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Workshop on thermodynamic modelling of solutions and alloys
Colphad Vol. 21, No. 2, pp. 247-263. 1997 0 1997 Published by Elsevier Science Ltd Printed in Great Britain. All rights reserved 0364-5916197 $17.00 + 0.00
Pergamon
PII SO364-5916(97)
Workshop
00023-o
on
THERMODYNAMIC
MODELLING ALLOYS
SchlolJ Ringberg,
GROUP
OF SOLUTIONS
March 7 O-16,1996
4: Modeling
of Oxides
Group Members
Royal Institute
M Hillert of Technology, Stockholm,
B Burton NIST, Gaithersburg,
Uppsala
:
USA
S K Saxena University, Uppsala,
S Degtcrov Ecole Polytechnique, Montreal,
University
Sweden
Sweden
Canada
K C Hari Kumar of Lcuven, Lcuvcn, The Netherlands
H Ohtani Tohoku University, Sendai, Japan F Aldinger Max Planck Institute, Stuttgart,
Germany
A Kussmaul Max Planck Institute, Stuttgart,
Germany
247
AND
M. HILLERT et al.
248
MODELING OF OXIDES M. Hillert, B. Burton, S. K. Saxena, S. Degterov, K.C. Hari Kumar, H. Ohtani, F. Aldinger A. Kussmaul (Assistant) October 15. 1996 Report of Group IV, Ringberg, March 1994
1 Introduction Group 4 was charged with the work on recommending appropriate solid solution formulations for oxides. We were to describe and discuss any problems in implementing the solid solution models in thermodynamic computer softwares (Chemsage, Thermo-Calc). The group concentrated on the study of solutions with mixing of cations on two or more sites. The solutions with one site, e.g., aluminous garnets (Ca,Mg,Fe,Mn)3AI2Si~Ol2 was considered to have been dealt with amply in literature and by Group one for liquids. Such solutions are easily handled by using the standard models, e.g. the Redlich-KisterMuggianu polynomial. For solutions with more than one sublattice, the Group adopted the Compound Energy Model (CEM). Several examples were considered: 1. Orthopyroxene (Mg,Fe)Ml [Fe,Mg]mSiZOh with binary mixing of exchangeable cations on two sites, 2. Spine1 (Zn ,Fez+,FeJ+)r [Zn,FeZ+, Fe3+]2004, 3.
Ti203 - FeTi03 - Fe203: ((Ti3+,Ti4+,F&+,Fe3+)(T?+,Ti‘t+,Fe2+,Fe3+))
4. Omphacite ((AI3+,Mg2+),(AI3+,Mg2+)’ D‘ral+,Ca2’],[Na*‘,Ca2+])Si4012 5. Clinopyroxene (Ca,Mg,Fe)m mg,Fe]Ml S&06. In all cases, it was demonstrated that the CEM could be used. Problems specific to the spine1 and the system Fe-Ti-0 are discussed in the text that follows.
1.1 Effect of Magnetic
Ordering
The following method will be used (1)
For one sublattice this reduces to
In line with opinions expressed by R-95, we recommend Inden’s model truncated to 3 terms. It has five model parameters: Tc B 1; m and n. It is recommended thatfand the
THERMODYNAMIC
MODELLING OF SOLUTIONS AND ALLOYS
exponents m and n be fixed for each phase, but may be optimised phase. Tc is described with Redlich-Kister,
(I/ (1 Lt.
’ “0 7-j +~~~y:y’.y; Tc = xxyiyj i jk ‘I i j
fl is more problematic
+ y: -y’.
L!..a+
J)Y.
-)
because it enters into an expression
249
for each new kind of
(3)
It@ + 1). G, is given for 1
mole of formula units. The effect of all atoms in a formula unit could be described
as xui
lncfli + 1) where Ui is the number of i atoms per formula unit. For practical reasons, a /3 value representing
the average effect of all i atoms could be used, yielding In(P(comp.)
+
1) 1-a;. In order to come closer to physical realities one may define CUi as the number of sites that may contain magnetic atoms. formula Fe304 and thus
CUi =
described with Redlich-Kister.
For spinels we may take magnetite with the real
3 as an example.
It is recommended
An alternative would be to replace
xU/
that p(comp.)
be
ln(pi + 1) by ln(Ji +
1) but those j? values could be much larger than individual /$. That does not matter for a compound but the composition
dependence
will be different.
