- Email: [email protected]
The thermodynamics of nitrogen austenites containing noble-metal substitutional solutes
J. Phys. Chem. Solids Vol. 48, No. 6, pp. 575-578, 1987 Printed in Gnat Britain..
0022-3697187 Q 1987 Pergamon
S3.W + 0.00
Journals Ltd.
THE THERMODYNAMICS OF NITROGEN AUSTENITES CONTAINING NOBLE-METAL SUBSTITUTIONAL SOLUTES C. Ko and REX B. MCLELLAN William Marsh Rice University, Department of Mechanical Engineering and Materials Science, Houston, Texas 77251, U.S.A. (Received 28 July 1986; accepted 20 November 1986)
Abstract-A gas-solid equilibration technique has been used to determine the partial thermodynamic functions of nitrogen dissolved in fee Fe-Rh solid solutions. The data span the temperature range 1370-1530K and the composition range (Mat.% Rh. The N-contents never exceeded 5 x IO-‘at.%. The data were analyzed in terms of the cell model for ternary solid solutions. Adherence to the model was good and revealed that Rh atoms produce shallow trapping sites for N in the fee lattice. Their depth, relative to the potential minima in the Fe-N system is -2.3 kJ/mol. Keywords: Thermodynamics,
nitrogen, austenites, equilibrium, cell models.
1. INTRODUCTION Austenitic solid solutions containing nitrogen and a substitutional solute element have recently been studied in some detail for such cases where the
substitutional element (V) is a transition metal dissolved in the fee Fe solvent (I’) matrix. The results obtained for N-austenites containing Co, Ni, Cr and Mn have been recently reviewed [l] and it has been shown that the thermodynamic properties of these systems, where the interstitial N-atoms (i) are essentially present at “infinite dilution”, are in good accord with the cell model [2] for ternary solid solutions and that the U-i, interaction energies derived from analyzing the thermodynamic data using this model are systematically correlatable with the electronic structure of the three interacting atomic species. Until recently there were no data pertaining to N-austenites containing a substitutional element other than one which was a member of the First Long Period of the transition elements. However, studies have recently been made of austenites containing the noble metals Pd [3] and Pt [4]. In the present work, results will be presented on thermodynamic measurements of Fe-Rh-N austenites and the results obtained for all the Fe-(noble metal)-N solutions summarized. The fee phase of iron dissolves up to 45 at. % Rh over a large range of temperature [5]. There is a low-temperature transformation from the ferromagnetic to the antiferromagnetic state in solutions in the composition range 0, = 0.48-0.52 (0, is the atomic ratio Rh/Fe) [6,7]. However, the solutions in the composition-temperature ranges studied in this work are uniform solid solutions. The meta-stable solutions can be formed by rapid cooling, and the X-ray lattice parameters have been measured by Chao, Duwez and Tsuei [8]. Their results are in
excellent accord with previous data [9]. These data have been utilized in this work in order to assess the effect of matrix expansion on the thermodynamic parameters of dissolved N-atoms. 2. EXPERIMENTAL
PROCEDURE
The equilibrate-quench-analyze method has been used to measure the solubility of N under N,-gas at
a pressure of 1.04 x 10’ Pa in F+Rh solid solutions containing O-6 at. % Rh in the temperature range 1370-l 530 K. Since the N-solubility under these conditions is small (i.e. 5 x 10-4-10-4 atom fraction of N), it is essential to ensure that the Fe-Rh solutions contain low levels of impurities. The binary solutions were made by arc melting MARZ grade Fe and Rh under purified argon. The molten solutions were agitated so as to promote homogenization. The films used in the equilibration experiments were produced by rolling and intermediate vacuum annealing. The final foil thickness was 2.0 x 10e4 m. The apparatus used to equilibrate, quench and analyze the foils for N-content has been described in detail in previous papers [lo, 111. The compositions of the Fe-Rh foils were determined by the electron microprobe. The results were found always to be identical to the nominal composition (given in Table 1) to within the accuracy of the microprobe system (* 5%).
