Thermodynamic properties of the Gd2O3-Y2O3-HfO2 system studied by high temperature Knudsen effusion mass spectrometry and optimized using the Barker lattice theory

Journal of Alloys and Compounds 791 (2019) 1207e1212

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Thermodynamic properties of the Gd2O3-Y2O3-HfO2 system studied by high temperature Knudsen effusion mass spectrometry and optimized using the Barker lattice theory A.L. Shilov*, V.L. Stolyarova, S.I. Lopatin, V.A. Vorozhtcov Saint Petersburg State University, Universitetskaya nab.7/9, 199034, Saint Petersburg, Russia

a r t i c l e i n f o

a b s t r a c t

Article history: Received 15 October 2018 Received in revised form 1 March 2019 Accepted 12 March 2019 Available online 14 March 2019

The vaporization processes and thermodynamic properties of twenty samples in the Gd2O3-Y2O3-HfO2 system in a wide concentration range were studied by high-temperature Knudsen effusion mass spectrometry. At the temperature 2500 K the main vapor species were GdO, YO, and O. From the measured values of their partial pressures activities of the Gd2O3 and Y2O3 as functions of concentration were determined. The HfO2 activity was calculated by the Gibbs-Duhem integration. The resulting excess Gibbs energy function exhibited negative deviations from ideal behavior. For the optimization of the obtained thermodynamic functions of the ternary Gd2O3-Y2O3-HfO2 system the generalized BarkerGuggenheim theory of associated solutions was applied. In an alternative computing procedure the same model was used, but the optimization was based on the available literature data for the corresponding binary Gd2O3-Y2O3, Gd2O3-HfO2, and Y2O3-HfO2 systems. Thus, the Barker lattice model may be used for estimation of the thermodynamic functions of a ternary system on the basis of the data for the corresponding binary sub-systems. © 2019 Published by Elsevier B.V.

Keywords: Knudsen effusion mass spectrometry Hafnia Yttria Gadolinia Lattice model Rare earth alloys and compounds Thermodynamic modeling

1. Introduction Refractory materials based on the Gd2O3-Y2O3-HfO2 system are considered promising for substitution of the nowadays-used ceramics, especially in such applications as thermal barrier coatings and molds for casting gas turbine blades [1]. Further development of such refractories requires a deeper insight into their hightemperature behavior. One of the problems to be solved is inequality of volatilities of components causing their selective evaporation and subsequent deterioration of the material performance. A detailed experimental study of the vapor phase above the samples of the Gd2O3-Y2O3-HfO2 system and derivation of its thermodynamic properties from the measured partial pressures of components were reported in our recent paper [2]. A brief account of these investigations, used as a basis for the present theoretical analysis, is given in the Experimental section. High temperature of vaporization and considerable difference in the partial pressures of components make difficulties also to the mass spectrometric study [2] of the Gd2O3-Y2O3-HfO2 system.

* Corresponding author. E-mail address: [email protected] (A.L. Shilov). https://doi.org/10.1016/j.jallcom.2019.03.182 0925-8388/© 2019 Published by Elsevier B.V.

Vapor pressure of HfO2 is too low in the considered temperature interval compared to Gd2O3 and Y2O3 and hence only the vapor species referring to gadolinium and yttrium oxides could be recorded simultaneously. The accuracy of the Gibbs energy determination is reduced also by possible interaction of the samples with the material of effusion cells at high temperatures, depletion of the samples surface with the more volatile components during the vaporization, and by the impossibility of oxygen partial pressure measurements in the given experimental conditions. For the samples with high HfO2 content, when the partial pressures of the detectable vapor species are very low, the errors may be most noticeable. As is generally known, optimization of the experimental data using methods of statistical thermodynamics can provide some idea of the probable bias and systematic errors in the data points and also makes possible extrapolation and interpretation of the results. In a number of mass spectrometric studies of oxide systems carried out earlier by the authors of the present article [3e6] dedicated mostly to glasses and glass-forming melts the optimization was based on the Barker-Guggenheim generalized theory of associated solutions [7]. In the present study this approach is applied to optimization of the Gd2O3-Y2O3-HfO2 system and the

