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The influence of long range order on the martensitic transformation in Cu–Zn
Materials Science and Engineering A273 – 275 (1999) 195 – 199 www.elsevier.com/locate/msea
The influence of long range order on the martensitic transformation in Cu–Zn J.E. Garce´s, M. Ahlers * Centro Ato´mico Bariloche, 8400, S.C. de Bariloche, Argentina
Abstract The transformation between the a (fcc) and b (bcc) equilibrium phases at elevated temperatures is compared with that between the long range ordered B2 phase and the martensite at low temperatures, taking the Cu – Zn system as a prototype. The low temperature vibrational entropy difference, as measured from the martensitic transformation, is extrapolated to elevated temperatures at which disordered a and b are stable. This makes it possible to compare the corresponding enthalpy differences and deduce pair interchange energies, which depend only on pair distance but not on the crystal structure. This result is consistent with the observation that in many alloys based on the noble metals the enthalpy difference between the equilibrium structures is controlled mainly by the electron concentration e/a although the much larger mixing enthalpy of each of these structures depends strongly on the specific system. It also permits to evaluate the short range order (SRO) contribution at elevated temperatures using the cluster variation method. It is shown that the Gibbs free energies due to SRO are almost the same in the a and b phases, although they are non-negligible. © 1999 Elsevier Science S.A. All rights reserved. Keywords: Martensitic transformation; Long range order; Cluster variation method; Equilibrium phases; Cu – Zn
1. Introduction The martensitic transformation in binary Cu–Zn occurs between a B2 (CsCl type) austenite (b phase) and a 9R monoclinic martensite with the inherited atomic distribution, which is quite well known, since by an adequate heat treatment prior to the transformation a quasiperfect long range ordered (LRO) state can be produced. Long range order has a profound effect on phase stability and is generally described in terms of pair interchange energies between A – B atom pairs in ineighbor position as: (i) (i) (i) w (i) AB = − 2V AB +V AA +V BB
(1)
where the V (i) AB are the corresponding pair interaction energies in a given crystal structure. Due to the small hysteresis between the martensitic transformation and retransformation an equilibrium M temperature T M can be 0 and an entropy difference DS * Corresponding author. Tel.: +54-2944-445157; fax: +54-2944445299. E-mail address: [email protected] (M. Ahlers)
determined experimentally with a good precision. In order to determine the LRO contribution and deduce the difference in pair interchange energies between austenite and martensite, two problems have to be solved: 1. On the one hand a comparison is to be made between an fcc disordered a phase with an fcc LRO martensite (3R). Instead, a monoclinic 9R is generally formed. It is therefore necessary to relate both, the 3R with the 9R LRO structure. This problem has been solved by studying the ternary Cu–Zn–Al alloys. The deviation from the cubic symmetry has been found to be negligible [1] and the enthalpy difference between 9R and 3R has been measured [2]. 2. On the other hand, the a and b phases retain some short range order (SRO) even at elevated temperatures. Its contribution to the enthalpy of mixing may be small, but not negligible compared with the enthalpy differences between the phases. It is therefore necessary to have an estimate of the influence of SRO, which can be calculated once the pair interchange energies are known. One of the aims of this work is to derive these energies, to calculate the
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SRO, and finally to estimate the LRO contribution to the stability of the martensitic phases. At elevated temperatures the a and b phases are separated by a two phase (a + b) field in the phase diagram. The temperature T a/b at which the Gibbs free energy difference between both phases is zero lies somewhere within the two phase field. Describing the free energy of mixing by a term which depends on the pair interchange energies and another which varies linearly with composition and includes the influence of the electron concentration [3], we have shown using the cluster variation method (CVM) that T a/b can be located in the middle of the two phase (a +b) region. In order to determine the enthalpy difference DH a/b at this temperature, the entropy difference DS a/b has to be known. In this paper the vibrational contribution to DS a/b will be derived by extrapolating to elevated temperatures the DS M obtained from the martensitic transformation. The enthalpy difference between disordered a and b differs from that associated with the martensitic transformation by a combination of pair interchange energies. A second objective of this paper therefore is to calculate the value of this combination, and derive from it the individual pair interchange energies which are used to estimate the SRO contribution at elevated temperatures. Here only the Cu – Zn system is being analysed as a prototype. But the results are applicable also to other noble metal alloys. 