- Email: [email protected]
Heat capacity of liquid La–Mg–Ni alloys
Journal of Alloys and Compounds 266 (1998) 216–223
L
Heat capacity of liquid La–Mg–Ni alloys J. Schmid, F. Sommer* ¨ Metallforschung and Institut f ur ¨ Metallkunde der Universitat ¨ Stuttgart, Seestr. 75, D-70174 Stuttgart, Germany Max-Planck-Institut f ur Received 28 July 1997
Abstract The heat capacity of liquid La–Mg–Ni alloys was measured using an adiabatic calorimeter. The heat capacities Cp (x,T ) exhibit large temperature and composition dependencies. The results indicate the existence of a maximum in the Cp (T )-curve in the undercooled melt. An association model was applied to calculate the thermodynamic mixing functions of ternary liquid La–Mg–Ni alloys using the model parameters of the three limiting binary systems and assuming an additional ternary association reaction. The calculations show a good agreement with the experimental enthalpy of mixing and the Cp (x,T )-data. 1998 Elsevier Science S.A. Keywords: Heat capacity; adiabatic calorimeter; Association model; La–Mg–Ni
1. Introduction
liquid ternary alloys. The extrapolation of the heat capacity in the undercooled melt is also discussed.
Ternary La–Mg–Ni alloys are known for their good glass formation by rapid quenching from the liquid state [1] and for their ability for hydrogen storage in the solid state [2,3]. The glass formation of liquid alloys is directly correlated to their thermodynamic properties and the tendency to form associates in the liquid state [4]. Some results already exist for the heat capacity in the amorphous, crystalline and undercooled liquid state obtained by differential scanning calorimetry (DSC) [5]. The composition dependence of the enthalpy of mixing of liquid magnesium and lanthanum-rich alloys at about 1030 K has been measured recently [6]. However, for a better knowledge of the temperature dependence of the thermodynamic mixing functions of liquid La–Mg–Ni alloys the heat capacity in the stable liquid state is needed. Therefore in this work we use an adiabatic calorimeter to determine Cp . The results of the temperature dependence of Cp in the stable liquid state measured for eight alloy compositions in the composition range of good glass formation are presented. The heat capacity of liquid aluminum and magnesium is also measured to show that the adiabatic calorimeter can be used to determine Cp of liquid alloys without any calibration procedure. An association model is further used to explain the observed thermodynamic properties of the *Corresponding author. Fax: 149 [email protected]
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2. Experimental The heat capacity of liquid La–Mg–Ni alloys, liquid Mg 85 Ni 15 and liquid aluminum and magnesium was determined using an adiabatic calorimeter. Details concerning the calorimeter setup and the measurement procedure have been described previously [7]. The outer three-zone furnace provides a constant temperature zone in a gas-tight alumina tube which is positioned inside the furnace and which contains adiabatic shieldings and the boron nitride (BN) sample container. The cylindrical sample container has vertically drilled holes for the heating elements inside its wall. Nickel rods provide the electrical support and the suspension of the sample container. The temperature of the sample is measured and controlled with a thermocouple inside the container. The measurements are performed in the isothermic mode which keeps the surroundings at a constant temperature T during the measurements. The alloy samples are prepared from La (purity: 99,9%), Mg (purity: 99,95%) and Ni (purity: 99,99%). The La samples were prepared and stored in an argon glove box. The samples are alloyed and casted under argon into a cylindrical copper mould. Mass losses could not be avoided due to the high vapor pressure of magnesium. The alloy compositions given in this work have been de-
J. Schmid, F. Sommer / Journal of Alloys and Compounds 266 (1998) 216 – 223
termined by chemical analysis. The cylindrical samples have a drilled blind hole for the stud of the BN container which contains the thermocouple. The sample and the BN container are heated for a short time (2–4 s) and the associated small temperature increase DT (1–1.5 K) is measured within 10–20 s. Adiabatic condition is maintained within this short measurement time [7]. The product of the specific heat of container, heater and sample c pCS and their mass m CS is given by:
S
D
DT DQ c CS T 1 ] m CS 5 ] p 2 DT
(1)
DQ is the electrical energy input supplied to the inner heater of the sample container. The heat capacity C lp of the liquid sample can be obtained from: m CS c CS p 2 m(BN)c p (BN) C pl 5 ]]]]]]] n
(2)
The product of the specific heat of container material and heater c p (BN) and their mass m(BN) is measured using a massive BN block with the dimensions of the sample container. n is the number of moles of the sample.
3. Results The results of the heat capacity measurement of liquid Al, Mg, Mg 85 Ni 15 and the liquid magnesium-rich La–Mg– Table 1 Heat capacity of liquid Al, Mg and Mg 85 Ni 15 Al T/K 940 945 948 958 960 970 975 978 985 995 1005
Cp / J mol 21 K 21 34,0 34,1 32,4 33,5 33,4 33,3 32,7 31,7 32,8 31,3 31,8
Mg 934 936 939 944 946 949 954
35,8 35,0 34,8 36,6 35,1 34,7 36,1
T/K 1008 1015 1025 1030 1038 1040 1055 1058 1065 1090
956 959 964 966 986 996
Cp / J mol 21 K 21 33,6 31,7 30,9 30,4 32,4 30,4 31,1 30,6 31,2 30,9
35,9 34,8 35,0 34,3 36,1 34,4
217
Ni alloys are given in Tables 1 and 2 and are shown in Figs. 1–5, respectively. The standard deviation of the obtained values is of the order of 63%. The composition dependence of the interpolated Cp -values of the ternary alloys at 1036 K are shown in Fig. 6. The C lp (T ) of liquid La 20 Mg 50 Ni 30 is shown in Fig. 3. The liquidus temperatures T L obtained from the measurements of C pl are very low in comparison to the melting temperatures T m of intermetallic phases of the basic binary systems. The phase equilibria of La–Mg–Ni alloys are not known. Due to the relatively small temperature difference between T L and the glass transition temperature T g , the magnesium rich liquid La–Mg–Ni alloys can be easily quenched into the glassy state [1].
