- Email: [email protected]
Evaluation of the viscosity for binary and ternary liquid alloys
Materials Science and Engineering A362 (2003) 223–227
Evaluation of the viscosity for binary and ternary liquid alloys S. Morioka Key Laboratory of Liquid Structure and Heredity of Materials, College of Materials Science and Engineering, Shandong University, Jinan, Shandong, China Received 18 April 2003; received in revised form 2 July 2003
Abstract The ‘gas-like’ model of viscosity for liquid alloys is examined in evaluation for the viscosity of binary and ternary liquid alloys of Ag, Au and Cu. The detail procedure to evaluate the viscosity of multi-component liquid alloys is given. The model predicts the overall experimental data of the viscosities for Ag–Au–Cu liquid alloys consistently with those for the component atomic liquid metals and the respective binary liquid alloys. © 2003 Elsevier B.V. All rights reserved. Keywords: Viscosity; Liquid metal; Alloy
1. Introduction The viscosity of liquid metals is of great importance in metallurgical processes as well as in the study for dynamics of atoms in the liquid metals [1,2]. High quality metals are mostly composed in forms of multi-component alloys, so that a predictable model to describe the viscosity for multi-component alloys is desired. The formula most frequently used in analysis for the viscosity of liquid metals is the Arrhenius type Evis η = A exp (1) RT where A is a constant, R the gas constant, T the absolute temperature, and Evis the activation energy of viscosity. The values of A and Evis are empirically determined as adjustable parameters by fitting the viscosity data. These parameters have been sought with the available viscosity data for mono-atomic metals [3]. The viscosity of liquid alloys is usually not the linear summation over component atomic viscosities, and hence a predictable model to describe the viscosity for multi-component alloys cannot be expected within this frame work. A predictable model to describe the viscosity of multi-component alloys, therefore, should be based on more fundamental physical quantities.
E-mail address: [email protected] (S. Morioka). 0921-5093/$ – see front matter © 2003 Elsevier B.V. All rights reserved. doi:10.1016/S0921-5093(03)00619-1
A more fundamental model to describe the viscosity may be the one based on the rate theory [4]: hNA Evis η= exp (2) V RT where h is Planck’s constant, NA the Avogadro’s number, and V the molar volume. As in the Arrhenius one, the values of Evis are empirically determined as adjustable parameters by fitting the viscosity data. The formula (2) is applicable only to describe the viscosity of mono-component liquids. For a liquid alloy, thus, one needs an additional consideration associated with the mixing of unlike atoms. Two kinds of models are proposed based on the rate theory to describe the viscosity for multi-component alloys: one derives a formula in terms of the superposition of component atomic viscosities with a correction due to the mixing of unlike atoms [5]. This correction term is related to a thermodynamic quantity which involves an arbitrary parameter. Besides, the model utilizes measured component atomic viscosities in evaluation for the viscosities of multi-component liquid alloys. Thus, the model is not based on the fundamental physical quantities. The other model regards the activation energy of an alloy as the sum of the activation energies of component atoms, the Gibbs energy of mixing and a kinetic correction associated with the mixing of unlike atoms [6]. The model does not involve an arbitrary parameter in this correction term, but it does involve some parameters in the activation energies of component atoms. These pa-
224
S. Morioka / Materials Science and Engineering A362 (2003) 223–227
rameters are not related to the known fundamental physical quantities. Recently, a model (gas-like model) to describe the viscosity for liquid metals was proposed [7], which is based on the gas-like kinetics with the viscous activation picture for liquid materials of Kittel [8]. The most important feature of this model is that it is capable to describe the viscosities consistently from mono-atomic liquid metals to multi-component liquid alloys with the parameters related to the known fundamental physical quantities. The model was applied to 11 cases of mono-atomic liquid metals, and described the overall experimental data of the viscosity in a wide range of temperature, predicting the mean free paths of an atom in accord with the experimental interatomic distances in the liquid metals. The model was extended to describe the viscosity for multi-component liquid alloys with the additional parameter associated with multi-component mixing. In this paper, we shall examine the gas-like model in the evaluation of the viscosity for binary and ternary liquid alloys in detail.
