Constant diffusional

When the backward transfer is taken into account, KAB becomes smaller than k as a result of partial restoration of the excited state. This effect is greater the larger is the kinetic rate constant kb, or the smaller is the diffusional constant ko. It is the most pronounced at minimal concentrations of A (y = 0). As this concentration increases, the effect is hindered as shown in Figure 3.85. According to MET... 

All the ko values listed in Table VII are much larger than the contact diffusional constants ko = 4nuD, so that the reactions are strongly in the diffusional limit, as was expected [248]. The same is true for the diffusional k(c), which is larger everywhere than the Stern-Volmer constant obtained in the contact approximation. The family of such curves compared in Figure 3.90... 

The physical (diffusional) constant kWi can readily be found from a long-time portion of the same A Bq layer thickness-time dependence but now plotted in the coordinates x- t (Fig. 1.5b). The slope of a plot of the squared thickness of the ApBq layer against time yields a value of k If reaction times and accordingly layer thicknesses are insufficiently large to neglect a linear portion of the ApBq layer thickness-time dependence without any noticeable error, equation (1.11) must be integrated with the use of initial condition x = x0 at t = 0 or x = x0 at t = t0, giving... 

The only reason for the complicated look of equation (1.25) in comparison with equation (1.8) is that in general the values of the chemical constants k()m and k()A2 as well as those of the diffusional constants k]m and kU2 are not equal to each other. Even the proportionality of the diffusional... 

The temperature dependence of the chemical and physical (diffusional) constants is likely to be described by an equation of the Arrhenius type (see Refs 126-128) ... 

It is obvious that from any experimental dependence of the total layer thickness upon time it is only possible to determine the sum of the chemical constants as well as the sum of physical (diffusional) constants. The former sum is to be found from an initial portion of this dependence plotted in the coordinates x - t, while the latter from its long-time portion plotted in the coordinates x2- t or x - /l/2. For their separate determination, it is necessary to measure the increases in thickness of the ApBq layer at its both interfaces with initial substances A and B. 

To determine the physical (diffusional) constants k]m and k]A2, it is necessary to establish the conditions under which the total thickness of the ApBq layer increases with time parabolically. Then, like the previous case, the increases, Axm and AxA2, in thickness of the layer at its interfaces 1 and 2 are to be measured. The values of the physical (diffusional) constants k m and k A2 are computed from the equations... 

It is evident that the separate determination of the chemical and physical (diffusional) constants is much more difficult in comparison with conventional experiments where only the total thickness of the growing ApBq layer is measured, with the further search for its growth law by means of mathematical treatment of the results using linear, parabolic, logarithmic and other dependences (see, for example, Refs 6, 7, 13). However, the former procedure ultimately gives a more complete description of the interaction in the examined reaction couple A-B compared to the latter. 

It should be noted that the value of each of the chemical constants kom and k0A2 depends on the physical-chemical properties of two reacting phases. The value of kom depends on the nature of substance A and the compound ApBq, while the value of k0A2 depends on the nature of substance B and the compound ApBq. Both physical (diffusional) constants depend only on the nature of the chemical compound ApBq and are therefore characteristic of this compound layer wherever it grows. However, as will be demostrated in the next chapters, the stoichiometry of adjacent phases must also be taken into account when estimating the growth rate of the ApBq layer in various reaction couples of the A-B binary system. 

The sum of the chemical constants k0Bi and k0A2 as well as the sum of the physical (diffusional) constants kim and kh42 can also be determined from the decrease in thickness of the layers of initial substances A and B. Indeed, using equations (1.1) and (1.2), it can readily be shown that the... 

It can easily be understood that this method has practically no advantage in comparison with direct measuring the thickness of the growing ApBq layer. Its disadvantages are obvious. Firstly, the diffusional constants are calculated using the differential forms of kinetic equations. This usually produces a larger error than the calculations with the use of the integrated equations. Secondly, the amount of initial substances A and B consumed in the reaction of compound formation is in most cases much less than their... 

Table 1.1. Evaluation of diffusional constants for the AlSb layer of the Al-Sb binary system using the experimental data by B.Ya. Pines and E.F. Chaikovskiy9,144...

Clearly, when calculating the physical (diffusional) constants, first of all it is necessary to check out whether the layer indeed grows in the diffusion controlled regime where the conditions 0sbi isbiA and k0A12 kXA 2/x must be satisfied. For this, the experimental data should be treated using the parabolic dependence (see equation (1.33))... 

