Predicting Phase Stability of Materials Based on …
571
value χ(SiO 4
4− ), as recommended [20], was accepted equal to χ(O
2− ) = 3.7 [19]; m
= 2, the number of formula units in the pseudobinary approximation of components.
Since the anionic sublattice of the crystal structure of oxyorthosilicates contains
SiO 4
4− and O
2− anions that are not bonded to the Si atom [15] and the substitution
limits are calculated per 1 mol of the replaceable ion, the oxyorthosilicate formulas
will be presented below as a pseudobinary compound Ln[(SiO 4 ) 0.5 O 0.5 ]; n = 6, the
coordination number of the replaceable structural unit in the pseudobinary approximation of the structure; z m = 3, z x = 3, the formal charges of the replaced and
general structural units in the components: z m = 3, z x = 4 · 0.5 + 2 · 0.5 = 3; δ
is a dimensional parameter, which for each system is characterized by the relative
difference of cube roots of unit cell volumes taken from [15, 21, 22], calculated by
the formula:
δ =
V
1/3
Ln −V
1/3
Sc
/V
1/3
Sc ;
α = 1.73, the reduced Madelung constant calculated by the Hoppe formula [23]:
(α/ n)
2
+ α = 1.81, where n = 6 is the coordination number in the pseudobinary
approximation of the structure; ε is the difference in the degree of ionicity of the
chemical bond in the components of the systems. D is the interatomic distance of the
cation––tetrahedral anion, D = 3.111, calculated from the data [22].
3 Calculation Results
Some initial data and calculation results are summarized in Tables 1 and 2 and
Fig. 1. The Table 1 shows that as the number of REE in the Tb–Lu row increases, the
contributions of Q δ values to the total mixing energy become smaller (from 34.5 to
10.8 kJ/mol), which is explained by smaller differences in the size of substitutable
structural units—scandium and REE.
Contributions due to different degrees of ionicity of chemical bond in the components of the Q ε systems are substantially smaller (in most cases, 2–3 times smaller)
than in Q δ , and they can be neglected in this case. This agrees with the recommendation not to take them into account, provided that ε ≤ 0.05 [16–18] (in this case,
ε ≤ 0.015). Therefore, it is accepted that Q mix = Q δ .
In all systems, the size parameter (δ) does not exceed 0.1 with its maximum
value of 0.0535 (Table 1). This, according to [16–18], makes it possible to use the
approximation of regular solutions when calculating the decomposition temperatures
(stability) of solid solutions. In this case, the curve showing the dependence of the
decomposition temperatures on the system composition will be nearly symmetric.
Therefore, to calculate the maximum (critical) decomposition temperatures, the
571
value χ(SiO 4
4− ), as recommended [20], was accepted equal to χ(O
2− ) = 3.7 [19]; m
= 2, the number of formula units in the pseudobinary approximation of components.
Since the anionic sublattice of the crystal structure of oxyorthosilicates contains
SiO 4
4− and O
2− anions that are not bonded to the Si atom [15] and the substitution
limits are calculated per 1 mol of the replaceable ion, the oxyorthosilicate formulas
will be presented below as a pseudobinary compound Ln[(SiO 4 ) 0.5 O 0.5 ]; n = 6, the
coordination number of the replaceable structural unit in the pseudobinary approximation of the structure; z m = 3, z x = 3, the formal charges of the replaced and
general structural units in the components: z m = 3, z x = 4 · 0.5 + 2 · 0.5 = 3; δ
is a dimensional parameter, which for each system is characterized by the relative
difference of cube roots of unit cell volumes taken from [15, 21, 22], calculated by
the formula:
δ =
V
1/3
Ln −V
1/3
Sc
/V
1/3
Sc ;
α = 1.73, the reduced Madelung constant calculated by the Hoppe formula [23]:
(α/ n)
2
+ α = 1.81, where n = 6 is the coordination number in the pseudobinary
approximation of the structure; ε is the difference in the degree of ionicity of the
chemical bond in the components of the systems. D is the interatomic distance of the
cation––tetrahedral anion, D = 3.111, calculated from the data [22].
3 Calculation Results
Some initial data and calculation results are summarized in Tables 1 and 2 and
Fig. 1. The Table 1 shows that as the number of REE in the Tb–Lu row increases, the
contributions of Q δ values to the total mixing energy become smaller (from 34.5 to
10.8 kJ/mol), which is explained by smaller differences in the size of substitutable
structural units—scandium and REE.
Contributions due to different degrees of ionicity of chemical bond in the components of the Q ε systems are substantially smaller (in most cases, 2–3 times smaller)
than in Q δ , and they can be neglected in this case. This agrees with the recommendation not to take them into account, provided that ε ≤ 0.05 [16–18] (in this case,
ε ≤ 0.015). Therefore, it is accepted that Q mix = Q δ .
In all systems, the size parameter (δ) does not exceed 0.1 with its maximum
value of 0.0535 (Table 1). This, according to [16–18], makes it possible to use the
approximation of regular solutions when calculating the decomposition temperatures
(stability) of solid solutions. In this case, the curve showing the dependence of the
decomposition temperatures on the system composition will be nearly symmetric.
Therefore, to calculate the maximum (critical) decomposition temperatures, the