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The maximum solid solubility of the transition metals in palladium
International Journal of Hydrogen Energy 27 (2002) 329–332
www.elsevier.com/locate/ijhydene
The maximum solid solubility of the transition metals in palladium S.S. Fang, G.W. Lin, J.L. Zhang, Z.Q. Zhou ∗ Institute of Hydrogen Storage Materials, Shanghai University, Shanghai 200072, China
Abstract The maximum solid solubility limit (Cmax ) of transition metals dissolved in palladium can be described as an equation in semi-empirical theories with parameters such as electronegativity di1erence, atomic diameter and covalent electrons. It has been found that the electronegativity di1erence and the covalent electron number mainly a1ected the Cmax of transition metals in palladium. The atomic size parameter had the smallest e1ect on the Cmax . ? 2002 International Association for Hydrogen Energy. Published by Elsevier Science Ltd. All rights reserved. Keywords: Palladium alloys; Solid solubility; Electronegativity di1erence; Atomic size parameters; Covalent electron
1. Introduction In the future, hydrogen will replace fossil fuels in both electrical power generation and automotive industry. However, hydrogen carries certain explosive dangers when it reaches an approximate 4% concentration in air or oxygen. In order to avoid these dangers, it is necessary to use hydrogen sensor to monitor and measure the concentration of hydrogen. Palladium alloy (Pd–Ag, Pd–Ni, Pd–Cr, etc.) is one of the hydrogen-sensing elements. The sensor relies on the reversible solubility of hydrogen in palladium alloy and, of course, on the composition of the alloy. In order to develop high performance palladium alloy for hydrogen sensor, it is neccessary to study the maximum solid solubility limit (Cmax ) of the solute in solvent (palladium). The Cmax of the solute in palladium is one of the theories of alloy phase formation to be studied. Many theories of alloy solubility rely on parameters in semi-empirical theories, such as valence, size or electronegativity for their predictions [1– 4]. In our previous work, Zhou et al. proposed a mathematical model that can be successfully applied
∗ Corresponding author. Tel.: +86-21-5638-2405. E-mail address: [email protected] (Z.Q. Zhou).
to predict the Cmax of transition metals in titanium by the three bond parameters, i.e., electronegativity di1erence and atomic size parameter and electron concentration [5]. The objective of this paper is to study if the model can be applied to the Cmax of transition metals in palladium and to End the main factor that a1ects the Cmax .
2. Results and discussion The Cmax data in this paper was read from equilibrium phase diagram [6]. For an eutectic phase diagram, the solid solubility of solute in solvent at the eutectic temperature was regarded as the Cmax of the alloy system. For a peritectic phase diagram, the solid solubility of solute in solvent at the peritectic temperature was not to be regarded as the Cmax of the alloy system; among them only the one that had the maximum solid solubility of solute in solvent was regarded as the Cmax of the alloy system. For a phase diagram showing complete liquid and solid solubility 100% was regarded as the Cmax of the alloy system. The data of Pauling electronegativity, atomic diameter and covalent electrons of solute and solvent was read from the Ref. [7].