2. The entropy difference DS a/b between the a and b equilibrium phases. The entropy difference DS a/b is the sum of three cona/b tributions: configurational (DS a/b conf), electronic (DS el ) a/b a/b and vibrational (DS vib ). DS conf will be considered in Section 4. DS a/b el , which is due to the conduction electrons, can be neglected since the electronic specific heats in a and b differ very little [3]. In order to derive the vibrational entropy differences, DS a/b vib at the elevated temperatures at which disordered a and b are stable from the measured vibrational entropy in the martensite, DS M, it is necessary to use more indirect extrapolation methods which will be discussed now. The vibrational entropy DS M has been measured below about 100°C by the martensitic transformation. It has been shown that it is due to the soft transversal (110)11( 0 phonon branch in the b phase [4], which is already fully excited at 80 K and leads to a constant DS M above 80 K. For several alloys which undergo a martensitic transformation the (110)11( 0 phonon branch has been measured by neutron diffraction. It has been shown that it increases slightly with temperature, including its long wave limit, the elastic constant c% =(c11 −c12)/2. It will be assumed in the following
that the temperature variation of c% reflects reasonably well the temperature change in the energy of the soft modes, and also that of the vibrational entropy difference DSvib, up to elevated temperatures. 1 dDSvib 1 dc% =− DSvib dT c% dT
(2)
The elastic constant c% does not show a simple temperature behavior: above MS c% increases first with temperature. In Cu–46.4%Zn, which does not transform martensitically, c% nevertheless increases up to 250°C and then starts to decrease influenced also by a decreasing LRO. The results for Cu–46.4%Zn [5] are characteristic also for ternary alloys which undergo a martensitic transformation. Measurements above the critical B2 ordering temperature T B2 C do not exist, and since an extrapolation of c% to higher temperatures is not possible, some other arguments have to be used. Therefore the extrapolation scheme will be divided into two parts: 1. First the relationship between DS M at the martensitic transformation temperature will be compared with that near T B2 C via the c% dependence followed by that above T B2 C . A collection of data for c% near MS in ternary Cu–Zn–Al and Cu–Al–Be [6] indicate that c% lies between 6 and 7 GPa. On the other hand c% near T B2 C is about 7.6 GPa for Cu–46.4%Zn [5] and 48.1%Zn [7] and still decreases rapidly with increasing temperature. Considering the uncertainties in the evaluation, it seems reasonable to conclude that DSvib does not differ notably between MS and a temperature slightly above T B2 C where LRO and SRO are no longer important. In the following both entropy differences are set equal. a/b 2. Of interest here is DS a/b . It has been shown vib at T M that DS increases with e/a [6]. On the other hand with increasing zinc concentration T a/b decreases, implying an increase in c%, i.e. a decrease in the temperature dependent part of DSvib. Both components therefore cancel to some extent. In order that they cancel completely it is necessary to have 1 dc% = − 4.10 − 4 K − 1 This is a reasonable value c% dT when compared with bcc Na ( − 5× 10 − 4 K − 1) [8], with the c44 in Cu–Zn, which behave normally, (−4× 10 − 4 K − 1) [9] and even with fcc Cu ( −4× 10 − 4 K − 1). It is suggested therefore that DS a/b vib can be supposed to be constant along the T a/b equilibrium temperatures. As a result of this discussion it is concluded that DS a/b vib is also similar to that for the martensitic transformation at low temperatures. For lack of more precise data, and of a good estimate of the error involved, both will be set equal in the following. An uncertainty of 9 10% would seem reasonable, which is already included in the measurement of DS M.