Table 1. Thermodynamic parameters Rh 0
1.04 2.49 4.04 6.03 575
R, (kJ/mole) -459.8 -460.37 -460.71 -461.68 -461.92
S”fk
6.59 6.44 6.34 6.18 6.06
r
0.9815 0.9610 0.9940 0.9557
516
C. Ko and REX B. MCLELLAN
9#C#
6.0 A-A-A
t
*
6.5
-.&-A
s
’
-----d,l.ox
’
I
’
1.0 10*/T
PO)
1
.
1.5
1
(K-l)
Fig. 1. Temperature-variation of the solubility Oi of N in Fe-Rh solid solutions in the form of plots of In(OiT7’4)vs reciprocal temperature. The nominal Rh-content (at. %) is given on each plot and each plot is displaced by an integer given in parentheses.
3. EXPERIMENTAL RESULTS
The results of the solubility measurements are seen in Fig. 1 in the form of plots of the quantity In (ei T7j4) vs l/T, where Oiis the atom ratio of the interstitial to the substitutional species N/(Fe + Rh). Each plot refers to the nominal value of 6, (given in at. % adjacent to each plot) and each plot is displaced by an integer (0, 1,. . . ) given in parentheses. It can be seen that the plots are linear and the scatter of the data points is small. Now the slope, 0, of plots of ln(fI,T7/4/lP’/2) vs 1/T, where 1 is a known constant [12] and P is the
N,-pressure, is given by [l 11: k@=E,D-iii,
(1)
where - Ef’ is the dissociation energy per atom of the N,-molecule at 0 K. Equation (1) is used to determine the partial molar enthalpy $ from the data of Fig. 1. The intercept of such plots at T = 00 is 3 p/k and the partial excess entropies thus determined are given in Table 1 together with the R/-values. The reference level for R, is the enthalpy of an N-atom at rest in a vacuum. The table also gives the correlation coefficient r. The first row entries in Table 1 give the thermodynamic parameters for the Fe-N binary austenite [lo]. The linearity of the plots in Fig. 1 indicates that the values of Ri and $7 are not temperaturedependent in the T-range studied to an extent exceeding the sensitivity of the experimental procedure. The values of f& (O-symbols, 1.h. scale) and 3 ;‘/k (A-symbols, r.h. scale) are plotted against 6. in Fig. 2. It is immediately clear that the variation of R, with Rh-content is slight. As 6, increases from O-0.06, i!& decreases by only w 2 kJ/mol (0.02 eV). 4. DISCUSSION
o
1
I
I
2
,
I
3 4 8, I Ioa
1
8
5
6
Fig. 2. Variation of the partial enthalpy and excess entropy of N in Fe-Rh-N solid solutions with the Rh-concentration 0“.
The linear variation of fli and 3:s with 0, as depicted in Fig. 2 is a general result common to the Fe-U-N austenites studied thus far [ 11.This behavior is in accord with the cell model for ternary solid solutions [2] and thus the present results may be discussed in terms of this model. The interstitial atoms can reside in any of (2 + 1) distinguishable interstitial sites (cells) whose first coordination shell contains n = 0, . . . , 2 U atoms and (Z - n) P-atoms (i.e. Fe-atoms). Thus, the statistical problem is to select the (Z -t 1) cell energy levels (El). The simplest
577
Thermodynamics of nitrogen austenites selection is the linear spectrum: E; = E, +
nE7,
(2)
where E,, is a reference energy. Having selected the set of E;, the thermodynamic functions of the interstitial atoms may be calculated from the cell partition functions Q,( V, T) and Q;( V, T) which are given by 121: Q,(k’, T)= Trg
(3)
and Q; (V, T) = Tr g’, where the elements of 2 and 2’ are:
E; exp (-.z;/kT),
where
Nn =(Z
Nz!
(1
-
e,)(“-n)e:
(5)
- n)!n!
and N = NV+ Nu is the number of Fe + Rh atoms. The partial enthalpies and excess entropies in terms of the partition functions are [2]:
-
44
(6)
and sy = (s:S),
+ k In Q,(V, T)
Q;K +TQV’Jz)
T)
+ @Lx.