1.2 Effect of pressure The problem of including the pressure effect in EC is closely linked to the pressure-volume relation for the end-member. At present, we use Murnaghan’s Equation of States (EoS) for expressing volume as a function of pressure. This EoS works well upto a pressure of about 10 to 15 GPa. It may be possible to extend this pressure range to 100 GPa. by using P-V-T data calculated from Birch-Mumaghan EoS which is recast in a form usable in ThermoCalc. Unfortunately, the Birch-Mumaghan EoS can not be used in deriving an expression for the Gibbs energy, Mumaghan’s EoS yields an expression for the Gibbs energy expressed as MG,=“G,,,(T=O)+G,P’eS(T,P)
(4)
where
A (l+nPK)‘-k G,p”(r
where
P) =
K(n - 1)
-1
1
+ PL
(9
M. HILLERT et al.
250 A = Jgo,o)-v,(o,oo)
(6)
n is sometimes
denoted K1.
1.3 Composition
dependencies
It would be helpful if n can be made constant for each phase.
A, L,q,
al and
K canbe
described with Redlich-Kister type of polynomial. This model is probably good up to 105 bar. To improve the model for P 1 105 bar, one could introduce a second region with new values of the model parameters but one should then try to avoid discontinuities between the two regions. This problem has, not yet been examined. In principle, there is an alternative MG,,, =
,+y;(
'G,#
P = O)+'G&"(Z:P))+EGr(T
P,comp.)
(7)
However, the composition dependence of EC is caused by the, variation of all model parameters with composition. It will probably be very difficult to make this alternative more attractive than the first one.
2 How to handle a large number of compound energies The compound energy model (CEM) was originally constructed to give the molar Gibbs energy for a phase with more than one sublattice under the assumption of random mixing of constituents within each sublattice. For the special case of two sublattices with the same number of sites on both and with all nearest neighbor bonds going between the two sublattices, CEM is identical to the classic bond energy model (BEM). However, it offers a possibility of writing a very general kind of program and in its programmed form it offers a convenient way of organizing the description of a phase with sublattices and defining the model parameters. It thus may be of value even for simple cases where the classical bond energy applies. A problem with both models, BEM and CEM, is that the number of model parameters increases rapidly with the number of constituents. In a system with two sublattices and m and n constituents on each one, there are rn. n bond energies in BEM and the same number of compound energies in CEM. For near stoichiometric phases, the number of mode1 parameters can be reduced by applying the Wagner-Schottky model, which is a dilute solution treatment whereas BEM and CEM in principle cover the whole composition range between the end-members of the solution. In very many applications the information is limited to a restricted range of composition and the energies of many end-members are not known from independent information. The number of significant parameters in BEM and CEM is then reduced and these models become equivalent to the Wagner-Schottky treatment but the model parameters may look different. From the very beginning BEM and
THERMODYNAMIC
MODELLING
OF SOLUTIONS
AND ALLOYS
251
CEM have parameters representing interactions between minor constituents and the questionis simply how can they be omitted when they would have a negligible effect. A practical way of doing this in CEM will now be discussed. In a simple, reciprocal phase (A,B),(D,E)t there are four compounds and all their energies are in principle independent.
According to CEM, the energy of the system is
and it is illustrated in Fig. 1.
The colon is used to separate constituents of different sublattices. Whenever not necessary for clarity, the colon may be omitted. The quantities yi and &” are site fractions. The composition components,
can be varied within a square and the phase thus has three independent just as a ternary system where the composition can be varied within a triangle.
Only three of the compound energies the components
s?i;j are needed to define references for the energy of
because one of the components
can be formed from the other three by the
reciprocal reaction AD + BE + AE + BD. That reaction has the following standard Gibbs energy of reaction,
In any actual case, the quantity A ‘G,,~B;DEwill have a value independent
of the choice of
references and it is thus convenient to choose it as the fourth model parameter rather than the fourth compound energy. It describes the shape of the surface in Fig. 1. One may know its value or may evaluate it from information through an assessment or one may even guess its value.
