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The thermodynamics of liquid Fe-based ternary solutions containing nitrogen
J. Phys. Ghan. Solti Vol. 54. No. 7. pp. 863466.1993 Printed ia Great Britain.
0022-3697193 s6.00 + 0.00 O1993PergttmonRuLtd
THE THERMODYNAMICS OF LIQUID Fe-BASED TERNARY SOLUTIONS CONTAINING NITROGEN R. B. MCLELLAN and M. L. WASZ Department of Mechanical Engineering and Materials Science, William Marsh Rice University, Houston, TX 77251-1892, U.S.A. (Received 17 September 1992; accepted in revised form 11 March 1993) Ahatraet-Ternary Fe-based liquid solutions containing nitrogen have been analyzed in terms of a first-order statistical model. The model enables interaction energies between the dissolved N-atoms and the matrix to be extracted from measured solubility data. Such interaction energies have been deduced from experimental thermodynamic data and shown to correlate well with the Allred-Rochow electronegativities of the atomic species in the ternary liquids. Keywords: Nitrogen, liquid ternary solutions, high N2 pressures, interaction parameters, electronegativity.
1. INTRODUCTION The thermodynamic
properties of the solid solutions
of N in f.c.c. Fe-based substitutional matrices (ternary N-austenites) have been studied in some detail [I]. In the case of the liquid ternary solutions, measurements have recently been performed at high Nr pressures where high N concentrations have been introduced into the condensed phase in equilibrium with Nr gas [2]. Such measurements are difficult, but fruitful since they can provide information with which statistical models may be evaluated at solute concentrations above the “infinite dilution” limit. Satir-Kolon and Feichtinger [2] measured the Nsolubility of Fe-based liquid solutions at 1873 K in equilibrium with N, gas at pressures up to 1O’Pa. These authors found deviations from Sievert’s law, and presented such observations in the form of the traditional Wagner interaction coefficients. The purpose of the present brief report is to show that the results may be interpreted in terms of a simple statistical model, and that the interatomic energies derived from an evaluation of the experimental data in terms of this model may be correlated with electronegativities. 2. MODEL FOR TJCRNARYLIQUID Let us make the usual assumption that at temperatures within 0.2 T, (T,,, = melting point), the liquid may be approximated by a quasi-crystal whose geometry with respect to short-range order does not
differ significantly from that of the solid. Furthermore, the Fe-based binary matrix will have an essentially random distribution, and the N-atoms will be located in the interstitial sites. In such solutions the interstitial (i) atoms are distributed in the interstitial sites (cells) according to the Fermi-Dirac distribution function [3], p; = [exp{[6p(n + 1) - ci]/kT} + I]-‘.
(1)
In this equation p; is the density of i-atoms in sites of type n, where n is defined as the number of U-atoms (U = Mn, Cr, . . . ) in the first atomic shell of the cell, and z - n is the number of Fe (V = solvent) atoms. The total number of atoms in the shell is z. The energy 6: is the energy required to insert an i-atom into a o-type cell, and 6: is a temperaturedependent energy whose evaluation may be carried out under certain circumstances [4]. However, in the low-density limit (p: # 1), the distribution approximates to Maxwell-Boltxmann statistics, and the corresponding solution of the equations (1) has been given previously [3]. Now, in general, when the atom fraction 8, of U-atoms in the binary matrix is greater than 0.05 and f?,, the atom ratio of interstitials is above the “infinite dilution” limit, it is not possible to solve the system of eqn (1) in a closed form. In the case of the solutions considered here [2], the values of 0, and ei are much larger than these limits, and it would seem that the cell approach would not be appropriate in 863
R. B. MCLELLANand M. L. WASZ
864
the high-temperature limit (Boltzmann limit). However, these are reasons for believing that the .$’ is so small that for all values of n, the values of ~p(n + l)/kT are all such that deviations from Maxwell-Boltzmann statistics will not be significant. If this is the case, the thermodynamics of the crystal are much simplified, and the cell occupation density is uniform such that all p,, = 8,, and the configurational enthalpy is
HC = i
L9,&n + 1)N”.