(7)
In these expressions, the symbol co stands for the partial quantity in the binary V - i (Fe-N) system and a is the thermal expansivity of the matrix (taken to be pure V). The terms involving the quantity I$ originate from the dilation of the Fe lattice effected by the substantial solution of Pt atoms. The quantity 4 is given by [13]: 4 = ~$3,AV/V~,
(8)
where pi is the partial molar volume of the N-atoms, B,, is the bulk modulus of y-Fe at the experimental temperature, V,” is the molar volume of Fe, and AV = vu - P$ The value of VU, the partial molar volume of the U-species in the V-U binary lattice is taken from the X-ray lattice parameter data of Chao et al. [8] (A V = 0.362 x 10m6m3/mol), and E0 is taken from recent elastic measurements [13]. The values P.C.S. 48,6-F
of r#~calculated from the above data were used to calculate the “constant volume” enthalpy Ry, i.e. Ri + &VU,and the result is shown in the lower dashed line (-l-l-) in Fig. 2. This value of & was then used to calculate Ri from eqn (6) by an iterative procedure in which ~7 was stepped in intervals of 0.1 kJ/mol. The “best-fit” value of E! is - 2.3 kJ/mol, and values of R, computed from this $-value and eqn (6) are given by the symbols (+) in Fig. 2. The “volume correction” @,a in the excess entropy is entirely negligible. The calculated values of 3:s taken from eqn (7) and E: = -2.3 kJ/mol are shown by the uppermost dashed line (---) in Fig. 2. It is clear that both the partial molar enthalpies and excess entropies are, as would be expected, in excellent accord with the predictions of the cell model. In both constant pressure (Ry) systems, the partial enthalpy decreases as the U-content of the solution increases. It can thus be seen that, as in the cases of U = Pd and U = Pt, the addition of Rh atoms produces sites for N-interstitials which are traps rather than antitraps. This is understood in the sense that, as shown in Fig. 2, (dRi/d6,) and (d&/de,) are both negative, i.e. the N-atoms are more strongly bonded as 6. increases. The trap depth ~7 is, however, small (- 2.3 kJ/mol) so that the traps are relatively shallow in comparison with those formed when the transition metal Cr is the U-species. The comparison is clear from the following values of E! (in kJ/mol): Cr(-18.8), Pd (-8.9), Pt (-7.1) and Rh (-2.3). The effect of Rh additions upon the diffusivity of N in the Fe-Rh-N ternary system will clearly be small. This may be demonstrated by considering the combined effect of lattice dilation and the production of trapping sites on DWei, the N-diffusivity in the V-U matrix. It was shown recently [14] that the effect of U-additions which cause a positive lattice dilation is to increase D-u-i over the value of D” (i.e. V-i binary system) at a given temperature. The actual value of the diffusivity ratio Idi’ = D”“/D”’ depends upon the dilation AV and the motion volume for interstitial diffusion, V”. The value of 1 observed in a constant-pressure diffusion experiment is the product of Adi’,and the factor A”““, which takes the production of trapping (or antitrapping) sites into account. Thus, 1=
~dil~sror
(9)
In using eqn (9), 1”“’ is calculated from the cell theory using the constant-volume representation and 1” may be estimated from A V, V” [ 151. However 1 may be estimated more directly by considering a simple constant-pressure ensemble where the cell energies are given by: a;= nE;+pe,CEo.
(10)
This spectrum replaces eqn (2). The “direct” cell energies in the constant-pressure case are written a;
C. Ko and REX B. MCLELLAN
578
(as opposed to E;) and the term ~6. represents a linear lattice dilation term which affects cells of all n-values equally. The constant-pressure partition function Q,(P, T) is defined in a manner analogous to ew (3). Now it is easy to show [16], that if 8” Q 1 and only the potential minima are perturbed by the U-additions, then 1 is given by 1 = [Q,(P,
V-l.