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main objective is a computation of the excess Gibbs energy (DGE) as a function of concentration and its comparison to the same function derived from the available literature data for the corresponding binary systems Gd2O3-Y2O3 [8], Gd2O3-HfO2 [9], and Y2O3-HfO2, [10]. The quasichemical lattice models are relatively simple but for modeling of inorganic oxide melts they are usually not enough flexible. Some modifications were suggested to enable a more detailed description of the thermodynamic behavior of a system, such as introduction of ‘equivalent fractions’ and variable coordination number of the lattice [11], or complex structural units with additional contact points [6]. However, their application requires specific information on the structural features of the melts, otherwise heavy arbitrary assumptions have to be made. Therefore, in the present study reliability of the experimental data and validity of the model were verified by comparing the results of computations employing the common set of model parameters but based on different experimental data: (1) on the activities of components measured in the Gd2O3-Y2O3-HfO2 system [2] and (2) on the activities of components measured in three binary systems: Gd2O3Y2O3 [9], Gd2O3-HfO2 [10], and Y2O3-HfO2, [11]. Since synthesis of the samples and measuring procedure for these binary systems were not identical to those for the ternary system under consideration the results of modeling can be considered to some extent independent and their comparison can help to reveal possible shortcomings in the measuring procedure. 2. Experimental The vaporization processes and thermodynamic properties of ceramics based on the Gd2O3-Y2O3-HfO2 system were investigated by the high temperature Knudsen effusion mass spectrometry. The samples for the vaporization studies were synthesized from pure gadolinium, yttrium, and hafnium oxides by pressing the appropriate blends of fine powders into pellets, which were then slowly heated up to 1973 K and after 1 h exposure at this temperature cooled down. For the vaporization studies, for X-ray fluorescence analysis (S8 Tiger wavelength dispersive spectrometer, Bruker), and for hot stage X-ray phase analysis (EMPYREAN diffractometer, PANalytical) the pellets were ground to powders in an agate mortar.

Compositions of the samples determined by the chemical analysis and used in the calculations are listed in Table 1. Experimental study of the vaporization processes and thermodynamic properties of the Gd2O3-Y2O3-HfO2 system were carried out with an MS-1301 magnetic mass spectrometer (Institute for Analytical Instrumentation of the Russian Academy of Sciences) equipped with a vaporization chamber enabling heating the samples up to 3000 K. The temperature was measured with an optical disappearing filament pyrometer EOP-66 (Pribor, Ukraine). The samples were vaporized form tungsten twin cells and the ions were produced by electron impact ionization at the energy 25 eV. The installation used was calibrated using the Au partial vapor pressure as the standard recommended by IUPAC [12]. Samples 3e5 mg in weight were put in one compartment of the cell and standard substance (pure oxide) in the other. Ion currents for a given mass to charge ratio referring to the sample Ii and to the standard Is were amplified by secondary electron multiplier. Partial vapor pressures pi were determined using the relation

pi ¼ ps ðIi Ti si gi Þ=ðIs Ts ss gs Þ;

(1)

where ps is the partial pressure of the standard, T is the absolute temperature, gi is the sensitivity coefficient of the secondary electron multiplier, and si is the ionization cross-section of the measured vapor species usually taken as the sum of the corresponding atomic cross-sections. Indices i and s refer to the sample and the standard, respectively. Analysis of the mass spectra indicated that at the temperature 2500 K the main vapor species over pure Gd2O3 and Y2O3 as well as over the samples of the Gd2O3-Y2O3-HfO2 system were GdO, YO, and O. Vapor species containing hafnium were not detected at this temperature above the samples of the Gd2O3-Y2O3-HfO2 system with the sensitivity of the mass spectrometer used being 106 mbar. Oxygen partial pressure could not be measured directly due to the high background level of the corresponding ion current produced by the residual gas in mass spectrometer. Thus, vaporization of the ternary system in the considered temperature range proceeds according to the dissociation reaction (2):

Table 1 Composition of the samples in the Gd2O3-Y2O3-HfO2 system established by the X-ray fluorescence analysis, Gd2O3, Y2O3, and HfO2 partial pressures (pi), activities (ai), and excess Gibbs energy (DGE) in the Gd2O3-Y2O3-HfO2 system at 2500 K according to Ref. [2]. pi, Pa