0360-3199/02/$ 20.00 ? 2002 International Association for Hydrogen Energy. Published by Elsevier Science Ltd. All rights reserved. PII: S 0 3 6 0 - 3 1 9 9 ( 0 1 ) 0 0 1 1 2 - 4
330
S.S. Fang et al. / International Journal of Hydrogen Energy 27 (2002) 329–332
Table 1 The data of electronegativity di1erence HX , atomic size parameter and covalent electrons n [6] Solvent
(HX )2
2 × 10−2
n2=3
Cmax
ln Cmax
ln Creg
Cr Mn Fe Co Ni Y Zr Nb Mo Ru Rh La Hf Ta Os Ir Pt
0.36 0.49 0.16 0.16 0.16 1 0.64 0.36 0.16 0 0 1.21 0.81 0.49 0 0 0
0.480 0.192 0.533 0.645 0.901 10.550 2.819 0.533 0.048 0.048 0.048 13.323 2.462 0.432 0.048 0.021 0.001
4.003 4.359 4.000 4.327 4.642 4.356 4.301 3.945 4.080 4.534 4.327 4.356 4.214 4.295 4.000 4.327 4.642
50 30 100 100 100 13 18 43 44 100 100 13 22.5 22.0 100 100 100
3.905 3.401 4.605 4.605 4.605 2.565 2.890 3.768 3.784 4.605 4.605 2.565 3.114 3.091 4.605 4.605 4.605
3.711 3.651 4.306 4.452 4.602 2.699 3.081 3.562 3.994 4.526 4.668 2.496 2.811 3.363 4.526 4.667 4.806
2.1. Factors controlling the Cmax of transition metals in palladium
ln Cmax = b0 + b1 (HX )2 + b2 2 + b3 n2=3 ;
(1)
where b1 , b2 and b3 are proportional coeLcients; b0 is a constant, HX is the electronegativity di1erence, is the atomic size parameter and n is the average of s and d electron numbers in the outermost shell of solvent and solute elements. HX and are given by HX = X0 − X;
(2)
= 1 − D=D0 :
(3)
Where X0 and X are the Pauling electronegativity of palladium (solvent) and transition metals (solute), respectively. D and D0 are the atomic diameter of transition metals and palladium, respectively. The coeLcient Cmax (Eq. (1)) of transition metals dissolved in palladium can be calculated by regression analysis from the data in Table 1 [6] according to our previous work [5]: ln Cmax = 0:65579 − 1:50294(HX )2 +409:32 + 0:89407n2=3 :
(4)
The correlation coeLcient of Eq. (4) is 0.97. Fig. 1 is a comparison between the calculated and experimental values
Pt Rh Ni Ir Ru Co Fe
4.5
Regression value
According to our previous work [5], three bond parameters were introduced in the equation of solid solubility by quasi-chemical treatment, so that the equation of Cmax should be a function of these three factors, i.e., electronegativity, atomic size parameter and covalent electrons. The equation of Cmax can be written as follows:
5.0
Mo
4.0
Cr
Mn Nb
3.5
Ta Zr
3.0
Y 2.5
Hf
La
2.5
3.0
3.5
4.0
4.5
5.0
Experimental value Fig. 1. A comparison between the calculated and experimental values of ln Cmax for Pd–B alloys.
of ln Cmax for Pd–B alloys where B represents the solute elements in the alloys. It can be seen from Fig. 1 and the correlative coeLcient that the values ln Cmax calculated from Eq. (4) are in good agreement with the experimental data. It is indicated that the mathematical model proposed in our previous work for the Cmax of Ti–B binary alloy can also be applied to the Cmax of Pd–B binary alloy. It is supposed from the results that the interaction among the three parameters is very little and the e1ect on the Cmax of B in Pd can be ignored in the mathematical model.
S.S. Fang et al. / International Journal of Hydrogen Energy 27 (2002) 329–332 Table 2 The statistic Fi of items in Eq. (4) i
Reg. coeLcient b(i)
Statistic Fi
b0 b1 b2 b3
0.65579 −1:50294 4.093 0.89407
0.0 10.98 2.24 8.37
In the third period, the Cmax of transition metals dissolved in palladium increases at Erst and then decreases; whereas in the fourth and Efth period, the Cmax of transition metals dissolved in palladium increases with the number (s + d). It can be deduced from Fig. 2 that the Cmax of transition metals dissolved in palladium has some relationship with the electronic structure, but it is diLcult to explain the rule in the Cmax change with the number (s + d) only from the electron theory.
According to the theory of regression analysis, the greater the value of statistic Fi of one item in a regression equation, the greater the e1ect of the item is. It is indicated by regression analysis from the data in Table 2 that the values of statistic Fi of the items (HX )2 , 2 and n2=3 are 10.98, 2.24 and 8.37, respectively. Therefore it can be deduced from Table 2 that the factor of electronegativity (HX ) has the largest e1ect on the Cmax of the Pd–B binary alloy, the factor of covalent electrons (n2=3 ) has the larger e1ect and atomic size parameter (2 ) has the least e1ect because its statistic Fi value is only 2.24, which is less than that of (HX ) and (n2=3 ) (10.98 and 8.37, respectively).