(2)
It-0
Equation (2) has a simple physical basis. The system is entropy-dominated, and the contribution to H’ from each cell type is the product of the number of such cells, the cell energy level, and the number of i-atoms for each cell. The corresponding partial molar enthalpy is R = EO+ i n-0
2
(?I + 1>cp,
0
(3)
where E” is a ground state energy and N, is the number of i-atoms in a type-n cell. The reasons for believing that such a simple situation may hold for V-U-i solutions investigated [2] lie in the fact that even in the corresponding solid solution systems, the values of 60 are small compared to kTat temperatures where the solid phase is stable. In the case of the liquid, the contribution to SF which, in the solid, would correspond to shear deformation in inserting the i-atoms, disappear. Furthermore, the differences in Goldschmidt atomic radii, 6 = (r, - r,)/r,, are small for U = Cr, 6 = -0.03, and for U = Ni, 6 = +0.05. For U = Mn, a Goldschmidt radius is not defined, but the slope of the Vegard law plot for Fe-Mn solutions is essentially identical to that of Fe-Ni. A solubility equation may be deduced by equating the chemical potential of i-atoms derived from eqn (3) to that for i-atoms in the gas phase, pk. The latter chemical potential is given by [S] & = Ef + kT in AP”*,
I
= @PI’* exp{ -Qcy/kT},
Q = i (z _;)!n, n-0
(n + l)(l
(6)
- 0,)(z-n’e:.
Equation (5) will be used to evaluate the experimental solubilities and extract values of .$’ therefrom. 3. DATA FITTING It is now clear that the model, upon which eqn (5) is based, does not consider i-i interactions, i.e. the energy Q is not dependent upon Bi. The largest span of 0,, P data accrues from the binary Fe-N system since the measurements of Satir-Kolorz and Feichtinger (SKF) [2] may be combined with other measurements at lower pressures. Figure 1 shows a plot of in P vs ln(B,/l - ei) using the SKF data and earlier measurements of Gomersall et al. [6] (O), Blossey and Pehlke [7] (0) and Humbert and Elliott [8] (0). The units of P in this plot are atmospheres. The linearity depicted indicates that, within the limits of the experimental scatter, Ri is sensibly independent of 0,. Now let us turn to the data on the ternary solutions. The data for Fe-Cr-N extend up to 8,%0.13 and P = 10’Pa. Plots of 6,&l -(Ii) vs PI’* are given in the lower section of Fig. 2 corresponding to each value of 0, studied. These plots are sensibly linear which is not observed in plots of the atom fraction of N vs PI/* [2]. As seen in Fig. 2, the plots of @/(l - 6,) vs fi for the smaller values of t$ extrapolate through the origin within experimental accuracy. This is, of course, in accord with the behavior predicted by eqn (5). The data for the solution with 36.6At.% Cr show a “residual” N-content corresponding to BiN 0.03. This phenomenon is by no means unexpected, and is observed in other metal-interstitial systems where there is a large attractive interaction between the lattice and the
(4)
is the dissociation energy of the N, where -Ef molecule at 0 K, P is the N, pressure, and I is a constant independent of temperature. A detailed derivation of the chemical potential of N-atoms in solution has been given previously (31. The resulting solubility equation is &
where @ is a constant, and
(5)
4
4.5
5
-h +[
6
63
7
1
Fig. 1. Plot of the solubility 0, of N in liquid Fe-N solutions 1873K as a function of the N, gas pressure. The form of the plot is In P vs ln[0,/(1 - @,)I,where the pressure is 8iven in
atmospheres.
Ternary Fe-based liquid solutions containing N
865
. Fig. 2. Plots of the sclubility function 0,(1 - 0,) vs fi for N in liquid Fe-Cr-N solutions at 1873K. The composition 0, of the Fe-Cr binary matrix is indicated on each graph. The inset diagram shows the variation of ln[e,/(l - 0,)] with e, at constant pressure.
Fig. 4. Correlation between the cell energy cp and the electronegativity function f for liquid Fe+N solutions containing elements in the first transition series.