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3. The pair interchange energies As will be shown in the next section, the SRO contribution to the Gibbs free energy difference between a and b at T a/b is so small, compared with the other uncertainties, that it can be neglected. Then from T a/b, from the equilibrium temperature T M for the 0 martensitic transformation between fcc based 3R martensite and B2 ordered austenite, and from DS M = (1.2 9 0.1) J mol − 1 K − 1 [6], the enthalpy difference between disordered a and b on the one hand, and B2 and 9R on the other hand can be determined. For perfect order the relation is: M (1) (T a/bf− T M −3w (2) −2m (1) +3m (2))C 2Zn 0 )DS =(4w (3) M where f is the ratio between entropies, f = DS a/b vib /DS , which has been set equal to 1 here. Inserting the values leads to:
4w (1) − 3w (2) − 2m (1) +3m (2) =(5.6 9 0.6) kJ mol − 1 (4) Here only first and second neighbor pair contributions in the bcc and in the fcc phase (denoted by w (i) and m (i), respectively) are taken into account. The influence of more distant pairs has been neglected. Although their interchange energies are not zero, they are small compared to first and second neighbor pair contributions [10]. Moreover, their contribution in the two phases of the same atomic volume and electron concentration is expected to be nearly the same and therefore largely cancels in Eq. (4). The w (i) determine also the critical ordering temperature T B2 C , which can be described with a good approximation for an equiatomic composition by [11], again setting w (i) =0 for i \2: (1) kT B2 −1.5x2w (2). C (0.5)= 2x1w
(5)
In the Bragg–Williams approximation x1 =x2 = 1, but due to SRO, x1 is reduced to x1 =0.811. However, x2 = 1, which indicates that the influence of second neighbor SRO on T B2 C is very small, although it may not be exactly zero. Without further assumptions, the (1) measurement of T B2 and C permits only to relate w (2) w . In order not to introduce additional, less justified assumptions, a range of possible w (i) are used, which are listed in Table 1 and are discussed below. The w (2) are then calculated from Eq. (5) and m (1) −1.5m (2) from Eq. (4). In the following row the m (1) are linearly interpolated between w (1) and w (2) with respect to the pair distances which are known in bcc and fcc, because fcc and bcc have the same atomic volume. The m (2) are then derived. Two more rows in Table 1 contain the resulting 6m (1) + 3m (2) and 4w (1) +3w (2). These quantities determine that part of the enthalpy of mixing of disordered a and b, which depends on the pair inter-
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change energies only, assuming that they are not different from those determined from the LRO contribution. First of all, from Table 1 it can be noted that 4w (1) + 3w (2) is very similar to 6m (1) + 3m (2), independent of the w (1) that is chosen. This means that the enthalpies of mixing of a and b, which depend on the pair interchange energies, are closely the same. This is a very satisfactory result, which had already been suggested earlier [12], when the enthalpies of mixing for several alloy systems were compared. Although they depend strongly on the alloy considered, the much smaller difference between a and b is determined only by e/a. On its basis it had been deduced that m (1) can be interpolated between w (1) and w (2), thus postulating that the close neighbor pair interchange energies depend on the pair distance but not on the crystal structure, fcc, bcc or hexagonal [12]. The same conclusion follows from the present approach. It is therefore concluded that the derivation of Eq. (4) through an extrapolation of DS M is as good as using a structure independent enthalpy of formation given by: 4w (1) + 3w (2) = 6m (1) + 3m (2)
(6)
excluding the electronic part. It is clear that in both cases assumptions have to be made, which may result in an error in the numerical values in Eq. (4), or in a deviation from equality in Eq. (6). The important point to be made here is that these deviations are sufficiently small not to invalidate the general conclusions. The fact that both approaches lead to the same results is a further support for the validity of structure independent pair interchange energies. Some additional information is available to select the most likely combination of pair interchange energies in Table 1. In earlier work the w (1) = 7.96 kJ mol − 1 had been taken from the literature [13], which was based on the assumption that the enthalpy of mixing in the austenite is due only to the pair interchange energy. By using the different sets in Table 1, it has been found that the validity of Eq. (4) is quite independent of the exact combination of w (1) and w (2), provided T B2 C (Eq. (5)) and the structure independence of the pair interchange energies remain unaltered. Thus also earlier results [11,12] do not become obsolete. Table 1 Different sets of pair interchange energies for the a and b phases w (1) (kJ mol−1) w (2) (kJ mol−1) m (1)−1.5m (2) (kJ mol−1) m (1) (kJ mol−1) m (2) (kJ mol−1) 4w (1)+3w (2) (kJ mol−1) 6m (1)+3m (2) (kJ mol−1)
7.96 4.46 6.4 90.3 7.25 0.58 45.2 45.6
6.25 2.63 5.8 90.3 5.58 −0.13 32.8 33.0
5.00 1.29 5.39 0.3 4.29 −0.67 23.8 22.3
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From ab initio calculations, w (1) =5.42 kJ mol − 1 and w (2) = 0.83 kJ mol − 1 were derived [14], neglecting charge correlations between the atoms. The enthalpy of mixing that was derived was smaller than the observed one. This could be remedied by including the charge correlations [15]. Adding this effect [15] and assuming that it modifies only w (1) and w (2), while keeping T B2 C constant, an increase to w (1) =6.25 kJ mol − 1 and w (2) =2.63 kJ mol − 1 is obtained. This seems to be the most reasonable value. It leads to a small negative m (2) which is also consistent with the measurements of SRO by neutron diffraction [10]. Furthermore it agrees favorably with the measured enthalpies of mixing, which would lead to 6m (1) +3m (2) =35 92 kJ mol − 1, if any electronic contribution were absent. Since the enthalpy of mixing is determined experimentally with respect to hexagonal Zn, an additional contribution of 3.3 kJ mol − 1 should have to be added to refer it to fcc Zn. It seems quite reasonable to expect that the conduction electrons contribute also to the stability of the alloy, compared to the pure elements, and therefore a lower pair interchange contribution of 33.0 kJ mol − 1 would be quite reasonable. If in Eq. (3) a 10% deviation of M DS a/b were admitted, this would lead to a vib from DS shift of 0.42 kJ mol − 1 in w (2) (Table 1), which still is sufficiently small not to modify seriously the present results.