2.3. Cmax and Darken–Gurry criterion Fig. 3 shows the Darken–Gurry ellipse, the minor axis of which is D0 ± 15% (abscissa) and the major axis is X ± 0:4 (ordinate). The center of the ellipse is at (2.74, 2.2) that stands for the atomic diameter and electronegativity of palladium, respectively. From Fig. 3 and Table 1, it can seen that Cmax of some transition metals (Os, Ir, Pt, Ru, Rh etc.) in the ellipse in palladium is larger than 50%, which is in agreement with the Darken–Gurry method, but Cmax of other transition metals (Tc and Re) in the ellipse in palladium is not larger than 50%, which is not in agreement with the Darken–Gurry method. Contrariwise Cmax of some transition metals (Fe and Ni) out of the ellipse in palladium is larger than 50%, which is also not in agreement with the Darken–Gurry method, but Cmax of other transition metals out of the ellipse in palladium is less than 50%, which is in agreement with the Darken–Gurry method. According to the mathematical model mentioned above, the factor of covalent electrons has the larger e1ect on the Cmax of the Pd–B alloy, but the Darken–Gurry method did not consider the factor. That is the reason why the rules of variable Cmax of some transition metals (not all solute) in palladium can be judged according to the Darken–Gurry method.
2.2. Cmax and the covalent electron number (s + d) of solute element Hume–Rothery’s rules stated that in some alloy systems electron concentration had the biggest e1ect on the Cmax of solute in solvent [3]. Fig. shows the e1ect of the covalent electron number (s + d) of transition metal solute element on the Cmax of transition metals in palladium. It can be seen from Fig. 2 that the Cmax of transition metals in palladium is equal to 100% as the number (s + d) is greater than 7; but the change in the Cmax for element in di1erent period is di1erent as the number (s + d) increases from 3 to 7. 100 50 Sc
Prediction Value of C max Cr Mn V Ti
Fe
Co
Ni
Tc
Ru
Rh
Pd
Re
Os
Ir
Pt
7
8
Cmax,%
100 50
Nb Y
Mo
Zr
100 W
50 Hf La
0 2
3
4
Ta 5
331
6
9
10
11
s+d
Fig. 2. The e1ects of covalent electrons of transition solutes on the Cmax in palladium-based alloys.
332
S.S. Fang et al. / International Journal of Hydrogen Energy 27 (2002) 329–332
Table 3 The prediction value of Cmax and their atomic parameters Solvent
(HX )2
2 × 10−2
n2=3
Prediction values ln Creg
Cmax
Sc Ti V Tc W Re
0.81 0.49 0.36 0.09 0.25 0.09
2.818 0.480 0.005 0.005 0.085 0.000
3.483 4.262 3.694 4.165 4.362 4.492
2.565 2.996 4.061 3.912 3.099 2.773
14.4 24.8 34.5 69.7 47.5 69.7
3. Conclusions
3.0
The mathematical model proposed by Zhou et al. [5] in our previous work for the Cmax of Ti–B binary alloy can also be applied to the Cmax of Pd–B binary alloy. It can be represented as a function of electronegativity di1erence, atomic size parameter and covalent electrons:
2.8 2.6
Electronegativity
2.4 2.2
Ir
ln Cmax = b0 + b1 (HX )2 + b2 2 + b3 n2=3 :
Pd Pt
The factor of electronegativity (HX ) has the largest effect on the Cmax of the Pd–B binary alloy, the factor of covalent electrons (n2=3 ) has the larger e1ect and atomic size parameter (2 ) has the least.