If two of the constituents
are minor, e.g. B and E, then the value of A@AB:DE
has a small effect and it could perhaps be put to zero, which means that the surface in Fig. 1 is assumed to be planar and that the plane is a tangent plane to the real surface as illustrated in the BE-AD section in Fig. 2. This diagram illustrates that, having decided to use ‘GB:D, ~GA:Dand @B:E as references, one can give any value to ~?A:E or A LTI;AB;DE. The system will be very close to the tangent plane at the comer of the A:D end-member if B and E are both minor constituents, and that the tangent plane is not affected by the value of ~A:E
~~A~AB:DE
In a system with two sublattices with m and n constituents pound energies,
each, there will be rn. n com-
‘i;ii:kl, but m + n - 1 of them will be needed as references
The term -1 comes from the stoichiometric
for components.
constraint caused by a fixed number of sites of
the two sublattices. The number of parameters remaining will be rn. n - m - n + 1 and that is equal to (m - 1) . (n - 1). However, the number of independent reciprocal subsystems is also equal to (m - 1) (n - 1) All the remaining model parameters may thus be defined as standard
Gibbs energies
of independent
defining the model parameters
reciprocal
reactions
is strongly recommended.
A‘GV;u. That method
It may be emphasized
of
that there
M. HILLERT et a/.
252
are many ways of defining
the set of independent reciprocal subsystem and it is an important question how this would be best done. If there is only one major constituent on each sublattice, then one would still need m + n - 1 of the compound energies as references and the (m - 1). (n - 1) that remain may be defined through (m - 1) (n - 1) independent
reciprocal subsystems.
put to zero or any other arbitrary value. In a system with k sublattices and many constituents
Their A ‘lC;iitk,can be
there will be IIni compound
energies if ni represents the number of constituents on the i-th sublattice. The number of compound energies needed for defining the references is Z “i - k + 1 and one of them will be the reference for charges if there are ions. As already explained, it is proposed that the values of the remaining compound energies be defined through the standard Gibbs energy of independent reciprocal reactions. There may be a practical problem of finding a set of independent reciprocal reactions. The following procedure is recommended: Define the most important compound, say ABCD if there are four sublattices. Then arrange the remaining constituents in the order of decreasing importance, e.g. judged by the content. Replace one constituent at a time with a new one, obtaining e.g. AECD. Thus, the Z ni - k + 1 compounds have been defined that will be used as references and they will be the main model parameters. Then replace a second constituent with a new one and describe the corresponding compound energy through reciprocal reactions involving compounds obtained with one or no replacement e.g.
c AECF
= 4931
+ GABCF
+ CAECD
- %ABCD
All those compounds are already defined. AAer having treated all such alternatives, introduce a third constituent and describe the corresponding energy through a reciprocal reaction involving compounds obtained with two or one replacement, etc. If the constituents are introduced in the order of decreasing importance, then one can pay less and less attention to the corresponding A”Gi quantity in the assessment of the properties of the phase. The experimental information: often will be limited and somewhere down the line one may start to introduce arbitrary values for the corresponding A”Gi ,e.g. zero, which do not need to be adjusted during the assessment. It should not be necessary to reassess a subsystem which has been assessed previously using a reciprocal subsystem not appearing accord to the above procedure. Instead, one could rearrange the order of the constituents in a such a way that it does appear. However, if a compound appears in two different subsystems, that have both been assessed previously, it may be necessary to make a reassessment
3 Symmetry Constraints on Compound Energy Models The compound energy model (CEM) is often applied to complex systems in which ordering may, or may not, result in symmetry breaking. In the geological literature these two types of ordering phenomena are typically called convergent- and nonconvergent ordering, respectively [69Tho, 7OTho). Only the symmetry breaking (convergent) ordering
THERMODYNAMIC
MODELLING OF SOLUTIONS AND ALLOYS
253
process is associated with a phase transition. Some recommendations are made on the notation to be used for representing CEM models in a way that, at least partially, specifies the relevant crystallographic features (symmetry constraints) of the system to be described. Specific examples are used to illustrate problems that can arise when symmetry constraints are ignored. In particular, it is shown that if order parameter coupling [SSSal, 86&l, 87Dav] is not treated correctly the model may predict two transitions even though space group symmetry only permits one.
