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Thermodynamics of the α- and β-phases equilibria and ordering in CuZn system
OOOl-6160/89 $3.00+ 0.00 Copyright c 1989Pergamoo Press plc
Acfa metall. Vol. 37, No. I I, pp. 3085-3090,1989 Printed in Great Britain. All rights reserved
THERMODYNAMICS OF THE a- AND B-PHASES EQUILIBRIA AND ORDERING IN Cu-Zn SYSTEM ZHOU
XIAOWANG
and T. Y. HSU
(XU ZUYAO)
Department of Materials Science, Shanghai Jiao Tong University, Shanghai, P.R. China (Received 20 July 1988; in revised form 19 December 1988)
Abstract-For both the solid solutions CLand p, composed of Cu and Zn, approximated by the regular solutions, the interaction parameters E” and Es of components in the a and p phases are calculated with the experimental activities. By the application of lattice stability parameters AG&Y and A[z~ of Cu and Zn, which are obtained from the phase diagram of Cu-Zn system in this paper, a general formula for AGfi+ is derived. On the basis of Inden formula for the change of free energy AG@-flin the /l-p ordering transition, it is proposed, through discussing the critical temperature T, of ordering and ordering degree as a function of temperature, that the theoretical maximum ordering degree can not be obtained for an alloy with given composition. The maximum ordering degree attained is approximately independent of composition for alloys with Xzn = 0.354.65, so an approximate equation for ordering degree as a function of temperature is suggested. This equation is used to calculate the a/(a + p’) and p’/(a + 6’) phase boundaries of Cu-Zn system, and the results are in good agreement with the phase diagram. R&sum&A partir des activites experimentales, on calcule les parametres d’interaction E” et Ep des composants dans les phases a et b, dans le cas des deux solutions solides a et p composees de cuivre et de zinc, trait&es de facon approchte comme des solutions regulieres. En appliquant les parambtres de stabilite reticulaire AC{;’ et AC!;” du cuivre et du zinc-que l’on obtient dans cet article a partir du diagramme de phases du systeme Cu-Zn-on deduit une formule gentrale pour AGfl’“. A partir de la formule de Inden decrivant la variation d’energie libre AG 8-p dans la transition de mise en ordre /I-b’, on propose-a travers une discussion du degre d’ordre et de la temperature critique T, de mise en ordre en fonction de la temperature--que le degre d’ordre theorique maximal ne peut &tre obtenu pour un alliage de composition don&e. Le degre d’ordre maximal atteint est a peu pres indtpendant de la composition pour les alliages dont la concentration en zinc varie entre 0,35 et 0,65, ce qui permet de presenter une equation approchee du degrt d’ordre en fonction de la temperature. On utilise cette equation pour calculer les joints de phases a/(m + /?‘) et P’/(a + j’) du systtme Cu-Zn. Les resultats sont en bon accord avec le diagramme de phases. Zusammenfassung-Fiir die beiden aus Cu und Zn bestehenden Mischkristalle a und b werden die Wechselwirkungsparameter E’ und Efi der Komponenten in den Phasen a und b aus den experimentell erhaltenen Aktivitaten bestimmt. Mit Hilfe der Parameter der Gitterstabilitat AC!? und AC!? von Cu und Zn, erhalten in dieser Arbeit aus dem Phasendiagramm, wird eine Formel%r AGfl’“-abgeleitet. Ausgehend von der Inden-Gleichung filr die Anderung der freien Energie AGp’P bei der Ordnungseinstellung B-/I’ ergibt sich bei einer Diskussion der kritischenTemperatur T, der Ordnungseinstellung und des Ordnungsgrades in Abhangigkeit von der Temperatur, dal3 der theoretische maximale Ordnungsgrad bei einer Legierung einer gegebenen Zusammensetzung nicht erhalten werden kann. Der sich einstellende maximale Ordnungsgrad ist im Bereich Xz, = 0,35 - 0,65 naherungsweise unabhlngig von der Zusammensetzung; es liegt also eine Naherungsgleichung fur den Ordnungsgrad als Funktion der Temperatur nahe. Mit dieser Gleichung werden die Phasengrenzen a - (a + p’) und p’ - (a + p’) im Zustandsdiagramm des Cu-Zn-Systems berechnet; die Ergebnisse stimmen mit dem Phasendiagramm gut ilberein.
1. INTRODUCTION According to the regular solution model, Kaufman [ 1,2] calculated the lattice stability parameters, AGg;’ and Act;‘, but his values fail to agree with the phase diagram of Cu-Zn system, as will be pointed out below. By means of the experimental phase diagram and measured thermodynamic data, Spencer [3] also evaluated the AC{;” and AGg;“, and obtained the interaction parameters E’ and Efl of components in the u and /I phases as functions of composition and temperature. In Spencer’s evaluation, the experimental thermodynamic data used are too scattered, which may be caused by the different experimental conditions of various authors, to get the
accurate results. In particular, just as Spencer himself believes, the data for the /I’ are not reliable, since it is difficult to establish equilibrium in the alloy specimens used for calorimetric measurement near room temperature. Using Spencer’s values for the /I’ [3], we found that the T, calculated through ACT’” = 0 is about 300 K higher than the martensitic transformation temperature M, [4], which contradicts the property of the thermoelastic martensitic transformation in Cu-Zn alloys. According to the currently widely used approximation that the chemical interchange energk are independent of composition and temperature [5-81, we regard the interaction parameters of the c( and /? phases as constants. Based on the present phase diagram [9, lo] and selected reliable 3085
3086
ZHOU and HSU:
THERMODYNAMICS
OF ORDERING
IN Cu-Zn
data [9] on activities, the thermodynamic parameters of the LYand /3 are recalculated, and the p’ is treated by a more physically-realistic ordering model. Of the many theories [11-161 developed on the problem of thermodynamics of the 8-p’ ordering transition, the BWG model [ 111possesses particularly the characteristics of simplicity and clarity. Applying the revised BWG model, i.e. the BWG-CVM model [8], a simplified equation for ordering is suggested in this paper such that the calculated results are in good agreement with the present phase diagram [9, lo]. 2. CALCULATIONS
Fig. 1. a/(~ + 8) and P/(M+ 8) phase boundaries calculated with the data of Kaufman’s [l, 21 as compared with the
AND RESULTS
practical phase diagram [9, lo].
