Predicting Phase Stability of Materials Based on …
573
following equation was used:
T cr = Q mix
2kN,
where k is the Boltzmann constant and N is the Avogadro number. In order to calculate
the substitution limits for a given decomposition temperature of a solid solution (T d )
or the decomposition temperature for a given substitution limit [24], the Becker
equation was used [24]:
−(1−2x)/ln[x/(1−x)] = RT d /Q mix ,
where R is the universal gas constant, Q mix is a mixing energy (or interaction
parameter), and x is a substitution limit.
As can be seen from Table 1 and Fig. 1 (curve f), the values of maximum decomposition temperatures, as expected, become smaller as REE number increases. The
Becker equation was also used to calculate the decomposition temperatures (stability)
of solid solutions for the substitution limits x = 0.01, 0.03, 0.05, 0.10, and 0.20
(Table 2), and to build their dependencies (Fig. 1) on the REE number (curves a, b,
c, d, and e, respectively). The latter can be used to determine the substitution limit
of scandium for REE based on a given temperature or calculate the decomposition
temperature based on the substitution limit. In the first case, it is necessary to draw
an isotherm from a given temperature to the intersection with the vertical line for this
REE. The intersection point makes it possible to estimate the range of x values within
which the substitution limit lies. The substitution limit should be defined by interpolating the vertical segment between the closest to the intersection point dependencies
of the substitution limit on the REE number. In the second case, based on the given
composition the point is determined on the vertical line of the REE, and then the
horizontal line is drawn until its intersection with the temperature axis. More precise
results can be obtained if using the Becker equation.
It is commonly known that as the temperature decreases, the mobility of the structural units in solid solution becomes smaller due to a decrease in the diffusion rate,
while the solubility regions become narrower [17]. This happens until the diffusion
rate becomes so low that the decrease in the solubility regions practically ceases, i.e.,
spontaneous quenching occurs, and solid solutions become metastable. If we assume
that the hardening temperature is close to the minimum temperature at which the
interaction of the components in the solid phase begins that leads to the formation of
a solid solution, we can estimate the temperature of spontaneous hardening and the
region of metastability in the system. It was previously established (Table 3) that the
temperature during the synthesis of solid solutions of Sc 2–x Er x SiO 5 oxyorthosilicates
as a part of the preparation of multinanolayer films is in the range of 1173–1373 K,
573
following equation was used:
T cr = Q mix
2kN,
where k is the Boltzmann constant and N is the Avogadro number. In order to calculate
the substitution limits for a given decomposition temperature of a solid solution (T d )
or the decomposition temperature for a given substitution limit [24], the Becker
equation was used [24]:
−(1−2x)/ln[x/(1−x)] = RT d /Q mix ,
where R is the universal gas constant, Q mix is a mixing energy (or interaction
parameter), and x is a substitution limit.
As can be seen from Table 1 and Fig. 1 (curve f), the values of maximum decomposition temperatures, as expected, become smaller as REE number increases. The
Becker equation was also used to calculate the decomposition temperatures (stability)
of solid solutions for the substitution limits x = 0.01, 0.03, 0.05, 0.10, and 0.20
(Table 2), and to build their dependencies (Fig. 1) on the REE number (curves a, b,
c, d, and e, respectively). The latter can be used to determine the substitution limit
of scandium for REE based on a given temperature or calculate the decomposition
temperature based on the substitution limit. In the first case, it is necessary to draw
an isotherm from a given temperature to the intersection with the vertical line for this
REE. The intersection point makes it possible to estimate the range of x values within
which the substitution limit lies. The substitution limit should be defined by interpolating the vertical segment between the closest to the intersection point dependencies
of the substitution limit on the REE number. In the second case, based on the given
composition the point is determined on the vertical line of the REE, and then the
horizontal line is drawn until its intersection with the temperature axis. More precise
results can be obtained if using the Becker equation.
It is commonly known that as the temperature decreases, the mobility of the structural units in solid solution becomes smaller due to a decrease in the diffusion rate,
while the solubility regions become narrower [17]. This happens until the diffusion
rate becomes so low that the decrease in the solubility regions practically ceases, i.e.,
spontaneous quenching occurs, and solid solutions become metastable. If we assume
that the hardening temperature is close to the minimum temperature at which the
interaction of the components in the solid phase begins that leads to the formation of
a solid solution, we can estimate the temperature of spontaneous hardening and the
region of metastability in the system. It was previously established (Table 3) that the
temperature during the synthesis of solid solutions of Sc 2–x Er x SiO 5 oxyorthosilicates
as a part of the preparation of multinanolayer films is in the range of 1173–1373 K,