-DGE,

N

Content of oxides, mol.% Gd2O3

Y2O3

HfO2

Gd2O3

Y2O3

Gd2O3

Y2O3

1 2 3 4 5 6 8 9 10 11 12 13 14 15 16 18 19 20 21 22

49.9 44.7 33.5 24.8 16.6 7.5 26.9 20.1 13.2 40.0 30.1 19.7 47.3 34.9 23.3 57.2 43.1 28.4 12.9 50.0

0.3 5.2 16.7 24.9 33.7 42.3 13.3 20.1 26.6 20.0 30.1 40.5 23.4 34.8 46.7 28.5 43.6 57.9 73.8 50.2

49.9 49.5 49.5 49.3 49.9 49.2 59.1 59.8 59.0 40.0 39.6 39.6 29.5 29.4 29.9 13.7 14.0 13.9 13.9 e

1.6  100 7.0  101 5.5  101 3.3  101 2.2  101 1.5  101 2.9  101 3.2  101 9.6  102 4.3  101 4.1  101 2.9  101 4.3  101 3.6  101 2.1  101 5.3  101 5.2  101 3.5  101 2.2  101 6.8  101

e 8.8  103 4.8  102 4.8  102 8.7  102 1.6  101 2.2  102 6.0  102 3.2  102 3.9  102 8.1  102 7.6  102 3.0  102 5.3  102 6.1  102 4.1  101 8.5  101 9.7  102 1.7  102 1.2  102

1.3  101 1.6  101 5.5  102 3.8  102 2.4  102 1.1  102 9.0  102 3.2  102 1.6  102 2.0  101 8.2  102 3.9  102 1.7  101 1.5  101 4.6  102 3.5  101 3.1  101 9.7  102 1.1  102 2.5  101

e 1.7  103 3.8  102 4.2  102 5.3  102 1.4  101 4.5  103 2.2  102 3.8  102 6.6  103 4.2  102 7.0  102 2.2  102 4.8  102 1.2  101 2.8  102 1.2  101 3.6  101 1.9  101 3.3  10-

ai HfO2

kJ/mol

3.8  101 3.6  101 4.0  101

20.6 22.2 21.9

5.1  101 5.1  101 5.3  101 2.7  101 2.2  101 2.6  101 1.5  101 1.1  101 1.2  101 1.4  102 3.9  103 2.0  103

17.2 18.9 17.9 23.3 25.1 24.9 25.6 26.5 26.9 26.0 24.9 24.2

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Me2 O3 ðs; 1Þ ¼ 2MeOðgÞ þ OðgÞ;

(2)

where Me is Gd or Y. At first, pure substances were examined. Partial pressure p (GdO) of GdO over pure Gd2O3 was determined by Equation (1), where gold was taken as the vapor pressure standard. Partial pressure of oxygen p (O) over individual Gd2O3 were calculated using the formula

p ðOÞ ¼

sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi MðOÞ ; MðGdOÞ

1 2 p ðGdOÞ

=



(3)

where M(A) is the molecular weight of the vapor species A. Partial pressure of YO over individual Y2O3 was taken from the literature data [13] and the corresponding oxygen pressure was calculated by the equation similar to (3). The partial pressures of GdO and YO over the Gd2O3-Y2O3-HfO2 system were determined using Equation (1), while the oxygen partial pressure p(O) was calculated from the partial pressures of gadolinium oxide p(GdO) and yttrium oxide p(YO), according to Equation (4):

sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 MðOÞ 1 MðOÞ þ pðYOÞ : pðOÞ ¼ pðGdOÞ 2 MðGdOÞ 2 MðYOÞ

(4)

The Gd2O3 activities in the samples of the ternary system were determined from the comparison of the Gd2O3 vaporization equilibria in the samples, reaction (2), and in pure Gd2O3 at the same temperature. Since the equilibrium constant depends on temperature only, the following relation was deduced:

.  aðGd2 O3 Þ ¼ pðGdOÞ2 $pðOÞ p+ ðGdOÞ2 $p+ ðOÞ :

(5)

The results are listed in Table 1. The data on the phase diagram of the Gd2O3-Y2O3-HfO2 system given in Ref. [14] leads to conclusion that most of the measured values of the Gd2O3 and Y2O3 activities belong to the field of solid solutions so that the activity of the HfO2 component could be evaluated by Gibbs-Duhem integration. The obtained values of HfO2 activity and the resulting DGE as a function of composition are also given in Table 1.