2. The model Let us briefly explain the gas-like model [7] where the viscosity ηi for a liquid metal of atom i is given as i E vis ηi = ηig exp (3) RT where ηig is the viscosity in the gas phase 2 Mi RT i ηg = 3NA σi π
(4)
i , M and σ are the activation energy, the mass and where Evis i i the collision cross-section of the atom i, respectively. The activation energy in our model is regarded as the energy for the atom to surmount the potential energy barrier provided by one or so of the nearest-neighbor atoms because of the short-range order [2] in the liquid, and thus the activation energy for the atom i is speculated as [7] i Evis =
i Hevap
Zi
(5)
i where Hevap is the enthalpy of evaporation of the atom i, and Zi is an adjustable parameter to the experimental data of the viscosity of the atom i, but can be considered as a quantity related to the coordination number of nearest-neighbor atoms in the liquid metal. The collision cross-section of a hard-sphere picture for the atom in the liquid phase is modified by the parameter κi as
σi = π(κi di )2 where di is the diameter of the atom i.
(6)
As for a multi-component liquid alloy, the viscosity η is assumed as Evis (7) η = ηg exp RT with 2 ηg = 3NA σ
MRT π
(8)
where we take a simple average for the masses and collision cross-sections of component atoms in the alloy as followings: M= xi Mi (9) σ=
xi σi
(10)
where xi is the mole fraction of the atom i. The total activation energy in the alloy carries an excess term associated with the mixing of unlike atoms in addition to the activation energies of component atoms, so that the averaged activation energy of the alloy can be written as i Evis = xi Evis (11) + Emix where Emix is the additional activation energy associated with the multi-component mixing. The additional parameter associated with the multi-component mixing is implemented as mix H Emix = − (12) Zmix where mix H is the enthalpy of mixing, whereas Zmix the adjustable parameter to the experimental data of the viscosity for the multi-component liquid alloys. For a binary liquid alloy of atoms i and j, let us write Eq. (12) explicitly for later convenience: i–j Emix
i–j
=−
mix H i–j
Zmix
(13)
The negative sign of the right-hand side of Eq. (12) leads to the plausible physical picture that the viscosity increases (decreases) when the interaction between the unlike atoms is attractive (repulsive). The parameters, κi , Zi and Zmix introduced in the gas-like model are eventually obtained by adjusting to the respective experimental viscosity data. It was found in [7] that κi = 0.4–0.7 and Zi = 7.8–15.6 (10.3–40.4) for the normal metal (the semi-metal) considering 11 cases of mono-atomic liqi–j uid metals, and that Zmix = 1.2–5.6 considering three cases of the binary liquid alloys. Here we follow the classification for metals of [2] that the normal metals possess simpler symmetrical crystal structures like face-centered cubic, body-centered cubic, hexagonal-closed packed, etc. in the solid phase than those of the semi-metals. Our parameters are not arbitrarily adjustable parameters to fit the experimental viscosity data but can be related to the
S. Morioka / Materials Science and Engineering A362 (2003) 223–227
225
fundamental physical quantities: This means that κi can be considered as a correction factor due to the deviation from a hard-sphere picture of the atoms in the liquid phase, and Zi and Zmix can be considered as a quantity related to the coordination number of nearest-neighbor atoms in the liquid metals (see [7] for the details).
3. Ternary systems Now let us examine our model in evaluation for the viscosities of ternary liquid alloys. Since our model consistently describe the viscosities from component atomic liquid metals to ternary liquid alloys, we need a set of systematic viscosity data for the component atomic liquid metals as well as those for the respective binary liquid alloys. In fact, such a set of systematic viscosity data is very rare because one measures the viscosities of alloys of specific compositions that one’s interest is concentrated. Only a ternary system of Ag–Au–Cu can be found as a suitable set of systematic viscosity data with the respective binary alloys of Ag–Au, Ag–Cu, Au–Cu [9,10]. The viscosities of the component atomic liquid metals for this ternary system were also measured by the same group. We shall employ this set of viscosity data to examine our model. In order to evaluate the viscosity of our ternary liquid alloys by the gas-like model, one needs to know, at least, the parameters κi and Zi for the component atomic liquids as i–j well as the parameters Zmix for the respective binary liquid alloys. The parameters for the component atomic liquids of Ag–Au–Cu system were already obtained by the gas-like model in a previous work [7]. Here we use the same physical quantities and parameters for the component atomic liquids. They are summarized in Table 1, and Fig. 1 compares the calculated viscosities for the component atomic liquids with each set of experimental data. As for the parami–j eters Zmix of the respective binary liquid alloys, we made 2 a χ -parameter search for each set of the experimental viscosity. The enthalpies of mixing employed in evaluation of Eq. (13) are displayed at eleven different mole fractions in Table 2. The obtained parameters are summarized in Table 3, and Fig. 2 shows the results comparing with the experimental viscosity data at 1373 K for the respective binary liquid alloys.