3.1 Sublattice Notation To promote understanding of sublattice/CEM models we recommend some conventions in notation that clarify symmetry constraints on CEM models. Consider for example a system with one sublattice at high-T and two at low-T, the usual representations are: high-T: (A,B,C ,.... )t low-T: (A,B,C ,... )I/,(A,B,C ,.... )I/~ If sublattices 1 and 2 become identical at high-T via an order-disorder the low-T system we suggest:
transition
then for
([A,B,C ,...I[A,B,C ,...I) to emphasize that the low-T sublattices are derived from the high-T sublattice through a symmetry breaking transition. The essential recommendation is that parentheses be used for sublattices which will never be symmetrically equivalent, and that square brackets, braces, etc. . .. [. . .], { . . . }. . . be used for sublattices that can become equivalent through an order-disorder transition. For example, consider ordering between tetrahedrally and octahedrally coordinated cations in spinel: (A,B,C ,... )~(A,B,C
,... )y’
Now, if we consider ordering on the octahedral sites of an inverse spinel, (B)(A,B), such as Mn2Ti04:
With this notation one can immediately see that ordering of Mn 2+ and Ti4+ between (..) and ([..],[..I) sublattices does not break the symmetry of the high-T phase, but ordering Note that there are Mn2+ and Ti4+ on the octahedral and octahedral’ sites does. symmetries in the interaction parameters that describe interactions between sublattices that
M. HILLERTet a/.
254 may become equivalent.
For example, pair interactions between the act and act’ sites must
be invariant with respect to interchange of species, so for A = Mn*+ and B = Ti4+, we have: ocr-ocf’
act-act’ =&BA
&AB
tet-act EAB
(10)
f p-act
(11)
Therefore, the symmetry information in the notation is sufficient to know which ordering processes may or may not be associated with a phase tradition.
4 Examples 4.1 Ti20J - FeTi03 - FezOJ A recent calculation of the Ti,O, - FeTiO, phase diagram by Eriksson evaluated the sublattice model:
et. al. [96Eri],
(Ti ‘+.Ti”,Fe*‘) (Ti”,Ti4’,Fe2’)
which treats borh the ilmenite (FeTiO,) and (Ti,O,) solid solutions as though they have the R3 ilmenite structure. In reality however, T&O, has the R3c corundum structure, from which an ilmenite structure phase may form via a second-order order-disorder tradition. In fact, the system FeTiO, - Fe,O, exhibits just such a transition which suggests that one can model the entire Ti,O, - FqO, binary with a sublattice model of the type: ([Ti”,Ti4’,Fe2’,Fe3’J[Ti3’,Ti4*,Fe2’,Fe3’])
4.2 Order Parameter Coupling When ordering breaks the symmetry on more than one sublattice, then more than one long range order (LRO) parameter will be required to, properly describe the system. In the simplest case of two LRO parameters (51 and 52) they may be symmebicdy coupled or
symmetrically decoupled: 1. If ?,I and 52 are symmetrically
coupled: the system will have one phase transition
and
one T,, 2. If 51 and 52 are symmetrically two z-c’s,
decoupled the system will have two phase transitions and
THERMODYNAMIC
MODELLING OF SOLUTIONS AND ALLOYS
Note, that E,t and 52 are always energefically Consider
the two ordered
considered
structures
as ordered I:1 compounds
coupled through the interaction
represented
in Figures
in the system a4-bB.
255
parameters.
3a and 3b which
can be
If the @,*)-sites and (Cl,W)-
sites always remain symmetrically distinct then both ordered phases (Figs. 3a and 3 b) can be represented by the four-sublattice model:
For simplicity,
assume that species a and b are confined to the (o,*) sublattices, and species
A and B are confined to the (Cl,=) sublattices; this approximation does not have any qualitative effect upon order-parameter coupling Components of the full binary are aA and bB, but we will only consider the transition behavior of the pure 1:l compound. As shown in Figure 3c, the ordered phase represented in Fig. 3a requires two LRO parameters (51 and j-J for a satisfactory description, but because 51 and 52 are symmetrically coupled there is only one T,.‘, The phase represented
in Figure 3b also requires two E,‘sbut in this
case 5, and 52 are symmetrically decoupled so there are two Tc’s. (decoupling)
of E,t and 52 can be understood qualitatively
The coupling
by examining the near neighbor
(nn) environments of (0) and (*)-sites with respect to (Q-site coordination. In the structure depicted in Fig. 3a (o)-sites have two empty (0) nn’s plus zero (W) nn’s, and the (*)-sites have zero (Cl) no’s plus two (m) no’s. Therefore ordering of A and B on the (O,W)-sites breaks the symmetry of the (0, *)-sites and vice versa which implies: 51’0-52’0
51=Oo~2=O~oneT, Note that crystal structure analysis of the mineral omphacite [68Cla, 87Dav]
indicates that it exhibits cation ordering that is topologically equivalent to the schematic shown in Fig. 3a. In the structure depicted in Fig. 3b both (o)-sites and (*)-sites have two empty (Cl) nn’s plus (W) nn’s. Therefore, the environments of (o)- and (.)-sites remain identical when ordering occurs on the (Cl,= )- sites, so in general 51 # c2, and the system exhibits two rc’s.