3. Theory Barker's authentic derivation of the modeling approach was presented in Ref. [7]. It can be summarized as follows. The structural units A, B, … corresponding to the components of the system and occupying rA, rB, … sites of the lattice with the coordination number z are considered to have cA, cB, … contact points, respectively, in accordance with the number of neighboring lattice sites of each of them. The chosen parameters must satisfy the relation

cA ¼ rA z  2rA þ 2:

(6)

Making relevant approximations the grand partition function is constructed from which the quasi-chemical equations are then deduced. Finally, introducing auxiliary variables XAm a system of n non-linear equations may be written

X Am

X

hmv AB X Bv ¼ Q Am xA =2;

(7)

where xA is the mole fraction of the A component, QAm is the number of m-type contact points of the A component, hAB mn is the energy parameter of the model that are related to free energy of exchange UAB mn of the m- and n-type contact points of the A and B structural

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AB units as hAB mn ¼ exp(-Umn /kT). The number of the unknowns XAm is equal to the number of different types of contact points and hence to the number of equations in system (7). For a particular composition and a selected set of energy parameters hAB mn system (7) may be solved and the values of XAm found. Substitution of these values into Equation (8)

Nmv AB ¼ 2Xm A X Bv hmv AB N

h

i

h i

mA s vB ;

(8)

where NAB mn is the number of contact points of m- and n-types of the particles A and B, respectively and N is the total number of the particles in the system, permits calculation of the relative numbers of chemical bonds (pairs of contacts) of different type. The energy parameters hAB mn may be determined by the best-fit procedure using the appropriate experimental data. The values of the excess chemical potential of the components (DmE(x)) in the Barker's theory are

DmE A ¼ RT

hX

.  i  . X xA þ rA ðz=2  1ÞIn Q Am In X Am X Ap ri xi =rA ; m (9)

where XAp m is the solution of system (7) for pure component A. The optimal set of energy parameters may be chosen by multiple substitutions of their trial values into Equation (9) and comparison of the resulting DmE(x) with the corresponding experimental dependencies. 4. Results and discussion The experimental method used in Ref. [2] imposes rather strict limitations on the range of measurable pressures and hence on the available experimental temperature range characteristic for every substance. Large difference in the volatilities of components of the Gd2O3-Y2O3-HfO2 system leads to the diversity in the temperatures of the activity measurements, which had to be chosen individually for each group of samples. For the theoretical treatment all experimental data referring to chemical potential of components and excess Gibbs energies were translated to an average uniform temperature 2500 K. This operation can be substantiated by the fact that average accuracy of the vapor pressure measurements by the method under consideration is usually ±30% and that the lattice model is only and approximate representation of a real system. For this reason the modeling of the thermodynamic properties of the Gd2O3-Y2O3-HfO2 system was performed in the frame of the assumption that in the temperature range 2300e2800 K the activities of components are temperature-independent. The model for optimization can be described as follows. The samples of the Gd2O3-Y2O3-HfO2 system are represented as ensembles of the [Gd2O3], [Y2O3], and [HfO2] particles (structural units) distributed over the lattice with the coordination number z ¼ 3, each structural unit occupying two sites of the lattice and having two oxygen-type contact points and two element-type contact points. The energies hmn of the pair exchange reactions of the oxygen-oxygen and element-element contacts were assigned zero values. Thus, there are nine adjustable energy parameters characterizing the following pairs of contact points: Gd-O[Gd], Y-O [Y], Hf-O[Hf], Gd-O[Y], Gd-O[Hf], Y-O[Gd], Y-O[Hf], Hf-O[Gd], Hf-O [Y]. This set of parameters hmn with arbitrary values is put into the system of six non-linear Equation (7) with a composition {x(Gd2O3), x(Y2O3), x(HfO2)} on the right hand side corresponding to one of the samples in Table 1. Its solution {Xi} substituted into Equation (9) provides the values of the excess chemical potentials and excess P Gibbs function DGE ¼ xA DmEA at this composition. By repeating this procedure for all samples, comparing the obtained DGE values

