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Experimental determination and computation of phase diagrams from their thermodynamic functions
CALFHAD
vo1.3,
No.3,
pp.205~222.
Pergamon Press Ltd. 1979. Printed in Great Britain.
EXPERIMENTAL DETERMINATIONAND C~PUTATION DF PHASE DIAGRAMS FROM THEIR THE~ODY~XC FUNCTIONS by R. CASTANET, C. BERGMAN and J.-C. MATHIEU Centre de Thermodynamiqueet de Microcalorimetriedu CNRS 26, rue du 141e R.I.A., 13003 Marseille France
(This paper was presented at CALPHAD VIII Stockholm, Sweden, May 1979)
Most methods of determinationof phase diagrams are based on the observation of the variation versus temperature, of a given property of alloys of defined compositions.These methods, such as thermal analysis of e.m.f. dete~ination~ enable by direct m~sur~ents the vertical crossing (in a parallel direction to the temperature axisl of phase boundaries. The aim of this paper is to describe the method by which partial and integral enthalpies of formation of alloys vs. molar fraction are measured at a constant temperature i the phase boundaries are then obtained by the horizontal crossing of the phase diagram. In this paper, the systems concerned are mainly metal-semimetalalloys. In addition to the experimental phase diagram, the enthalpies of formation of the different phases of the system are simultaneouslyobtained with respect to composition and temperature.Combined with the values of the free enthalpies of formation obtained by a potential method, these results allow the computation of equilibrium diagrams. Lastly, the comparison of computed and experimentalresults enables the checking of the consistency of all the experimental informations. I. PRINCIPLE Let us consider a system of two components A and 5 whose melting temperaturesare TA and TD respectively.Fig. lb and Fig. Ic show the integral and partial enthalpy of formation. in a relatively simple case where the phase diagram (Fig. la] of the system has only one congruently melting compound. The enthalpies are referred to both liquid components at the equilibrium temperature T. In the two-phase regions BC and DE, the partial enthalpy of 6 is constant and the integral enthalpy is linear with respect to the mole fraction XB ; at the crossing of the phase boundary, the partial enthalpy of B undergoes a discontinuitywhich permits the localization of this boundary. Figure I 0
k I
1C
t 8
~
s
____-
----
c
E
E
la. Phase diagram of the A-B binary system. lb. Integral molar enthalpy of formation of A-B alloys at temperature T vs. mole fraction of El. Ic. Partial molar enthalpy of formation of component B at temperature T vs. mole fraction of B.
205
R. Castanet et al.
206
II.
EXPERIMENTALMETHOD
The apparatus used are Tian-Calvatmicrocalorimetersparticularlywall suited to the measurements of small heat quantities and therefore to the determinationof partial enthalpies. The experimentalmethod is a drop technique. which we first used to determine the enthalpy of formation of the silver-germaniumsystem at 1280 K. (11. The liquid component A. Fig. 2, is placed in a container C, the material of which can be Pyrex glass, silica, graphite, alumina, . . . depending on the temperature and the constituents of the alloy. The container is located at the bottom of the calorimeter cell at temperature T. The solid component 8, at room temperature T0, is introduced into the container C by means of the guiding pipe G and dissolves in pure A or in the mixture obtained by the previous additions of the component 8. In the blank pile is placed the same cell as in the laboratory pile. The whole device is under argon atmosphere, in order to prevent any pollution of the mixture.
Fig. 2. Calorimetric cell for measurements of enthalpies of mixing (direct drop method]. A : alloy, C : crucible, G : guiding pipe.
The calibrationof the calorimeter is carried out by dropping the solid component A at To into pure liquid A, before adding 8. The enthalpy variation of A and 8. which is necessary to, first of all, calibrate the apparatus and secondly to calculate hf(Xil referred to liquid components, is taken in tables of numerical data [2,31 or measured by comparisonwith that of N.E.S. a-alumina (41 of C.R.M.T. copper (5). The global thermal effect measured includes a term which is due to the heating of the component added to the bath and this term can, in some cases, represent the main part of the whole effect. The final result, that is the enthalpy of mixing referred to the two components at the temperature T of the calorimeter,may bear a high level of uncertainty.Therefore, it is necessary to use a device where the added component is preheated at the bath level before being dissolved (Fig. 31. The sample is blocked in the funnel I and the dissolutionis then started by lifting the piston J. The measured thermal effect in this case does not include the parasite term due to the heat-content.However, since this method is more sophisticated,we use it only when necessary. When the preheated component risks being attacked by the vapours of the bath. the funnel should be replaced by a valve F (Fig. 41 floating in a protecting low This method has been applied to vapour pressure liquid C at the experimental temperature. study the mercury-thalliunsystem (61 and can be used up to 600 K, above which it is difficult to find a protective liquid fullfilling the required conditions. The thermal phenomenon due to every corresponds
to
the
following
reaction
drop
of
the
component
B in the
solvent
bath
:
niB(s.Tol + Ni_,All.(T.xi_,l +
(Ni_,,+nilA1l.(Tl.xil
th where ni is the molar quantity of the component E added to the bath at the i addition i-l and Ni_, = n + f nithe molar quantity of bath after the [i-llth addition of metal 8. A
207
EXPERIMENTAL DETERMINATIONAND COIQUTATION OF PHASE DIAGRAMS
Figure
3
Figure
Calorimetric device for measurements of enthalpies of mixing in the case of alloys of high vapour pressure. F : valve, C : protecting liquid. B : alloy melt.
Calorimetric cell for measurements of enthalpies of mixing ( drop method with preheating). I : funnel, J : piston.
are the mole fractions of + i nil and x. 1-I 1 nA being the molar and (i-11th B additions respectively, placed in the cell before any addition of 6.
‘1
=In
1
i
/[nA
ith
at
The temperature
integral T can
molar enthalpy be written :
of
formation
with
i
enthalpy
corresponding
with 6 = 0 when T > TR the partial enthalpy of
and mixing
to
to
both
the
1 i-IA+ c
=
the
6 = 1 when of B with
reaction
liquid
Hi
the A
components
[21
n.
1 (11
with
hi
= HB(Tl
-
small In case ni is very to Xi can be also deduced
T < Tm. respe%
h;[xil=
after
component
*i
1 the
reference
B obtained of
cHi-hB1”hi
(Xi]
being
component quantity
1
hf
Hi
4
-
hi ni
HBIToI
+ 6HFus
compared
:
ni [31
to
N-l-1.
208
R. Castanet et al.
III.