2.1. Determination of E” and Es
The disordered a and fi phases may be regarded as regular solutions. The activity coefficient of component i in the regular solution 4, y!, should be RTlny?
= E@‘(l-Xf)*
(1)
where X, is the atom fraction of component i in the 4 phase. Substituting the experimental activities [9] into the above equation, the mean value of the interaction parameter of components in the CIphase is calculated as: E” = -29.047 (kJ/mol). Since the standard state of the experimental activity data [9] of Cu in the /I phase is the pure cc-Cu, which is necessary to be converted into that of the pure p-Cu, the following equation for chemical potentials of Cu with different standard states is adopted
= G$ + RT In X& + EB(l - X&,)’
(2)
where n, G, and a represent the chemical potential, free energy of pure component and activity, respectively, and the symbols in brackets refer the corresponding standard states. Simplifying equation (2) yields AGg;’ + RT In a&-c”1 = RT In X$ + E8(l - Xg,)‘.
(3)
Substituting equation (3) with different data of a& _ ‘“1and X& [9], the mean value of the interaction parameter of components in the p phase can be solved as: EB = -43.014 (kJ/mol). The scatter in E” and Efl for various composition are estimated to be 4% and 3% respectively in our calculation. With experimental data, Inden [9] determined the chemical interchange energies among nearest and next nearest neighbours in the /I phase as: W& = 955k and W&” = 535k (k = 13.8 x 10mz4J), where, (1) and (2) stand for nearest and next nearest interactions respectively. He considered only these two interactions. EB = -No(8 WC?,, + 6W&)/2 = -45.091 Since (kJ/mol) is close to the above result, the above calculated result of EB appears to be reliable. 2.2. Calculations of AG$;‘,
AG$;;” and AGfl’”
Adopting the experimental activity data of Cu and Zn at definite temperatures in the systems of CuZn
and Ag-Zn et al., Kaufman [1,2] calculated the lattice stability parameters AC&’ and AG$;’ of Cu and Zn at these temperatures, and then expressed them as linear functions of temperature through regression AG&” = -6276.30 + 3.34736T (Jjmol)
(4)
AG[;” = - 1046.05 + 0.83684T (J/ml).
(5)
and
Because the chemical potentials of components in the c( and p phases are equal when the CIand /I phases are in equilibrium the following equations hold AG&I*”= - RT In $,! - EpI(X;j,B)2+Efl(X&)z, (6) ( cl? ‘) and - E”(_@{)*+ E~(,@~)*.
(7)
The equilibrium phase boundaries X# and Xgk calculated through equations (6) and (7) with Kaufman’s AGC;’ and AC!;’ in equations (4) and (5) are shown in Fig. 1, together with those from the phase diagram [9, lo]. It is seen that the deviation of the calculated results from the phase diagram is very large. Owing to this, AC&;’ and AC&‘” should be recalculated in terms of the phase diagram. Because the published phase diagram [9, lo] of the Cu-Zn system only gives the equilibrium concentrations X>z and Xf: of the two phases within the temperature range 727-l 175 K, it is necessary to extrapolate the equilibrium concentration lines to low temperature region reasonably, so as the AG&+’and AGil” for the whole temperature range can be determined. Regression by linear function for X;t and X& from the phase diagram [9, lo] shown in Fig. 2 yields X2! = 0.5027-1.536 x 10m4T
(8)
A’!! = 0.582441.809 x lO-4 7’.
(9)
Substituting equations (8) and (9) into equations (6) and (7), calculation, regression and simplification give
ZHOU and HSU:
THERMODYNAMICS
OF ORDERING
3087
IN Cu-Zn
12OOr 115011004050 Y
1000-
+
950-
-
approximation cl
900
-
850
-
800
-
750
-
w3
B
Linear
I 240
-7500;
1 480
~,, 700 0.30
0.34
regression
I 720
0.42
0.460.50
Fig. 4. Two approximations of AG{;” compared with that of Kaufman’s-
X .?n
Fig. 2. Linear regression for the X2: and Xgt of the phase diagram. Phase diagram from Refs [9, lo]. Linear regression.
AGC;‘”= -7232.40 + 3.14348T (Jjmol)
(10)
AC&;‘= -325.08-0.797137 - 8.1704 x 10m4T2(J/mol).
I 1200
K)
T ( 0.38
I 960
(11)
In order to estimate the error of the data calculated, the AG&+’and AG&‘” shown in Figs 4 and 5, are also derived in the same way, from the X2! and Xgt taken as the tangents to the equilibrium concentration lines of the phase diagram shown in Fig. 3 at 727 K. It can be seen from Figs 4 and 5 that different approximations do not bring too large divergence as long as the Xgl and .Ygt are obtained from the shaded regions shown in Fig. 3, while the results calculated in terms of Kaufman’s data all deviate greatly. Because of the occurence of the martensitic transformation in alloys with X,, = 0.378880.4128 [4], the alloys should be in region of the single a phase at low temperature, i.e. the relationship of X9; and Xg with
temperature is singular, as shown in Fig. 6(a) and (b). If the influence of the concavity of the equilibrium concentration lines toward the Cu-rich side due to ordering is removed, we believe that Fig. 6(b) is more reasonable than Fig. 6(a). Therefore, it is certain that the actual Xg and Xg lie in the shaded region shown in Fig. 3. According to the results discussed above, the general formula for the driving force AGfl-” of phase transformation is expressed as follows AGB’” = Xc,AG~~~+X,,AG~I;“+X,,X,,(E”-EB) = - 7232.40 + 20874.32X,, - 13967X;, + (3.14348 - 3.94061X&T - 8.170 x 10-4Xz,T2. 2.3. Thermodynamics
(12)
of ordering transition
j?-Cu-Zn alloys will undergo the B -+fi’ (B2 structure) ordering transistion below a critical temperature T, [17]. For the thermodynamics of the ordering transition, introducing the concept of the chemical interchange energies and applying the BWG model [ll], Inden and Kikuchi supposed that the ordering of the b.c.c. binary alloys occurs at four sublattice sites, and derived the free energy for the ordered solid
-600 a E ;
t” “2
-
-1200-
-1800-
a -2400
X Z”
Fig. 3. Two linear approximattions for the a/(a + p) and B/(a + /3) phase boundaries. Linear regression approximation. --- Tangent approximation Phase diagram from Refs [9, IO].