Table 1 The physical quantities and the parameters employed for the mono-atomic liquid metals Element i
di (Å)
Mi (g/mol)
i Hevap (kJ/mol)
κi
Zi
Ag Au Cu
2.90 2.88 2.56
107.9 197.0 63.54
258 343 307
0.488 0.411 0.670
10.2 14.8 8.8
The values of d and those of M and Hevap are taken from [8] and [2], respectively.
Fig. 1. The viscosities of mono-atomic liquid metals of Ag, Au and Cu. The solid curves are the calculated results for each set of the experimental data. Table 2 i−j The enthalpies of mixing (J/mol), mix H, of the binary liquid alloys xAu/Cu
Ag–Au
Ag–Cu
Au–Cu
0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0
0 −1476 −2624 −3444 −3937 −4101 −3937 −3444 −2624 −1476 0
0 1769 3050 3888 4321 4384 4105 3510 2618 1444 0
0 −1436 −2616 −3512 −4102 −4370 −4287 −3830 −2976 −1708 0
The values for Ag–Au and Ag–Cu liquid alloys are the same employed in [7], and those for Au–Cu liquid alloys are taken from [11]. Table 3 The obtained parameters for the binary liquid alloys i–j
Alloy i–j
Zmix
Ag–Au Ag–Cu Au–Cu
5.2 6.5 32
There were minor numerical errors in the previous evaluation [7] of Zmix for Ag–Au and Ag–Cu binary liquid alloys, so that the values for them in the present work slightly differ from the previous results.
The formula Eq. (7) of the viscosity to describe our ternary alloys requires the quantities given by Eqs. (9)–(11). M = xAg MAg + xAu MAu + xCu MCu
(14)
σ = xAg σAg + xAu σAu + xCu σCu
(15)
Ag–Au–Cu
Evis
Ag
Ag−Au−Cu
Au Cu = xAg Evis + xAu Evis + xCu Evis + Emix
(16)
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S. Morioka / Materials Science and Engineering A362 (2003) 223–227
Fig. 2. The viscosities of Ag–Au, Ag–Cu and Au–Cu binary liquid alloys. The solid curves are the calculated results for each set of the experimental data.
where σAg may be evaluated by Eq. (6) with the parameters Ag κAg and Evis may be evaluated by Eq. (5) with the parameters ZAg . Thus the only unknown quantity is the last term of Ag–Au–Cu which is the activation energy Eq. (16), namely Emix associated with mixing of the ternary components. Since the main correlation of the atoms even in a multi-component liquid alloy is of the two-atomic type, so that the activation energy associated with mixing of Ag, Au and Cu can be approximated by the sum of the activation energies associated with mixing of the respective binary components as Ag–Au–Cu
Emix
Ag–Au
Emix
Ag−Cu
+ Emix
Au–Cu + Emix
Fig. 3. The iso-thermal viscosities at 1373 K for Ag–Au–Cu liquid alloys. The solid curves correspond to the predicted results for 3.62, 4.10 and 4.67 mPa s, respectively. The points on the binary axes are interpolated from the experimental data for the respective binary liquid alloys [9].
figures that the model predicts the overall experimental data of the ternary liquid alloys. Fig. 3 strongly suggests that the predictions for the viscosity of the ternary liquid alloys would be improved by some better adjustments for Ag–Cu and Au–Cu binary liquid alloys. Thus some deviations of the predicted viscosity from the experimental data would not be characteristic of the ternary mixing. This would also apply to the predicted temperature dependence of the viscosity for the case 4 in Fig. 4 where the concentration of Cu is relatively large. The temperature dependence of the pre-
(17)
with Ag–Au
Ag–Au
Emix
=−
mix
H
(18)
Ag−Au Zmix Ag–Cu
Au–Cu . All of and the similar formulas for Emix and Emix Ag−Au may be evaluated by Eq. (13) with the parameters Emix Ag–Au Zmix obtained earlier, so that the evaluation for the viscosity of our ternary liquid alloy does not need any additional parameters. Therefore, the viscosity of multi-component liquid alloys can be predicted with the parameters of the component atomic liquids as well as those of the respective binary liquids. Employing κAg , ZAg and the physical quantities summaAg–Au and the enthalpies of rized in Table 1 as well as Zmix mixing summarized in Tables 2 and 3, one can evaluate the viscosity of our ternary liquid alloys. Our predictions for the viscosity of Ag–Au–Cu liquid alloys are shown in Fig. 3 of the iso-thermal viscosity at 1373 K, and in Fig. 4 of the viscosity dependence on temperature. We see from these
Fig. 4. The temperature dependence of the viscosity of Ag–Au–Cu liquid alloys. The solid curves correspond to the predicted results for each set of the experimental data.