Fig. 1. Short range order contribution to the Gibbs free energy in the b phase as function of zinc concentration for different sets of pair interchange energies: (a) w (1) =5.00 kJ mol − 1, w (2) =1.29 kJ mol − 1 and m (1) =4.29 kJ mol − 1 (b): w (1) =6.25 kJ mol − 1, w (2) = 2.63 kJ mol − 1 and m (1) =5.58 kJ mol − 1 and (c): w (1) =7.96 kJ mol − 1, w (2) =4.46 kJ mol − 1 and m (1) =7.25 kJ mol − 1. The curves at each of the sets (a) to (c) correspond to 975, 1025, 1075, 1125 and 1175 K from the lowest to the uppermost curve, respectively.
2) that the difference in Gibbs free energy between a and b due to SRO is reduced. The reduction is such that T a/b can be used as the temperature Te between the
4. The short range order contribution The SRO has been determined by the cluster variation method (CVM) [16], using for bcc and fcc the tetrahedron approximation which includes first and second neighbor interchange energies for bcc and first neighbors only for fcc. Contributions from second neighbors are small in fcc. Fig. 1 shows the SRO contribution to the Gibbs free energy for the bcc phase as a function of Zn concentration at various temperatures, using the three w (1) combinations listed in Table 1. It can be noted from the figure that there is a considerable SRO contribution. If now the SRO contribution in the a phase were negligible, using the argument that LRO has not been observed in a, then a large influence on T a/b could be expected, leading to a shift from the equilibrium temperature Te between a and b in the absence of SRO given by: T a/b − Te DG bSRO −DG aSRO DG bSRO = : Te DH a/b T a/b · DS M
(7)
which would imply a shift of 20 – 50%, for the three sets of pair interchange energies, which in no way could be neglected. These values are reduced when SRO is present in the a phase. In fact, using the corresponding m (1) from Table 1 and neglecting m (2) it is found (Fig.
Fig. 2. Short range order contributions to the difference in Gibbs free energy between the b and a phases as function of zinc concentration for different sets of pair interchange energies: (a) w (1) = 5.00 kJ mol − 1, w (2) =1.29 kJ mol − 1 and m (1) =4.29 kJ mol − 1 (b): w (1) = 6.25 kJ mol − 1, w (2) =2.63 kJ mol − 1 and m (1) =5.58 kJ mol − 1 and (c): w (1) =7.96 kJ mol − 1, w (2) =4.46 kJ mol − 1 and m (1) = 7.25 kJ mol − 1. The curves at each of the sets (a) to (c) correspond to 975, 1025, 1075, 1125 and 1175 K from the lowest to the uppermost curve, respectively.
J.E. Garce´s, M. Ahlers / Materials Science and Engineering A273–275 (1999) 195–199
are a function only of the pair distance, thus supporting further the earlier conclusions which had been obtained in a different way.
completely disordered phases, independent of the exact value of m (1), once the structure insensivity of the pair interchange energies is accepted.
5. Conclusions
199
References
1. It is shown that at elevated temperatures, SRO has a negligible effect on the temperatures at which the difference in Gibbs free energy between a and b is zero. 2. The corresponding entropy difference between a and b has been estimated by extrapolating the entropy difference for the martensitic transformation to higher temperatures. 3. By comparing the enthalpy difference between disordered a and b with that for the martensitic transformation between B2 ordered b austenite and fcc based 3R martensite, the long range order contribution to the martensitic transformation has been obtained. 4. The results are consistent with the assumption that the pair interchange energies for two closest neighbors pairs are practically structure independent, and
.