Fig. 3 and the determinaton of co because of the “residual effect” discussed previously. Now, Fig. 3 gives points calculated from the best interstitial solute [9, lo]. Thus, the deviation from fits of the raw data in the representation of Fig. 2. Sievert’s law has to do with the fact that as ei Thus, the calculated points given in Fig. 3 do not increases, the fractional site filling ei/(l - 0,) begins reflect the inherent scatter in the raw data. The error to deviate from the atomic fraction of solute 13~. in the intercepts of Fig. 3, and thus in the estimated The inset graph in Fig. 2 shows the variation of CO-values, has been calculated from the regression ln[t$/(l - fJi)] with 0, at a constant pressure (81 atm). analysis and is given by the error bars in Fig. 4, which In these graphs the units of pressure are, for depicts a correlation between 6: and a function f convenience, atmospheres. The data for Fe-Ni-N containing the electronegativities of the atomic and Fe-Mn-N have been treated in a similar species in the solution. For U = Ni, only half of the manner. symmetrical error bar is shown. In order to estimate the values of up, note that for a given value of P, eqn (5) may be written in the form 4. DISCUSSION
(7) where K is a constant. Thus, plots of Q -* ln[eJ( 1 - fIi)] vs Q -’ should be linear, and 6” may be obtained from the intercept. These plots are shown in Fig. 3. The constant K in eqn (7) is a function of P. The plots shown in Fig. 3 were constructed by choosing values of fi from the data given in the plots exemplified by Fig. 2, i.e. by choosing a value of fi and calculating fI,/(l - 0,) from the least-squares regressions. The data for the FeCr-N system for U = 0.366 were not included in
Fig. 3. Plots of Q-l ln[0,/(1 -0,)]
vs Q-l for liquid Fe-(Mn, Cr, Ni)-N solutions at 1873 K.
It has been shown previously [l l] that, provided the atomic mismatch 6 is small, the energy 6: for liquid solutions may be correlated to the “chemical” differences between the free atoms. This difference may be expressed by the function
f
=(x”-x,)2-(x,--,Y,
(8)
where the xs are the Allred-Rochow [12, 131 negatives of the species involved. The data taken from the present work are denoted by open circles in Fig. 4. Despite the large error bars, the degree of adherence to the correlation is quite remarkable. Data for 6: have been previously calculated [ll] for other U-elements in the transition series elements, and have been included in Fig. 4. The points for U = Cr are taken from Schiirmann and Kunze [14,15] (0 symbol), Pehlke and Elliott [16] (0 symbol) and Btiek [17] (0 symbol). Data for U = Mn were taken from Shiirmann and Kunze [ 14,15](@ symbol) and Pehlke and Elliott [16] (0 symbol). For U = Co, the cp values were taken from [ 141 and [ 151 (e symbol), Reference [16] (0 symbol), and [17] (V symbol). For
866
R. B. MCLELLANand M. L. WUZ
U = Ni, the data point (IJ) was taken from [14] and [15]. These data sets were all measured at P = 1 atm. The significance of the correlation given in Fig. 4 is clearly open to question because the species in the solution are not “free atoms,” but components in a condensed phase. Nevertheless, the correlation clearly exists and enables the energies of V-U-i-type liquid solutions to be estimated in a simple way from the electronegativities. Perhaps the more important conclusion is that the analysis indicates that, in the solutions considered, deviations from a random i-atom distribution may be ignored. This radical simplification is certainly not generally valid, and is clearly not true in solid solutions at lower temperatures. However, a simple thought experiment is useful. Because of the large number of energy levels involved, a strict solution eqn (1) has not yet been obtained; but if we take as an extremum example, U = Cr and 6, = 0.36, we can consider the crystal to be a defect lattice containing N, O-cells and N, 6-type cells. This effectively reduces the problem to a system of only two energy levels, 6: and 7~p, and ignores the intermediate cells. In this hypothetical system, N6= N(l - 19,) 0: = 1.194 x IO-‘N. If the high-temperature limit were strictly correct, the ratio N,/Ns would be 8.36 x 102. The solution of the two-level problem has been given previously [18] and, using the CP-value of Cr taken from Fig. 4, it is easy to calculate N,/N, = 8.28 x lo2 at T = 2000 K. This calculation thus lends much credence to the assumption made in the present work. It should, however, be pointed out that the strictly high-temperature approximation may provide a useful model for discussing the thermodynamic proper-
ties of a solution, but it should be treated with great caution when considering i-atom diffusion since the diffusivities may be dominated by i-atoms in lower energy sites of the lattice. Ack~owledgemenf-The authors are grateful for the support of the Robert A. Welch Foundation.