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A.L. Shilov et al. / Journal of Alloys and Compounds 791 (2019) 1207e1212 Table 3 The energy parameters hmn obtained in the DGE best-fit calculations based on the ternary Gd2O3-Y2O3-HfO2 system experimental data and binary Gd2O3-Y2O3, Gd2O3HfO2, and Y2O3-HfO2 systems data.

Fig. 1. Comparison of excess Gibbs energy, kJ/mol, as function of concentration interpolated using experimental data for the ternary Gd2O3-Y2O3-HfO2 system [2], solid lines, and for corresponding binary systems Gd2O3-Y2O3 [8], Gd2O3-HfO2 [9], and Y2O3-HfO2 [10], dashed lines.

with the corresponding experimental ones listed in Table 1, and running through all possible sets of energy parameters the best-fit DGE function can be found. The results are shown in Fig. 1 in the form of constant DGE contours. The values of the energy parameters hmn obtained as a result of the interpolation procedure are put in Table 3. Since the exactness of interpolation for the samples with high content of HfO2 is rather low it was interesting to compare the obtained DGE functions with the ones derived from the data for the binary Gd2O3-Y2O3 [8], Gd2O3-HfO2 [9], and Y2O3-HfO2 [10] systems found in the literature, Table 2. As it was mentioned above, for the correct comparison all the values were re-calculated to the temperature 2500 K. The same procedure and program code could be used for the interpolation of the binary systems since the mole fraction of the third component could be set at a value approaching zero. The energies of the Gd-O[Gd], Y-O[Y], Hf-O[Hf] contact pairs (bonds) were taken the same as in the ternary Gd2O3-Y2O3-HfO2 system, the energies of Gd-O[Y], Gd-O[Hf], Y-O[Gd], Y-O[Hf], Hf-O[Gd], Hf-O[Y] contacts were determined in the best-fit procedure. The resulting DGE functions for the corresponding binary systems are shown in Fig. 2 and the DGE function for the ternary Gd2O3-Y2O3-HfO2 system

Type of bond

Bond energy hmn, kJ/mol Ternary system data

Binary systems data

Gd-O[Gd] Gd-O[Y] Gd-O[Hf] Y-O[Gd] Hf-O[Gd] Y-O[Y] Y-O[Hf] Hf-O[Y] Hf-O[Hf]

178.6 203.9 191.0 198.2 160.5 178.5 193.3 159.8 145.3

178.6 205.9 207.0 204.2 170.0 178.5 207.2 160.0 145.3

calculated with the hmn energy parameters determined for the binary subsystems is plotted in Fig. 1. Thus within the frame of the employed method the modeling was performed using two considerably independent approaches. Using the first approach the modeling was based on the experimental results obtained singly for the ternary Gd2O3-Y2O3-HfO2 system. In the frame of the second approach in which the same model was used the experimental data obtained in the studies of Kablov et al. [8], Sevastyanov et al. [9], and Belov and Semenov [10] for the corresponding binary systems were employed. The obtained thermodynamic functions are superimposed in Fig. 1. It demonstrates that Barker's model provides an easy way of estimation of thermodynamic functions of a ternary system using the experimental data obtained for the corresponding binary systems. To a certain degree the discrepancy of the two sets of curves in Fig. 1 can be attributed to the possible difference in the methods of samples synthesis and the experimental apparatus used by the authors of the studies [2,8e10]. In both cases the excess Gibbs energy function demonstrates negative deviation from ideal behavior. It correlates well with the values of the energy parameters given in Table 3: the energies of the “bonds” characterizing pure components Gd-O[Gd], Y-O[Y], and Hf-O[Hf] are lower than the energies of the mixes “bonds” Gd-O[Y], Gd-O[Hf], Y-O[Gd], Y-O [Hf], Hf-O[Gd], and Hf-O[Y]. The stronger negative deviations of DGE for the results obtained using the second approach are also in full agreement with the data in Table 3, since all the hmn values based on the binary systems data are more negative that those based on the ternary system data. As it follows from Equation (8) for the relative numbers of bonds in the system, the modeling based on the generalized theory of