LIQUID-SOLID EQUILIBRIA
Men the temperature of the calorimeter is higher than the melting points of the two components,it is possible to study the system either by successive drops of B or by drop of A into the melt with direct reference to both liquid components. The two sets of measurements are totally independent and even calibrationsare not related. The agreement between these two series can therefore be considered as a good test of reliability.Fig. 5 shows the integral molar enthalpy of formation of Cu-Ge liquid alloys [71 at 1298 K obtained by adding pure copper and pure germanium to the bath. We can observe that the two ranges of measurement overlap and that in the concentrated solutions, the two series m of results are in agreement.
I Figure 5 Molar integral enthalpy of formation of Cu-Ge alloys referred to solid Cu and liquid Ge at 1298 K. Full circles : by adding Ge into the bath. Half-filled circles : by adding Cu into the bath.
ca10r~8,~et,hIS”~~~~~a~~~~ yphe melting point of the component 8, it is also possible to carry out the measurement by adding A(s.T,l to the component BIs,T). For the first additions, the component A warms up. melts and dissolves progressivelythe component 8. Beyond the B-rich liquidus. the resultant of the successive additions of metal A is in all respects identical to the preceeding case. It is here possible to refer, mla either to the solid component A and the liquid B or to the two liquid components by estimating the enthalpy of melting of B at the working temperature T. In the first case, t$ is merely equal to hG(T1 - hg[T,). In the second case, the reference state of 8 is an hypotheticalsupercooled state. hfus B (T) = hG(l,Tl - hB(s.T1 where hG(l.Tl can be obtained by extrapolatingthe enthalpy of the liquid below its melting point. Fig. 6 shows an example of this type of measurement in the case of Ge-Te alloys (81 with reference to liquid tellurium and solid germanium.
Figure 6 Molar enthalpy of formation hf of the Ge-Te system with reference to liquid tellurium and solid germanium versus mole fraction of germaniumxGe. The results have been obtained by successive additions of Te(s.298 K1 to Ge(s,Tl in the temrange 1028 < T/K < 1150. The limits of the two-phase region Liquid + Ge(s,purel which correspond to the region where the enthalpy of formation is linear with respect to the mole fraction, are represented by circles on Fig. 7. These results are not in agreement with the liquidus given by Hansen and Anderko [91. perature
EXPERI~NT~
209
DRTE~INATION AND CO~UTATION OF PHASE DIAGRAMS
%m
Figure 7 Phase diagram of the Ge-Te system. Open circles : our results (81.
Lastly, when the experimental temperature is lower than the melting points of the two components, it is no longer possible to obtain directly the integral snthalpy of formation of the liquid phase. However, it is always possible to measure the partial enthalpies of the two components by dropping them in a liquid master alloy prepared outside the calorimeter.The integral enthalpy of formation is then calculated by the aid of the following relation : hf =xh f A + II-xl hf B This has been done at 1380 K for Pd-Si alloys (101 in the two eutectic regions (see Fig. 5) on both side of the compound Pd2Si (Tm = 1671 Ki. Results are presented in Fig. 9. The method of determinationof phase boundaries of which some examples have been given here concerning binary alloys, applies also to the ternary ones fill Fig. IO shows the partial molar enthalpy of bismuth at 737 K, h& in an initial binary solution of tellurium and antimony at xsb = 0.0123. The sudden decrease of h& for XBi 'L 0.13 (XBi, ternary molar fraction of the solution Te-Sb-Bil corresponds to the intersection of the plane T = 737 K with the tellurium-rich liquidus surface at the ratio XBb,/ XTe = 0.012%. Fig. 11 shows the evolution with XBi of the integral molar enthalpy of formation of the ternary system referring to liquid bismuth and to liquid binary Te-Sb initial solutions, for some values of the ratio Xsb / XTe*
Fkt51 + fSi)
The phase boundaries determined in this way are shown on Fig. 12. Measurements of more concentratedsolutions such as the one shown on Fig. 13, also enabled us to determine the limits of the solid phase (Bi2_$b2+ )Te%. In fact, we can see (fig. 121 that tKese limits do not include the previous formula thus confirming the nonexistence of a continuous solid solutions at 737 K between Bi2Te3 and So2Te3.
60
70
80
90
it
Figure 8 Phase diagram of the Pd-Si system according to 1341.
R. Castanet
210
et az.
Figure
9
Partial and integral enthalpies of formation of Pd-Si liquid alloys at 1380 K referred to pure solid components. o
(hpfdl
e
f (hSil
Curve using
Figure
experimental
points.
h$d calculated from hfSi Gibbs-Ouhem relation.
IO
Partial molar enthalpy of Bi at 737 K for initial mole fraction xSb c 0.0123 in the Sb-Te binary.
m
0
a05
al0
25L+
a15
7
Figure
11
Integral molar enthalpy of formation at K of the ternary Bi-Sb-Te system referred to pure liquid Bi and liquid binary Sb-Te for increasing xSb / xTe
737
ratio.
EXPERIMENTAL DETERMINATIONAND COMPUTATION OF PHASE DIAGRAMS
211
Figure 12 Phase boundaries of the Bi-SbTe system in the Te-rich corner at 737 K. 1 and 2 : solid solution. 3 : liquidus surface.
Figure 13 Integral molar enthalpy of formation at 737 K of the ternary Ei-Sb-Te system referred to pure liquid Bi and liquid binary Sb-Te = 0.04661. (',b
This observation is in agreement with the results of Abrikosov and Poretskaia (121 whose conclusions regarding section XTe = 0.6 are shown on figure 14.
Figure 14 Intersectionof the ternary Bi-Sb-Te phase diagram by the = 0.6 plane according to 2% . 12. Open circle : our result (Ill at 737 K.
R. Castanet et al.
212
IV. SOLID PHASES In many cases, the method can lead to the determination of the boundaries and the enthalpies of formation of solid compounds and intermediatephases. For example, the phase diagrams of the Sn-Te and Pb-Te systems according to Hansen and Anderko (91 and the isothermal enthalpies of formation at 737 K measured in the laboratory (131 are shown on the next figures.
+-Figure 15 Phase diagram of the Sn-Te and Pb-Te systems according to (9;
b
I I&
1
Figure 16 + Integral molar enthalpy of formation of the Pb-Te alloys at 737 K with reference to both pure liquid components.
According to the published diagrams, SnTe is a straight line compound and PbTe is stable from 0.33 c xTR < 0.57 at T=737 K Our results, on the contrary [Fig. 16 and 171 show that PbTe is a straig t line compound since the enthalpy of formation of the Pb-Te alloys has a very strong minimum for xTe = 0.5. In the case of Sn-Te (Fig. 18 and 191 the enthalpy of < 0.53 due to formation is not linear with respect to mole fraction in the range 0.49 < x the weak solubilities of Te and Sn in the compound. However, the liquid / lI&id + solid boundaries of both systems agree well with the Hansen's phase diagram.