[1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16]
F. Saule, M. Ahlers, Acta Metall. Mater. 43 (1995) 2373. M. Ahlers, J.L. Pelegrina, Acta Metall. Mater. 40 (1992) 3213. T.B. Massalski, V. Mizutami, Prog. Mat. Sci. 22 (1978) 151. R. Romero, M. Ahlers, J. Phys. Condens. Matter 1 (1989) 3191. A. Prasetyo, F. Reynand, H. Warlimont, Acta Metall. 24 (1979) 1009. R. Romero, J.L. Pelegrina, Phys. Rev. B 50 (1994) 9046. P.L. Young, A. Bienenstock, J. Appl. Phys. 42 (1971) 3008. See Landolt-Bornstein, Neue Serie III, 2, Springer 1969. G. Guenin, M. Morin, P.F. Gobin, W. Dejonghe, L. Delaey, Scripta Metall. 11 (1977) 1071. L. Reinhard, B. Scho¨nfeld, G. Kostorz, W. Bu¨hrer, Zeitschr. Metallkde. 84 (1993) 251. M. Ahlers, Phil. Mag. A 70 (1994) 247. M. Ahlers, Zeitschr. Physik B 99 (1996) 491. G. Inden, Zeitschr. Metallkde 66 (1975) 577. P.E.A. Turchi, F.J. Pinski, D.D. Johnson, D.M. Nicholson, G.M. Stocks, J.B. Staunton, Phys. Rev. Lett. 67 (1993) 1779. D.D. Johnson, F.J. Pinski, Phys. Rev. B 48 (1993) 11553. R. Kikuchi, Phys. Rev. 81 (1951) 988.
The influence of long range order on the martensitic transformation in Cu–Zn J.E. Garce´s, M. Ahlers * Centro Ato´mico Bariloche, 8400, S.C. de Bariloche, Argentina
Abstract The transformation between the a (fcc) and b (bcc) equilibrium phases at elevated temperatures is compared with that between the long range ordered B2 phase and the martensite at low temperatures, taking the Cu – Zn system as a prototype. The low temperature vibrational entropy difference, as measured from the martensitic transformation, is extrapolated to elevated temperatures at which disordered a and b are stable. This makes it possible to compare the corresponding enthalpy differences and deduce pair interchange energies, which depend only on pair distance but not on the crystal structure. This result is consistent with the observation that in many alloys based on the noble metals the enthalpy difference between the equilibrium structures is controlled mainly by the electron concentration e/a although the much larger mixing enthalpy of each of these structures depends strongly on the specific system. It also permits to evaluate the short range order (SRO) contribution at elevated temperatures using the cluster variation method. It is shown that the Gibbs free energies due to SRO are almost the same in the a and b phases, although they are non-negligible. © 1999 Elsevier Science S.A. All rights reserved. Keywords: Martensitic transformation; Long range order; Cluster variation method; Equilibrium phases; Cu – Zn
1. Introduction The martensitic transformation in binary Cu–Zn occurs between a B2 (CsCl type) austenite (b phase) and a 9R monoclinic martensite with the inherited atomic distribution, which is quite well known, since by an adequate heat treatment prior to the transformation a quasiperfect long range ordered (LRO) state can be produced. Long range order has a profound effect on phase stability and is generally described in terms of pair interchange energies between A – B atom pairs in ineighbor position as: (i) (i) (i) w (i) AB = − 2V AB +V AA +V BB
(1)
where the V (i) AB are the corresponding pair interaction energies in a given crystal structure. Due to the small hysteresis between the martensitic transformation and retransformation an equilibrium M temperature T M can be 0 and an entropy difference DS * Corresponding author. Tel.: +54-2944-445157; fax: +54-2944445299. E-mail address: [email protected] (M. Ahlers)
determined experimentally with a good precision. In order to determine the LRO contribution and deduce the difference in pair interchange energies between austenite and martensite, two problems have to be solved: 1. On the one hand a comparison is to be made between an fcc disordered a phase with an fcc LRO martensite (3R). Instead, a monoclinic 9R is generally formed. It is therefore necessary to relate both, the 3R with the 9R LRO structure. This problem has been solved by studying the ternary Cu–Zn–Al alloys. The deviation from the cubic symmetry has been found to be negligible [1] and the enthalpy difference between 9R and 3R has been measured [2]. 2. On the other hand, the a and b phases retain some short range order (SRO) even at elevated temperatures. Its contribution to the enthalpy of mixing may be small, but not negligible compared with the enthalpy differences between the phases. It is therefore necessary to have an estimate of the influence of SRO, which can be calculated once the pair interchange energies are known. One of the aims of this work is to derive these energies, to calculate the