Ru,Rh,Os 2.0 Tc Re Ni Mo Co Fe W Nb Cr V
1.8 1.6
Mn Ta Ti
References
Zr
1.4 Hf
Sc Y
1.2
La 1.0 2.0
2.2
2.4
2.6
2.8
3.0
3.2
3.4
3.6
3.8
4.0
Atomic diameter
Fig. 3. The Darken–Gurry map with an ellipse about the solvent palladium. The vertical line is the tangent to the ellipse at the termini of the minor axis (±15% of palladium’s diameter).
2.4. Predicting of unknown Cmax To our knowledge, it is not very clear that the maximum solid solution limits of some transition metals such as Sc, Ti, V, Tc, Re and W that solved in palladium, whose solid solution lines in the phase diagram are dashed. Eq. (4) can be applied to predict the unknown Cmax of transition metals dissolved in palladium. Table 3 shows the results of Cmax according to Eq. (4).
[1] Bennett LH. Parameters in semi-empirical theories of alloy phase formation. In: Bennett LH, editor. Theory of alloy phase formation. The Metallurgical Society of AIME, New York, USA, 1980. p. 390. [2] Miedema AM, de Chatel PF. A semi-empirical approach to the heat of formation problem. In: Bennett LH, editor. Theory of alloy phase formation. The Metallurgical Society of AIME, New York, USA, 1980. p. 344. [3] Bennett LH, editor. Theory of alloy phase formation. A Publication of The Metallurgical Society of AIME, New York, USA, 1980. p. 1–39. [4] Darken LS, Gurry RW. Physical chemistry of metals. New York: McGraw-Hill Co., 1953. p. 74 –92. [5] Zhou ZQ, Zhang JL, Fang SS, et al. Proceedings of the 97’ Materials Symposiums in China, Sponsored by Chinese Materials Research Society (C-MRS), Shanghai, 1997. p. 37– 40 [in Chinese]. [6] Massalski TB. Binary alloy phase diagrams. Metals Park, OH 44073; ASM, October 1986. [7] Ji-mei Xiao, Energetics of Alloys. Shanghai Publishing Company of Science and Technology, Shanghai, China, 1985. p. 296 [in Chinese].
www.elsevier.com/locate/ijhydene
The maximum solid solubility of the transition metals in palladium S.S. Fang, G.W. Lin, J.L. Zhang, Z.Q. Zhou ∗ Institute of Hydrogen Storage Materials, Shanghai University, Shanghai 200072, China
Abstract The maximum solid solubility limit (Cmax ) of transition metals dissolved in palladium can be described as an equation in semi-empirical theories with parameters such as electronegativity di1erence, atomic diameter and covalent electrons. It has been found that the electronegativity di1erence and the covalent electron number mainly a1ected the Cmax of transition metals in palladium. The atomic size parameter had the smallest e1ect on the Cmax . ? 2002 International Association for Hydrogen Energy. Published by Elsevier Science Ltd. All rights reserved. Keywords: Palladium alloys; Solid solubility; Electronegativity di1erence; Atomic size parameters; Covalent electron
1. Introduction In the future, hydrogen will replace fossil fuels in both electrical power generation and automotive industry. However, hydrogen carries certain explosive dangers when it reaches an approximate 4% concentration in air or oxygen. In order to avoid these dangers, it is necessary to use hydrogen sensor to monitor and measure the concentration of hydrogen. Palladium alloy (Pd–Ag, Pd–Ni, Pd–Cr, etc.) is one of the hydrogen-sensing elements. The sensor relies on the reversible solubility of hydrogen in palladium alloy and, of course, on the composition of the alloy. In order to develop high performance palladium alloy for hydrogen sensor, it is neccessary to study the maximum solid solubility limit (Cmax ) of the solute in solvent (palladium). The Cmax of the solute in palladium is one of the theories of alloy phase formation to be studied. Many theories of alloy solubility rely on parameters in semi-empirical theories, such as valence, size or electronegativity for their predictions [1– 4]. In our previous work, Zhou et al. proposed a mathematical model that can be successfully applied
∗ Corresponding author. Tel.: +86-21-5638-2405. E-mail address: [email protected] (Z.Q. Zhou).
to predict the Cmax of transition metals in titanium by the three bond parameters, i.e., electronegativity di1erence and atomic size parameter and electron concentration [5]. The objective of this paper is to study if the model can be applied to the Cmax of transition metals in palladium and to End the main factor that a1ects the Cmax .