lWFHtFNCES
1. Ko C. and McLellan R. B., Acfo Metali. 31, 1821 (1983). 2. Satir-Kolon A. H. and Feichtinger H. K., 2. Metafkde sz, 697 (1991). 3. McLellan R. B., Acra Metall. 30, 317 (1982). 4. Miinster A., Sratierische Thermo&amik. SpringerVerlag, Berlin (1969). 5. McLcllan R. B.. in Treari.te in Materials Science (Edited by H. Herman), p. 393. Academic Press, New York (1975). 6. Gomersall E., McLean A. and Ward R. G., Trans. Metall. Sot. AIME N2, 1311 (1968). I. Blossey R. G. and Pehlke R. D., Trans. Metall. Sot. AZME 236,566 (1966). 8. Humbert J. C. and Elliott J. F., Trans. Metall. Sot. AIME 218, 1076 (1980). 9. Dunn W. W. and McLellan R. B., Mel. Trans. 2, 1079 (1971). 10. Swartz J. C., Trans. Met. Sot. AIME 245, 1083 (1%9). 11. McLellan R. B., Scripta Metall. 12, 559 (1978). 12. Allred A. L. and Rochow E. G., J. Inorg. Nucl. Chem. 5, 264 (1958). 13. Allred A. L. and Rochow E. G., J. Inorg. Nucl. Chem. 5, 269 (1958). 14. Schiirmann E. and Kunze H. D., Arch. Eisenhtittenw. 38, 585 (1967). 15. Schiirmann E. and Kunze H. D., Arch. Eisenhlittenw. 38, 658 (1967). 16. Pehlke R. D. and Elliott J. F., Trans. Met. Sot. AIME 218, 1088 (1960). 17. Btiek Z., Chemical Metallurgy of Iron and Steel. Iron and Steel Institute, London (1973). 18. McLellan R. B., Acta Metall. 27, 1655 (1979).
0022-3697193 s6.00 + 0.00 O1993PergttmonRuLtd
THE THERMODYNAMICS OF LIQUID Fe-BASED TERNARY SOLUTIONS CONTAINING NITROGEN R. B. MCLELLAN and M. L. WASZ Department of Mechanical Engineering and Materials Science, William Marsh Rice University, Houston, TX 77251-1892, U.S.A. (Received 17 September 1992; accepted in revised form 11 March 1993) Ahatraet-Ternary Fe-based liquid solutions containing nitrogen have been analyzed in terms of a first-order statistical model. The model enables interaction energies between the dissolved N-atoms and the matrix to be extracted from measured solubility data. Such interaction energies have been deduced from experimental thermodynamic data and shown to correlate well with the Allred-Rochow electronegativities of the atomic species in the ternary liquids. Keywords: Nitrogen, liquid ternary solutions, high N2 pressures, interaction parameters, electronegativity.
1. INTRODUCTION The thermodynamic
properties of the solid solutions
of N in f.c.c. Fe-based substitutional matrices (ternary N-austenites) have been studied in some detail [I]. In the case of the liquid ternary solutions, measurements have recently been performed at high Nr pressures where high N concentrations have been introduced into the condensed phase in equilibrium with Nr gas [2]. Such measurements are difficult, but fruitful since they can provide information with which statistical models may be evaluated at solute concentrations above the “infinite dilution” limit. Satir-Kolon and Feichtinger [2] measured the Nsolubility of Fe-based liquid solutions at 1873 K in equilibrium with N, gas at pressures up to 1O’Pa. These authors found deviations from Sievert’s law, and presented such observations in the form of the traditional Wagner interaction coefficients. The purpose of the present brief report is to show that the results may be interpreted in terms of a simple statistical model, and that the interatomic energies derived from an evaluation of the experimental data in terms of this model may be correlated with electronegativities. 2. MODEL FOR TJCRNARYLIQUID Let us make the usual assumption that at temperatures within 0.2 T, (T,,, = melting point), the liquid may be approximated by a quasi-crystal whose geometry with respect to short-range order does not
differ significantly from that of the solid. Furthermore, the Fe-based binary matrix will have an essentially random distribution, and the N-atoms will be located in the interstitial sites. In such solutions the interstitial (i) atoms are distributed in the interstitial sites (cells) according to the Fermi-Dirac distribution function [3], p; = [exp{[6p(n + 1) - ci]/kT} + I]-‘.