EXPERIMENTAL DETERMINATIONAND COMPUTATION OF PHASE DIAGRAMS
213
Figure 17 Relative partial molar enthalpies of Pb and Te in liquid Pb-Te alloys at 737 K referred to purrsliquid components.
Figure 18 Integral molar enthalpy of formation of the Sn-Te alloys at 737 K with reference to both pure liquid components.
Figure 19 Relative partial molar enthalpies of Sn and Te in liquid Sn-Te alloys at 737 K referred to pure liquid components.
214
R. Castanet et al.
The low-temperatureresults obtained in the Ag-Te system (141 adding silver in liquid tellurium are shown on Fig. 20. At T = 745 K, the minimum of the hf = f[xl curve OccURi for xTe = 0.335 indicating that the solid phase in equilibrium is Ag2Te. At T = 726 K, the minimum is located at xTe = 0.353, correspondingto the incongruentlymelting compound Ag,,64Te in agreement with the Kracek's phase diagram [I51 as shown on Fig. 21. At higher temperature,due to the high vapor pressure of tellurium, it was impossible to add silver into liquid tellurium in the calorimeter,but the strong associations in the liquid make possible the opposite that it to add tellurium on solid or liquid silver [161. Figure 20 Integral molar enthalpy of formation of the Ag-Te alloys referred to solid Ag and liquid Te. 1. 745 K [liquid in equilibrium with Ag2Tel. 2. 726 K (liquid in equilibrium with Agl 64Tel.
The and For due
results are representedon Fig. 22. 23 and 24. the correspondingphase boundaries on Fig. 21. xTe > 0.333tAg2Tel.the results are scattered to the existence of free tellurium.
The previous examples have demonstrated the possibility to determine the solidus using direct reaction calorimetry. Simultaneously, the method provides the enthalpies of formation of the solid phase with respect to mole fraction in the cases of bertholides (Fig. 16. Sn-Te system). If these results are performed at different temperatures,we can calculate the heat capacity of formation of the compound. In case of the Ag-Te system (Fig. 251. the value is negative : Cf = - 3.8 J K-lmol-l. From the enthalpy of fo&ation of the liquid compound Ag2Te at 1235 K and the solid one at 1210 K, its melting enthalpy has been calculated : Atim= 3.0 ? 0.6 kJ mol-'. When the enthalpies of formation of the liquid or the solid phase are temperature dependent, the measurements have to be performed as near as possible of the melting point. For this reason, Au-Te melts
for one
the enthalpies of formation of the [17,16) were determined at 728 K the solid phase and at 733 K for the liquid [Fig. 261. In the peculiar case of the E-Se
system (191, we have obtained the enthalpies of formation at the same temperature,T = 666 K, since after a set of additions of selenium into
Figure 21 diagram of the Ag-Te system. Full circles : our results at high temperature adding Te into the bath (16) and at low temperature adding Ag into the bath Phase
(141.
215
EXPERIMENTAL DETERMINATIONAND COMPUTATION OF PHASE DIAGRAMS
+
Figure
22
Integral molar enthalpy of formation of the alloys at 1302 K [half-filled circlesl, 1255 K (open circles1 and 1235 K (full cfrclesl referred to bath liquid components.
Ag-Te
Figure 23 Relative partial molar enthalpy of Ag in liquid Ag-Te alloys in the temperature range 1235-1302 K referred to pure liquid Ag.
+- Figure 24 Integral molar enthalpy of formation of the alloys referred to solid Ag and liquid Te.
Ag-Te
m 11210 K1 1 o 11185 o [1155 to e (985 IO. The points boundaries
I o A, as
(1115
K) : Lo i
e (1165 l
(1050
Kl : K1 i
8, C. 0 lead to the phase indicated on Fig. 21.
R.
216
Castanet
et
al.
m 0
po
900
A
lloo Figure
25
Enthalpy
of
formation
of
AgC.667TaC 333 with respect to temoeraiure referred to pure sblid Ag and Te.
Figure
26
5
Integral molar enthalpy of formation of the Au-Te alloys at 728 K [open circles1 and
0
-5
Lurm* t -10
Figure 27 Integral molar enthalpy of formation of the Bi-Se alloys at 888 K. A : alleys in thermodynamicalequilibrium. B : supercooled liquid alloys.
737 K (others).
at
EXPERI~NT~
217
DETE~INATION AND CO~UTATION OF PHASE DIAGRAMS
the melt, we did not obtain (Fig. 271 solid Bi2SeS but a metastable single-phasemelt. This phenomenon of isothermal supercooling yielded a value of the enthalpy of melting of BiC.4SeO.8 below its melting point. Thus. the method allowed us to determine the enthalpies of phase transitions.With this method and because it deals with measur~ents in equilibrium,we think that the results obtained are more valuable than those by DTA.
V. COHERENCY CRITERION We have seen that our calorimetricmeasurements yield the experimental phase diagram, the partial and integral enthalpies of fo~ation of all the present phases and the excess heat capacities. On the other hand, these measurements are related with potential ones as Knudsen-cellmass spectrometry. liquid or solid cells potentiometry,available in the laboratory, for obtaining the free enthalpies of formation of these phases. The excess free enthalpies of formation of each liquid or solid phase can be represented by the analytical expression :
gf,E = = f
(aJ
=
[bJ
sf.E =
(CJ
where h
Cf = IdI P Then, the phase equilibria can be calculated by the classical method of minimizing the free enthalpy of formation. This has been done in our laboratory in the following cases.
and
For the gold-silicon system, the free enthalpies of formation of the liquid alloys I201 have been measured in the temperature range 1550 * T/K < 2000 bY the "intensitv ratiomethod" (211 with a Knudsen-cellmass-spectrometerand the enthalpies of formation i22J by direct reaction calorimetry at T = 1373 K. The excess heat capacity of the liquid phase has been deduced from the measurements of Chen and Turnbull I23J. This set of thermodynamicresults yield a computed equilibrium diagram in agreement with that of Predel and Bankstahl (241 as shown on Fig. 28. That is a test of the good thermodynamic consistency of the whole informations used. The solid circle on Fig. 28 was obtained from the jump of the partial enthalpy of Si vs. mole fraction as shown on Fig. 23.
Figure 28 Phase diagram of the Au-Si system. Thick line and circles : according to literature. Full circle : according to our work [ZZJ. Thin lines : according to ideal behaviour of the liquid phase. ~:. Oashed lines : computed phase diagram.