4. Discussion The Cp (T ) for liquid aluminum and magnesium agree with the literature data [8]. These results show that the Cp of liquid metals and alloys can be determined with the used adiabatic calorimeter without any calibration procedure. The overall uncertainty is about 63–4%. The Cp (T )-data of high melting liquid metals (T m .950 K) are found to be constant within their experimental uncertainty [8] as we have observed for liquid magnesium (Fig. 2). However, the Cp (T ) of liquid aluminum tends to increase with decreasing temperature in the vicinity of about 80 K above T m (Fig. 1). Several metals (such as indium, tin and bismuth [9]) exhibit a similar Cp (T ) dependence and experimental results also show that this temperature dependence continues into the undercooled liquid state. The increase in Cp with decreasing temperature arises because of structural contributions and could be viewed as structural freezing [10–12]. The loss in entropy resulting from increasing Cp leads to a decrease in the entropy difference between the undercooled liquid and the stable crystalline state. The Cp -values of the liquid La–Mg–Ni alloys exhibit large values and a strong temperature dependence. There exists a systematic temperature and composition dependence of C lp with maximum values of about 55 J mol 21 K 21 near to La 30 Mg 45 Ni 25 at 1036 K. The excess heat capacity DC pl is given by the difference between the measured heat capacity of the liquid alloy and the mechanical mixture of the heat capacities of the pure liquid components.
OxC 3
C lp (x, T ) 5 DC lp (x, T ) 1
i
l p,i
(T )
(3)
i 51
Mg 85 Ni 15 940 950 960 970
38,8 39,0 38,3 37,9
980 990 1000 1010
38,2 37,6 37,3 36,9
The C lp,i (T ) of the pure components in the liquid and undercooled liquid state was calculated using known C pl values at their melting temperature [8], assuming constant values in the whole temperature range. The values for
J. Schmid, F. Sommer / Journal of Alloys and Compounds 266 (1998) 216 – 223
218
Table 2 Heat capacity of liquid La–Mg–Ni alloys La 47 Mg 41 Ni 12 T/K 1005 1010 1015 1025 1030 1035 1040
Cp / J mol 21 K 21 49,5 48,7 49,4 46,7 49,5 50,0 46,9
T/K 1045 1050 1055 1065 1090 1115
Cp / J mol 21 K 21 46,0 49,3 49,1 45,4 45,7 43,1
La 30 Mg 50 Ni 20 1005 1010 1013 1015 1020 1025
54,7 58,1 55,9 53,9 55,7 53,9
1030 1035 1040 1045 1050 1055
52,9 54,4 53,3 51,9 53,3 51,5
La 20 Mg 50 Ni 30 984 986 989 994 999 1004 1009 1014
54,6 53,4 53,9 54,5 54,5 54,2 52,5 52,4
1019 1029 1034 1039 1044 1054 1059 1069
53,7 52,4 52,5 49,9 51,7 50,1 47,9 47,2
La 20 Mg 48 Ni 32 980 985 990 995 1000 1005
51,6 51,9 52,3 51,3 50,4 50,0
1010 1015 1025 1040 1055
49,6 47,8 49,6 47,4 47,0
La 10 Mg 53 Ni 37 960 965 970 980 990 995 1005 1010 1015
53,9 52,7 52,4 52,0 53,1 50,0 50,5 48,6 51,1
1020 1030 1040 1050 1060 1070 1080 1090 1100
47,8 47,1 48,7 46,9 49,0 46,6 44,3 45,4 46,6
La 38 Mg 35 Ni 27 1025 1028 1030 1035 1040 1045
53,0 54,8 55,7 55,9 54,9 50,6
1050 1055 1057 1060 1065 1090
52,4 49,6 50,1 46,0 50,2 48,2
La 20 Mg 60 Ni 20 998 1003 1005 1008 1010 1013 1015 1020 1023
55,0 55,2 57,0 53,8 55,9 55,8 55,7 54,1 52,2
1025 1030 1035 1040 1045 1050 1055 1065
52,1 49,8 50,1 50,8 47,1 48,1 49,2 50,2
Fig. 1. Heat capacity of liquid aluminum (– – – [8]).
Fig. 2. Heat capacity of liquid magnesium (– – – [8]).
Fig. 3. Heat capacity of the liquid La 20 Mg 50 Ni 30 alloy (– – – heat capacity of the mechanical mixture of the liquid or undercooled liquid components).
J. Schmid, F. Sommer / Journal of Alloys and Compounds 266 (1998) 216 – 223
Fig. 4. Heat capacity of liquid La 60 Mg 40 –Mg 55 Ni 45 alloys. (j 980 K, s 1000 K, m 1020 K, , 1040 K, ♦ 1060 K, h 1080 K; – heat capacity of the mechanical mixture of the liquid or undercooled liquid components).
Fig. 5. Heat capacity of liquid La 43 Mg 30 Ni 27 –Mg 85 Ni 15 alloys. (j 980 K, s 1000 K, m 1020 K, , 1040 K, ♦ 1060 K, h 1080 K; – heat capacity of the mechanical mixture of the liquid or undercooled liquid components).
Fig. 6. Heat capacity of liquid La–Mg–Ni alloys at T51036 K (d measurement compositions).
219
Fig. 7. The excess heat capacity of liquid La 60 Mg 40 –Mg 55 Ni 45 alloys. (j 980 K, (j) extrapolated value, m 1020 K, h 1080 K; ? ? ? 980 K, — 1020 K, – – 1080 K calculated using the association model).
DC lp (T ) as a function of temperature and composition are shown in Figs. 7 and 8. The excess heat capacity and the temperature dependence is large in comparison to the results of the binary bordering systems. The maximum DCp values center around the composition La 30 Mg 45 Ni 25 (see Figs. 7 and 8). The Cp values of the good glass-forming liquid La 20 Mg 50 Ni 30 alloy are shown in Fig. 9 together with the heat capacity of the crystalline C ps and the undercooled liquid state. The value at T L is larger than those of the undercooled liquid state above the glass transition temperature. The C lp (T ) curve should therefore exhibit a maximum in the undercooled liquid state [5]. The large C pl values of the liquid La 20 Mg 50 Ni 30 alloy above T L correspond not to a specific behavior of this glass-forming alloy as Fig. 4 Fig. 5 Fig. 6 show. The enthalpy of melting DH m of this alloy was measured by DTA [5]. It has been shown in [5] that the experimentally obtained C lp of La 20 Mg 50 Ni 30 and
Fig. 8. The excess heat capacity of liquid La 67 Ni 33 –Mg 85 Ni 15 alloys. (j 980 K, (j) extrapolated value, m 1020 K, h 1080 K; ? ? ? 980 K, — 1020 K, – – 1080 K calculated using the association model).