The value of p may be obtained by noting that, from eqns (2) and (10) and the definitions of Q,(V, T) and
470-
Cf
QAP, T),
PI
QzV’,T) -----=exp QAK T)
PO. 1 -kT I ’
(12)
Now it is also easy to show from classical thermodynamics [17] that the chemical potential pi of the interstitial species is given by:
$ 4e5 .S If
:/A, 460
Rh
01234567691 8”
6
pi = % + kT ln Qz(p, T) 0.
=E,+kTlneZ(v,++&
(13)
and p may be identified with 4. Expanding Qz( V, T) up to n = 3 and using 13~4 1 shows that eqn (11) may be written
in the simple form:
Fig. 3. Variation of the partial enthalpy and excess entropy of N in Fe-U-N solid solutions for various U-elements. The inset diagram shows the variation of the cell interaction energy E! with the electronic specific heat coefficient when U is a noble metal.
Acknowledgemenf-The authors are grateful for the support provided by the National Science Foundation under the Metallurgy Program (Grant No. DMR-78-01306).
(14) REFERENCES
1 when C$> 0 the “volume” factor exp (qbO,/kT) leads to an increase in 1, and for trapping sites the denominator in expression (14) gives rise to an offsetting decrease in 1 as 8, increases. For Fe-Rh-N solutions, using the parameters generated here (4 = 16.16 kJ/ mol from eqn (8) and ~7 = -2.3 kJ/mol) eqn (14) gives I = 1.054 at 8, = 0.05 and 1 = 0.97 at 0” = 0.1. Thus this approximation suggests that the Ndiffusivity in Fe-Rh-N solutions should be virtually invariant to the Rh-content. It should be carefully noted that this conclusion is based upon the assumption that volume changes uniformly affect trap depths in the simple linear fashion taken in eqn (10) and that the positions of the potential maxima are not affected. Because of the small magnitude of the &values for Fe-(noble metal)-N-systems it is probable that meaningful calculations based upon currently advancing electronic models for interstitial solutions cannot be made [16] in the sense that the uncertainties inherent in such calculations are invariably of the same order as the &P-values themselves. It is interesting to note that the &p-values for the noble metals vary in a systematic way as the density of states at the Fermi surface increases. The inset diagram in Fig. 3 shows the $-values plotted against y, the electronic specific heat coefficient [18].
2. 3. 4. 5. 6.
Ko C. and McLellan R. B., Acra metall. 31, 1821 (1983). McLellan R. B., Acta metall. 30, 317 (1982). Ko C. and McLellan R. B., Acta metall. 34,2273 (1986). Ko C. and McLellan R. B., J. Phys. Chem. Solids, in press. Moffatt W. G., The Handbook of Binary Phase Diagrams. General Electric Co., Schenectedy, New York (1978). Bergevin F. and Muldawer L., Compt. Rend. 252, 1347 (1961).
7. Kouvel J. S. and Hartelius C. C., J. appl. Phys. Suppl. 33, 1343 (1962). 8. Chao C. C., Duwez P. and Tsuei C. C., J. appl. Phys. 42, 4282 (1971). 9. Lommel J. M. and Kouvel J. S., J. appl. Phys. 38, 128 (1967). 10. McLellan R. B. and Ferraro R. J., Acra metall. 28, 417 (1980). 11. Yoshihara M. and McLellan R. B., Acta metall. 29,471 (1981).
12. McL&lan R. B., Mater. Sci. Eng. 9, 121 (1972). 13. Yoshihara M. and McLellan R. B., J. Phys. Chem. Soli& 42, 767 (1981). 14. McLellan R. B. and Ishikawa T.. Scriofa . metall. 20, 99 (1986). 15. Ishikawa T. and McLellan R. B., Acta merail. 34, 1825 (1986). 16. Yoshihara M. and McLellan R. B., Acia metall., 35, 197 (1987). 17. McLellan R. B. and Kirchheim R., J. Phys. Chem. Solids 42, 157 (1981). 18. Gschneider K. A., Solid St. Physics (Edited by F. Seitz and D. Tumbull), Vol. 16, p. 275. Academic Press, New York (1964).