This is similar to the experimental
transformation
situation
in Heusler alloys that exhibit
the
sequence: L2t +B2 + BCC, except that in the Heusler alloy all sublattices
at high-T so that the sublattice description should be become equivalent ([a,b,..],[a,b,..]‘,[a,b,..]“, [a,b,..]“‘). An example that would strictly correspond to the sublattice structure depicted in Fig. 3b is ordering on the A- and B-sites of an (A)(B)03 perovskite [(A)-sites are 6coordinated coordinated by oxide ions].
by oxide ions, and (B)-sites
are (ideally)
l2-
Assuming equal contents of a, b, A, and B, site occupancies are as given in Table 2. Analytically, order parameter coupling/decoupling in the structures represented in Fig. 3 can be understood as follows in the next two sections.
M. HILLERT et al.
256 4.2.1
Coupled Ordering
AI-I= =AHM~ +AHM~ +~MIM~ where:
hy42 AHM~=T ‘Z 1,2
[(l+s2X~+e2)+(1-52Xl-42)]=~(1+~:)
03)
mMlM2-7 - 1~(~+5,)[2,,3(l-SZ)]+hZ2,4~[(1+12X1-51)] 4
=wofl+ w, [&,3 + z1.4 +(z1,4 4
%22[2
(14)
- Z1,3j5152]
- Q&52)]
AHMIM~ =
(15)
4
where WMIM c Was + WbA and AZ= 2, ,3 - Z1,4. Note that the bilinear term _ WMIM2=&52) 4
. only present because Z,,3 # ZIP. The excess Gibbs energy is: IS
Minimizing with respect to 51and 52 yields:
a@ =
o =
as2
wMIzMMl -<2__-
wMIM2;MIM2 5, _&-&I
2
a2
respectively. Consider the first of the above equations, if it is solved for 52 (with 51s 0):
and a similar analysis of the second equation indicates:
THERMODYNAMIC tJ=ozt,
MODELLING OF SOLUTIONS AND ALLOYS
257
=o
.‘.<, = 0 c3 62 = 0
This implies that there is only one T,. even though there are two LRO parameters that take independent values in the range 0 < T < T,. The essential point here is that Ht,,t~~ is bilinear in
4.2.2 Decoupled Ordering Now consider the structure represented in Fig. 3b. In this case, the bilinear term is not present because: Z1,J = z,,4 = &,) = z&t = 2 * AZ = 0 The derivation of hH~t* for the structure depicted in Fig. 3b is exactly the same as for the structure shown in Fig. 3a, but because AZ = 21,~ - Zt,4 = 0, the AHhrltM2term is a constant:
AH MIM2
h4,,2(q3 =
+ z1.4) 4
= 2WMI
M2
Clearly, there is no coupling of 51and 52 because A&t,, independent of 52 and there may be two Tc’s.
4.3
f f& or 52). Therefore, 5) is
The Problem associated with Pseudo End-Member (PEM)
a) The number of PEM
For solutions with more than two sublattices and/or with mixing of more than two cations on each crystallographic site, the number of PEM grows with each addition. Fig. 4 shows the crystal structure of an amphibole. If now, we were to count all the possible endmembers, we will have 1536 of them! We note that in counting the numbers, we have already exercised the crystal-chemical constraints; for example Na is not allowed in Ml, M2 and M3 sites or Al not permitted on A and M4 site etc.. It is quite clear that we need to reduce the number of independent PEM to a manageable few. b) Cation Site Preference Data
It will be important to tabulate data on cation site preference for oxides. This is already shown above and in Fig. 4 for amphibole. We may adopt the scheme of writing chemical
258
M. HILLERT et a/.
formula such that the cations are listed in the preference order. For example orthopyroxene, we write (Mg,Fe),, [Fe,Mg]Mz to indicate that Fe prefers the M2 site.