J. Schmid, F. Sommer / Journal of Alloys and Compounds 266 (1998) 216 – 223
220
1 DH 5 ]] k nj
O
j 51
OO
1
O
i 51
O S O
1 DS 5 ]] k nj
O
k9 1 k 9 k 9 reg n 9k n l9 0 ] C k,l ]] 1 n 9i DH i k9 2 k51 l 51 i 51 n 9i k9
2R
(n 9i ln z i 1 n 9i DS 0i )
i 51
D
2
(6)
(7)
j 51
where k is the number of components, n is the number of moles of the components, k9 is the number of species (monomer, associates), n i9 is the number of moles and z the composition of the species respectively. DHi 8 and DSi 8 are the enthalpy and entropy of formation of the associates with DGi 85DHi 82T DSi 8,0. The monomers (say A) are described as associates of the type A,B 0 ,C 0 ,... and the Gibbs energy of formation DGA,B 0 ,C 0 50. The interaction parameters between the species k and l are represented by reg C reg k,l and it is assumed that C k,k 50. The equilibrium values of n 9i are determined by the law of mass action: Fig. 9. Heat capacity of liquid, undercooled liquid (s [5]) and crystalline (– ? – [5]) La 20 Mg 50 Ni 30 alloy.
their extrapolated temperature dependence of the undercooled liquid state given in Fig. 9 meet the thermodynamic relation 2 DH c (T c ) 5 DH m (T m ) 1 DH*
(4)
with Tc
DH* 5
E (C 2 C )dT l p
s p
(5)
Tm s
where the heat capacity of crystalline (C p ) and undercooled liquid La 20 Mg 50 Ni 30 , the crystallization enthalpy (DH c) and the crystallization temperature (T c ) were obtained by DSC [5].
5. Calculation of the thermodynamic properties of liquid and undercooled liquid La–Mg–Ni alloys Alloy melts with strong compound forming tendency exhibit chemical short range order (CSRO). The CSRO depends on composition and temperature. The association model given by Sommer [13] describes the relation between CSRO and the thermodynamic mixing functions. The model assumes the existence of associates, each with defined stoichiometry but undefined lifetime and free atoms in dynamic equilibrium with them. This equilibrium is governed by the law of mass action. The composition of these associates is often similar to that of corresponding intermetallic compounds. For the enthalpy and entropy of mixing of binary and higher component ( j .2) liquid alloys the following relations result [20].
z ig i9 0 0 exp[2(DH i 2 T DS i ) /RT ] 5 ]]]] k (z jg j9 )e i, j
P
(8)
j 51
g 9 is the activity coefficient of the respective species k and e i, j is the stoichiometric factor of component j in associate i. The parameters DH i0 and DS i0 and the interaction parameters are determined by fitting the experimental data, such as DH, DCp and activities, by solving Eqs. (6)–(8) iteratively. These parameters were determined for liquid La–Mg [14], La–Ni [15] and Mg–Ni [14] alloys assuming the existence of La 1 Mg 3 , La 1 Ni 2 and Mg 2 Ni 1 associates, respectively. The stoichiometries La 1 Mg 3 and Mg 2 Ni 1 correspond to intermetallic compounds with the same stoichiometry. For the model description of the thermodynamic properties of liquid La–Ni alloys it was necessary 0 to assume temperature dependent DH La - and DSLa 1 Ni 2 1 Ni 2 values [15]. The interaction parameters of liquid Mg–Ni alloys were revised by taking into account additionally the C lp data in Table 1 and the activity data of [16] in a new fitting procedure. The association model parameters of the basic binary systems are given in Table 3. For the calculation of the thermodynamic properties of the ternary liquid La–Mg–Ni alloys one has to consider three binary associates and the three pure components. It is assumed that the associates interact only with the free atoms but that there is no interaction between the different associates themselves. The equilibrium values of the mol numbers of the different species, n 9i , are determined by laws of mass action according to Eq. (8) with the model parameters listed in Table 3. DH(x,T ) and DS(x,T ) of liquid and undercooled liquid La–Mg–Ni alloys were calculated according to Eqs. (6) and (7). The obtained DH(x)-values at 1036 K are less negative than the measured values [6] (see also Fig. 10 Fig.
J. Schmid, F. Sommer / Journal of Alloys and Compounds 266 (1998) 216 – 223
221
Table 3 Association model parameters of the basic binary systems (in kJ mol 21 ) Associate
La–Mg La 1 Mg 3
La–Ni La 1 Ni 2
Mg–Ni Mg 2 Ni 1
DH 0A i B j
268.0
2108.0 10.2462*(T21376 K) 23.578 10 24 *(T 2 21376 2 K 2 ) / 2 11.3 10 27 *(T 3 21376 3 K 3 ) / 3 20.028 10.2462* ln(T / 1376 K) 23.578 10 24 *(T21376 K) 11.3 10 27 *(T 2 21376 2 K 2 ) / 2 286.0 237.0 247.0 DH at 1376 K: [17] DC p (T ): [15]
248.4
DS 0A i B j
C reg A 1 ,B 1 C reg A 1 ,A i B j C reg B 1 ,A i B j Data used for fitting / reference
20.0513
215.3 221.1 224.0 DH at 970 K, 985 K and 1060 K: [14]
20.0243
231.0 210.0 233.0 a Mg at 1373 K: [18] a Mg at 1073 K: [16] DH at 1120 K: [19] DCp (T ): present study
11). The difference between experimental as well as the extrapolated values and calculated results show systematic deviations due to ternary interactions [5]. The deviations of about 23 kJ mol 21 between DH exp and DH calc center around the composition La 30 Mg 50 Ni 20 . The maxima in C pl (x,T ) show up near this composition (see Fig. 6). The influence of ternary interactions is described in a further evaluation by an additional ternary association reaction with a La 3 Mg 4 Ni 2 stoichiometry. The composition of the ternary La 3 Mg 4 Ni 2 associate is close to the composition range of maximum deviations between DH exp and DH calc and of maximum Cp (x,T ). It is assumed that this ternary associate interacts only with the three components but not with the three binary associates. The first
set of parameters was fixed on the basis of the experimental DH(x) at 1036 K. The results show a good agreement to DH exp . The excess heat capacity DC pl of liquid and the undercooled liquid ternary alloys can be obtained from the calculated dependencies on temperature and composition of DH:
Fig. 10. Enthalpy of mixing of liquid and undercooled liquid (La 52 Ni 48 – ) 12x Mg x alloy at 1036 K [6] calculated using the association model. – – – Calculation using the associates La 1 Ni 2 , La 1 Mg 3 and Mg 2 Ni 1 — Calculation using the associates La 1 Ni 2 , La 1 Mg 3 , Mg 2 Ni 1 and La 3 Mg 4 Ni 2 d experimental results.
Fig. 11. Enthalpy of mixing of liquid and undercooled liquid (Mg 79 Ni 21 – ) 12x La x alloy at 1036 K [6] calculated using the association model. – – – calculation using the associates La 1 Ni 2 , La 1 Mg 3 and Mg 2 Ni 1 — calculation using the associates La 1 Ni 2 , La 1 Mg 3 , Mg 2 Ni 1 and La 3 Mg 4 Ni 2 , d experimental results.