-30001)
-
I 240
I 480
I
720
I
960
,\ 1200
T t Kl
Fig. 5. Two approximations of AG{,” compared with that of Kaufman’s
3088 (
T
ZHOU and HSU: (a)
THERMODYNAMICS
o+B
B
IN Cu-Zn
where symbol x is replaced by the order parameter 9 = x = (P[‘] - Plf1)/2 = Ptl - x0, to represent ordering degree.‘For alloys of given composition, define rl 2 0, then rl should satisfy
(b)
u
OF ORDERING
T
>)
when 1, < 0.5
(22)
0 < q < 1 - x,, when xB> 0.5.
(23)
O
X *n
Fig. 6. Two relationships of Xyi and X9: with temperature. (a) The relationship of Xgi and X& with temperature is not singular, alloys are in the single j region at both high and low temperature, and alloys have no martensitic transformation (b) The relationship of X9! and X{k with temperature is singular, alloys are in the single /I region at high temperature but in the single a region at low temperature, and alloys have the martensitic transformation.
solutions as follows [18-201 Gr = U,, - N0~,xb(4 W9 + 3 ~$9 - &Kg Wbb’- 6 W$9x2 + 3 ~$7~’ + z2)]/2 + N,,kT c(Pt
In P,” + Pf In Pf)/4
L
(13)
When equation (21) is used to calculate the ordering temperature T,., however, the results obtained is larger than the experimental ones [8]. This is believed to be caused mainly by the fact that, in the BWG model, only the long range order among atoms is considered while the short range order neglected [8]. Comparing the results of calculation in terms of the CVM model [14, 151 with those from the BWG model, Inden noticed that merely replacing T in equation (21) by T/x will yield results in good agreement with experimental ones [8], and he found x = 0.67 in Cu-Zn alloys [8]. Therefore, for Cu-Zn alloys, we have AGB’K = - 18415.5q2 - RT[2Xc,ln
where N, = 6.023 x 1O23(l/mol), U,, is the internal energy for pure components, and Pf the occupation probabilities of component i on the sublattice L (L = I, II, III, IV), while x, y and z stand for ordering degree, which are defined respectively as x = (Pfi + Pg - Pg* - P94
(14)
y = (Pf’ - P92
(15)
z = (Pfi - P92.
(16)
On B2 ordering,
+ 2x,, ln xz, - (rl + xc,) ln(? + xc,) -(&,-?)ln(Xz,-rl) - (rl + x&ln(rl
Py + Py = 2x,
(17)
Pjfl + Pp = 2&
(18)
p[‘l + pb’l= 1
(19)
PiI + Pgl = 1.
(20)
+ &,)
- (&, - q) ln (&, - q M.34. The ordering temperature (aA2GB’~/~~2)],,=, =0 as T, =
is
obtained
(24) from
2968.1 (25)
1/xc, + 1ix,,
PL = Pi’ # Pi” = Py . For simplic-
ity, the sublattices I and II are combined as the sublattice [l] and the sublattices III and IV as the sublattice [2], then Ptl = Pf, = Pi # Pyl = Pfil’= Pi”. Considering N,x,, atoms of a and Noxb atoms of b are distributed among N,/2 sites of the sublattice [l] and N,/2 sites of the sublattice [2], it is apparent that the following geometric relations for parameters should be satisfied
Xc,
The calculated results in terms of equation (25) shown in Fig. 7 are in relative good agreement with the phase diagram [9, lo]. The equilibrium ordering degree at a certain temperature is solved from
and the calculated ordering degree for alloys of various compositions is shown in Fig. 8.
Simplifying equation (13) with equations (17))(20), and defining the complete disordered state as Pt] = PFl= x0 and Prl = Ppl = xb, the change of free energy in the ordering transition is derived as ’
AGB’F = N,,(3 Wb’,’ - 4 W$‘)tl*
660 t
- RT]~x, ln L + 2xb ln zh
640
- (‘I + z,)ln(rl + x,)
t 6OOo.44 I
-(zh-q)ln(X,--) - (‘I + zJln(?
+ xb)
-(x,--‘lYnk-4W
0.45I
0.46,
0.47,
0.46,
X2.
(21)
Fig. 7. Critical ordering temperature T,.
0.49I
ZHOU and HSU:
THERMODYNAMICS
OF ORDERING
IN Cu-Zn
3089
cally by a method given here for the convenience of calculation by computer. Supposing that the j?’ phase with composition X& decomposes into the /?; phase with composition X& and the CIphase with composition X&, the driving force for the phase transformation should be
240
-
According to the regular solution model, the following formulae can also be listed: Gfl= X&G& + X%Gg, + RT(X&,ln X$
Fig. 8. Relationship between ordering degree and temperature for alloys with various compositions. Calculated results according to equations (24) and (26): (1) X,, = 0.35, (2) X,, = 0.40, (3) X,, = 0.45, (4) X,, = 0.50. Approximate equation: (5). X,, = 0.354.65.