0022-3697187 Q 1987 Pergamon
S3.W + 0.00
Journals Ltd.
THE THERMODYNAMICS OF NITROGEN AUSTENITES CONTAINING NOBLE-METAL SUBSTITUTIONAL SOLUTES C. Ko and REX B. MCLELLAN William Marsh Rice University, Department of Mechanical Engineering and Materials Science, Houston, Texas 77251, U.S.A. (Received 28 July 1986; accepted 20 November 1986)
Abstract-A gas-solid equilibration technique has been used to determine the partial thermodynamic functions of nitrogen dissolved in fee Fe-Rh solid solutions. The data span the temperature range 1370-1530K and the composition range (Mat.% Rh. The N-contents never exceeded 5 x IO-‘at.%. The data were analyzed in terms of the cell model for ternary solid solutions. Adherence to the model was good and revealed that Rh atoms produce shallow trapping sites for N in the fee lattice. Their depth, relative to the potential minima in the Fe-N system is -2.3 kJ/mol. Keywords: Thermodynamics,
nitrogen, austenites, equilibrium, cell models.
1. INTRODUCTION Austenitic solid solutions containing nitrogen and a substitutional solute element have recently been studied in some detail for such cases where the
substitutional element (V) is a transition metal dissolved in the fee Fe solvent (I’) matrix. The results obtained for N-austenites containing Co, Ni, Cr and Mn have been recently reviewed [l] and it has been shown that the thermodynamic properties of these systems, where the interstitial N-atoms (i) are essentially present at “infinite dilution”, are in good accord with the cell model [2] for ternary solid solutions and that the U-i, interaction energies derived from analyzing the thermodynamic data using this model are systematically correlatable with the electronic structure of the three interacting atomic species. Until recently there were no data pertaining to N-austenites containing a substitutional element other than one which was a member of the First Long Period of the transition elements. However, studies have recently been made of austenites containing the noble metals Pd [3] and Pt [4]. In the present work, results will be presented on thermodynamic measurements of Fe-Rh-N austenites and the results obtained for all the Fe-(noble metal)-N solutions summarized. The fee phase of iron dissolves up to 45 at. % Rh over a large range of temperature [5]. There is a low-temperature transformation from the ferromagnetic to the antiferromagnetic state in solutions in the composition range 0, = 0.48-0.52 (0, is the atomic ratio Rh/Fe) [6,7]. However, the solutions in the composition-temperature ranges studied in this work are uniform solid solutions. The meta-stable solutions can be formed by rapid cooling, and the X-ray lattice parameters have been measured by Chao, Duwez and Tsuei [8]. Their results are in
excellent accord with previous data [9]. These data have been utilized in this work in order to assess the effect of matrix expansion on the thermodynamic parameters of dissolved N-atoms. 2. EXPERIMENTAL
PROCEDURE
The equilibrate-quench-analyze method has been used to measure the solubility of N under N,-gas at
a pressure of 1.04 x 10’ Pa in F+Rh solid solutions containing O-6 at. % Rh in the temperature range 1370-l 530 K. Since the N-solubility under these conditions is small (i.e. 5 x 10-4-10-4 atom fraction of N), it is essential to ensure that the Fe-Rh solutions contain low levels of impurities. The binary solutions were made by arc melting MARZ grade Fe and Rh under purified argon. The molten solutions were agitated so as to promote homogenization. The films used in the equilibration experiments were produced by rolling and intermediate vacuum annealing. The final foil thickness was 2.0 x 10e4 m. The apparatus used to equilibrate, quench and analyze the foils for N-content has been described in detail in previous papers [lo, 111. The compositions of the Fe-Rh foils were determined by the electron microprobe. The results were found always to be identical to the nominal composition (given in Table 1) to within the accuracy of the microprobe system (* 5%).