S
D
Tl 2 T2 DH(x, T l ) 2 DH(x, T 2 ) l DC p x, T l 1 ]] 5 ]]]]]] 2 Tl 2 T2
(9)
The calculated heat capacity obtained with the first set of parameters from Eq. (9) was generally too low in comparison to the experimental DCp (x,T ). Hence, it was assumed that the enthalpy and entropy of formation of the
222
J. Schmid, F. Sommer / Journal of Alloys and Compounds 266 (1998) 216 – 223
ternary La 3 Mg 4 Ni 2 associate can not be regarded as independent of temperature. If DHi 8 and DSi 8 are fixed at a temperature T l the following relationship for T ,T l can be used [13]: DCp,i 5 A 1 BT 1 CT 2
(10)
B DH 0i (T ) 5 DH 0i (T l ) 1 A(T 2 T l ) 1 ] (T 2 2 T l2 ) 2 C 1 ] (T 3 2 T l3 ) 3
(11)
T C DS 0i (T ) 5 DS 0i (T l ) 1 A ln ] 1 B(T 2 T l ) 1 ] (T 2 2 T l2 ) Tl 2 (12) The temperature dependence given in Eqs. (10)–(12) has also been assumed for the binary associate LaNi 2 [15]. The model parameters of Eqs. (6)–(8) are determined in a first step at 1200 K. In a following trial and error process the temperature dependence of the formation of the ternary associate below 1200 K was fixed according to Eqs. (10)–(12) on the basis of the heat capacity data given in Table 2. The resulting model parameters are given in Table 4. The systematic deviations between DH exp and DH calc found by using only binary associates [6] completely vanish if the ternary association reaction with the stoichiometry La 3 Mg 4 Ni 2 is assumed (see Figs. 10 and 11). The presence of ternary associates in liquid La–Mg–Ni alloys results in a more negative enthalpy of mixing by about 3 kJ mol 21 near the composition La 30 Mg 40 Ni 30 . The calculated DC pl (x,T ) are shown in Figs. 7 and 8 and are in good agreement with the experimental data within the limit of measurement accuracy. The calculated DC lp (x,T ) of liquid and undercooled liquid La 20 Mg 50 Ni 30 in Fig. 12 exhibits a maximum in the undercooled liquid state and corresponds reasonably to the experimental values and the extrapolated C lp shown in Fig. 9. The temperature dependent CSRO described with the Table 4 Association model parameters of the ternary system (in kJ mol 21 ) Associate DH 0A i B j C k
DS 0A i B j C k
reg A 1 ,A i B j C k reg B 1 ,A i B j C k reg C 1 ,A i B j C k
C C C Data used for fitting / reference
La–Mg–Ni La 3 Mg 4 Ni 2 2205.5 20.7799*(T21200 K) 12.8 10 23 *(T 2 21200 2 K 2 ) / 2 21.842 10 26 *(T 3 21200 3 K 3 ) / 3 20.029 20.7799* ln(T / 1200 K) 12.8 10 23 *(T21200 K) 21.842 10 26 *(T 2 21200 2 K 2 ) / 2 235.0 240.0 295.0 DH at 1036 K: [6] DCp (T ): present study
Fig. 12. Calculated DC 1p of liquid and undercooled liquid La 20 Mg 50 Ni 30 alloy (– – – heat capacity of the mechanical mixture of liquid and undercooled liquid components, s [5]).
association model originates therefore the large values and l the maximum of C p (T ). The CSRO and the association tendency are generally small at high temperatures and large at low temperatures. The change in CSRO in these temperature ranges is small and therefore also the temperature dependence of C lp . The change in CSRO is large in the temperature range in between and exhibits a maximum. The results of the La 20 Mg 50 Ni 30 and Al 30 La 50 Ni 20 [5] l alloys provide the first experimental evidence that C p (x,T ) of alloys exhibiting metallic bonding show a maximum in the liquid or undercooled liquid state. The maximum in C lp (T ) of the undercooled liquid alloys can be explained within the model description by considering the temperature dependence of the mole numbers of the assumed species. For the undercooled liquid La 20 Mg 50 Ni 30 alloy the mole number of the La 1 Ni 2 associate decreases and the slope of the temperature dependence of the mole number of the La 3 Mg 4 Ni 2 associate changes below 900 K.
6. Conclusion The CSRO in liquid compound forming ternary alloys can be expressed in terms of binary and ternary association reactions where associates and free atoms are in a dynamic equilibrium. The observed maximum in C lp (T ) can be explained by a temperature dependent enthalpy and entropy of formation of the assumed associates.
References [1] A. Inoue, M. Kohinata, A. Tsai, T. Masumoto, Mat. Trans. JIM 30 (1989) 378. [2] M. Khrussanova, P. Peshev, Mat. Res. Bull. 22 (1987) 1653. [3] M. Khrussanova, P. Peshev, J. Less-Common Met. 131 (1987) 379. [4] F. Sommer, in: S. Steeb, H. Warlimont (Eds.), Rapidly Quenched Metals, Elsevier B.V., Amsterdam, 1985, p. 153. [5] F. Sommer, Mater. Sci. Eng. A 226–228 (1997) 757.
J. Schmid, F. Sommer / Journal of Alloys and Compounds 266 (1998) 216 – 223 [6] H. Feufel, M. Krishnaja, F. Sommer, B. Predel, J. Phase Equilibria 15 (1994) 303. [7] M. Bienzle, F. Sommer, Z. Metallkde. 85 (1994) 11. [8] A.T. Dinsdale, Calphad 15 (1991) 317. [9] J.H. Perepezko, J.S. Paik, J. Non-Cryst. Solids 61 (1984) 113. [10] R.N. Singh, F. Sommer, Phys. Chem. Liq. 28 (1994) 129. [11] F. Sommer, J. Alloys Comp. 220 (1995) 174. [12] J. Agren, B. Cheynet, M.T. Clavaguera-Mora et al., Calphad 19 (1995) 449. [13] F. Sommer, J. Non-Cryst. Solids 117–118 (1990) 505.
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[14] R. Agarwal, H. Feufel, F. Sommer, J. Alloys Comp. 217 (1995) 59. [15] H. Feufel, F. Schuller, J. Schmid, F. Sommer, J. Alloys Comp. 257 (1997) 234. [16] K. Micke, H. Ipser, Monatsh. Chem. 127 (1996) 7. [17] S. Watanabe, O.J. Kleppa, J. Chem. Thermodynamics 15 (1983) 633. [18] I.T. Sryvalin, O.A. Esin, B.M. LePinskich, Russ. J. Phys. Chem. 38(5) (1964) 637. [19] F. Sommer, J.J. Lee, B. Predel, Z. Metallkde. 74 (1983) 100. [20] H.-G. Krull, R.N. Singh, F. Sommer, will be published.