218
R. Castanet et aZ.
m
+ Figure 29 Relative partial molar enthalpy of Si in Cu-Si alloys (half-filledcircles) and in Au-Si alloys fopen circles) referred to pure liquid Si.
Figure 30 Phase diagram of according to ref D our results by rimetrie, o our results by analysis
The phase diagram of the goldsilicon system was rather simple since there is no mutual solid solubility of the pure metals and no intermediate phases. In the case of the Au-Te system, we measured some phase boundaries by direct reaction calorimetry (0, Fig. 301 and some others by d.t.a. (0, Fig. 301. The results agree well with the phase diagram compiled in 191. Furthermore,we computed the equilibrium diagram fmm our enthalpic values and the values of the free enthalpy measured by Predel et al. (35). The computed diagram is in agreement with the experimentalresults as can be seen on Pig. 31. In the tellurium-thalliumsystem, the presence of three solid compounds, TlSTe3, TlTe and T12Te3 makes the phase diagram more complex. The partial and integral enthalpies of formation of the Tl-Te alloys have been measured at 18 temperatures ranged from 561 to 740 R (251.
Figure
31 +
Phase diagram of the Au-Te system. Thick lines : according to ref. 191. Thin lines : computed from thermodynamic functions.
the Au-Te system (91 direct reaction calodifferential thermal
m3
lzoo T 5; Iwo
I
Sal
0
EXPERIMENTAL DETERMINATIONAND COMPUTATION OF PHASE DIAGRAMS
219
Figure 32 Integral molar enthalpy of formation of the Te-Tl alloys at 717 and 718 K referred to both liquid components.
The results obtained at 717 and 718 K are shown on Fig. 32 and 33. We can see that the melting point of T1STe3, the mole fraction of which xTe = 0.375 corresponds to the enthalpic minimum at 717 K, is located between 717 and 718 K. The difference between the integral enthalpies of formation at 717 and 718 K at xTe = 0.375 corresponds to its enthalpy of melting.
I
We have reported on Fig. 34 the whole results dealing with the phase boundaries. In view of the boundaries of the miscibility gap at 740 K, it seems that the critical temperature deduced from our measurements is lower than that given by Wobst I261. From our calorimetricmeasurements and the potentiometric ones of Nakarmra et al. I.271,
Figure 33 Partial molar enthalpy of Te in the Te-Tl alloys at 717 and 718 K referred to liquid tellurium.
Figure 34 Phase diagram of the Te-Tl system. Thick line : ref (91. Half-filled circles : ref (261. Open circles : our experimentalwork (251.
R. Castanet et al.
220
we were able to settle an analytical relation representing the free enthalpy of formation of the liquid versus molar fraction and temperature.The concentrationrange of validity of this expression is 0.333 < xTe < 1 because of the peculiar shape [that of a strong heteroassociated system1 of the entropy of formation. However, since the excess heat capacity is zero in this temperaturerange, the coefficientsci and di of the equation [al are zero too. Using this expression,we were able to compute the liquid-Te(slequilibrium. The results are not in agreement with those compiled by Elliott (281 as shown on figure 35 but agrees well with our experimentalvalue measured at 643 K. In the same manner, assuming the compound T15Te3 to be a daltonide,we have computed the equilibrium liquid + T15Te3(s]. First, the values of the thermodynamicfunctions of the solid compiled by Mills (291 were used for computing this equilibrium.However, the computed melting point of the compound T15Te3 was much higher than the experimentalone and, calculatingthe enthalpy of formation of the solid from that of the liquid 125) and its enthalpy of fusion [30). we adjusted the entropv of formation of the solid on-its melting pgint. The Fig. 36 shows that the computed results agree well with our expem 1 rimental ones. In the same way, the liquid-TlTe and liquid-T12Te3 equilibria have been computed from the values of the thermodynamic 7 functions of the corresponding solids obtained by Vasilev [311 and Terpilowski (321. These values are in agreement concerning the enthalpies of formation but do not agree concerning the entropies of formation. As for the former equilibrium, the agreement between computed and experimentalphase diagram was poor and we have adjusted the entropy of formation of Tl-Te (resp. T12Te3)on the peritectic level TlTe + T15Te3 + liquid
I
-1
Figure 35 Phase diagram of the Te-Tl system. Thick lines : according to ref [91 Thin lines : computed phase diagram (331.
Figure 36 Phase diagram of the Te-Tl system. Thick lines and open circles : our experimentalwork (251. Thin lines : computed phase diagram (331.
[resp. T12Te3 + TlTe'+ liquid). The results are All these determinationsallowed us to propose computed values to phase diagram from pure tellurium to T15Te3. in agreement with our experimentalboundaries which differ in some points from the Elliot's diagram [Fig. 351 and the set of thermodynamicfunctions that we proposed for the tellurium-thalliumsystem is self-consistent[331.
221
EXPERIMENTAL DETERMINATIONAND COMPUTATION OF PHASE DIAGRAMS
VI. CONCLUSION In conclusion, we can say that the method described can be considered as a powerful tool of investigation from a thermodynamic point of view since it leads simultaneously to the experimental determination of the phase boundaries and to the values of the enthalpies the enthalpies of transition and the heat capacities of formation of any phase of formation, Coupled with a potential method, it enables providing its formation involves a liquid phase. to compute equilibrium diagram and to propose a coherent set of thermodynamic functions.
showing librium. better,
The main limitation is that it is not possible to investigate a metatectic decomposition that is those which give two other In this case, complementary measurements can be performed by by enthalpimetry.
the solid phases solid phases in equithermal analysis or.
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of
[U.S.I. O.C.,
Laffitte, Castanet,
Ber.
267
properties
unpublished
Castanet, Anderko,
Paris,
M. Gleiser, properties
Eur. Stand. Washington,
Bergman,
Abrikosov,
Sci.
343. 1974.
1966.
103.
15.
1133. 61,
1977,
-6,
1976,
1976,
561.
1000. 271.
1403.
Engng.,
-32,
1976,
93.
3646. 1975,
Kehiaian,
191. Z.
Metallkde,
-66,
1977,
342.
222
6. Castanet et al.
(261
M. Wobst, Scripta Vet., 2, 1971, 563.
(271
Y. Nakamura, M. Shimoji, Trans. Faraday Sot.. -67, 1971, 1270.
(261
R.P. Elliott, "Constitutionof binary alloys", First Suppl., MC Graw-Hill, New York, 1965.
[291
K.C. Mills, "Thermodynamicdata for inorganic sulphides, selenides and tellurides", Buttexworths,London, 1974.