Table 1. Thermodynamic parameters Rh 0
1.04 2.49 4.04 6.03 575
R, (kJ/mole) -459.8 -460.37 -460.71 -461.68 -461.92
S”fk
6.59 6.44 6.34 6.18 6.06
r
0.9815 0.9610 0.9940 0.9557
516
C. Ko and REX B. MCLELLAN
9#C#
6.0 A-A-A
t
*
6.5
-.&-A
s
’
-----d,l.ox
’
I
’
1.0 10*/T
PO)
1
.
1.5
1
(K-l)
Fig. 1. Temperature-variation of the solubility Oi of N in Fe-Rh solid solutions in the form of plots of In(OiT7’4)vs reciprocal temperature. The nominal Rh-content (at. %) is given on each plot and each plot is displaced by an integer given in parentheses.
3. EXPERIMENTAL RESULTS
The results of the solubility measurements are seen in Fig. 1 in the form of plots of the quantity In (ei T7j4) vs l/T, where Oiis the atom ratio of the interstitial to the substitutional species N/(Fe + Rh). Each plot refers to the nominal value of 6, (given in at. % adjacent to each plot) and each plot is displaced by an integer (0, 1,. . . ) given in parentheses. It can be seen that the plots are linear and the scatter of the data points is small. Now the slope, 0, of plots of ln(fI,T7/4/lP’/2) vs 1/T, where 1 is a known constant [12] and P is the
N,-pressure, is given by [l 11: k@=E,D-iii,
(1)
where - Ef’ is the dissociation energy per atom of the N,-molecule at 0 K. Equation (1) is used to determine the partial molar enthalpy $ from the data of Fig. 1. The intercept of such plots at T = 00 is 3 p/k and the partial excess entropies thus determined are given in Table 1 together with the R/-values. The reference level for R, is the enthalpy of an N-atom at rest in a vacuum. The table also gives the correlation coefficient r. The first row entries in Table 1 give the thermodynamic parameters for the Fe-N binary austenite [lo]. The linearity of the plots in Fig. 1 indicates that the values of Ri and $7 are not temperaturedependent in the T-range studied to an extent exceeding the sensitivity of the experimental procedure. The values of f& (O-symbols, 1.h. scale) and 3 ;‘/k (A-symbols, r.h. scale) are plotted against 6. in Fig. 2. It is immediately clear that the variation of R, with Rh-content is slight. As 6, increases from O-0.06, i!& decreases by only w 2 kJ/mol (0.02 eV). 4. DISCUSSION
o
1
I
I
2
,
I
3 4 8, I Ioa
1
8
5
6
Fig. 2. Variation of the partial enthalpy and excess entropy of N in Fe-Rh-N solid solutions with the Rh-concentration 0“.
The linear variation of fli and 3:s with 0, as depicted in Fig. 2 is a general result common to the Fe-U-N austenites studied thus far [ 11.This behavior is in accord with the cell model for ternary solid solutions [2] and thus the present results may be discussed in terms of this model. The interstitial atoms can reside in any of (2 + 1) distinguishable interstitial sites (cells) whose first coordination shell contains n = 0, . . . , 2 U atoms and (Z - n) P-atoms (i.e. Fe-atoms). Thus, the statistical problem is to select the (Z -t 1) cell energy levels (El). The simplest
577
Thermodynamics of nitrogen austenites selection is the linear spectrum: E; = E, +
nE7,
(2)
where E,, is a reference energy. Having selected the set of E;, the thermodynamic functions of the interstitial atoms may be calculated from the cell partition functions Q,( V, T) and Q;( V, T) which are given by 121: Q,(k’, T)= Trg
(3)
and Q; (V, T) = Tr g’, where the elements of 2 and 2’ are:
E; exp (-.z;/kT),
where
Nn =(Z
Nz!
(1
-
e,)(“-n)e:
(5)
- n)!n!
and N = NV+ Nu is the number of Fe + Rh atoms. The partial enthalpies and excess entropies in terms of the partition functions are [2]:
-
44
(6)
and sy = (s:S),
+ k In Q,(V, T)
Q;K +TQV’Jz)
T)
+ @Lx.