L
Heat capacity of liquid La–Mg–Ni alloys J. Schmid, F. Sommer* ¨ Metallforschung and Institut f ur ¨ Metallkunde der Universitat ¨ Stuttgart, Seestr. 75, D-70174 Stuttgart, Germany Max-Planck-Institut f ur Received 28 July 1997
Abstract The heat capacity of liquid La–Mg–Ni alloys was measured using an adiabatic calorimeter. The heat capacities Cp (x,T ) exhibit large temperature and composition dependencies. The results indicate the existence of a maximum in the Cp (T )-curve in the undercooled melt. An association model was applied to calculate the thermodynamic mixing functions of ternary liquid La–Mg–Ni alloys using the model parameters of the three limiting binary systems and assuming an additional ternary association reaction. The calculations show a good agreement with the experimental enthalpy of mixing and the Cp (x,T )-data. 1998 Elsevier Science S.A. Keywords: Heat capacity; adiabatic calorimeter; Association model; La–Mg–Ni
1. Introduction
liquid ternary alloys. The extrapolation of the heat capacity in the undercooled melt is also discussed.
Ternary La–Mg–Ni alloys are known for their good glass formation by rapid quenching from the liquid state [1] and for their ability for hydrogen storage in the solid state [2,3]. The glass formation of liquid alloys is directly correlated to their thermodynamic properties and the tendency to form associates in the liquid state [4]. Some results already exist for the heat capacity in the amorphous, crystalline and undercooled liquid state obtained by differential scanning calorimetry (DSC) [5]. The composition dependence of the enthalpy of mixing of liquid magnesium and lanthanum-rich alloys at about 1030 K has been measured recently [6]. However, for a better knowledge of the temperature dependence of the thermodynamic mixing functions of liquid La–Mg–Ni alloys the heat capacity in the stable liquid state is needed. Therefore in this work we use an adiabatic calorimeter to determine Cp . The results of the temperature dependence of Cp in the stable liquid state measured for eight alloy compositions in the composition range of good glass formation are presented. The heat capacity of liquid aluminum and magnesium is also measured to show that the adiabatic calorimeter can be used to determine Cp of liquid alloys without any calibration procedure. An association model is further used to explain the observed thermodynamic properties of the *Corresponding author. Fax: 149 [email protected]
711
121
1280;
e-mail:
0925-8388 / 98 / $19.00 1998 Elsevier Science S.A. All rights reserved. PII S0925-8388( 97 )00460-X
2. Experimental The heat capacity of liquid La–Mg–Ni alloys, liquid Mg 85 Ni 15 and liquid aluminum and magnesium was determined using an adiabatic calorimeter. Details concerning the calorimeter setup and the measurement procedure have been described previously [7]. The outer three-zone furnace provides a constant temperature zone in a gas-tight alumina tube which is positioned inside the furnace and which contains adiabatic shieldings and the boron nitride (BN) sample container. The cylindrical sample container has vertically drilled holes for the heating elements inside its wall. Nickel rods provide the electrical support and the suspension of the sample container. The temperature of the sample is measured and controlled with a thermocouple inside the container. The measurements are performed in the isothermic mode which keeps the surroundings at a constant temperature T during the measurements. The alloy samples are prepared from La (purity: 99,9%), Mg (purity: 99,95%) and Ni (purity: 99,99%). The La samples were prepared and stored in an argon glove box. The samples are alloyed and casted under argon into a cylindrical copper mould. Mass losses could not be avoided due to the high vapor pressure of magnesium. The alloy compositions given in this work have been de-
J. Schmid, F. Sommer / Journal of Alloys and Compounds 266 (1998) 216 – 223
termined by chemical analysis. The cylindrical samples have a drilled blind hole for the stud of the BN container which contains the thermocouple. The sample and the BN container are heated for a short time (2–4 s) and the associated small temperature increase DT (1–1.5 K) is measured within 10–20 s. Adiabatic condition is maintained within this short measurement time [7]. The product of the specific heat of container, heater and sample c pCS and their mass m CS is given by:
S
D
DT DQ c CS T 1 ] m CS 5 ] p 2 DT
(1)
DQ is the electrical energy input supplied to the inner heater of the sample container. The heat capacity C lp of the liquid sample can be obtained from: m CS c CS p 2 m(BN)c p (BN) C pl 5 ]]]]]]] n
(2)
The product of the specific heat of container material and heater c p (BN) and their mass m(BN) is measured using a massive BN block with the dimensions of the sample container. n is the number of moles of the sample.
3. Results The results of the heat capacity measurement of liquid Al, Mg, Mg 85 Ni 15 and the liquid magnesium-rich La–Mg– Table 1 Heat capacity of liquid Al, Mg and Mg 85 Ni 15 Al T/K 940 945 948 958 960 970 975 978 985 995 1005
Cp / J mol 21 K 21 34,0 34,1 32,4 33,5 33,4 33,3 32,7 31,7 32,8 31,3 31,8
Mg 934 936 939 944 946 949 954
35,8 35,0 34,8 36,6 35,1 34,7 36,1
T/K 1008 1015 1025 1030 1038 1040 1055 1058 1065 1090
956 959 964 966 986 996
Cp / J mol 21 K 21 33,6 31,7 30,9 30,4 32,4 30,4 31,1 30,6 31,2 30,9
35,9 34,8 35,0 34,3 36,1 34,4
217
Ni alloys are given in Tables 1 and 2 and are shown in Figs. 1–5, respectively. The standard deviation of the obtained values is of the order of 63%. The composition dependence of the interpolated Cp -values of the ternary alloys at 1036 K are shown in Fig. 6. The C lp (T ) of liquid La 20 Mg 50 Ni 30 is shown in Fig. 3. The liquidus temperatures T L obtained from the measurements of C pl are very low in comparison to the melting temperatures T m of intermetallic phases of the basic binary systems. The phase equilibria of La–Mg–Ni alloys are not known. Due to the relatively small temperature difference between T L and the glass transition temperature T g , the magnesium rich liquid La–Mg–Ni alloys can be easily quenched into the glassy state [1].