+ X$ In Xg”) + Es Xc, X$ G” = X&G& + Xt, G’& + RT(X& In X& + X;, In Xi,) + E” X&, Xi,
(&)‘+(&J=’
(30)
Gfll = X&G& + X2”G$ + RT(X& In X&
+ X& In X&) + EP1X& X& It is seen from Fig. 8 that, ordering degree increases rapidly with decreasing temperature, and, when T = T,/2, q has approached the maximum value that the alloy can have with a definite composition. However, when calculating the phase diagram using the theoretical maximum ordering degree of alloys, i.e. equations (22) and (23), the results do not agree with the phase diagram, which is probably due to the failure of alloys to attain the maximum degree of ordering allowed by their compositions. The driving force for increasing the degree of ordering should be represented by F = -(aAGfl+f/aq). If it is assumed that r) has approached its maximum value, or rl = %nax- 0.001, at the temperature of T = 300 K, the values of Fare calculated through equation (24) as: F = 2480 (J/mol) for q,,, = 0.35 and F = 941 (Jjmol) for I],,, = 0.5. This can be concluded that the larger the theoretical maximum ordering degree, the more difficult it is to be attained. The correction for the maximum ordering degree is needed to make the relationship between the actual )I,,,~~ and X,, flatter, as shown in Fig. 9. For alloys of X,, = 0.354.65, the maximum ordering degree I],,, is taken approximately as independent of composition. For convenience of calculation, an approximate expression for ordering degree as a function of temperature is suggested according to Fig. 8
(29)
(31)
while Gr = Gfl+ AGfl’fi Gfi’= Gfl1+ AG+fli
(32) (33)
and E B= E BI(E is regarded as independent of composition). Combining equations (lo), (1 l), (24), (27x33) it is possible to calculate the driving force AGf-fli+z. ]AGp-fli+=] reaches its maximum when the j?’ phase with any composition in the ct + p’ two-phase region decomposes into the 8; with composition X$t and the c( with composition X2:. Supposing the composition of the /J’ be X$ = 0.40, the equilibrium concentrations Xgt and Xy[ can be obtained through solving for the extremum points by computer. Because the values of X# and X$t at 700 K calculated from the previous linear regression equations (8) and (9) are about 0.019 and 0.009 larger than those in the phase diagram, X;r + 0.019 and Xl! + 0.009 are substituted into formulae instead of Xy! and Xgt for this correction, and the final calculated results are shown in Fig. 10. It is seen from Fig. 10 that the X2: calculated according to the theory is in extremely good agreement with the phase diagram of Hansen’s [lo]. The present phase diagram has not determined the Xg formally [9], and the calculation here provides a theoretical base.
(27)
and the calculated results are shown in Fig. 8 [curve (S)] for qmax= 0.32. 2.4. Calculation of phase boundaries X2{ and XgA The phase diagrams [9, lo] proposed by various authors are different in the phase boundaries X2! and X[c, and the maximum ordering degree of rlmax= 0.32 estimated above must be checked as well. Therefore, the Xgf and Xc: are calculated theoreti-
Fig. 9. The relationship between the maximum ordering degree and composition of alloys.
ZHOU and HSU:
THERMODYNAMICS
OF ORDERING
IN Cu-Zn
2. The lattice stability parameters of Cu and Zn are calculated as AGC;‘”= -7232.40 + 3.14348T (Jjmol) and AGg;” = -325.08-0.79713T-8.1704 x 10-4T2 (Jjmol). 3. The critical ordering temperature T, of /?Cu-Zn alloys is derived as
700-
650-
2968.1
T, =
1/XC” + 1lxz, 0.29 0.32 0.35 0.38 0.41 0.44 0.47 0.50 %n
4. The equation for ordering degree as a function of temperature for the 8 phase with Xz, = 0.35-0.65 is suggested to be
Fig. 10. The calculated results of the phase boundaries Xyi and X@ compared with those of the phase diagram.
3. DISCUSSION
Avoiding using the scattered thermodynamic data of various authors, this paper has derived the lattice stability parameters AGt;’ and AG$;” from the phase diagram. The results are satisfactory because all the activity information between 727 and 1175 K provided by the phase diagram is fully utilized. Kaufman used values of different binary systems for regression, his results do not suit for the Cu-Zn system completely. Spencer’s evaluation also contained doubtful experimental data, and the results for the 8’ fail to calculate the To. Different from Ref. [3], this paper regards the interaction parameters as independent of composition and temperature, and the calculated results of X2: as well as XQ are in relative agreement with the present phase diagram. The problem of ordering is treated by using the BWGCVM model. In the calculation of the phase diagram, it has been suggested that the theoretical maximum ordering degree cannot be attained actually for alloys of Xz, = 0.35-0.65, and qmaxmay be regarded as independent of composition. That the rlmaxcannot reach the theoretical maximum value may be due to the effect of kinetics, which should be clarified in the future.
(&2)+(&>‘=‘. 5. The calculated equilibrium phase boundaries of the tl f/l’ two-phase region are in good agreement with Hansen’s phase diagram of the Cu-Zn system. REFERENCES I. L. Kaufman and H. Bernstein, Computer Calculations of Phase Diagrams, p. 46. Academic Press, New York
(1970). 2. L. Kaufman, Bull. Am. Phys. Sot. 4, 181 (1959). 3. P. J. Spencer. Calphad 10, 175 (1986). 4. H. Pops and T. B. Massalski, Trans. metall. Sot. A.I.M.E.
230, 1662 (1964).
5. S. C. Singh, Y. Murakami
and L. Delaey, Scripta
metall. 12, 435 (1978).
6. R. Rapacioli and M. Ahlers, Scripta metall. 11, 1147 (1977). 7. A. A. Arab and M. Ahlers, Acta metal. 36, 2627 (1988). 8. G. Inden, Z. Metallk. 66, 577, 648 (1975). 9. Selected Values of the Thermodynamic Properties of Binary Alloys, p. 812. Am. Sot. Metals, Metals Park, Ohio (1973). 10. M. Hansen and K. Anderko, Constitution of Binary Alloys, 2nd edn, p. 649. McGraw-Hill, New York (1958). 11. E. A. Guggenheim, Mixtures, p. 101. University Press, Oxford (1952). 12. C. H. P. Lupis and J. F. Elliott, Acta metall. 15, 265 (1967). 13. J.-C. Mathieu, F. Durand and E. Bonnier, J. them. Phys. 11, 1289 (1965). 14. R. Kikuchi and H. Sato, Acta metall. 22, 1099 (1974). 15. R. Kikuchi and C. M. Van Baal, Scripta metall. 8, 425 (1974).
4. CONCLUSIONS
Thermodynamic calculation reported in this paper yields the following conclusions; 1. The interaction parameters of components in the CIand fl phases are determined as E’= -29.047 (kJ/mol) and EB = -43.014 (kJ/mol).