(7)
In these expressions, the symbol co stands for the partial quantity in the binary V - i (Fe-N) system and a is the thermal expansivity of the matrix (taken to be pure V). The terms involving the quantity I$ originate from the dilation of the Fe lattice effected by the substantial solution of Pt atoms. The quantity 4 is given by [13]: 4 = ~$3,AV/V~,
(8)
where pi is the partial molar volume of the N-atoms, B,, is the bulk modulus of y-Fe at the experimental temperature, V,” is the molar volume of Fe, and AV = vu - P$ The value of VU, the partial molar volume of the U-species in the V-U binary lattice is taken from the X-ray lattice parameter data of Chao et al. [8] (A V = 0.362 x 10m6m3/mol), and E0 is taken from recent elastic measurements [13]. The values P.C.S. 48,6-F
of r#~calculated from the above data were used to calculate the “constant volume” enthalpy Ry, i.e. Ri + &VU,and the result is shown in the lower dashed line (-l-l-) in Fig. 2. This value of & was then used to calculate Ri from eqn (6) by an iterative procedure in which ~7 was stepped in intervals of 0.1 kJ/mol. The “best-fit” value of E! is - 2.3 kJ/mol, and values of R, computed from this $-value and eqn (6) are given by the symbols (+) in Fig. 2. The “volume correction” @,a in the excess entropy is entirely negligible. The calculated values of 3:s taken from eqn (7) and E: = -2.3 kJ/mol are shown by the uppermost dashed line (---) in Fig. 2. It is clear that both the partial molar enthalpies and excess entropies are, as would be expected, in excellent accord with the predictions of the cell model. In both constant pressure (Ry) systems, the partial enthalpy decreases as the U-content of the solution increases. It can thus be seen that, as in the cases of U = Pd and U = Pt, the addition of Rh atoms produces sites for N-interstitials which are traps rather than antitraps. This is understood in the sense that, as shown in Fig. 2, (dRi/d6,) and (d&/de,) are both negative, i.e. the N-atoms are more strongly bonded as 6. increases. The trap depth ~7 is, however, small (- 2.3 kJ/mol) so that the traps are relatively shallow in comparison with those formed when the transition metal Cr is the U-species. The comparison is clear from the following values of E! (in kJ/mol): Cr(-18.8), Pd (-8.9), Pt (-7.1) and Rh (-2.3). The effect of Rh additions upon the diffusivity of N in the Fe-Rh-N ternary system will clearly be small. This may be demonstrated by considering the combined effect of lattice dilation and the production of trapping sites on DWei, the N-diffusivity in the V-U matrix. It was shown recently [14] that the effect of U-additions which cause a positive lattice dilation is to increase D-u-i over the value of D” (i.e. V-i binary system) at a given temperature. The actual value of the diffusivity ratio Idi’ = D”“/D”’ depends upon the dilation AV and the motion volume for interstitial diffusion, V”. The value of 1 observed in a constant-pressure diffusion experiment is the product of Adi’,and the factor A”““, which takes the production of trapping (or antitrapping) sites into account. Thus, 1=
~dil~sror
(9)
In using eqn (9), 1”“’ is calculated from the cell theory using the constant-volume representation and 1” may be estimated from A V, V” [ 151. However 1 may be estimated more directly by considering a simple constant-pressure ensemble where the cell energies are given by: a;= nE;+pe,CEo.
(10)
This spectrum replaces eqn (2). The “direct” cell energies in the constant-pressure case are written a;
C. Ko and REX B. MCLELLAN
578
(as opposed to E;) and the term ~6. represents a linear lattice dilation term which affects cells of all n-values equally. The constant-pressure partition function Q,(P, T) is defined in a manner analogous to ew (3). Now it is easy to show [16], that if 8” Q 1 and only the potential minima are perturbed by the U-additions, then 1 is given by 1 = [Q,(P,
V-l.