4. Discussion The Cp (T ) for liquid aluminum and magnesium agree with the literature data [8]. These results show that the Cp of liquid metals and alloys can be determined with the used adiabatic calorimeter without any calibration procedure. The overall uncertainty is about 63–4%. The Cp (T )-data of high melting liquid metals (T m .950 K) are found to be constant within their experimental uncertainty [8] as we have observed for liquid magnesium (Fig. 2). However, the Cp (T ) of liquid aluminum tends to increase with decreasing temperature in the vicinity of about 80 K above T m (Fig. 1). Several metals (such as indium, tin and bismuth [9]) exhibit a similar Cp (T ) dependence and experimental results also show that this temperature dependence continues into the undercooled liquid state. The increase in Cp with decreasing temperature arises because of structural contributions and could be viewed as structural freezing [10–12]. The loss in entropy resulting from increasing Cp leads to a decrease in the entropy difference between the undercooled liquid and the stable crystalline state. The Cp -values of the liquid La–Mg–Ni alloys exhibit large values and a strong temperature dependence. There exists a systematic temperature and composition dependence of C lp with maximum values of about 55 J mol 21 K 21 near to La 30 Mg 45 Ni 25 at 1036 K. The excess heat capacity DC pl is given by the difference between the measured heat capacity of the liquid alloy and the mechanical mixture of the heat capacities of the pure liquid components.
OxC 3
C lp (x, T ) 5 DC lp (x, T ) 1
i
l p,i
(T )
(3)
i 51
Mg 85 Ni 15 940 950 960 970
38,8 39,0 38,3 37,9
980 990 1000 1010
38,2 37,6 37,3 36,9
The C lp,i (T ) of the pure components in the liquid and undercooled liquid state was calculated using known C pl values at their melting temperature [8], assuming constant values in the whole temperature range. The values for
J. Schmid, F. Sommer / Journal of Alloys and Compounds 266 (1998) 216 – 223
218
Table 2 Heat capacity of liquid La–Mg–Ni alloys La 47 Mg 41 Ni 12 T/K 1005 1010 1015 1025 1030 1035 1040
Cp / J mol 21 K 21 49,5 48,7 49,4 46,7 49,5 50,0 46,9
T/K 1045 1050 1055 1065 1090 1115
Cp / J mol 21 K 21 46,0 49,3 49,1 45,4 45,7 43,1
La 30 Mg 50 Ni 20 1005 1010 1013 1015 1020 1025
54,7 58,1 55,9 53,9 55,7 53,9
1030 1035 1040 1045 1050 1055
52,9 54,4 53,3 51,9 53,3 51,5
La 20 Mg 50 Ni 30 984 986 989 994 999 1004 1009 1014
54,6 53,4 53,9 54,5 54,5 54,2 52,5 52,4
1019 1029 1034 1039 1044 1054 1059 1069
53,7 52,4 52,5 49,9 51,7 50,1 47,9 47,2
La 20 Mg 48 Ni 32 980 985 990 995 1000 1005
51,6 51,9 52,3 51,3 50,4 50,0
1010 1015 1025 1040 1055
49,6 47,8 49,6 47,4 47,0
La 10 Mg 53 Ni 37 960 965 970 980 990 995 1005 1010 1015
53,9 52,7 52,4 52,0 53,1 50,0 50,5 48,6 51,1
1020 1030 1040 1050 1060 1070 1080 1090 1100
47,8 47,1 48,7 46,9 49,0 46,6 44,3 45,4 46,6
La 38 Mg 35 Ni 27 1025 1028 1030 1035 1040 1045
53,0 54,8 55,7 55,9 54,9 50,6
1050 1055 1057 1060 1065 1090
52,4 49,6 50,1 46,0 50,2 48,2
La 20 Mg 60 Ni 20 998 1003 1005 1008 1010 1013 1015 1020 1023
55,0 55,2 57,0 53,8 55,9 55,8 55,7 54,1 52,2
1025 1030 1035 1040 1045 1050 1055 1065
52,1 49,8 50,1 50,8 47,1 48,1 49,2 50,2
Fig. 1. Heat capacity of liquid aluminum (– – – [8]).
Fig. 2. Heat capacity of liquid magnesium (– – – [8]).
Fig. 3. Heat capacity of the liquid La 20 Mg 50 Ni 30 alloy (– – – heat capacity of the mechanical mixture of the liquid or undercooled liquid components).
J. Schmid, F. Sommer / Journal of Alloys and Compounds 266 (1998) 216 – 223
Fig. 4. Heat capacity of liquid La 60 Mg 40 –Mg 55 Ni 45 alloys. (j 980 K, s 1000 K, m 1020 K, , 1040 K, ♦ 1060 K, h 1080 K; – heat capacity of the mechanical mixture of the liquid or undercooled liquid components).
Fig. 5. Heat capacity of liquid La 43 Mg 30 Ni 27 –Mg 85 Ni 15 alloys. (j 980 K, s 1000 K, m 1020 K, , 1040 K, ♦ 1060 K, h 1080 K; – heat capacity of the mechanical mixture of the liquid or undercooled liquid components).
Fig. 6. Heat capacity of liquid La–Mg–Ni alloys at T51036 K (d measurement compositions).
219
Fig. 7. The excess heat capacity of liquid La 60 Mg 40 –Mg 55 Ni 45 alloys. (j 980 K, (j) extrapolated value, m 1020 K, h 1080 K; ? ? ? 980 K, — 1020 K, – – 1080 K calculated using the association model).
DC lp (T ) as a function of temperature and composition are shown in Figs. 7 and 8. The excess heat capacity and the temperature dependence is large in comparison to the results of the binary bordering systems. The maximum DCp values center around the composition La 30 Mg 45 Ni 25 (see Figs. 7 and 8). The Cp values of the good glass-forming liquid La 20 Mg 50 Ni 30 alloy are shown in Fig. 9 together with the heat capacity of the crystalline C ps and the undercooled liquid state. The value at T L is larger than those of the undercooled liquid state above the glass transition temperature. The C lp (T ) curve should therefore exhibit a maximum in the undercooled liquid state [5]. The large C pl values of the liquid La 20 Mg 50 Ni 30 alloy above T L correspond not to a specific behavior of this glass-forming alloy as Fig. 4 Fig. 5 Fig. 6 show. The enthalpy of melting DH m of this alloy was measured by DTA [5]. It has been shown in [5] that the experimentally obtained C lp of La 20 Mg 50 Ni 30 and
Fig. 8. The excess heat capacity of liquid La 67 Ni 33 –Mg 85 Ni 15 alloys. (j 980 K, (j) extrapolated value, m 1020 K, h 1080 K; ? ? ? 980 K, — 1020 K, – – 1080 K calculated using the association model).