2. Results and discussion The Cmax data in this paper was read from equilibrium phase diagram [6]. For an eutectic phase diagram, the solid solubility of solute in solvent at the eutectic temperature was regarded as the Cmax of the alloy system. For a peritectic phase diagram, the solid solubility of solute in solvent at the peritectic temperature was not to be regarded as the Cmax of the alloy system; among them only the one that had the maximum solid solubility of solute in solvent was regarded as the Cmax of the alloy system. For a phase diagram showing complete liquid and solid solubility 100% was regarded as the Cmax of the alloy system. The data of Pauling electronegativity, atomic diameter and covalent electrons of solute and solvent was read from the Ref. [7].
0360-3199/02/$ 20.00 ? 2002 International Association for Hydrogen Energy. Published by Elsevier Science Ltd. All rights reserved. PII: S 0 3 6 0 - 3 1 9 9 ( 0 1 ) 0 0 1 1 2 - 4
330
S.S. Fang et al. / International Journal of Hydrogen Energy 27 (2002) 329–332
Table 1 The data of electronegativity di1erence HX , atomic size parameter and covalent electrons n [6] Solvent
(HX )2
2 × 10−2
n2=3
Cmax
ln Cmax
ln Creg
Cr Mn Fe Co Ni Y Zr Nb Mo Ru Rh La Hf Ta Os Ir Pt
0.36 0.49 0.16 0.16 0.16 1 0.64 0.36 0.16 0 0 1.21 0.81 0.49 0 0 0
0.480 0.192 0.533 0.645 0.901 10.550 2.819 0.533 0.048 0.048 0.048 13.323 2.462 0.432 0.048 0.021 0.001
4.003 4.359 4.000 4.327 4.642 4.356 4.301 3.945 4.080 4.534 4.327 4.356 4.214 4.295 4.000 4.327 4.642
50 30 100 100 100 13 18 43 44 100 100 13 22.5 22.0 100 100 100
3.905 3.401 4.605 4.605 4.605 2.565 2.890 3.768 3.784 4.605 4.605 2.565 3.114 3.091 4.605 4.605 4.605
3.711 3.651 4.306 4.452 4.602 2.699 3.081 3.562 3.994 4.526 4.668 2.496 2.811 3.363 4.526 4.667 4.806
2.1. Factors controlling the Cmax of transition metals in palladium
ln Cmax = b0 + b1 (HX )2 + b2 2 + b3 n2=3 ;
(1)
where b1 , b2 and b3 are proportional coeLcients; b0 is a constant, HX is the electronegativity di1erence, is the atomic size parameter and n is the average of s and d electron numbers in the outermost shell of solvent and solute elements. HX and are given by HX = X0 − X;
(2)
= 1 − D=D0 :
(3)
Where X0 and X are the Pauling electronegativity of palladium (solvent) and transition metals (solute), respectively. D and D0 are the atomic diameter of transition metals and palladium, respectively. The coeLcient Cmax (Eq. (1)) of transition metals dissolved in palladium can be calculated by regression analysis from the data in Table 1 [6] according to our previous work [5]: ln Cmax = 0:65579 − 1:50294(HX )2 +409:32 + 0:89407n2=3 :
(4)
The correlation coeLcient of Eq. (4) is 0.97. Fig. 1 is a comparison between the calculated and experimental values
Pt Rh Ni Ir Ru Co Fe
4.5
Regression value
According to our previous work [5], three bond parameters were introduced in the equation of solid solubility by quasi-chemical treatment, so that the equation of Cmax should be a function of these three factors, i.e., electronegativity, atomic size parameter and covalent electrons. The equation of Cmax can be written as follows:
5.0
Mo
4.0
Cr
Mn Nb
3.5
Ta Zr
3.0
Y 2.5
Hf
La
2.5
3.0
3.5
4.0
4.5
5.0
Experimental value Fig. 1. A comparison between the calculated and experimental values of ln Cmax for Pd–B alloys.