(301
R. Castanet, J.-P. Bms. M. Laffitte, J. Chim. Phys., -65, 1966, 1535. V.P. Vasilev, A.V. Nikolskaia. Ya. I. Gerasimov, A.F. Kuznetsov, Izv. Akad. Nauk SSSR. Neor. Mater.. 4, 1966, 1040.
[311
(341
J. Terpilowski, E. Zaleska. Roczn. Chem., 37. 1963, 193. R. Castanet, M. Gilbert, C. Bergman, 2. Metallkde.-69, 1976, 298. W.G. Moffatt, "Binary phase diagram handbook", General Electric, Schenectady,1976.
[351
8. Predel, J. Piehl, Z. Netallkde,-66, 1975, 33.
[321 (331
vo1.3,
No.3,
pp.205~222.
Pergamon Press Ltd. 1979. Printed in Great Britain.
EXPERIMENTAL DETERMINATIONAND C~PUTATION DF PHASE DIAGRAMS FROM THEIR THE~ODY~XC FUNCTIONS by R. CASTANET, C. BERGMAN and J.-C. MATHIEU Centre de Thermodynamiqueet de Microcalorimetriedu CNRS 26, rue du 141e R.I.A., 13003 Marseille France
(This paper was presented at CALPHAD VIII Stockholm, Sweden, May 1979)
Most methods of determinationof phase diagrams are based on the observation of the variation versus temperature, of a given property of alloys of defined compositions.These methods, such as thermal analysis of e.m.f. dete~ination~ enable by direct m~sur~ents the vertical crossing (in a parallel direction to the temperature axisl of phase boundaries. The aim of this paper is to describe the method by which partial and integral enthalpies of formation of alloys vs. molar fraction are measured at a constant temperature i the phase boundaries are then obtained by the horizontal crossing of the phase diagram. In this paper, the systems concerned are mainly metal-semimetalalloys. In addition to the experimental phase diagram, the enthalpies of formation of the different phases of the system are simultaneouslyobtained with respect to composition and temperature.Combined with the values of the free enthalpies of formation obtained by a potential method, these results allow the computation of equilibrium diagrams. Lastly, the comparison of computed and experimentalresults enables the checking of the consistency of all the experimental informations. I. PRINCIPLE Let us consider a system of two components A and 5 whose melting temperaturesare TA and TD respectively.Fig. lb and Fig. Ic show the integral and partial enthalpy of formation. in a relatively simple case where the phase diagram (Fig. la] of the system has only one congruently melting compound. The enthalpies are referred to both liquid components at the equilibrium temperature T. In the two-phase regions BC and DE, the partial enthalpy of 6 is constant and the integral enthalpy is linear with respect to the mole fraction XB ; at the crossing of the phase boundary, the partial enthalpy of B undergoes a discontinuitywhich permits the localization of this boundary. Figure I 0
k I
1C
t 8
~
s
____-
----
c
E
E
la. Phase diagram of the A-B binary system. lb. Integral molar enthalpy of formation of A-B alloys at temperature T vs. mole fraction of El. Ic. Partial molar enthalpy of formation of component B at temperature T vs. mole fraction of B.
205
R. Castanet et al.
206
II.
EXPERIMENTALMETHOD
The apparatus used are Tian-Calvatmicrocalorimetersparticularlywall suited to the measurements of small heat quantities and therefore to the determinationof partial enthalpies. The experimentalmethod is a drop technique. which we first used to determine the enthalpy of formation of the silver-germaniumsystem at 1280 K. (11. The liquid component A. Fig. 2, is placed in a container C, the material of which can be Pyrex glass, silica, graphite, alumina, . . . depending on the temperature and the constituents of the alloy. The container is located at the bottom of the calorimeter cell at temperature T. The solid component 8, at room temperature T0, is introduced into the container C by means of the guiding pipe G and dissolves in pure A or in the mixture obtained by the previous additions of the component 8. In the blank pile is placed the same cell as in the laboratory pile. The whole device is under argon atmosphere, in order to prevent any pollution of the mixture.
Fig. 2. Calorimetric cell for measurements of enthalpies of mixing (direct drop method]. A : alloy, C : crucible, G : guiding pipe.
The calibrationof the calorimeter is carried out by dropping the solid component A at To into pure liquid A, before adding 8. The enthalpy variation of A and 8. which is necessary to, first of all, calibrate the apparatus and secondly to calculate hf(Xil referred to liquid components, is taken in tables of numerical data [2,31 or measured by comparisonwith that of N.E.S. a-alumina (41 of C.R.M.T. copper (5). The global thermal effect measured includes a term which is due to the heating of the component added to the bath and this term can, in some cases, represent the main part of the whole effect. The final result, that is the enthalpy of mixing referred to the two components at the temperature T of the calorimeter,may bear a high level of uncertainty.Therefore, it is necessary to use a device where the added component is preheated at the bath level before being dissolved (Fig. 31. The sample is blocked in the funnel I and the dissolutionis then started by lifting the piston J. The measured thermal effect in this case does not include the parasite term due to the heat-content.However, since this method is more sophisticated,we use it only when necessary. When the preheated component risks being attacked by the vapours of the bath. the funnel should be replaced by a valve F (Fig. 41 floating in a protecting low This method has been applied to vapour pressure liquid C at the experimental temperature. study the mercury-thalliunsystem (61 and can be used up to 600 K, above which it is difficult to find a protective liquid fullfilling the required conditions. The thermal phenomenon due to every corresponds
to
the
following
reaction
drop
of
the
component
B in the
solvent
bath
:
niB(s.Tol + Ni_,All.(T.xi_,l +
(Ni_,,+nilA1l.(Tl.xil
th where ni is the molar quantity of the component E added to the bath at the i addition i-l and Ni_, = n + f nithe molar quantity of bath after the [i-llth addition of metal 8. A
207
EXPERIMENTAL DETERMINATIONAND COIQUTATION OF PHASE DIAGRAMS
Figure
3
Figure
Calorimetric device for measurements of enthalpies of mixing in the case of alloys of high vapour pressure. F : valve, C : protecting liquid. B : alloy melt.
Calorimetric cell for measurements of enthalpies of mixing ( drop method with preheating). I : funnel, J : piston.
are the mole fractions of + i nil and x. 1-I 1 nA being the molar and (i-11th B additions respectively, placed in the cell before any addition of 6.
‘1
=In
1
i
/[nA
ith
at
The temperature
integral T can
molar enthalpy be written :
of
formation
with
i
enthalpy
corresponding
with 6 = 0 when T > TR the partial enthalpy of
and mixing
to
to
both
the
1 i-IA+ c
=
the
6 = 1 when of B with
reaction
liquid
Hi
the A
components
[21
n.