J. Schmid, F. Sommer / Journal of Alloys and Compounds 266 (1998) 216 – 223
220
1 DH 5 ]] k nj
O
j 51
OO
1
O
i 51
O S O
1 DS 5 ]] k nj
O
k9 1 k 9 k 9 reg n 9k n l9 0 ] C k,l ]] 1 n 9i DH i k9 2 k51 l 51 i 51 n 9i k9
2R
(n 9i ln z i 1 n 9i DS 0i )
i 51
D
2
(6)
(7)
j 51
where k is the number of components, n is the number of moles of the components, k9 is the number of species (monomer, associates), n i9 is the number of moles and z the composition of the species respectively. DHi 8 and DSi 8 are the enthalpy and entropy of formation of the associates with DGi 85DHi 82T DSi 8,0. The monomers (say A) are described as associates of the type A,B 0 ,C 0 ,... and the Gibbs energy of formation DGA,B 0 ,C 0 50. The interaction parameters between the species k and l are represented by reg C reg k,l and it is assumed that C k,k 50. The equilibrium values of n 9i are determined by the law of mass action: Fig. 9. Heat capacity of liquid, undercooled liquid (s [5]) and crystalline (– ? – [5]) La 20 Mg 50 Ni 30 alloy.
their extrapolated temperature dependence of the undercooled liquid state given in Fig. 9 meet the thermodynamic relation 2 DH c (T c ) 5 DH m (T m ) 1 DH*
(4)
with Tc
DH* 5
E (C 2 C )dT l p
s p
(5)
Tm s
where the heat capacity of crystalline (C p ) and undercooled liquid La 20 Mg 50 Ni 30 , the crystallization enthalpy (DH c) and the crystallization temperature (T c ) were obtained by DSC [5].
5. Calculation of the thermodynamic properties of liquid and undercooled liquid La–Mg–Ni alloys Alloy melts with strong compound forming tendency exhibit chemical short range order (CSRO). The CSRO depends on composition and temperature. The association model given by Sommer [13] describes the relation between CSRO and the thermodynamic mixing functions. The model assumes the existence of associates, each with defined stoichiometry but undefined lifetime and free atoms in dynamic equilibrium with them. This equilibrium is governed by the law of mass action. The composition of these associates is often similar to that of corresponding intermetallic compounds. For the enthalpy and entropy of mixing of binary and higher component ( j .2) liquid alloys the following relations result [20].
z ig i9 0 0 exp[2(DH i 2 T DS i ) /RT ] 5 ]]]] k (z jg j9 )e i, j
P
(8)
j 51
g 9 is the activity coefficient of the respective species k and e i, j is the stoichiometric factor of component j in associate i. The parameters DH i0 and DS i0 and the interaction parameters are determined by fitting the experimental data, such as DH, DCp and activities, by solving Eqs. (6)–(8) iteratively. These parameters were determined for liquid La–Mg [14], La–Ni [15] and Mg–Ni [14] alloys assuming the existence of La 1 Mg 3 , La 1 Ni 2 and Mg 2 Ni 1 associates, respectively. The stoichiometries La 1 Mg 3 and Mg 2 Ni 1 correspond to intermetallic compounds with the same stoichiometry. For the model description of the thermodynamic properties of liquid La–Ni alloys it was necessary 0 to assume temperature dependent DH La - and DSLa 1 Ni 2 1 Ni 2 values [15]. The interaction parameters of liquid Mg–Ni alloys were revised by taking into account additionally the C lp data in Table 1 and the activity data of [16] in a new fitting procedure. The association model parameters of the basic binary systems are given in Table 3. For the calculation of the thermodynamic properties of the ternary liquid La–Mg–Ni alloys one has to consider three binary associates and the three pure components. It is assumed that the associates interact only with the free atoms but that there is no interaction between the different associates themselves. The equilibrium values of the mol numbers of the different species, n 9i , are determined by laws of mass action according to Eq. (8) with the model parameters listed in Table 3. DH(x,T ) and DS(x,T ) of liquid and undercooled liquid La–Mg–Ni alloys were calculated according to Eqs. (6) and (7). The obtained DH(x)-values at 1036 K are less negative than the measured values [6] (see also Fig. 10 Fig.
J. Schmid, F. Sommer / Journal of Alloys and Compounds 266 (1998) 216 – 223
221
Table 3 Association model parameters of the basic binary systems (in kJ mol 21 ) Associate
La–Mg La 1 Mg 3
La–Ni La 1 Ni 2
Mg–Ni Mg 2 Ni 1
DH 0A i B j
268.0
2108.0 10.2462*(T21376 K) 23.578 10 24 *(T 2 21376 2 K 2 ) / 2 11.3 10 27 *(T 3 21376 3 K 3 ) / 3 20.028 10.2462* ln(T / 1376 K) 23.578 10 24 *(T21376 K) 11.3 10 27 *(T 2 21376 2 K 2 ) / 2 286.0 237.0 247.0 DH at 1376 K: [17] DC p (T ): [15]
248.4
DS 0A i B j
C reg A 1 ,B 1 C reg A 1 ,A i B j C reg B 1 ,A i B j Data used for fitting / reference
20.0513
215.3 221.1 224.0 DH at 970 K, 985 K and 1060 K: [14]
20.0243
231.0 210.0 233.0 a Mg at 1373 K: [18] a Mg at 1073 K: [16] DH at 1120 K: [19] DCp (T ): present study
11). The difference between experimental as well as the extrapolated values and calculated results show systematic deviations due to ternary interactions [5]. The deviations of about 23 kJ mol 21 between DH exp and DH calc center around the composition La 30 Mg 50 Ni 20 . The maxima in C pl (x,T ) show up near this composition (see Fig. 6). The influence of ternary interactions is described in a further evaluation by an additional ternary association reaction with a La 3 Mg 4 Ni 2 stoichiometry. The composition of the ternary La 3 Mg 4 Ni 2 associate is close to the composition range of maximum deviations between DH exp and DH calc and of maximum Cp (x,T ). It is assumed that this ternary associate interacts only with the three components but not with the three binary associates. The first
set of parameters was fixed on the basis of the experimental DH(x) at 1036 K. The results show a good agreement to DH exp . The excess heat capacity DC pl of liquid and the undercooled liquid ternary alloys can be obtained from the calculated dependencies on temperature and composition of DH:
Fig. 10. Enthalpy of mixing of liquid and undercooled liquid (La 52 Ni 48 – ) 12x Mg x alloy at 1036 K [6] calculated using the association model. – – – Calculation using the associates La 1 Ni 2 , La 1 Mg 3 and Mg 2 Ni 1 — Calculation using the associates La 1 Ni 2 , La 1 Mg 3 , Mg 2 Ni 1 and La 3 Mg 4 Ni 2 d experimental results.
Fig. 11. Enthalpy of mixing of liquid and undercooled liquid (Mg 79 Ni 21 – ) 12x La x alloy at 1036 K [6] calculated using the association model. – – – calculation using the associates La 1 Ni 2 , La 1 Mg 3 and Mg 2 Ni 1 — calculation using the associates La 1 Ni 2 , La 1 Mg 3 , Mg 2 Ni 1 and La 3 Mg 4 Ni 2 , d experimental results.