(1)
In this equation p; is the density of i-atoms in sites of type n, where n is defined as the number of U-atoms (U = Mn, Cr, . . . ) in the first atomic shell of the cell, and z - n is the number of Fe (V = solvent) atoms. The total number of atoms in the shell is z. The energy 6: is the energy required to insert an i-atom into a o-type cell, and 6: is a temperaturedependent energy whose evaluation may be carried out under certain circumstances [4]. However, in the low-density limit (p: # 1), the distribution approximates to Maxwell-Boltxmann statistics, and the corresponding solution of the equations (1) has been given previously [3]. Now, in general, when the atom fraction 8, of U-atoms in the binary matrix is greater than 0.05 and f?,, the atom ratio of interstitials is above the “infinite dilution” limit, it is not possible to solve the system of eqn (1) in a closed form. In the case of the solutions considered here [2], the values of 0, and ei are much larger than these limits, and it would seem that the cell approach would not be appropriate in 863
R. B. MCLELLANand M. L. WASZ
864
the high-temperature limit (Boltzmann limit). However, these are reasons for believing that the .$’ is so small that for all values of n, the values of ~p(n + l)/kT are all such that deviations from Maxwell-Boltzmann statistics will not be significant. If this is the case, the thermodynamics of the crystal are much simplified, and the cell occupation density is uniform such that all p,, = 8,, and the configurational enthalpy is
HC = i
L9,&n + 1)N”.
(2)
It-0
Equation (2) has a simple physical basis. The system is entropy-dominated, and the contribution to H’ from each cell type is the product of the number of such cells, the cell energy level, and the number of i-atoms for each cell. The corresponding partial molar enthalpy is R = EO+ i n-0
2
(?I + 1>cp,
0
(3)
where E” is a ground state energy and N, is the number of i-atoms in a type-n cell. The reasons for believing that such a simple situation may hold for V-U-i solutions investigated [2] lie in the fact that even in the corresponding solid solution systems, the values of 60 are small compared to kTat temperatures where the solid phase is stable. In the case of the liquid, the contribution to SF which, in the solid, would correspond to shear deformation in inserting the i-atoms, disappear. Furthermore, the differences in Goldschmidt atomic radii, 6 = (r, - r,)/r,, are small for U = Cr, 6 = -0.03, and for U = Ni, 6 = +0.05. For U = Mn, a Goldschmidt radius is not defined, but the slope of the Vegard law plot for Fe-Mn solutions is essentially identical to that of Fe-Ni. A solubility equation may be deduced by equating the chemical potential of i-atoms derived from eqn (3) to that for i-atoms in the gas phase, pk. The latter chemical potential is given by [S] & = Ef + kT in AP”*,
I
= @PI’* exp{ -Qcy/kT},
Q = i (z _;)!n, n-0
(n + l)(l
(6)
- 0,)(z-n’e:.