S. Morioka / Materials Science and Engineering A362 (2003) 223–227 Ag–Au
dicted viscosity also suggests that Zmix slightly depends on temperature. Our results are comparable with those predicted in [5]. It should be emphasized that our model predicts the viscosities consistently from the component atomic liquid metals to the ternary liquid alloys without any arbitrary parameters. 4. Conclusions 1. The model predicts the overall viscosities for the ternary liquid alloys of Ag, Au, and Cu consistently with those for the component atomic liquid metals and the respective binary liquid alloys without any arbitrary parameters. 2. The parameters Zmix slightly depend on temperature. Acknowledgements The author would like to express his gratitude to Prof. Bian Xiufang for his continuous encouragement. He also
227
appreciates useful comments of Prof. T. Iida. The present work was supported by the National Nature Science Foundation of China (Grant No. 50071028).
References [1] J.R. Wilson, Meth. Rev. 10 (1965) 381. [2] T. Iida, R.I.L. Guthrie, The Physical Properties of Liquid Metals, Oxford University Press, Oxford, 1988. [3] R.P. Chhabra, D.K. Sheth, Z. Metallkd. 81 (1990) 264. [4] S. Glasstone, K.J. Laidler, H. Eyring, Theory of Rate Processes, McGraw-Hill, New York, 1941. [5] M. Kucharski, Z. Metallkd. 79 (1988) 264. [6] S. Seetharaman, S. Du, Metall. Mater. Trans. B 25B (1994) 589. [7] S. Morioka, X. Bian, M. Sun, Z. Metallkd. 93 (2002) 288. [8] C. Kittel, Introduction to Solid State Physics, Wiley, New York, 1986. [9] E. Gebhardt, M. Becker, Z. Metallkd. 42 (1951) 111; E. Gebhardt, G. Wörwag, Z. Metallkd. 42 (1951) 358. [10] E. Gebhardt, G. Wörwag, Z. Metallkd. 43 (1952) 106. [11] R. Hultgren, P.D. Desai, D.T. Hawkins, M. Gleiser, K.K. Kelley, Selected Values of the Thermodynamic Properties of Binary Alloys, ASM, Metal Park, OH, 1973.
Evaluation of the viscosity for binary and ternary liquid alloys S. Morioka Key Laboratory of Liquid Structure and Heredity of Materials, College of Materials Science and Engineering, Shandong University, Jinan, Shandong, China Received 18 April 2003; received in revised form 2 July 2003
Abstract The ‘gas-like’ model of viscosity for liquid alloys is examined in evaluation for the viscosity of binary and ternary liquid alloys of Ag, Au and Cu. The detail procedure to evaluate the viscosity of multi-component liquid alloys is given. The model predicts the overall experimental data of the viscosities for Ag–Au–Cu liquid alloys consistently with those for the component atomic liquid metals and the respective binary liquid alloys. © 2003 Elsevier B.V. All rights reserved. Keywords: Viscosity; Liquid metal; Alloy
1. Introduction The viscosity of liquid metals is of great importance in metallurgical processes as well as in the study for dynamics of atoms in the liquid metals [1,2]. High quality metals are mostly composed in forms of multi-component alloys, so that a predictable model to describe the viscosity for multi-component alloys is desired. The formula most frequently used in analysis for the viscosity of liquid metals is the Arrhenius type Evis η = A exp (1) RT where A is a constant, R the gas constant, T the absolute temperature, and Evis the activation energy of viscosity. The values of A and Evis are empirically determined as adjustable parameters by fitting the viscosity data. These parameters have been sought with the available viscosity data for mono-atomic metals [3]. The viscosity of liquid alloys is usually not the linear summation over component atomic viscosities, and hence a predictable model to describe the viscosity for multi-component alloys cannot be expected within this frame work. A predictable model to describe the viscosity of multi-component alloys, therefore, should be based on more fundamental physical quantities.