1 (11
with
hi
= HB(Tl
-
small In case ni is very to Xi can be also deduced
T < Tm. respe%
h;[xil=
after
component
*i
1 the
reference
B obtained of
cHi-hB1”hi
(Xi]
being
component quantity
1
hf
Hi
4
-
hi ni
HBIToI
+ 6HFus
compared
:
ni [31
to
N-l-1.
208
R. Castanet et al.
III.
LIQUID-SOLID EQUILIBRIA
Men the temperature of the calorimeter is higher than the melting points of the two components,it is possible to study the system either by successive drops of B or by drop of A into the melt with direct reference to both liquid components. The two sets of measurements are totally independent and even calibrationsare not related. The agreement between these two series can therefore be considered as a good test of reliability.Fig. 5 shows the integral molar enthalpy of formation of Cu-Ge liquid alloys [71 at 1298 K obtained by adding pure copper and pure germanium to the bath. We can observe that the two ranges of measurement overlap and that in the concentrated solutions, the two series m of results are in agreement.
I Figure 5 Molar integral enthalpy of formation of Cu-Ge alloys referred to solid Cu and liquid Ge at 1298 K. Full circles : by adding Ge into the bath. Half-filled circles : by adding Cu into the bath.
ca10r~8,~et,hIS”~~~~~a~~~~ yphe melting point of the component 8, it is also possible to carry out the measurement by adding A(s.T,l to the component BIs,T). For the first additions, the component A warms up. melts and dissolves progressivelythe component 8. Beyond the B-rich liquidus. the resultant of the successive additions of metal A is in all respects identical to the preceeding case. It is here possible to refer, mla either to the solid component A and the liquid B or to the two liquid components by estimating the enthalpy of melting of B at the working temperature T. In the first case, t$ is merely equal to hG(T1 - hg[T,). In the second case, the reference state of 8 is an hypotheticalsupercooled state. hfus B (T) = hG(l,Tl - hB(s.T1 where hG(l.Tl can be obtained by extrapolatingthe enthalpy of the liquid below its melting point. Fig. 6 shows an example of this type of measurement in the case of Ge-Te alloys (81 with reference to liquid tellurium and solid germanium.
Figure 6 Molar enthalpy of formation hf of the Ge-Te system with reference to liquid tellurium and solid germanium versus mole fraction of germaniumxGe. The results have been obtained by successive additions of Te(s.298 K1 to Ge(s,Tl in the temrange 1028 < T/K < 1150. The limits of the two-phase region Liquid + Ge(s,purel which correspond to the region where the enthalpy of formation is linear with respect to the mole fraction, are represented by circles on Fig. 7. These results are not in agreement with the liquidus given by Hansen and Anderko [91. perature
EXPERI~NT~
209
DRTE~INATION AND CO~UTATION OF PHASE DIAGRAMS
%m
Figure 7 Phase diagram of the Ge-Te system. Open circles : our results (81.
Lastly, when the experimental temperature is lower than the melting points of the two components, it is no longer possible to obtain directly the integral snthalpy of formation of the liquid phase. However, it is always possible to measure the partial enthalpies of the two components by dropping them in a liquid master alloy prepared outside the calorimeter.The integral enthalpy of formation is then calculated by the aid of the following relation : hf =xh f A + II-xl hf B This has been done at 1380 K for Pd-Si alloys (101 in the two eutectic regions (see Fig. 5) on both side of the compound Pd2Si (Tm = 1671 Ki. Results are presented in Fig. 9. The method of determinationof phase boundaries of which some examples have been given here concerning binary alloys, applies also to the ternary ones fill Fig. IO shows the partial molar enthalpy of bismuth at 737 K, h& in an initial binary solution of tellurium and antimony at xsb = 0.0123. The sudden decrease of h& for XBi 'L 0.13 (XBi, ternary molar fraction of the solution Te-Sb-Bil corresponds to the intersection of the plane T = 737 K with the tellurium-rich liquidus surface at the ratio XBb,/ XTe = 0.012%. Fig. 11 shows the evolution with XBi of the integral molar enthalpy of formation of the ternary system referring to liquid bismuth and to liquid binary Te-Sb initial solutions, for some values of the ratio Xsb / XTe*
Fkt51 + fSi)
The phase boundaries determined in this way are shown on Fig. 12. Measurements of more concentratedsolutions such as the one shown on Fig. 13, also enabled us to determine the limits of the solid phase (Bi2_$b2+ )Te%. In fact, we can see (fig. 121 that tKese limits do not include the previous formula thus confirming the nonexistence of a continuous solid solutions at 737 K between Bi2Te3 and So2Te3.
60
70
80
90
it
Figure 8 Phase diagram of the Pd-Si system according to 1341.
R. Castanet
210
et az.
Figure
9
Partial and integral enthalpies of formation of Pd-Si liquid alloys at 1380 K referred to pure solid components. o
(hpfdl
e
f (hSil
Curve using
Figure
experimental
points.
h$d calculated from hfSi Gibbs-Ouhem relation.
IO
Partial molar enthalpy of Bi at 737 K for initial mole fraction xSb c 0.0123 in the Sb-Te binary.
m
0
a05
al0
25L+
a15
7
Figure
11
Integral molar enthalpy of formation at K of the ternary Bi-Sb-Te system referred to pure liquid Bi and liquid binary Sb-Te for increasing xSb / xTe
737
ratio.
EXPERIMENTAL DETERMINATIONAND COMPUTATION OF PHASE DIAGRAMS
211
Figure 12 Phase boundaries of the Bi-SbTe system in the Te-rich corner at 737 K. 1 and 2 : solid solution. 3 : liquidus surface.
Figure 13 Integral molar enthalpy of formation at 737 K of the ternary Ei-Sb-Te system referred to pure liquid Bi and liquid binary Sb-Te = 0.04661. (',b
This observation is in agreement with the results of Abrikosov and Poretskaia (121 whose conclusions regarding section XTe = 0.6 are shown on figure 14.
Figure 14 Intersectionof the ternary Bi-Sb-Te phase diagram by the = 0.6 plane according to 2% . 12. Open circle : our result (Ill at 737 K.
R. Castanet et al.
212
IV. SOLID PHASES In many cases, the method can lead to the determination of the boundaries and the enthalpies of formation of solid compounds and intermediatephases. For example, the phase diagrams of the Sn-Te and Pb-Te systems according to Hansen and Anderko (91 and the isothermal enthalpies of formation at 737 K measured in the laboratory (131 are shown on the next figures.