of ln Cmax for Pd–B alloys where B represents the solute elements in the alloys. It can be seen from Fig. 1 and the correlative coeLcient that the values ln Cmax calculated from Eq. (4) are in good agreement with the experimental data. It is indicated that the mathematical model proposed in our previous work for the Cmax of Ti–B binary alloy can also be applied to the Cmax of Pd–B binary alloy. It is supposed from the results that the interaction among the three parameters is very little and the e1ect on the Cmax of B in Pd can be ignored in the mathematical model.
S.S. Fang et al. / International Journal of Hydrogen Energy 27 (2002) 329–332 Table 2 The statistic Fi of items in Eq. (4) i
Reg. coeLcient b(i)
Statistic Fi
b0 b1 b2 b3
0.65579 −1:50294 4.093 0.89407
0.0 10.98 2.24 8.37
In the third period, the Cmax of transition metals dissolved in palladium increases at Erst and then decreases; whereas in the fourth and Efth period, the Cmax of transition metals dissolved in palladium increases with the number (s + d). It can be deduced from Fig. 2 that the Cmax of transition metals dissolved in palladium has some relationship with the electronic structure, but it is diLcult to explain the rule in the Cmax change with the number (s + d) only from the electron theory.
According to the theory of regression analysis, the greater the value of statistic Fi of one item in a regression equation, the greater the e1ect of the item is. It is indicated by regression analysis from the data in Table 2 that the values of statistic Fi of the items (HX )2 , 2 and n2=3 are 10.98, 2.24 and 8.37, respectively. Therefore it can be deduced from Table 2 that the factor of electronegativity (HX ) has the largest e1ect on the Cmax of the Pd–B binary alloy, the factor of covalent electrons (n2=3 ) has the larger e1ect and atomic size parameter (2 ) has the least e1ect because its statistic Fi value is only 2.24, which is less than that of (HX ) and (n2=3 ) (10.98 and 8.37, respectively).
2.3. Cmax and Darken–Gurry criterion Fig. 3 shows the Darken–Gurry ellipse, the minor axis of which is D0 ± 15% (abscissa) and the major axis is X ± 0:4 (ordinate). The center of the ellipse is at (2.74, 2.2) that stands for the atomic diameter and electronegativity of palladium, respectively. From Fig. 3 and Table 1, it can seen that Cmax of some transition metals (Os, Ir, Pt, Ru, Rh etc.) in the ellipse in palladium is larger than 50%, which is in agreement with the Darken–Gurry method, but Cmax of other transition metals (Tc and Re) in the ellipse in palladium is not larger than 50%, which is not in agreement with the Darken–Gurry method. Contrariwise Cmax of some transition metals (Fe and Ni) out of the ellipse in palladium is larger than 50%, which is also not in agreement with the Darken–Gurry method, but Cmax of other transition metals out of the ellipse in palladium is less than 50%, which is in agreement with the Darken–Gurry method. According to the mathematical model mentioned above, the factor of covalent electrons has the larger e1ect on the Cmax of the Pd–B alloy, but the Darken–Gurry method did not consider the factor. That is the reason why the rules of variable Cmax of some transition metals (not all solute) in palladium can be judged according to the Darken–Gurry method.