E-mail address: [email protected] (S. Morioka). 0921-5093/$ – see front matter © 2003 Elsevier B.V. All rights reserved. doi:10.1016/S0921-5093(03)00619-1
A more fundamental model to describe the viscosity may be the one based on the rate theory [4]: hNA Evis η= exp (2) V RT where h is Planck’s constant, NA the Avogadro’s number, and V the molar volume. As in the Arrhenius one, the values of Evis are empirically determined as adjustable parameters by fitting the viscosity data. The formula (2) is applicable only to describe the viscosity of mono-component liquids. For a liquid alloy, thus, one needs an additional consideration associated with the mixing of unlike atoms. Two kinds of models are proposed based on the rate theory to describe the viscosity for multi-component alloys: one derives a formula in terms of the superposition of component atomic viscosities with a correction due to the mixing of unlike atoms [5]. This correction term is related to a thermodynamic quantity which involves an arbitrary parameter. Besides, the model utilizes measured component atomic viscosities in evaluation for the viscosities of multi-component liquid alloys. Thus, the model is not based on the fundamental physical quantities. The other model regards the activation energy of an alloy as the sum of the activation energies of component atoms, the Gibbs energy of mixing and a kinetic correction associated with the mixing of unlike atoms [6]. The model does not involve an arbitrary parameter in this correction term, but it does involve some parameters in the activation energies of component atoms. These pa-
224
S. Morioka / Materials Science and Engineering A362 (2003) 223–227
rameters are not related to the known fundamental physical quantities. Recently, a model (gas-like model) to describe the viscosity for liquid metals was proposed [7], which is based on the gas-like kinetics with the viscous activation picture for liquid materials of Kittel [8]. The most important feature of this model is that it is capable to describe the viscosities consistently from mono-atomic liquid metals to multi-component liquid alloys with the parameters related to the known fundamental physical quantities. The model was applied to 11 cases of mono-atomic liquid metals, and described the overall experimental data of the viscosity in a wide range of temperature, predicting the mean free paths of an atom in accord with the experimental interatomic distances in the liquid metals. The model was extended to describe the viscosity for multi-component liquid alloys with the additional parameter associated with multi-component mixing. In this paper, we shall examine the gas-like model in the evaluation of the viscosity for binary and ternary liquid alloys in detail.
2. The model Let us briefly explain the gas-like model [7] where the viscosity ηi for a liquid metal of atom i is given as i E vis ηi = ηig exp (3) RT where ηig is the viscosity in the gas phase 2 Mi RT i ηg = 3NA σi π
(4)
i , M and σ are the activation energy, the mass and where Evis i i the collision cross-section of the atom i, respectively. The activation energy in our model is regarded as the energy for the atom to surmount the potential energy barrier provided by one or so of the nearest-neighbor atoms because of the short-range order [2] in the liquid, and thus the activation energy for the atom i is speculated as [7] i Evis =
i Hevap
Zi
(5)
i where Hevap is the enthalpy of evaporation of the atom i, and Zi is an adjustable parameter to the experimental data of the viscosity of the atom i, but can be considered as a quantity related to the coordination number of nearest-neighbor atoms in the liquid metal. The collision cross-section of a hard-sphere picture for the atom in the liquid phase is modified by the parameter κi as
σi = π(κi di )2 where di is the diameter of the atom i.