for
c) Calculation of the Enthalpy of PEM It would be quite desirable to calculate the gibbs energy of PEM either from first principles or by some other empirical method. One possibility is that we use the polyhedral approach as discussed below. We may consider that each crystal is composed of several polyhedra and a certain characteristic enthalpy can be assigned to such coordination unit. Thus one may consider that an orthopyroxene (MgMtMg&Si20,j consists of an octahedron of the MgMt type, a second octahedron of the Mgm
type and the third one of a Si type. We add
a second mineral to this data set e.g. an amphibole called anthophyllite
Mg$Si8022(OH)2
by comparing the structure shown in Fig. 5 with an orthopyroxene. We find that the Ml, M2 and M3 octahedra in amphibole are very similar to the Ml site of orthopyroxene; the M4 octahedron in amphibole is similar to M2 in orthopyroxene etc. Using such arguments, we build a data set with a series of compounds (e.g. silicates or spinels) for which we have enthalpy data. We convert the enthalpy data to lattice enthalpy, e.g. for enstatite, where AHL refers to the reaction: Mg*+ (gas) + Si 4+ + 30*- (gas) = MgSi03 (crystal) Such lattice enthalpy data can be regressed for finding the polyhedral contributions; enstatite the equation is a (Mg-Ml) + b (MpM2)
for
+ 2 e (Si) = AH, (enstatite)
We write similar equations for all solids and obtain the polyhedral energies for several silicates. Table 4 shows such energies for the polyhedra which may be used to calculate AHL for any silicate PEM of suitable composition. Let us use the data in Table 1 to calculate the PEM’s for ortbopyroxene (Mg,Fe)Mt [Fe,MglM2Si206. We need the enthalpies for the ordered, PEM MgMtFe&i206
and for the antiordered MgMZFeMtSi20,j.
From the table we have (KJ/mol): MgMt = -3911.4; FeMt - 4032.3; 2*Si -26282.0 MgM2 -3857.9; FeM2 -4015.6. Converting back to Ah’_‘_ (6*0-2 = 5400,2*Si = 20856.6, Fez+ 2842 and Mg*+ = 2349), we have mf for MgMlFe&i206 as -2762.0 and for FeMtMg&i20,j as - 2725.0. These values lie between that of MgMtMgM2Si206(-3095.45)
and FeMlFe&i,O,(-2389.35).
The data presented above are illustrative of how we can approach the solution to the PEM energy problem. We expect that such databases, one each for different types of compounds (silicates, spinels, sulfides etc.), can be used with advantage in future for assessing solid solution data from phase equilibrium experiments. It is necessary that we build such databases by determining the enthalpy of, as many different types of polyhedra as occur in oxides and silicates.
THERMODYNAMIC
MODELLING OF SOLUTIONS AND ALLOYS
References [69Tho] J.B. Thompson, Am. Mineral. 54,341 (1969) [7OTho] J.B. Thompson, Am. Mineral. 55,628 (1970) [85&l] E. Salje Phys. Chem. Minerals 12,93 (1985) [86Sal] E. Salje and V. Devarajan, Phase Transitions 6,235 (1986) [87Dav] P.M. Davidson and B.P. Burton Am. Mineral. 72,337 (1987) [96Eri] G. Eriksson, A. Pelton, E. Woermann and A. Ender, Ber. Bunsenges. Chem,to appear, 1996 [68CIa] J.R. Clark and J.J. Papike, Am. Mineral. 53,840
259
Phys.
Text for figures: Fig. 1. Gibbs energy diagram for a reciprocal system at constant T and P. The Gibbs energy surface is curved but sections parallel to the sides are straight. Fig. 2. Diagonal section through Fig. 1. Alloys close to the AD comer fall very close to the tangent, which is independent
of the value of ‘t?,,
Fig.3. Two ordered structures are shown in Figures 3a and 3b. Figure 3c shows that the ordered phase represented in Fig. 3a requires two LRO parameters (51 and 52) for a satisfactory
description,
but because
T,‘. The phase represented symmetrically
coupled there is only one
in Figure 3b also requires two 5’s but in this case ??,tand 52 are
decoupled so there are two Tc’s.