2.2. Cmax and the covalent electron number (s + d) of solute element Hume–Rothery’s rules stated that in some alloy systems electron concentration had the biggest e1ect on the Cmax of solute in solvent [3]. Fig. shows the e1ect of the covalent electron number (s + d) of transition metal solute element on the Cmax of transition metals in palladium. It can be seen from Fig. 2 that the Cmax of transition metals in palladium is equal to 100% as the number (s + d) is greater than 7; but the change in the Cmax for element in di1erent period is di1erent as the number (s + d) increases from 3 to 7. 100 50 Sc
Prediction Value of C max Cr Mn V Ti
Fe
Co
Ni
Tc
Ru
Rh
Pd
Re
Os
Ir
Pt
7
8
Cmax,%
100 50
Nb Y
Mo
Zr
100 W
50 Hf La
0 2
3
4
Ta 5
331
6
9
10
11
s+d
Fig. 2. The e1ects of covalent electrons of transition solutes on the Cmax in palladium-based alloys.
332
S.S. Fang et al. / International Journal of Hydrogen Energy 27 (2002) 329–332
Table 3 The prediction value of Cmax and their atomic parameters Solvent
(HX )2
2 × 10−2
n2=3
Prediction values ln Creg
Cmax
Sc Ti V Tc W Re
0.81 0.49 0.36 0.09 0.25 0.09
2.818 0.480 0.005 0.005 0.085 0.000
3.483 4.262 3.694 4.165 4.362 4.492
2.565 2.996 4.061 3.912 3.099 2.773
14.4 24.8 34.5 69.7 47.5 69.7
3. Conclusions
3.0
The mathematical model proposed by Zhou et al. [5] in our previous work for the Cmax of Ti–B binary alloy can also be applied to the Cmax of Pd–B binary alloy. It can be represented as a function of electronegativity di1erence, atomic size parameter and covalent electrons:
2.8 2.6
Electronegativity
2.4 2.2
Ir
ln Cmax = b0 + b1 (HX )2 + b2 2 + b3 n2=3 :
Pd Pt
The factor of electronegativity (HX ) has the largest effect on the Cmax of the Pd–B binary alloy, the factor of covalent electrons (n2=3 ) has the larger e1ect and atomic size parameter (2 ) has the least.
Ru,Rh,Os 2.0 Tc Re Ni Mo Co Fe W Nb Cr V
1.8 1.6
Mn Ta Ti
References
Zr
1.4 Hf
Sc Y
1.2
La 1.0 2.0
2.2
2.4
2.6
2.8
3.0
3.2
3.4
3.6
3.8
4.0
Atomic diameter
Fig. 3. The Darken–Gurry map with an ellipse about the solvent palladium. The vertical line is the tangent to the ellipse at the termini of the minor axis (±15% of palladium’s diameter).
2.4. Predicting of unknown Cmax To our knowledge, it is not very clear that the maximum solid solution limits of some transition metals such as Sc, Ti, V, Tc, Re and W that solved in palladium, whose solid solution lines in the phase diagram are dashed. Eq. (4) can be applied to predict the unknown Cmax of transition metals dissolved in palladium. Table 3 shows the results of Cmax according to Eq. (4).
[1] Bennett LH. Parameters in semi-empirical theories of alloy phase formation. In: Bennett LH, editor. Theory of alloy phase formation. The Metallurgical Society of AIME, New York, USA, 1980. p. 390. [2] Miedema AM, de Chatel PF. A semi-empirical approach to the heat of formation problem. In: Bennett LH, editor. Theory of alloy phase formation. The Metallurgical Society of AIME, New York, USA, 1980. p. 344. [3] Bennett LH, editor. Theory of alloy phase formation. A Publication of The Metallurgical Society of AIME, New York, USA, 1980. p. 1–39. [4] Darken LS, Gurry RW. Physical chemistry of metals. New York: McGraw-Hill Co., 1953. p. 74 –92. [5] Zhou ZQ, Zhang JL, Fang SS, et al. Proceedings of the 97’ Materials Symposiums in China, Sponsored by Chinese Materials Research Society (C-MRS), Shanghai, 1997. p. 37– 40 [in Chinese]. [6] Massalski TB. Binary alloy phase diagrams. Metals Park, OH 44073; ASM, October 1986. [7] Ji-mei Xiao, Energetics of Alloys. Shanghai Publishing Company of Science and Technology, Shanghai, China, 1985. p. 296 [in Chinese].