As x0 can be taken with the plus or minus sign, the experimental data should produce a straight line in the coordinates either (x + x0) - t/(x - x0) or (x - x0) - / (x + x0). This form of presenting the experimental results was used by N.A. Kolobov and M.M. Samokhvalov,17 who found the values of the reaction-diffusion constants for the Si02 layer in the oxidation of silicon by oxygen, listed in Table 1.2. The temperature dependence of both the chemical constant k0 and the physical (diffusional) constant k is well described by the Arrhenius relation (see equation (1.34)) with the activation energy 155 and 120 kJ mol1, respectively. 

The value of the physical (diffusional) constant 1g"t1esral) can be calculated using integrated equations (1.60) or (1.61). The superscript integral is used to distinguish between the value of the constant calculated from the integrated equation and that found from the differential equation. The latter... 

Table 1.4. Experimental values of the diffusional constants for the NiBi3 layer growing between nickel and bismuth150...

The temperature dependence of the reaction-diffusion coefficient (diffusional constant) of bismuth atoms in (or, rather, across) the growing NiBi3 layer is described by the Arrhenius relation (see equation (1.34)). As... 

S.S. Kristy and J.B. Condon176 found a value of D0 of the order of 10 23 m2 s 1 at 700°C and arrived at the conclusion that this value could not have any noticeable effect on the kinetics of reaction of silicon with oxygen. This is undoubtly the case in view of much greater values of the diffusional constant, ku listed in Table 1.2. From the work of J.A. Costello and R.E. Tressler,177 it can be concluded that the diffusion coefficient of oxygen atoms found by radiotracers becomes comparable with the growth-rate constant of the Si02 layer at temperatures above 1300°C. 

Thus, the value of cAi in the denominator of equation (1.70) is numerically equal to the content of A in AB, while cA2 is practically zero. Consequently, this equation yields k]A2 = DA. Therefore, the physical (diffusional) constant k A2 is identified with the reaction-diffusion coefficient, Da, of component A in the lattice of any chemical compound. 

Section 1.4 klA2 was used to denote the physical (diffusional) constant relating to reaction (1.2). From equations (1.2) and (2.12), it follows... 

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. ... 

The layer-growth kinetics were found to be parabolic for both compounds (Fig. 2.18), indicative of diffusion control. This is an expectable result since the layer thickness varied from about 10 pm to 300 pm for the Al12Mg17 intermetallic compound and from about 80 pm to more than 900 pm for the Al3Mg2 intermetallic compound. Diffusional constants were calculated using parabolic equations of the type x2 = 2k t. The temperature dependence of the diffusional constants was found to obey the Arrhenius relation ... 

The layer thickness-time kinetic relationships are in general rather complicated, not merely parabolic. Depending on the values of the chemical and physical (diffusional) constants, their different portions can be described by linear, paralinear, asymptotic, parabolic and other laws. 

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. 

Therefore, firstly, the physical (diffusional) constants become strongly dependent upon the layer thickness and consequently vary with annealing time. Secondly, the chemical constants also become time-dependent due to the variation of the boundary concentrations of the components in that layer. 

For the purpose of further discussion, the averaged values of the diffusional constants presented in Table 3.10 will be used. Their temperature dependences obey the Arrhenius relation, as seen in Fig.3.16 and Table 3.11. 

Table 3.9. Diffusional constants, k, for the Ni-Zn and Co-Zn binary systems calculated from the experimental data according to parabolic equations of the type kx = (xf where x, and x0 are the layer thickness at time U and t = 0, res-... 

Fig. 3.16. Temperature dependence of the diffusional constants for the Co-Zn binary system. 1, Co-bordering layer 2, Zn-bordering layer.

The homogeneity ranges of the intermetallics of the Ni-Zn and Co-Zn binary systems are known to vary very little in the 250-400 °C temperature range, if at all (see Figs 3.10 and 3.11). Their widths are presented in Table 3.12. The values of the diffusional constants at a temperature of 250 °C, calculated from appropriate temperature dependences (see Table 3.11), are also included in the table. 

Table 3.12. Comparison between the width of the homogeneity range and the diffusional constant for the intermetallic compounds of the Ni-Zn and Co-Zn binary systems at a temperature of 250°C...

Simultaneously, the ArBs layer thickness increases at interface 3 as a result of the phase transformation of A/Bn into ArBs by reaction (4.18). The diffusional constant klB2 characterises partial chemical reaction (4.1) in which 5 diffusing B atoms take part. Again, according to equation (4.18) the loss of r AtBn molecules (rn-ls) B atoms results in the formation of l ArBs molecules. Therefore, the growth rate of the ArBs layer at interface 3 is... 

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