Equation (5) will be used to evaluate the experimental solubilities and extract values of .$’ therefrom. 3. DATA FITTING It is now clear that the model, upon which eqn (5) is based, does not consider i-i interactions, i.e. the energy Q is not dependent upon Bi. The largest span of 0,, P data accrues from the binary Fe-N system since the measurements of Satir-Kolorz and Feichtinger (SKF) [2] may be combined with other measurements at lower pressures. Figure 1 shows a plot of in P vs ln(B,/l - ei) using the SKF data and earlier measurements of Gomersall et al. [6] (O), Blossey and Pehlke [7] (0) and Humbert and Elliott [8] (0). The units of P in this plot are atmospheres. The linearity depicted indicates that, within the limits of the experimental scatter, Ri is sensibly independent of 0,. Now let us turn to the data on the ternary solutions. The data for Fe-Cr-N extend up to 8,%0.13 and P = 10’Pa. Plots of 6,&l -(Ii) vs PI’* are given in the lower section of Fig. 2 corresponding to each value of 0, studied. These plots are sensibly linear which is not observed in plots of the atom fraction of N vs PI/* [2]. As seen in Fig. 2, the plots of @/(l - 6,) vs fi for the smaller values of t$ extrapolate through the origin within experimental accuracy. This is, of course, in accord with the behavior predicted by eqn (5). The data for the solution with 36.6At.% Cr show a “residual” N-content corresponding to BiN 0.03. This phenomenon is by no means unexpected, and is observed in other metal-interstitial systems where there is a large attractive interaction between the lattice and the
(4)
is the dissociation energy of the N, where -Ef molecule at 0 K, P is the N, pressure, and I is a constant independent of temperature. A detailed derivation of the chemical potential of N-atoms in solution has been given previously (31. The resulting solubility equation is &
where @ is a constant, and
(5)
4
4.5
5
-h +[
6
63
7
1
Fig. 1. Plot of the solubility 0, of N in liquid Fe-N solutions 1873K as a function of the N, gas pressure. The form of the plot is In P vs ln[0,/(1 - @,)I,where the pressure is 8iven in
atmospheres.
Ternary Fe-based liquid solutions containing N
865
. Fig. 2. Plots of the sclubility function 0,(1 - 0,) vs fi for N in liquid Fe-Cr-N solutions at 1873K. The composition 0, of the Fe-Cr binary matrix is indicated on each graph. The inset diagram shows the variation of ln[e,/(l - 0,)] with e, at constant pressure.
Fig. 4. Correlation between the cell energy cp and the electronegativity function f for liquid Fe+N solutions containing elements in the first transition series.
Fig. 3 and the determinaton of co because of the “residual effect” discussed previously. Now, Fig. 3 gives points calculated from the best interstitial solute [9, lo]. Thus, the deviation from fits of the raw data in the representation of Fig. 2. Sievert’s law has to do with the fact that as ei Thus, the calculated points given in Fig. 3 do not increases, the fractional site filling ei/(l - 0,) begins reflect the inherent scatter in the raw data. The error to deviate from the atomic fraction of solute 13~. in the intercepts of Fig. 3, and thus in the estimated The inset graph in Fig. 2 shows the variation of CO-values, has been calculated from the regression ln[t$/(l - fJi)] with 0, at a constant pressure (81 atm). analysis and is given by the error bars in Fig. 4, which In these graphs the units of pressure are, for depicts a correlation between 6: and a function f convenience, atmospheres. The data for Fe-Ni-N containing the electronegativities of the atomic and Fe-Mn-N have been treated in a similar species in the solution. For U = Ni, only half of the manner. symmetrical error bar is shown. In order to estimate the values of up, note that for a given value of P, eqn (5) may be written in the form 4. DISCUSSION
(7) where K is a constant. Thus, plots of Q -* ln[eJ( 1 - fIi)] vs Q -’ should be linear, and 6” may be obtained from the intercept. These plots are shown in Fig. 3. The constant K in eqn (7) is a function of P. The plots shown in Fig. 3 were constructed by choosing values of fi from the data given in the plots exemplified by Fig. 2, i.e. by choosing a value of fi and calculating fI,/(l - 0,) from the least-squares regressions. The data for the FeCr-N system for U = 0.366 were not included in
Fig. 3. Plots of Q-l ln[0,/(1 -0,)]
vs Q-l for liquid Fe-(Mn, Cr, Ni)-N solutions at 1873 K.