+-Figure 15 Phase diagram of the Sn-Te and Pb-Te systems according to (9;
b
I I&
1
Figure 16 + Integral molar enthalpy of formation of the Pb-Te alloys at 737 K with reference to both pure liquid components.
According to the published diagrams, SnTe is a straight line compound and PbTe is stable from 0.33 c xTR < 0.57 at T=737 K Our results, on the contrary [Fig. 16 and 171 show that PbTe is a straig t line compound since the enthalpy of formation of the Pb-Te alloys has a very strong minimum for xTe = 0.5. In the case of Sn-Te (Fig. 18 and 191 the enthalpy of < 0.53 due to formation is not linear with respect to mole fraction in the range 0.49 < x the weak solubilities of Te and Sn in the compound. However, the liquid / lI&id + solid boundaries of both systems agree well with the Hansen's phase diagram.
EXPERIMENTAL DETERMINATIONAND COMPUTATION OF PHASE DIAGRAMS
213
Figure 17 Relative partial molar enthalpies of Pb and Te in liquid Pb-Te alloys at 737 K referred to purrsliquid components.
Figure 18 Integral molar enthalpy of formation of the Sn-Te alloys at 737 K with reference to both pure liquid components.
Figure 19 Relative partial molar enthalpies of Sn and Te in liquid Sn-Te alloys at 737 K referred to pure liquid components.
214
R. Castanet et al.
The low-temperatureresults obtained in the Ag-Te system (141 adding silver in liquid tellurium are shown on Fig. 20. At T = 745 K, the minimum of the hf = f[xl curve OccURi for xTe = 0.335 indicating that the solid phase in equilibrium is Ag2Te. At T = 726 K, the minimum is located at xTe = 0.353, correspondingto the incongruentlymelting compound Ag,,64Te in agreement with the Kracek's phase diagram [I51 as shown on Fig. 21. At higher temperature,due to the high vapor pressure of tellurium, it was impossible to add silver into liquid tellurium in the calorimeter,but the strong associations in the liquid make possible the opposite that it to add tellurium on solid or liquid silver [161. Figure 20 Integral molar enthalpy of formation of the Ag-Te alloys referred to solid Ag and liquid Te. 1. 745 K [liquid in equilibrium with Ag2Tel. 2. 726 K (liquid in equilibrium with Agl 64Tel.
The and For due
results are representedon Fig. 22. 23 and 24. the correspondingphase boundaries on Fig. 21. xTe > 0.333tAg2Tel.the results are scattered to the existence of free tellurium.
The previous examples have demonstrated the possibility to determine the solidus using direct reaction calorimetry. Simultaneously, the method provides the enthalpies of formation of the solid phase with respect to mole fraction in the cases of bertholides (Fig. 16. Sn-Te system). If these results are performed at different temperatures,we can calculate the heat capacity of formation of the compound. In case of the Ag-Te system (Fig. 251. the value is negative : Cf = - 3.8 J K-lmol-l. From the enthalpy of fo&ation of the liquid compound Ag2Te at 1235 K and the solid one at 1210 K, its melting enthalpy has been calculated : Atim= 3.0 ? 0.6 kJ mol-'. When the enthalpies of formation of the liquid or the solid phase are temperature dependent, the measurements have to be performed as near as possible of the melting point. For this reason, Au-Te melts
for one
the enthalpies of formation of the [17,16) were determined at 728 K the solid phase and at 733 K for the liquid [Fig. 261. In the peculiar case of the E-Se
system (191, we have obtained the enthalpies of formation at the same temperature,T = 666 K, since after a set of additions of selenium into
Figure 21 diagram of the Ag-Te system. Full circles : our results at high temperature adding Te into the bath (16) and at low temperature adding Ag into the bath Phase
(141.
215
EXPERIMENTAL DETERMINATIONAND COMPUTATION OF PHASE DIAGRAMS
+
Figure
22
Integral molar enthalpy of formation of the alloys at 1302 K [half-filled circlesl, 1255 K (open circles1 and 1235 K (full cfrclesl referred to bath liquid components.
Ag-Te
Figure 23 Relative partial molar enthalpy of Ag in liquid Ag-Te alloys in the temperature range 1235-1302 K referred to pure liquid Ag.
+- Figure 24 Integral molar enthalpy of formation of the alloys referred to solid Ag and liquid Te.
Ag-Te
m 11210 K1 1 o 11185 o [1155 to e (985 IO. The points boundaries
I o A, as
(1115
K) : Lo i
e (1165 l
(1050
Kl : K1 i
8, C. 0 lead to the phase indicated on Fig. 21.
R.
216
Castanet
et
al.
m 0
po
900
A
lloo Figure
25
Enthalpy
of
formation
of
AgC.667TaC 333 with respect to temoeraiure referred to pure sblid Ag and Te.
Figure
26
5
Integral molar enthalpy of formation of the Au-Te alloys at 728 K [open circles1 and
0
-5
Lurm* t -10
Figure 27 Integral molar enthalpy of formation of the Bi-Se alloys at 888 K. A : alleys in thermodynamicalequilibrium. B : supercooled liquid alloys.
737 K (others).
at
EXPERI~NT~
217
DETE~INATION AND CO~UTATION OF PHASE DIAGRAMS
the melt, we did not obtain (Fig. 271 solid Bi2SeS but a metastable single-phasemelt. This phenomenon of isothermal supercooling yielded a value of the enthalpy of melting of BiC.4SeO.8 below its melting point. Thus. the method allowed us to determine the enthalpies of phase transitions.With this method and because it deals with measur~ents in equilibrium,we think that the results obtained are more valuable than those by DTA.
V. COHERENCY CRITERION We have seen that our calorimetricmeasurements yield the experimental phase diagram, the partial and integral enthalpies of fo~ation of all the present phases and the excess heat capacities. On the other hand, these measurements are related with potential ones as Knudsen-cellmass spectrometry. liquid or solid cells potentiometry,available in the laboratory, for obtaining the free enthalpies of formation of these phases. The excess free enthalpies of formation of each liquid or solid phase can be represented by the analytical expression :
gf,E = = f
(aJ
=
[bJ
sf.E =
(CJ
where h
Cf = IdI P Then, the phase equilibria can be calculated by the classical method of minimizing the free enthalpy of formation. This has been done in our laboratory in the following cases.
and
For the gold-silicon system, the free enthalpies of formation of the liquid alloys I201 have been measured in the temperature range 1550 * T/K < 2000 bY the "intensitv ratiomethod" (211 with a Knudsen-cellmass-spectrometerand the enthalpies of formation i22J by direct reaction calorimetry at T = 1373 K. The excess heat capacity of the liquid phase has been deduced from the measurements of Chen and Turnbull I23J. This set of thermodynamicresults yield a computed equilibrium diagram in agreement with that of Predel and Bankstahl (241 as shown on Fig. 28. That is a test of the good thermodynamic consistency of the whole informations used. The solid circle on Fig. 28 was obtained from the jump of the partial enthalpy of Si vs. mole fraction as shown on Fig. 23.