16. L. D. Landau and E. M. Lifshitz, Statistical Physics, 3rd edn, Part I, pp. 451478. Pergamon Press, Oxford (1980). 17. F. W. Jones and C. Sykes, Proc. R. Sot., p. 440 (1937). 18. G. Inden, Acta metail. 22, 945 (1974). 19. G. Inden and W. Pitsch. Z. Metallk. 62. 627 (1971). 20. V. V. Geichenki, V. M. Danilenko and A.‘A. Smirnov, Fizika Metall. 13, 321 (1962).
Acfa metall. Vol. 37, No. I I, pp. 3085-3090,1989 Printed in Great Britain. All rights reserved
THERMODYNAMICS OF THE a- AND B-PHASES EQUILIBRIA AND ORDERING IN Cu-Zn SYSTEM ZHOU
XIAOWANG
and T. Y. HSU
(XU ZUYAO)
Department of Materials Science, Shanghai Jiao Tong University, Shanghai, P.R. China (Received 20 July 1988; in revised form 19 December 1988)
Abstract-For both the solid solutions CLand p, composed of Cu and Zn, approximated by the regular solutions, the interaction parameters E” and Es of components in the a and p phases are calculated with the experimental activities. By the application of lattice stability parameters AG&Y and A[z~ of Cu and Zn, which are obtained from the phase diagram of Cu-Zn system in this paper, a general formula for AGfi+ is derived. On the basis of Inden formula for the change of free energy AG@-flin the /l-p ordering transition, it is proposed, through discussing the critical temperature T, of ordering and ordering degree as a function of temperature, that the theoretical maximum ordering degree can not be obtained for an alloy with given composition. The maximum ordering degree attained is approximately independent of composition for alloys with Xzn = 0.354.65, so an approximate equation for ordering degree as a function of temperature is suggested. This equation is used to calculate the a/(a + p’) and p’/(a + 6’) phase boundaries of Cu-Zn system, and the results are in good agreement with the phase diagram. R&sum&A partir des activites experimentales, on calcule les parametres d’interaction E” et Ep des composants dans les phases a et b, dans le cas des deux solutions solides a et p composees de cuivre et de zinc, trait&es de facon approchte comme des solutions regulieres. En appliquant les parambtres de stabilite reticulaire AC{;’ et AC!;” du cuivre et du zinc-que l’on obtient dans cet article a partir du diagramme de phases du systeme Cu-Zn-on deduit une formule gentrale pour AGfl’“. A partir de la formule de Inden decrivant la variation d’energie libre AG 8-p dans la transition de mise en ordre /I-b’, on propose-a travers une discussion du degre d’ordre et de la temperature critique T, de mise en ordre en fonction de la temperature--que le degre d’ordre theorique maximal ne peut &tre obtenu pour un alliage de composition don&e. Le degre d’ordre maximal atteint est a peu pres indtpendant de la composition pour les alliages dont la concentration en zinc varie entre 0,35 et 0,65, ce qui permet de presenter une equation approchee du degrt d’ordre en fonction de la temperature. On utilise cette equation pour calculer les joints de phases a/(m + /?‘) et P’/(a + j’) du systtme Cu-Zn. Les resultats sont en bon accord avec le diagramme de phases. Zusammenfassung-Fiir die beiden aus Cu und Zn bestehenden Mischkristalle a und b werden die Wechselwirkungsparameter E’ und Efi der Komponenten in den Phasen a und b aus den experimentell erhaltenen Aktivitaten bestimmt. Mit Hilfe der Parameter der Gitterstabilitat AC!? und AC!? von Cu und Zn, erhalten in dieser Arbeit aus dem Phasendiagramm, wird eine Formel%r AGfl’“-abgeleitet. Ausgehend von der Inden-Gleichung filr die Anderung der freien Energie AGp’P bei der Ordnungseinstellung B-/I’ ergibt sich bei einer Diskussion der kritischenTemperatur T, der Ordnungseinstellung und des Ordnungsgrades in Abhangigkeit von der Temperatur, dal3 der theoretische maximale Ordnungsgrad bei einer Legierung einer gegebenen Zusammensetzung nicht erhalten werden kann. Der sich einstellende maximale Ordnungsgrad ist im Bereich Xz, = 0,35 - 0,65 naherungsweise unabhlngig von der Zusammensetzung; es liegt also eine Naherungsgleichung fur den Ordnungsgrad als Funktion der Temperatur nahe. Mit dieser Gleichung werden die Phasengrenzen a - (a + p’) und p’ - (a + p’) im Zustandsdiagramm des Cu-Zn-Systems berechnet; die Ergebnisse stimmen mit dem Phasendiagramm gut ilberein.
1. INTRODUCTION According to the regular solution model, Kaufman [ 1,2] calculated the lattice stability parameters, AGg;’ and Act;‘, but his values fail to agree with the phase diagram of Cu-Zn system, as will be pointed out below. By means of the experimental phase diagram and measured thermodynamic data, Spencer [3] also evaluated the AC{;” and AGg;“, and obtained the interaction parameters E’ and Efl of components in the u and /I phases as functions of composition and temperature. In Spencer’s evaluation, the experimental thermodynamic data used are too scattered, which may be caused by the different experimental conditions of various authors, to get the
accurate results. In particular, just as Spencer himself believes, the data for the /I’ are not reliable, since it is difficult to establish equilibrium in the alloy specimens used for calorimetric measurement near room temperature. Using Spencer’s values for the /I’ [3], we found that the T, calculated through ACT’” = 0 is about 300 K higher than the martensitic transformation temperature M, [4], which contradicts the property of the thermoelastic martensitic transformation in Cu-Zn alloys. According to the currently widely used approximation that the chemical interchange energk are independent of composition and temperature [5-81, we regard the interaction parameters of the c( and /? phases as constants. Based on the present phase diagram [9, lo] and selected reliable 3085
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data [9] on activities, the thermodynamic parameters of the LYand /3 are recalculated, and the p’ is treated by a more physically-realistic ordering model. Of the many theories [11-161 developed on the problem of thermodynamics of the 8-p’ ordering transition, the BWG model [ 111possesses particularly the characteristics of simplicity and clarity. Applying the revised BWG model, i.e. the BWG-CVM model [8], a simplified equation for ordering is suggested in this paper such that the calculated results are in good agreement with the present phase diagram [9, lo]. 2. CALCULATIONS
Fig. 1. a/(~ + 8) and P/(M+ 8) phase boundaries calculated with the data of Kaufman’s [l, 21 as compared with the
AND RESULTS
practical phase diagram [9, lo].