The value of p may be obtained by noting that, from eqns (2) and (10) and the definitions of Q,(V, T) and
470-
Cf
QAP, T),
PI
QzV’,T) -----=exp QAK T)
PO. 1 -kT I ’
(12)
Now it is also easy to show from classical thermodynamics [17] that the chemical potential pi of the interstitial species is given by:
$ 4e5 .S If
:/A, 460
Rh
01234567691 8”
6
pi = % + kT ln Qz(p, T) 0.
=E,+kTlneZ(v,++&
(13)
and p may be identified with 4. Expanding Qz( V, T) up to n = 3 and using 13~4 1 shows that eqn (11) may be written
in the simple form:
Fig. 3. Variation of the partial enthalpy and excess entropy of N in Fe-U-N solid solutions for various U-elements. The inset diagram shows the variation of the cell interaction energy E! with the electronic specific heat coefficient when U is a noble metal.
Acknowledgemenf-The authors are grateful for the support provided by the National Science Foundation under the Metallurgy Program (Grant No. DMR-78-01306).
(14) REFERENCES
1 when C$> 0 the “volume” factor exp (qbO,/kT) leads to an increase in 1, and for trapping sites the denominator in expression (14) gives rise to an offsetting decrease in 1 as 8, increases. For Fe-Rh-N solutions, using the parameters generated here (4 = 16.16 kJ/ mol from eqn (8) and ~7 = -2.3 kJ/mol) eqn (14) gives I = 1.054 at 8, = 0.05 and 1 = 0.97 at 0” = 0.1. Thus this approximation suggests that the Ndiffusivity in Fe-Rh-N solutions should be virtually invariant to the Rh-content. It should be carefully noted that this conclusion is based upon the assumption that volume changes uniformly affect trap depths in the simple linear fashion taken in eqn (10) and that the positions of the potential maxima are not affected. Because of the small magnitude of the &values for Fe-(noble metal)-N-systems it is probable that meaningful calculations based upon currently advancing electronic models for interstitial solutions cannot be made [16] in the sense that the uncertainties inherent in such calculations are invariably of the same order as the &P-values themselves. It is interesting to note that the &p-values for the noble metals vary in a systematic way as the density of states at the Fermi surface increases. The inset diagram in Fig. 3 shows the $-values plotted against y, the electronic specific heat coefficient [18].
2. 3. 4. 5. 6.
Ko C. and McLellan R. B., Acra metall. 31, 1821 (1983). McLellan R. B., Acta metall. 30, 317 (1982). Ko C. and McLellan R. B., Acta metall. 34,2273 (1986). Ko C. and McLellan R. B., J. Phys. Chem. Solids, in press. Moffatt W. G., The Handbook of Binary Phase Diagrams. General Electric Co., Schenectedy, New York (1978). Bergevin F. and Muldawer L., Compt. Rend. 252, 1347 (1961).
7. Kouvel J. S. and Hartelius C. C., J. appl. Phys. Suppl. 33, 1343 (1962). 8. Chao C. C., Duwez P. and Tsuei C. C., J. appl. Phys. 42, 4282 (1971). 9. Lommel J. M. and Kouvel J. S., J. appl. Phys. 38, 128 (1967). 10. McLellan R. B. and Ferraro R. J., Acra metall. 28, 417 (1980). 11. Yoshihara M. and McLellan R. B., Acta metall. 29,471 (1981).
12. McL&lan R. B., Mater. Sci. Eng. 9, 121 (1972). 13. Yoshihara M. and McLellan R. B., J. Phys. Chem. Soli& 42, 767 (1981). 14. McLellan R. B. and Ishikawa T.. Scriofa . metall. 20, 99 (1986). 15. Ishikawa T. and McLellan R. B., Acta merail. 34, 1825 (1986). 16. Yoshihara M. and McLellan R. B., Acia metall., 35, 197 (1987). 17. McLellan R. B. and Kirchheim R., J. Phys. Chem. Solids 42, 157 (1981). 18. Gschneider K. A., Solid St. Physics (Edited by F. Seitz and D. Tumbull), Vol. 16, p. 275. Academic Press, New York (1964).