(6)
As for a multi-component liquid alloy, the viscosity η is assumed as Evis (7) η = ηg exp RT with 2 ηg = 3NA σ
MRT π
(8)
where we take a simple average for the masses and collision cross-sections of component atoms in the alloy as followings: M= xi Mi (9) σ=
xi σi
(10)
where xi is the mole fraction of the atom i. The total activation energy in the alloy carries an excess term associated with the mixing of unlike atoms in addition to the activation energies of component atoms, so that the averaged activation energy of the alloy can be written as i Evis = xi Evis (11) + Emix where Emix is the additional activation energy associated with the multi-component mixing. The additional parameter associated with the multi-component mixing is implemented as mix H Emix = − (12) Zmix where mix H is the enthalpy of mixing, whereas Zmix the adjustable parameter to the experimental data of the viscosity for the multi-component liquid alloys. For a binary liquid alloy of atoms i and j, let us write Eq. (12) explicitly for later convenience: i–j Emix
i–j
=−
mix H i–j
Zmix
(13)
The negative sign of the right-hand side of Eq. (12) leads to the plausible physical picture that the viscosity increases (decreases) when the interaction between the unlike atoms is attractive (repulsive). The parameters, κi , Zi and Zmix introduced in the gas-like model are eventually obtained by adjusting to the respective experimental viscosity data. It was found in [7] that κi = 0.4–0.7 and Zi = 7.8–15.6 (10.3–40.4) for the normal metal (the semi-metal) considering 11 cases of mono-atomic liqi–j uid metals, and that Zmix = 1.2–5.6 considering three cases of the binary liquid alloys. Here we follow the classification for metals of [2] that the normal metals possess simpler symmetrical crystal structures like face-centered cubic, body-centered cubic, hexagonal-closed packed, etc. in the solid phase than those of the semi-metals. Our parameters are not arbitrarily adjustable parameters to fit the experimental viscosity data but can be related to the
S. Morioka / Materials Science and Engineering A362 (2003) 223–227
225
fundamental physical quantities: This means that κi can be considered as a correction factor due to the deviation from a hard-sphere picture of the atoms in the liquid phase, and Zi and Zmix can be considered as a quantity related to the coordination number of nearest-neighbor atoms in the liquid metals (see [7] for the details).
3. Ternary systems Now let us examine our model in evaluation for the viscosities of ternary liquid alloys. Since our model consistently describe the viscosities from component atomic liquid metals to ternary liquid alloys, we need a set of systematic viscosity data for the component atomic liquid metals as well as those for the respective binary liquid alloys. In fact, such a set of systematic viscosity data is very rare because one measures the viscosities of alloys of specific compositions that one’s interest is concentrated. Only a ternary system of Ag–Au–Cu can be found as a suitable set of systematic viscosity data with the respective binary alloys of Ag–Au, Ag–Cu, Au–Cu [9,10]. The viscosities of the component atomic liquid metals for this ternary system were also measured by the same group. We shall employ this set of viscosity data to examine our model. In order to evaluate the viscosity of our ternary liquid alloys by the gas-like model, one needs to know, at least, the parameters κi and Zi for the component atomic liquids as i–j well as the parameters Zmix for the respective binary liquid alloys. The parameters for the component atomic liquids of Ag–Au–Cu system were already obtained by the gas-like model in a previous work [7]. Here we use the same physical quantities and parameters for the component atomic liquids. They are summarized in Table 1, and Fig. 1 compares the calculated viscosities for the component atomic liquids with each set of experimental data. As for the parami–j eters Zmix of the respective binary liquid alloys, we made 2 a χ -parameter search for each set of the experimental viscosity. The enthalpies of mixing employed in evaluation of Eq. (13) are displayed at eleven different mole fractions in Table 2. The obtained parameters are summarized in Table 3, and Fig. 2 shows the results comparing with the experimental viscosity data at 1373 K for the respective binary liquid alloys.
Table 1 The physical quantities and the parameters employed for the mono-atomic liquid metals Element i
di (Å)
Mi (g/mol)
i Hevap (kJ/mol)
κi
Zi
Ag Au Cu
2.90 2.88 2.56
107.9 197.0 63.54
258 343 307
0.488 0.411 0.670
10.2 14.8 8.8
The values of d and those of M and Hevap are taken from [8] and [2], respectively.
Fig. 1. The viscosities of mono-atomic liquid metals of Ag, Au and Cu. The solid curves are the calculated results for each set of the experimental data. Table 2 i−j The enthalpies of mixing (J/mol), mix H, of the binary liquid alloys xAu/Cu
Ag–Au
Ag–Cu
Au–Cu
0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0
0 −1476 −2624 −3444 −3937 −4101 −3937 −3444 −2624 −1476 0
0 1769 3050 3888 4321 4384 4105 3510 2618 1444 0
0 −1436 −2616 −3512 −4102 −4370 −4287 −3830 −2976 −1708 0
The values for Ag–Au and Ag–Cu liquid alloys are the same employed in [7], and those for Au–Cu liquid alloys are taken from [11]. Table 3 The obtained parameters for the binary liquid alloys i–j
Alloy i–j
Zmix
Ag–Au Ag–Cu Au–Cu
5.2 6.5 32
There were minor numerical errors in the previous evaluation [7] of Zmix for Ag–Au and Ag–Cu binary liquid alloys, so that the values for them in the present work slightly differ from the previous results.