2.1. Determination of E” and Es
The disordered a and fi phases may be regarded as regular solutions. The activity coefficient of component i in the regular solution 4, y!, should be RTlny?
= E@‘(l-Xf)*
(1)
where X, is the atom fraction of component i in the 4 phase. Substituting the experimental activities [9] into the above equation, the mean value of the interaction parameter of components in the CIphase is calculated as: E” = -29.047 (kJ/mol). Since the standard state of the experimental activity data [9] of Cu in the /I phase is the pure cc-Cu, which is necessary to be converted into that of the pure p-Cu, the following equation for chemical potentials of Cu with different standard states is adopted
= G$ + RT In X& + EB(l - X&,)’
(2)
where n, G, and a represent the chemical potential, free energy of pure component and activity, respectively, and the symbols in brackets refer the corresponding standard states. Simplifying equation (2) yields AGg;’ + RT In a&-c”1 = RT In X$ + E8(l - Xg,)‘.
(3)
Substituting equation (3) with different data of a& _ ‘“1and X& [9], the mean value of the interaction parameter of components in the p phase can be solved as: EB = -43.014 (kJ/mol). The scatter in E” and Efl for various composition are estimated to be 4% and 3% respectively in our calculation. With experimental data, Inden [9] determined the chemical interchange energies among nearest and next nearest neighbours in the /I phase as: W& = 955k and W&” = 535k (k = 13.8 x 10mz4J), where, (1) and (2) stand for nearest and next nearest interactions respectively. He considered only these two interactions. EB = -No(8 WC?,, + 6W&)/2 = -45.091 Since (kJ/mol) is close to the above result, the above calculated result of EB appears to be reliable. 2.2. Calculations of AG$;‘,
AG$;;” and AGfl’”
Adopting the experimental activity data of Cu and Zn at definite temperatures in the systems of CuZn
and Ag-Zn et al., Kaufman [1,2] calculated the lattice stability parameters AC&’ and AG$;’ of Cu and Zn at these temperatures, and then expressed them as linear functions of temperature through regression AG&” = -6276.30 + 3.34736T (Jjmol)
(4)
AG[;” = - 1046.05 + 0.83684T (J/ml).
(5)
and
Because the chemical potentials of components in the c( and p phases are equal when the CIand /I phases are in equilibrium the following equations hold AG&I*”= - RT In $,! - EpI(X;j,B)2+Efl(X&)z, (6) ( cl? ‘) and - E”(_@{)*+ E~(,@~)*.
(7)
The equilibrium phase boundaries X# and Xgk calculated through equations (6) and (7) with Kaufman’s AGC;’ and AC!;’ in equations (4) and (5) are shown in Fig. 1, together with those from the phase diagram [9, lo]. It is seen that the deviation of the calculated results from the phase diagram is very large. Owing to this, AC&;’ and AC&‘” should be recalculated in terms of the phase diagram. Because the published phase diagram [9, lo] of the Cu-Zn system only gives the equilibrium concentrations X>z and Xf: of the two phases within the temperature range 727-l 175 K, it is necessary to extrapolate the equilibrium concentration lines to low temperature region reasonably, so as the AG&+’and AGil” for the whole temperature range can be determined. Regression by linear function for X;t and X& from the phase diagram [9, lo] shown in Fig. 2 yields X2! = 0.5027-1.536 x 10m4T
(8)
A’!! = 0.582441.809 x lO-4 7’.
(9)
Substituting equations (8) and (9) into equations (6) and (7), calculation, regression and simplification give
ZHOU and HSU:
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12OOr 115011004050 Y
1000-
+
950-
-
approximation cl
900
-
850
-
800
-
750
-
w3
B
Linear
I 240
-7500;
1 480
~,, 700 0.30
0.34
regression
I 720
0.42
0.460.50
Fig. 4. Two approximations of AG{;” compared with that of Kaufman’s-
X .?n
Fig. 2. Linear regression for the X2: and Xgt of the phase diagram. Phase diagram from Refs [9, lo]. Linear regression.
AGC;‘”= -7232.40 + 3.14348T (Jjmol)
(10)
AC&;‘= -325.08-0.797137 - 8.1704 x 10m4T2(J/mol).
I 1200
K)
T ( 0.38
I 960
(11)
In order to estimate the error of the data calculated, the AG&+’and AG&‘” shown in Figs 4 and 5, are also derived in the same way, from the X2! and Xgt taken as the tangents to the equilibrium concentration lines of the phase diagram shown in Fig. 3 at 727 K. It can be seen from Figs 4 and 5 that different approximations do not bring too large divergence as long as the Xgl and .Ygt are obtained from the shaded regions shown in Fig. 3, while the results calculated in terms of Kaufman’s data all deviate greatly. Because of the occurence of the martensitic transformation in alloys with X,, = 0.378880.4128 [4], the alloys should be in region of the single a phase at low temperature, i.e. the relationship of X9; and Xg with
temperature is singular, as shown in Fig. 6(a) and (b). If the influence of the concavity of the equilibrium concentration lines toward the Cu-rich side due to ordering is removed, we believe that Fig. 6(b) is more reasonable than Fig. 6(a). Therefore, it is certain that the actual Xg and Xg lie in the shaded region shown in Fig. 3. According to the results discussed above, the general formula for the driving force AGfl-” of phase transformation is expressed as follows AGB’” = Xc,AG~~~+X,,AG~I;“+X,,X,,(E”-EB) = - 7232.40 + 20874.32X,, - 13967X;, + (3.14348 - 3.94061X&T - 8.170 x 10-4Xz,T2. 2.3. Thermodynamics
(12)
of ordering transition
j?-Cu-Zn alloys will undergo the B -+fi’ (B2 structure) ordering transistion below a critical temperature T, [17]. For the thermodynamics of the ordering transition, introducing the concept of the chemical interchange energies and applying the BWG model [ll], Inden and Kikuchi supposed that the ordering of the b.c.c. binary alloys occurs at four sublattice sites, and derived the free energy for the ordered solid
-600 a E ;
t” “2
-
-1200-
-1800-
a -2400
X Z”
Fig. 3. Two linear approximattions for the a/(a + p) and B/(a + /3) phase boundaries. Linear regression approximation. --- Tangent approximation Phase diagram from Refs [9, IO].