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SRO, and finally to estimate the LRO contribution to the stability of the martensitic phases. At elevated temperatures the a and b phases are separated by a two phase (a + b) field in the phase diagram. The temperature T a/b at which the Gibbs free energy difference between both phases is zero lies somewhere within the two phase field. Describing the free energy of mixing by a term which depends on the pair interchange energies and another which varies linearly with composition and includes the influence of the electron concentration [3], we have shown using the cluster variation method (CVM) that T a/b can be located in the middle of the two phase (a +b) region. In order to determine the enthalpy difference DH a/b at this temperature, the entropy difference DS a/b has to be known. In this paper the vibrational contribution to DS a/b will be derived by extrapolating to elevated temperatures the DS M obtained from the martensitic transformation. The enthalpy difference between disordered a and b differs from that associated with the martensitic transformation by a combination of pair interchange energies. A second objective of this paper therefore is to calculate the value of this combination, and derive from it the individual pair interchange energies which are used to estimate the SRO contribution at elevated temperatures. Here only the Cu – Zn system is being analysed as a prototype. But the results are applicable also to other noble metal alloys. 2. The entropy difference DS a/b between the a and b equilibrium phases. The entropy difference DS a/b is the sum of three cona/b tributions: configurational (DS a/b conf), electronic (DS el ) a/b a/b and vibrational (DS vib ). DS conf will be considered in Section 4. DS a/b el , which is due to the conduction electrons, can be neglected since the electronic specific heats in a and b differ very little [3]. In order to derive the vibrational entropy differences, DS a/b vib at the elevated temperatures at which disordered a and b are stable from the measured vibrational entropy in the martensite, DS M, it is necessary to use more indirect extrapolation methods which will be discussed now. The vibrational entropy DS M has been measured below about 100°C by the martensitic transformation. It has been shown that it is due to the soft transversal (110)11( 0 phonon branch in the b phase [4], which is already fully excited at 80 K and leads to a constant DS M above 80 K. For several alloys which undergo a martensitic transformation the (110)11( 0 phonon branch has been measured by neutron diffraction. It has been shown that it increases slightly with temperature, including its long wave limit, the elastic constant c% =(c11 −c12)/2. It will be assumed in the following
that the temperature variation of c% reflects reasonably well the temperature change in the energy of the soft modes, and also that of the vibrational entropy difference DSvib, up to elevated temperatures. 1 dDSvib 1 dc% =− DSvib dT c% dT
(2)
The elastic constant c% does not show a simple temperature behavior: above MS c% increases first with temperature. In Cu–46.4%Zn, which does not transform martensitically, c% nevertheless increases up to 250°C and then starts to decrease influenced also by a decreasing LRO. The results for Cu–46.4%Zn [5] are characteristic also for ternary alloys which undergo a martensitic transformation. Measurements above the critical B2 ordering temperature T B2 C do not exist, and since an extrapolation of c% to higher temperatures is not possible, some other arguments have to be used. Therefore the extrapolation scheme will be divided into two parts: 1. First the relationship between DS M at the martensitic transformation temperature will be compared with that near T B2 C via the c% dependence followed by that above T B2 C . A collection of data for c% near MS in ternary Cu–Zn–Al and Cu–Al–Be [6] indicate that c% lies between 6 and 7 GPa. On the other hand c% near T B2 C is about 7.6 GPa for Cu–46.4%Zn [5] and 48.1%Zn [7] and still decreases rapidly with increasing temperature. Considering the uncertainties in the evaluation, it seems reasonable to conclude that DSvib does not differ notably between MS and a temperature slightly above T B2 C where LRO and SRO are no longer important. In the following both entropy differences are set equal. a/b 2. Of interest here is DS a/b . It has been shown vib at T M that DS increases with e/a [6]. On the other hand with increasing zinc concentration T a/b decreases, implying an increase in c%, i.e. a decrease in the temperature dependent part of DSvib. Both components therefore cancel to some extent. In order that they cancel completely it is necessary to have 1 dc% = − 4.10 − 4 K − 1 This is a reasonable value c% dT when compared with bcc Na ( − 5× 10 − 4 K − 1) [8], with the c44 in Cu–Zn, which behave normally, (−4× 10 − 4 K − 1) [9] and even with fcc Cu ( −4× 10 − 4 K − 1). It is suggested therefore that DS a/b vib can be supposed to be constant along the T a/b equilibrium temperatures. As a result of this discussion it is concluded that DS a/b vib is also similar to that for the martensitic transformation at low temperatures. For lack of more precise data, and of a good estimate of the error involved, both will be set equal in the following. An uncertainty of 9 10% would seem reasonable, which is already included in the measurement of DS M.