It has been shown previously [l l] that, provided the atomic mismatch 6 is small, the energy 6: for liquid solutions may be correlated to the “chemical” differences between the free atoms. This difference may be expressed by the function
f
=(x”-x,)2-(x,--,Y,
(8)
where the xs are the Allred-Rochow [12, 131 negatives of the species involved. The data taken from the present work are denoted by open circles in Fig. 4. Despite the large error bars, the degree of adherence to the correlation is quite remarkable. Data for 6: have been previously calculated [ll] for other U-elements in the transition series elements, and have been included in Fig. 4. The points for U = Cr are taken from Schiirmann and Kunze [14,15] (0 symbol), Pehlke and Elliott [16] (0 symbol) and Btiek [17] (0 symbol). Data for U = Mn were taken from Shiirmann and Kunze [ 14,15](@ symbol) and Pehlke and Elliott [16] (0 symbol). For U = Co, the cp values were taken from [ 141 and [ 151 (e symbol), Reference [16] (0 symbol), and [17] (V symbol). For
866
R. B. MCLELLANand M. L. WUZ
U = Ni, the data point (IJ) was taken from [14] and [15]. These data sets were all measured at P = 1 atm. The significance of the correlation given in Fig. 4 is clearly open to question because the species in the solution are not “free atoms,” but components in a condensed phase. Nevertheless, the correlation clearly exists and enables the energies of V-U-i-type liquid solutions to be estimated in a simple way from the electronegativities. Perhaps the more important conclusion is that the analysis indicates that, in the solutions considered, deviations from a random i-atom distribution may be ignored. This radical simplification is certainly not generally valid, and is clearly not true in solid solutions at lower temperatures. However, a simple thought experiment is useful. Because of the large number of energy levels involved, a strict solution eqn (1) has not yet been obtained; but if we take as an extremum example, U = Cr and 6, = 0.36, we can consider the crystal to be a defect lattice containing N, O-cells and N, 6-type cells. This effectively reduces the problem to a system of only two energy levels, 6: and 7~p, and ignores the intermediate cells. In this hypothetical system, N6= N(l - 19,) 0: = 1.194 x IO-‘N. If the high-temperature limit were strictly correct, the ratio N,/Ns would be 8.36 x 102. The solution of the two-level problem has been given previously [18] and, using the CP-value of Cr taken from Fig. 4, it is easy to calculate N,/N, = 8.28 x lo2 at T = 2000 K. This calculation thus lends much credence to the assumption made in the present work. It should, however, be pointed out that the strictly high-temperature approximation may provide a useful model for discussing the thermodynamic proper-
ties of a solution, but it should be treated with great caution when considering i-atom diffusion since the diffusivities may be dominated by i-atoms in lower energy sites of the lattice. Ack~owledgemenf-The authors are grateful for the support of the Robert A. Welch Foundation.
lWFHtFNCES
1. Ko C. and McLellan R. B., Acfo Metali. 31, 1821 (1983). 2. Satir-Kolon A. H. and Feichtinger H. K., 2. Metafkde sz, 697 (1991). 3. McLellan R. B., Acra Metall. 30, 317 (1982). 4. Miinster A., Sratierische Thermo&amik. SpringerVerlag, Berlin (1969). 5. McLcllan R. B.. in Treari.te in Materials Science (Edited by H. Herman), p. 393. Academic Press, New York (1975). 6. Gomersall E., McLean A. and Ward R. G., Trans. Metall. Sot. AIME N2, 1311 (1968). I. Blossey R. G. and Pehlke R. D., Trans. Metall. Sot. AZME 236,566 (1966). 8. Humbert J. C. and Elliott J. F., Trans. Metall. Sot. AIME 218, 1076 (1980). 9. Dunn W. W. and McLellan R. B., Mel. Trans. 2, 1079 (1971). 10. Swartz J. C., Trans. Met. Sot. AIME 245, 1083 (1%9). 11. McLellan R. B., Scripta Metall. 12, 559 (1978). 12. Allred A. L. and Rochow E. G., J. Inorg. Nucl. Chem. 5, 264 (1958). 13. Allred A. L. and Rochow E. G., J. Inorg. Nucl. Chem. 5, 269 (1958). 14. Schiirmann E. and Kunze H. D., Arch. Eisenhtittenw. 38, 585 (1967). 15. Schiirmann E. and Kunze H. D., Arch. Eisenhlittenw. 38, 658 (1967). 16. Pehlke R. D. and Elliott J. F., Trans. Met. Sot. AIME 218, 1088 (1960). 17. Btiek Z., Chemical Metallurgy of Iron and Steel. Iron and Steel Institute, London (1973). 18. McLellan R. B., Acta Metall. 27, 1655 (1979).