-30001)
-
I 240
I 480
I
720
I
960
,\ 1200
T t Kl
Fig. 5. Two approximations of AG{,” compared with that of Kaufman’s
3088 (
T
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B
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where symbol x is replaced by the order parameter 9 = x = (P[‘] - Plf1)/2 = Ptl - x0, to represent ordering degree.‘For alloys of given composition, define rl 2 0, then rl should satisfy
(b)
u
OF ORDERING
T
>)
when 1, < 0.5
(22)
0 < q < 1 - x,, when xB> 0.5.
(23)
O
X *n
Fig. 6. Two relationships of Xyi and X9: with temperature. (a) The relationship of Xgi and X& with temperature is not singular, alloys are in the single j region at both high and low temperature, and alloys have no martensitic transformation (b) The relationship of X9! and X{k with temperature is singular, alloys are in the single /I region at high temperature but in the single a region at low temperature, and alloys have the martensitic transformation.
solutions as follows [18-201 Gr = U,, - N0~,xb(4 W9 + 3 ~$9 - &Kg Wbb’- 6 W$9x2 + 3 ~$7~’ + z2)]/2 + N,,kT c(Pt
In P,” + Pf In Pf)/4
L
(13)
When equation (21) is used to calculate the ordering temperature T,., however, the results obtained is larger than the experimental ones [8]. This is believed to be caused mainly by the fact that, in the BWG model, only the long range order among atoms is considered while the short range order neglected [8]. Comparing the results of calculation in terms of the CVM model [14, 151 with those from the BWG model, Inden noticed that merely replacing T in equation (21) by T/x will yield results in good agreement with experimental ones [8], and he found x = 0.67 in Cu-Zn alloys [8]. Therefore, for Cu-Zn alloys, we have AGB’K = - 18415.5q2 - RT[2Xc,ln
where N, = 6.023 x 1O23(l/mol), U,, is the internal energy for pure components, and Pf the occupation probabilities of component i on the sublattice L (L = I, II, III, IV), while x, y and z stand for ordering degree, which are defined respectively as x = (Pfi + Pg - Pg* - P94
(14)
y = (Pf’ - P92
(15)
z = (Pfi - P92.
(16)
On B2 ordering,
+ 2x,, ln xz, - (rl + xc,) ln(? + xc,) -(&,-?)ln(Xz,-rl) - (rl + x&ln(rl
Py + Py = 2x,
(17)
Pjfl + Pp = 2&
(18)
p[‘l + pb’l= 1
(19)
PiI + Pgl = 1.
(20)
+ &,)
- (&, - q) ln (&, - q M.34. The ordering temperature (aA2GB’~/~~2)],,=, =0 as T, =
is
obtained
(24) from
2968.1 (25)
1/xc, + 1ix,,
PL = Pi’ # Pi” = Py . For simplic-
ity, the sublattices I and II are combined as the sublattice [l] and the sublattices III and IV as the sublattice [2], then Ptl = Pf, = Pi # Pyl = Pfil’= Pi”. Considering N,x,, atoms of a and Noxb atoms of b are distributed among N,/2 sites of the sublattice [l] and N,/2 sites of the sublattice [2], it is apparent that the following geometric relations for parameters should be satisfied
Xc,
The calculated results in terms of equation (25) shown in Fig. 7 are in relative good agreement with the phase diagram [9, lo]. The equilibrium ordering degree at a certain temperature is solved from
and the calculated ordering degree for alloys of various compositions is shown in Fig. 8.
Simplifying equation (13) with equations (17))(20), and defining the complete disordered state as Pt] = PFl= x0 and Prl = Ppl = xb, the change of free energy in the ordering transition is derived as ’
AGB’F = N,,(3 Wb’,’ - 4 W$‘)tl*
660 t
- RT]~x, ln L + 2xb ln zh
640
- (‘I + z,)ln(rl + x,)
t 6OOo.44 I
-(zh-q)ln(X,--) - (‘I + zJln(?
+ xb)
-(x,--‘lYnk-4W
0.45I
0.46,
0.47,
0.46,
X2.
(21)
Fig. 7. Critical ordering temperature T,.
0.49I
ZHOU and HSU:
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3089
cally by a method given here for the convenience of calculation by computer. Supposing that the j?’ phase with composition X& decomposes into the /?; phase with composition X& and the CIphase with composition X&, the driving force for the phase transformation should be
240
-
According to the regular solution model, the following formulae can also be listed: Gfl= X&G& + X%Gg, + RT(X&,ln X$
Fig. 8. Relationship between ordering degree and temperature for alloys with various compositions. Calculated results according to equations (24) and (26): (1) X,, = 0.35, (2) X,, = 0.40, (3) X,, = 0.45, (4) X,, = 0.50. Approximate equation: (5). X,, = 0.354.65.