S
D
Tl 2 T2 DH(x, T l ) 2 DH(x, T 2 ) l DC p x, T l 1 ]] 5 ]]]]]] 2 Tl 2 T2
(9)
The calculated heat capacity obtained with the first set of parameters from Eq. (9) was generally too low in comparison to the experimental DCp (x,T ). Hence, it was assumed that the enthalpy and entropy of formation of the
222
J. Schmid, F. Sommer / Journal of Alloys and Compounds 266 (1998) 216 – 223
ternary La 3 Mg 4 Ni 2 associate can not be regarded as independent of temperature. If DHi 8 and DSi 8 are fixed at a temperature T l the following relationship for T ,T l can be used [13]: DCp,i 5 A 1 BT 1 CT 2
(10)
B DH 0i (T ) 5 DH 0i (T l ) 1 A(T 2 T l ) 1 ] (T 2 2 T l2 ) 2 C 1 ] (T 3 2 T l3 ) 3
(11)
T C DS 0i (T ) 5 DS 0i (T l ) 1 A ln ] 1 B(T 2 T l ) 1 ] (T 2 2 T l2 ) Tl 2 (12) The temperature dependence given in Eqs. (10)–(12) has also been assumed for the binary associate LaNi 2 [15]. The model parameters of Eqs. (6)–(8) are determined in a first step at 1200 K. In a following trial and error process the temperature dependence of the formation of the ternary associate below 1200 K was fixed according to Eqs. (10)–(12) on the basis of the heat capacity data given in Table 2. The resulting model parameters are given in Table 4. The systematic deviations between DH exp and DH calc found by using only binary associates [6] completely vanish if the ternary association reaction with the stoichiometry La 3 Mg 4 Ni 2 is assumed (see Figs. 10 and 11). The presence of ternary associates in liquid La–Mg–Ni alloys results in a more negative enthalpy of mixing by about 3 kJ mol 21 near the composition La 30 Mg 40 Ni 30 . The calculated DC pl (x,T ) are shown in Figs. 7 and 8 and are in good agreement with the experimental data within the limit of measurement accuracy. The calculated DC lp (x,T ) of liquid and undercooled liquid La 20 Mg 50 Ni 30 in Fig. 12 exhibits a maximum in the undercooled liquid state and corresponds reasonably to the experimental values and the extrapolated C lp shown in Fig. 9. The temperature dependent CSRO described with the Table 4 Association model parameters of the ternary system (in kJ mol 21 ) Associate DH 0A i B j C k
DS 0A i B j C k
reg A 1 ,A i B j C k reg B 1 ,A i B j C k reg C 1 ,A i B j C k
C C C Data used for fitting / reference
La–Mg–Ni La 3 Mg 4 Ni 2 2205.5 20.7799*(T21200 K) 12.8 10 23 *(T 2 21200 2 K 2 ) / 2 21.842 10 26 *(T 3 21200 3 K 3 ) / 3 20.029 20.7799* ln(T / 1200 K) 12.8 10 23 *(T21200 K) 21.842 10 26 *(T 2 21200 2 K 2 ) / 2 235.0 240.0 295.0 DH at 1036 K: [6] DCp (T ): present study
Fig. 12. Calculated DC 1p of liquid and undercooled liquid La 20 Mg 50 Ni 30 alloy (– – – heat capacity of the mechanical mixture of liquid and undercooled liquid components, s [5]).
association model originates therefore the large values and l the maximum of C p (T ). The CSRO and the association tendency are generally small at high temperatures and large at low temperatures. The change in CSRO in these temperature ranges is small and therefore also the temperature dependence of C lp . The change in CSRO is large in the temperature range in between and exhibits a maximum. The results of the La 20 Mg 50 Ni 30 and Al 30 La 50 Ni 20 [5] l alloys provide the first experimental evidence that C p (x,T ) of alloys exhibiting metallic bonding show a maximum in the liquid or undercooled liquid state. The maximum in C lp (T ) of the undercooled liquid alloys can be explained within the model description by considering the temperature dependence of the mole numbers of the assumed species. For the undercooled liquid La 20 Mg 50 Ni 30 alloy the mole number of the La 1 Ni 2 associate decreases and the slope of the temperature dependence of the mole number of the La 3 Mg 4 Ni 2 associate changes below 900 K.
6. Conclusion The CSRO in liquid compound forming ternary alloys can be expressed in terms of binary and ternary association reactions where associates and free atoms are in a dynamic equilibrium. The observed maximum in C lp (T ) can be explained by a temperature dependent enthalpy and entropy of formation of the assumed associates.
References [1] A. Inoue, M. Kohinata, A. Tsai, T. Masumoto, Mat. Trans. JIM 30 (1989) 378. [2] M. Khrussanova, P. Peshev, Mat. Res. Bull. 22 (1987) 1653. [3] M. Khrussanova, P. Peshev, J. Less-Common Met. 131 (1987) 379. [4] F. Sommer, in: S. Steeb, H. Warlimont (Eds.), Rapidly Quenched Metals, Elsevier B.V., Amsterdam, 1985, p. 153. [5] F. Sommer, Mater. Sci. Eng. A 226–228 (1997) 757.
J. Schmid, F. Sommer / Journal of Alloys and Compounds 266 (1998) 216 – 223 [6] H. Feufel, M. Krishnaja, F. Sommer, B. Predel, J. Phase Equilibria 15 (1994) 303. [7] M. Bienzle, F. Sommer, Z. Metallkde. 85 (1994) 11. [8] A.T. Dinsdale, Calphad 15 (1991) 317. [9] J.H. Perepezko, J.S. Paik, J. Non-Cryst. Solids 61 (1984) 113. [10] R.N. Singh, F. Sommer, Phys. Chem. Liq. 28 (1994) 129. [11] F. Sommer, J. Alloys Comp. 220 (1995) 174. [12] J. Agren, B. Cheynet, M.T. Clavaguera-Mora et al., Calphad 19 (1995) 449. [13] F. Sommer, J. Non-Cryst. Solids 117–118 (1990) 505.
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[14] R. Agarwal, H. Feufel, F. Sommer, J. Alloys Comp. 217 (1995) 59. [15] H. Feufel, F. Schuller, J. Schmid, F. Sommer, J. Alloys Comp. 257 (1997) 234. [16] K. Micke, H. Ipser, Monatsh. Chem. 127 (1996) 7. [17] S. Watanabe, O.J. Kleppa, J. Chem. Thermodynamics 15 (1983) 633. [18] I.T. Sryvalin, O.A. Esin, B.M. LePinskich, Russ. J. Phys. Chem. 38(5) (1964) 637. [19] F. Sommer, J.J. Lee, B. Predel, Z. Metallkde. 74 (1983) 100. [20] H.-G. Krull, R.N. Singh, F. Sommer, will be published.