Figure 28 Phase diagram of the Au-Si system. Thick line and circles : according to literature. Full circle : according to our work [ZZJ. Thin lines : according to ideal behaviour of the liquid phase. ~:. Oashed lines : computed phase diagram.
218
R. Castanet et aZ.
m
+ Figure 29 Relative partial molar enthalpy of Si in Cu-Si alloys (half-filledcircles) and in Au-Si alloys fopen circles) referred to pure liquid Si.
Figure 30 Phase diagram of according to ref D our results by rimetrie, o our results by analysis
The phase diagram of the goldsilicon system was rather simple since there is no mutual solid solubility of the pure metals and no intermediate phases. In the case of the Au-Te system, we measured some phase boundaries by direct reaction calorimetry (0, Fig. 301 and some others by d.t.a. (0, Fig. 301. The results agree well with the phase diagram compiled in 191. Furthermore,we computed the equilibrium diagram fmm our enthalpic values and the values of the free enthalpy measured by Predel et al. (35). The computed diagram is in agreement with the experimentalresults as can be seen on Pig. 31. In the tellurium-thalliumsystem, the presence of three solid compounds, TlSTe3, TlTe and T12Te3 makes the phase diagram more complex. The partial and integral enthalpies of formation of the Tl-Te alloys have been measured at 18 temperatures ranged from 561 to 740 R (251.
Figure
31 +
Phase diagram of the Au-Te system. Thick lines : according to ref. 191. Thin lines : computed from thermodynamic functions.
the Au-Te system (91 direct reaction calodifferential thermal
m3
lzoo T 5; Iwo
I
Sal
0
EXPERIMENTAL DETERMINATIONAND COMPUTATION OF PHASE DIAGRAMS
219
Figure 32 Integral molar enthalpy of formation of the Te-Tl alloys at 717 and 718 K referred to both liquid components.
The results obtained at 717 and 718 K are shown on Fig. 32 and 33. We can see that the melting point of T1STe3, the mole fraction of which xTe = 0.375 corresponds to the enthalpic minimum at 717 K, is located between 717 and 718 K. The difference between the integral enthalpies of formation at 717 and 718 K at xTe = 0.375 corresponds to its enthalpy of melting.
I
We have reported on Fig. 34 the whole results dealing with the phase boundaries. In view of the boundaries of the miscibility gap at 740 K, it seems that the critical temperature deduced from our measurements is lower than that given by Wobst I261. From our calorimetricmeasurements and the potentiometric ones of Nakarmra et al. I.271,
Figure 33 Partial molar enthalpy of Te in the Te-Tl alloys at 717 and 718 K referred to liquid tellurium.
Figure 34 Phase diagram of the Te-Tl system. Thick line : ref (91. Half-filled circles : ref (261. Open circles : our experimentalwork (251.
R. Castanet et al.
220
we were able to settle an analytical relation representing the free enthalpy of formation of the liquid versus molar fraction and temperature.The concentrationrange of validity of this expression is 0.333 < xTe < 1 because of the peculiar shape [that of a strong heteroassociated system1 of the entropy of formation. However, since the excess heat capacity is zero in this temperaturerange, the coefficientsci and di of the equation [al are zero too. Using this expression,we were able to compute the liquid-Te(slequilibrium. The results are not in agreement with those compiled by Elliott (281 as shown on figure 35 but agrees well with our experimentalvalue measured at 643 K. In the same manner, assuming the compound T15Te3 to be a daltonide,we have computed the equilibrium liquid + T15Te3(s]. First, the values of the thermodynamicfunctions of the solid compiled by Mills (291 were used for computing this equilibrium.However, the computed melting point of the compound T15Te3 was much higher than the experimentalone and, calculatingthe enthalpy of formation of the solid from that of the liquid 125) and its enthalpy of fusion [30). we adjusted the entropv of formation of the solid on-its melting pgint. The Fig. 36 shows that the computed results agree well with our expem 1 rimental ones. In the same way, the liquid-TlTe and liquid-T12Te3 equilibria have been computed from the values of the thermodynamic 7 functions of the corresponding solids obtained by Vasilev [311 and Terpilowski (321. These values are in agreement concerning the enthalpies of formation but do not agree concerning the entropies of formation. As for the former equilibrium, the agreement between computed and experimentalphase diagram was poor and we have adjusted the entropy of formation of Tl-Te (resp. T12Te3)on the peritectic level TlTe + T15Te3 + liquid
I
-1
Figure 35 Phase diagram of the Te-Tl system. Thick lines : according to ref [91 Thin lines : computed phase diagram (331.
Figure 36 Phase diagram of the Te-Tl system. Thick lines and open circles : our experimentalwork (251. Thin lines : computed phase diagram (331.
[resp. T12Te3 + TlTe'+ liquid). The results are All these determinationsallowed us to propose computed values to phase diagram from pure tellurium to T15Te3. in agreement with our experimentalboundaries which differ in some points from the Elliot's diagram [Fig. 351 and the set of thermodynamicfunctions that we proposed for the tellurium-thalliumsystem is self-consistent[331.
221
EXPERIMENTAL DETERMINATIONAND COMPUTATION OF PHASE DIAGRAMS
VI. CONCLUSION In conclusion, we can say that the method described can be considered as a powerful tool of investigation from a thermodynamic point of view since it leads simultaneously to the experimental determination of the phase boundaries and to the values of the enthalpies the enthalpies of transition and the heat capacities of formation of any phase of formation, Coupled with a potential method, it enables providing its formation involves a liquid phase. to compute equilibrium diagram and to propose a coherent set of thermodynamic functions.
showing librium. better,
The main limitation is that it is not possible to investigate a metatectic decomposition that is those which give two other In this case, complementary measurements can be performed by by enthalpimetry.
the solid phases solid phases in equithermal analysis or.
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R.
[21
R. Hultgren, P.O. Oesai. O.T. Hawkins, “Selected values of the thermodynamic Metals Park, Ohio, U.S.A., 1973.
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M.
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Acad.
(31
I. Barin, O.Knacke, “Thermochemical Verlag, Berlin, 1973.
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