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3. The pair interchange energies As will be shown in the next section, the SRO contribution to the Gibbs free energy difference between a and b at T a/b is so small, compared with the other uncertainties, that it can be neglected. Then from T a/b, from the equilibrium temperature T M for the 0 martensitic transformation between fcc based 3R martensite and B2 ordered austenite, and from DS M = (1.2 9 0.1) J mol − 1 K − 1 [6], the enthalpy difference between disordered a and b on the one hand, and B2 and 9R on the other hand can be determined. For perfect order the relation is: M (1) (T a/bf− T M −3w (2) −2m (1) +3m (2))C 2Zn 0 )DS =(4w (3) M where f is the ratio between entropies, f = DS a/b vib /DS , which has been set equal to 1 here. Inserting the values leads to:
4w (1) − 3w (2) − 2m (1) +3m (2) =(5.6 9 0.6) kJ mol − 1 (4) Here only first and second neighbor pair contributions in the bcc and in the fcc phase (denoted by w (i) and m (i), respectively) are taken into account. The influence of more distant pairs has been neglected. Although their interchange energies are not zero, they are small compared to first and second neighbor pair contributions [10]. Moreover, their contribution in the two phases of the same atomic volume and electron concentration is expected to be nearly the same and therefore largely cancels in Eq. (4). The w (i) determine also the critical ordering temperature T B2 C , which can be described with a good approximation for an equiatomic composition by [11], again setting w (i) =0 for i \2: (1) kT B2 −1.5x2w (2). C (0.5)= 2x1w
(5)
In the Bragg–Williams approximation x1 =x2 = 1, but due to SRO, x1 is reduced to x1 =0.811. However, x2 = 1, which indicates that the influence of second neighbor SRO on T B2 C is very small, although it may not be exactly zero. Without further assumptions, the (1) measurement of T B2 and C permits only to relate w (2) w . In order not to introduce additional, less justified assumptions, a range of possible w (i) are used, which are listed in Table 1 and are discussed below. The w (2) are then calculated from Eq. (5) and m (1) −1.5m (2) from Eq. (4). In the following row the m (1) are linearly interpolated between w (1) and w (2) with respect to the pair distances which are known in bcc and fcc, because fcc and bcc have the same atomic volume. The m (2) are then derived. Two more rows in Table 1 contain the resulting 6m (1) + 3m (2) and 4w (1) +3w (2). These quantities determine that part of the enthalpy of mixing of disordered a and b, which depends on the pair inter-
197
change energies only, assuming that they are not different from those determined from the LRO contribution. First of all, from Table 1 it can be noted that 4w (1) + 3w (2) is very similar to 6m (1) + 3m (2), independent of the w (1) that is chosen. This means that the enthalpies of mixing of a and b, which depend on the pair interchange energies, are closely the same. This is a very satisfactory result, which had already been suggested earlier [12], when the enthalpies of mixing for several alloy systems were compared. Although they depend strongly on the alloy considered, the much smaller difference between a and b is determined only by e/a. On its basis it had been deduced that m (1) can be interpolated between w (1) and w (2), thus postulating that the close neighbor pair interchange energies depend on the pair distance but not on the crystal structure, fcc, bcc or hexagonal [12]. The same conclusion follows from the present approach. It is therefore concluded that the derivation of Eq. (4) through an extrapolation of DS M is as good as using a structure independent enthalpy of formation given by: 4w (1) + 3w (2) = 6m (1) + 3m (2)
(6)
excluding the electronic part. It is clear that in both cases assumptions have to be made, which may result in an error in the numerical values in Eq. (4), or in a deviation from equality in Eq. (6). The important point to be made here is that these deviations are sufficiently small not to invalidate the general conclusions. The fact that both approaches lead to the same results is a further support for the validity of structure independent pair interchange energies. Some additional information is available to select the most likely combination of pair interchange energies in Table 1. In earlier work the w (1) = 7.96 kJ mol − 1 had been taken from the literature [13], which was based on the assumption that the enthalpy of mixing in the austenite is due only to the pair interchange energy. By using the different sets in Table 1, it has been found that the validity of Eq. (4) is quite independent of the exact combination of w (1) and w (2), provided T B2 C (Eq. (5)) and the structure independence of the pair interchange energies remain unaltered. Thus also earlier results [11,12] do not become obsolete. Table 1 Different sets of pair interchange energies for the a and b phases w (1) (kJ mol−1) w (2) (kJ mol−1) m (1)−1.5m (2) (kJ mol−1) m (1) (kJ mol−1) m (2) (kJ mol−1) 4w (1)+3w (2) (kJ mol−1) 6m (1)+3m (2) (kJ mol−1)
7.96 4.46 6.4 90.3 7.25 0.58 45.2 45.6
6.25 2.63 5.8 90.3 5.58 −0.13 32.8 33.0
5.00 1.29 5.39 0.3 4.29 −0.67 23.8 22.3
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From ab initio calculations, w (1) =5.42 kJ mol − 1 and w (2) = 0.83 kJ mol − 1 were derived [14], neglecting charge correlations between the atoms. The enthalpy of mixing that was derived was smaller than the observed one. This could be remedied by including the charge correlations [15]. Adding this effect [15] and assuming that it modifies only w (1) and w (2), while keeping T B2 C constant, an increase to w (1) =6.25 kJ mol − 1 and w (2) =2.63 kJ mol − 1 is obtained. This seems to be the most reasonable value. It leads to a small negative m (2) which is also consistent with the measurements of SRO by neutron diffraction [10]. Furthermore it agrees favorably with the measured enthalpies of mixing, which would lead to 6m (1) +3m (2) =35 92 kJ mol − 1, if any electronic contribution were absent. Since the enthalpy of mixing is determined experimentally with respect to hexagonal Zn, an additional contribution of 3.3 kJ mol − 1 should have to be added to refer it to fcc Zn. It seems quite reasonable to expect that the conduction electrons contribute also to the stability of the alloy, compared to the pure elements, and therefore a lower pair interchange contribution of 33.0 kJ mol − 1 would be quite reasonable. If in Eq. (3) a 10% deviation of M DS a/b were admitted, this would lead to a vib from DS shift of 0.42 kJ mol − 1 in w (2) (Table 1), which still is sufficiently small not to modify seriously the present results.