The formula Eq. (7) of the viscosity to describe our ternary alloys requires the quantities given by Eqs. (9)–(11). M = xAg MAg + xAu MAu + xCu MCu
(14)
σ = xAg σAg + xAu σAu + xCu σCu
(15)
Ag–Au–Cu
Evis
Ag
Ag−Au−Cu
Au Cu = xAg Evis + xAu Evis + xCu Evis + Emix
(16)
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S. Morioka / Materials Science and Engineering A362 (2003) 223–227
Fig. 2. The viscosities of Ag–Au, Ag–Cu and Au–Cu binary liquid alloys. The solid curves are the calculated results for each set of the experimental data.
where σAg may be evaluated by Eq. (6) with the parameters Ag κAg and Evis may be evaluated by Eq. (5) with the parameters ZAg . Thus the only unknown quantity is the last term of Ag–Au–Cu which is the activation energy Eq. (16), namely Emix associated with mixing of the ternary components. Since the main correlation of the atoms even in a multi-component liquid alloy is of the two-atomic type, so that the activation energy associated with mixing of Ag, Au and Cu can be approximated by the sum of the activation energies associated with mixing of the respective binary components as Ag–Au–Cu
Emix
Ag–Au
Emix
Ag−Cu
+ Emix
Au–Cu + Emix
Fig. 3. The iso-thermal viscosities at 1373 K for Ag–Au–Cu liquid alloys. The solid curves correspond to the predicted results for 3.62, 4.10 and 4.67 mPa s, respectively. The points on the binary axes are interpolated from the experimental data for the respective binary liquid alloys [9].
figures that the model predicts the overall experimental data of the ternary liquid alloys. Fig. 3 strongly suggests that the predictions for the viscosity of the ternary liquid alloys would be improved by some better adjustments for Ag–Cu and Au–Cu binary liquid alloys. Thus some deviations of the predicted viscosity from the experimental data would not be characteristic of the ternary mixing. This would also apply to the predicted temperature dependence of the viscosity for the case 4 in Fig. 4 where the concentration of Cu is relatively large. The temperature dependence of the pre-
(17)
with Ag–Au
Ag–Au
Emix
=−
mix
H
(18)
Ag−Au Zmix Ag–Cu
Au–Cu . All of and the similar formulas for Emix and Emix Ag−Au may be evaluated by Eq. (13) with the parameters Emix Ag–Au Zmix obtained earlier, so that the evaluation for the viscosity of our ternary liquid alloy does not need any additional parameters. Therefore, the viscosity of multi-component liquid alloys can be predicted with the parameters of the component atomic liquids as well as those of the respective binary liquids. Employing κAg , ZAg and the physical quantities summaAg–Au and the enthalpies of rized in Table 1 as well as Zmix mixing summarized in Tables 2 and 3, one can evaluate the viscosity of our ternary liquid alloys. Our predictions for the viscosity of Ag–Au–Cu liquid alloys are shown in Fig. 3 of the iso-thermal viscosity at 1373 K, and in Fig. 4 of the viscosity dependence on temperature. We see from these
Fig. 4. The temperature dependence of the viscosity of Ag–Au–Cu liquid alloys. The solid curves correspond to the predicted results for each set of the experimental data.
S. Morioka / Materials Science and Engineering A362 (2003) 223–227 Ag–Au
dicted viscosity also suggests that Zmix slightly depends on temperature. Our results are comparable with those predicted in [5]. It should be emphasized that our model predicts the viscosities consistently from the component atomic liquid metals to the ternary liquid alloys without any arbitrary parameters. 4. Conclusions 1. The model predicts the overall viscosities for the ternary liquid alloys of Ag, Au, and Cu consistently with those for the component atomic liquid metals and the respective binary liquid alloys without any arbitrary parameters. 2. The parameters Zmix slightly depend on temperature. Acknowledgements The author would like to express his gratitude to Prof. Bian Xiufang for his continuous encouragement. He also
227
appreciates useful comments of Prof. T. Iida. The present work was supported by the National Nature Science Foundation of China (Grant No. 50071028).
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