, the ApBq and AtBn layers will grow until one of initial substances A or B is consumed completely. The ArBs layer can then form and grow either in the ApBq-ArBs-AiBn-B system or in the A-ApBq-ArBs-AiBn system. 

The results presented in this chapter were obtained assuming that the rate of transformation of ApBq into ArBs is not less than the rate of diffusion of the A atoms across the ArBs layer. For thick layers, this seems to be sufficiently substantiated. In thin layers of chemical compounds, especially if the structures of the ApBq and ArBs crystal lattices differ considerably, rearrangement of the ApBq into ArBs lattice may be a rate-determining step. Therefore, the growth kinetics of the ArBs layer will be dependent upon the rate of this transformation. 

If two layers ApBq and ArBs are formed, the kinetic dependence is more complicated and can scarcely be described by any single analytical function. One of its most probable variants is shown schematically in Fig. 5.20. Both layers are assumed to form from the very beginning of interaction between substances A and B. Their growth kinetics are described by the system of differential equations (2.27). 

Fig. 2.3. Initial stage of formation of the ApBq (line 1) and ArBs (line 2) compound layers in the course of interaction of elementary substances A and B.

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. 

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.

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