Table 2 Experimental excess Gibbs energies (DGE) as functions of composition xi for the binary Gd2O3-Y2O3 [9], Gd2O3-HfO2 [10], and Y2O3-HfO2 [11] systems measured at the temperatures 2630, 2610, and 2843 K, respectively, and converted to 2500 K. Gd2O3-Y2O3

Gd2O3-HfO2

Y2O3-HfO2

x(Gd2O3), mole fract.

eDGE, kJ/mol

x(Gd2O3), mole fract.

eDGE, kJ/mol

x(Y2O3), mole fract.

eDGE, kJ/mol

0.12 0.16 0.21 0.27 0.34 0.35 0.37 0.41 0.44 0.47 0.49

15.4 19.4 24.2 27.9 31.1 31.5 31.9 32.5 32.7 32.2 31.8

0.006 0.016 0.026 0.047 0.066 0.088 0.131 0.172 0.196 0.226 0.247 0.27

0.9 2.0 3.1 8.2 11.5 12.2 19.5 18.4 25.4 24.2 28.0 35.5

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

0 13.3 19.3 24.7 24.8 24.4 21.4 17.8 12.7 7.4 0

A.L. Shilov et al. / Journal of Alloys and Compounds 791 (2019) 1207e1212

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Fig. 3. Relative number Ni-O-k/N of bonds between the atoms of i and k types through bridging oxygen atoms in the samples of the Gd2O3-Y2O3-HfO2 system at 2500 K: a e along the x(HfO2) ¼ 0.5 node, b e along the x(HfO2) ¼ 0.67 node. Numbers indicate the following bonds: 1 e Gd-O-Hf, 2 e Y-O-Hf, 3 e Gd -O- Gd, 4 e Y-O-Y, 5 e Hf-O-Hf, 6 e Gd -O-Y.

Fig. 2. Experimental points given in Refs. [8e10] and the corresponding DGE functions optimized using the accepted model for the systems: a - Gd2O3-Y2O3, b - Gd2O3-HfO2, and c - Y2O3-HfO2.

indicates that at the increase of HfO2 in the system the relative number of the Gd-O-Y and Y-O-Y bonds decreases while the relative number of the Hf-O-Hf bonds increases. It should be mentioned that earlier the correctness of the calculations of the relative number of bonds in oxide systems was confirmed for the glasses and melts in the B2O3-SiO2 system when the results were obtained using also Infrared Spectroscopy method [15].

5. Conclusions associated solutions provides a unique possibility to correlate the thermodynamic behavior of a system and the corresponding relative number of bonds when the second coordination sphere is taken into consideration. Composition dependence of the number of Gd-O-Hf, Y-O-Hf, Gd -O- Gd, Y-O-Y, Hf-O-Hf, and Gd -O-Y bonds for the x(HfO2) ¼ 0.5 and x(HfO2) ¼ 0.67 nodes shown in Fig. 3, is an example of such computations. Comparison of Fig. 3 a and b

Optimization of the thermodynamic functions of the Gd2O3Y2O3-HfO2 system based on the generalized theory of associated solutions was performed using two independent approaches, in which the excess Gibbs energy functions were interpolated on the basis of: 1) experimental data for the ternary Gd2O3-Y2O3-HfO2 system and 2) independent data for the Gd2O3-Y2O3, Gd2O3-HfO2, and Y2O3-HfO2 binary systems. Compatibility of the obtained

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thermodynamic functions with the corresponding energy parameters of the model proves that this theory can be applied for the estimation of the excess Gibbs energy functions of ternary systems using the experimental data on the corresponding binary subsystems. The results of the thermodynamic modeling of solid solutions based on the generalized theory of associated solutions at the temperatures 2500 K prove that it can be applied not only to glasses and glass-forming melts but to high refractory oxides as well. Acknowledgements This work was supported by the Russian Foundation for Basic Research Project No. 16-03-00940.

[5]

[6]

[7] [8]

[9]

[10]

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