+ X$ In Xg”) + Es Xc, X$ G” = X&G& + Xt, G’& + RT(X& In X& + X;, In Xi,) + E” X&, Xi,
(&)‘+(&J=’
(30)
Gfll = X&G& + X2”G$ + RT(X& In X&
+ X& In X&) + EP1X& X& It is seen from Fig. 8 that, ordering degree increases rapidly with decreasing temperature, and, when T = T,/2, q has approached the maximum value that the alloy can have with a definite composition. However, when calculating the phase diagram using the theoretical maximum ordering degree of alloys, i.e. equations (22) and (23), the results do not agree with the phase diagram, which is probably due to the failure of alloys to attain the maximum degree of ordering allowed by their compositions. The driving force for increasing the degree of ordering should be represented by F = -(aAGfl+f/aq). If it is assumed that r) has approached its maximum value, or rl = %nax- 0.001, at the temperature of T = 300 K, the values of Fare calculated through equation (24) as: F = 2480 (J/mol) for q,,, = 0.35 and F = 941 (Jjmol) for I],,, = 0.5. This can be concluded that the larger the theoretical maximum ordering degree, the more difficult it is to be attained. The correction for the maximum ordering degree is needed to make the relationship between the actual )I,,,~~ and X,, flatter, as shown in Fig. 9. For alloys of X,, = 0.354.65, the maximum ordering degree I],,, is taken approximately as independent of composition. For convenience of calculation, an approximate expression for ordering degree as a function of temperature is suggested according to Fig. 8
(29)
(31)
while Gr = Gfl+ AGfl’fi Gfi’= Gfl1+ AG+fli
(32) (33)
and E B= E BI(E is regarded as independent of composition). Combining equations (lo), (1 l), (24), (27x33) it is possible to calculate the driving force AGf-fli+z. ]AGp-fli+=] reaches its maximum when the j?’ phase with any composition in the ct + p’ two-phase region decomposes into the 8; with composition X$t and the c( with composition X2:. Supposing the composition of the /J’ be X$ = 0.40, the equilibrium concentrations Xgt and Xy[ can be obtained through solving for the extremum points by computer. Because the values of X# and X$t at 700 K calculated from the previous linear regression equations (8) and (9) are about 0.019 and 0.009 larger than those in the phase diagram, X;r + 0.019 and Xl! + 0.009 are substituted into formulae instead of Xy! and Xgt for this correction, and the final calculated results are shown in Fig. 10. It is seen from Fig. 10 that the X2: calculated according to the theory is in extremely good agreement with the phase diagram of Hansen’s [lo]. The present phase diagram has not determined the Xg formally [9], and the calculation here provides a theoretical base.
(27)
and the calculated results are shown in Fig. 8 [curve (S)] for qmax= 0.32. 2.4. Calculation of phase boundaries X2{ and XgA The phase diagrams [9, lo] proposed by various authors are different in the phase boundaries X2! and X[c, and the maximum ordering degree of rlmax= 0.32 estimated above must be checked as well. Therefore, the Xgf and Xc: are calculated theoreti-
Fig. 9. The relationship between the maximum ordering degree and composition of alloys.
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2. The lattice stability parameters of Cu and Zn are calculated as AGC;‘”= -7232.40 + 3.14348T (Jjmol) and AGg;” = -325.08-0.79713T-8.1704 x 10-4T2 (Jjmol). 3. The critical ordering temperature T, of /?Cu-Zn alloys is derived as
700-
650-
2968.1
T, =
1/XC” + 1lxz, 0.29 0.32 0.35 0.38 0.41 0.44 0.47 0.50 %n
4. The equation for ordering degree as a function of temperature for the 8 phase with Xz, = 0.35-0.65 is suggested to be
Fig. 10. The calculated results of the phase boundaries Xyi and X@ compared with those of the phase diagram.
3. DISCUSSION
Avoiding using the scattered thermodynamic data of various authors, this paper has derived the lattice stability parameters AGt;’ and AG$;” from the phase diagram. The results are satisfactory because all the activity information between 727 and 1175 K provided by the phase diagram is fully utilized. Kaufman used values of different binary systems for regression, his results do not suit for the Cu-Zn system completely. Spencer’s evaluation also contained doubtful experimental data, and the results for the 8’ fail to calculate the To. Different from Ref. [3], this paper regards the interaction parameters as independent of composition and temperature, and the calculated results of X2: as well as XQ are in relative agreement with the present phase diagram. The problem of ordering is treated by using the BWGCVM model. In the calculation of the phase diagram, it has been suggested that the theoretical maximum ordering degree cannot be attained actually for alloys of Xz, = 0.35-0.65, and qmaxmay be regarded as independent of composition. That the rlmaxcannot reach the theoretical maximum value may be due to the effect of kinetics, which should be clarified in the future.
(&2)+(&>‘=‘. 5. The calculated equilibrium phase boundaries of the tl f/l’ two-phase region are in good agreement with Hansen’s phase diagram of the Cu-Zn system. REFERENCES I. L. Kaufman and H. Bernstein, Computer Calculations of Phase Diagrams, p. 46. Academic Press, New York
(1970). 2. L. Kaufman, Bull. Am. Phys. Sot. 4, 181 (1959). 3. P. J. Spencer. Calphad 10, 175 (1986). 4. H. Pops and T. B. Massalski, Trans. metall. Sot. A.I.M.E.
230, 1662 (1964).
5. S. C. Singh, Y. Murakami
and L. Delaey, Scripta
metall. 12, 435 (1978).
6. R. Rapacioli and M. Ahlers, Scripta metall. 11, 1147 (1977). 7. A. A. Arab and M. Ahlers, Acta metal. 36, 2627 (1988). 8. G. Inden, Z. Metallk. 66, 577, 648 (1975). 9. Selected Values of the Thermodynamic Properties of Binary Alloys, p. 812. Am. Sot. Metals, Metals Park, Ohio (1973). 10. M. Hansen and K. Anderko, Constitution of Binary Alloys, 2nd edn, p. 649. McGraw-Hill, New York (1958). 11. E. A. Guggenheim, Mixtures, p. 101. University Press, Oxford (1952). 12. C. H. P. Lupis and J. F. Elliott, Acta metall. 15, 265 (1967). 13. J.-C. Mathieu, F. Durand and E. Bonnier, J. them. Phys. 11, 1289 (1965). 14. R. Kikuchi and H. Sato, Acta metall. 22, 1099 (1974). 15. R. Kikuchi and C. M. Van Baal, Scripta metall. 8, 425 (1974).
4. CONCLUSIONS
Thermodynamic calculation reported in this paper yields the following conclusions; 1. The interaction parameters of components in the CIand fl phases are determined as E’= -29.047 (kJ/mol) and EB = -43.014 (kJ/mol).
16. L. D. Landau and E. M. Lifshitz, Statistical Physics, 3rd edn, Part I, pp. 451478. Pergamon Press, Oxford (1980). 17. F. W. Jones and C. Sykes, Proc. R. Sot., p. 440 (1937). 18. G. Inden, Acta metail. 22, 945 (1974). 19. G. Inden and W. Pitsch. Z. Metallk. 62. 627 (1971). 20. V. V. Geichenki, V. M. Danilenko and A.‘A. Smirnov, Fizika Metall. 13, 321 (1962).