Fig. 1. Short range order contribution to the Gibbs free energy in the b phase as function of zinc concentration for different sets of pair interchange energies: (a) w (1) =5.00 kJ mol − 1, w (2) =1.29 kJ mol − 1 and m (1) =4.29 kJ mol − 1 (b): w (1) =6.25 kJ mol − 1, w (2) = 2.63 kJ mol − 1 and m (1) =5.58 kJ mol − 1 and (c): w (1) =7.96 kJ mol − 1, w (2) =4.46 kJ mol − 1 and m (1) =7.25 kJ mol − 1. The curves at each of the sets (a) to (c) correspond to 975, 1025, 1075, 1125 and 1175 K from the lowest to the uppermost curve, respectively.
2) that the difference in Gibbs free energy between a and b due to SRO is reduced. The reduction is such that T a/b can be used as the temperature Te between the
4. The short range order contribution The SRO has been determined by the cluster variation method (CVM) [16], using for bcc and fcc the tetrahedron approximation which includes first and second neighbor interchange energies for bcc and first neighbors only for fcc. Contributions from second neighbors are small in fcc. Fig. 1 shows the SRO contribution to the Gibbs free energy for the bcc phase as a function of Zn concentration at various temperatures, using the three w (1) combinations listed in Table 1. It can be noted from the figure that there is a considerable SRO contribution. If now the SRO contribution in the a phase were negligible, using the argument that LRO has not been observed in a, then a large influence on T a/b could be expected, leading to a shift from the equilibrium temperature Te between a and b in the absence of SRO given by: T a/b − Te DG bSRO −DG aSRO DG bSRO = : Te DH a/b T a/b · DS M
(7)
which would imply a shift of 20 – 50%, for the three sets of pair interchange energies, which in no way could be neglected. These values are reduced when SRO is present in the a phase. In fact, using the corresponding m (1) from Table 1 and neglecting m (2) it is found (Fig.
Fig. 2. Short range order contributions to the difference in Gibbs free energy between the b and a phases as function of zinc concentration for different sets of pair interchange energies: (a) w (1) = 5.00 kJ mol − 1, w (2) =1.29 kJ mol − 1 and m (1) =4.29 kJ mol − 1 (b): w (1) = 6.25 kJ mol − 1, w (2) =2.63 kJ mol − 1 and m (1) =5.58 kJ mol − 1 and (c): w (1) =7.96 kJ mol − 1, w (2) =4.46 kJ mol − 1 and m (1) = 7.25 kJ mol − 1. The curves at each of the sets (a) to (c) correspond to 975, 1025, 1075, 1125 and 1175 K from the lowest to the uppermost curve, respectively.
J.E. Garce´s, M. Ahlers / Materials Science and Engineering A273–275 (1999) 195–199
are a function only of the pair distance, thus supporting further the earlier conclusions which had been obtained in a different way.
completely disordered phases, independent of the exact value of m (1), once the structure insensivity of the pair interchange energies is accepted.
5. Conclusions
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References
1. It is shown that at elevated temperatures, SRO has a negligible effect on the temperatures at which the difference in Gibbs free energy between a and b is zero. 2. The corresponding entropy difference between a and b has been estimated by extrapolating the entropy difference for the martensitic transformation to higher temperatures. 3. By comparing the enthalpy difference between disordered a and b with that for the martensitic transformation between B2 ordered b austenite and fcc based 3R martensite, the long range order contribution to the martensitic transformation has been obtained. 4. The results are consistent with the assumption that the pair interchange energies for two closest neighbors pairs are practically structure independent, and
.
[1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16]
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