Fig.4. Structure of an amphibole. Double chains of Si04 tetrahedra are viewed along the c axis. The M 1, M2 and M3 cations form chains of edge-sharing octahedra between the apices of the tetrahedra while the larger M4 site does so between the bases. Site occupancy: A (Na‘,Ca*‘,Va), Ml, M2 or M3 (Fe2’,Mg2’,Fe’*,A13’), M4 (Ca2’, Na’, FeZ*,Mg2’), T (Si”, Al”) (OH, F) O,, .
Table 1: Nearest Neighbor Coordination
Table 2: Site Occupancies
Numbers for structure 1
for Structures 1 and 2
M. HILLERT et al.
260
Table 3: Nearest Neighbor Coordination Numbers for Structure 2
Table 4: Regressed lattice enthalpy of the polyhedral unite (W/mole of formula unit) Polyhedral
Ca+2
Tw nsmrt
_?CK, d”“,
.,..&I.“.
c ..,
Olivine Ml Olivine M2
-3525.4 -3531.9
Pyroxene Ml Pyroxene M2
-3545.3
Amphibole Feldspar Talc 4-coord 6-coord Kaol (03H3) Zoisite
.,,
.
A *._
I
I
_A,,?n
A1+3
Fe+2
II
Na+’
.lAh7 C
I”
,
K+’
1
._,.”
I
I
I
-387 -3911.4 -3857.0
-3889.
(coU7\
.....L......-
-3011
-3555.8 -3595.2
.All~lf-im.=
Fe+3
Mgf2
, -‘I”,‘.>
,
-,4>,.‘l I
I
Si+4
H20
THERMODYNAMIC
AE
MODELLING OF SOLUTIONS AND ALLOYS
AD
Fig. 1. Gibbs energy diagram for a reciprocal system at constant T and P. The Gibbs energy surface is curved but sections parallel to the sides are straight.
BE
AD
Fig. 2. Diagonal section through Fig. 1. Alloys close to the AD corner fall very close to the tangent, which is independent of the value of ‘GBE.
261
262
M. HILLERT et a/.
a
1
b
‘5 d T
Figure 3
-
T,
T,'
THERMODYNAMIC
Figure 4
MODELLING OF SOLUTIONS AND ALLOYS
263
Pergamon
PII SO364-5916(97)
Workshop
00023-o
on
THERMODYNAMIC
MODELLING ALLOYS
SchlolJ Ringberg,
GROUP
OF SOLUTIONS
March 7 O-16,1996
4: Modeling
of Oxides
Group Members
Royal Institute
M Hillert of Technology, Stockholm,
B Burton NIST, Gaithersburg,
Uppsala
:
USA
S K Saxena University, Uppsala,
S Degtcrov Ecole Polytechnique, Montreal,
University
Sweden
Sweden
Canada
K C Hari Kumar of Lcuven, Lcuvcn, The Netherlands
H Ohtani Tohoku University, Sendai, Japan F Aldinger Max Planck Institute, Stuttgart,
Germany
A Kussmaul Max Planck Institute, Stuttgart,
Germany
247
AND
M. HILLERT et al.
248
MODELING OF OXIDES M. Hillert, B. Burton, S. K. Saxena, S. Degterov, K.C. Hari Kumar, H. Ohtani, F. Aldinger A. Kussmaul (Assistant) October 15. 1996 Report of Group IV, Ringberg, March 1994
1 Introduction Group 4 was charged with the work on recommending appropriate solid solution formulations for oxides. We were to describe and discuss any problems in implementing the solid solution models in thermodynamic computer softwares (Chemsage, Thermo-Calc). The group concentrated on the study of solutions with mixing of cations on two or more sites. The solutions with one site, e.g., aluminous garnets (Ca,Mg,Fe,Mn)3AI2Si~Ol2 was considered to have been dealt with amply in literature and by Group one for liquids. Such solutions are easily handled by using the standard models, e.g. the Redlich-KisterMuggianu polynomial. For solutions with more than one sublattice, the Group adopted the Compound Energy Model (CEM). Several examples were considered: 1. Orthopyroxene (Mg,Fe)Ml [Fe,Mg]mSiZOh with binary mixing of exchangeable cations on two sites, 2. Spine1 (Zn ,Fez+,FeJ+)r [Zn,FeZ+, Fe3+]2004, 3.
Ti203 - FeTi03 - Fe203: ((Ti3+,Ti4+,F&+,Fe3+)(T?+,Ti‘t+,Fe2+,Fe3+))
4. Omphacite ((AI3+,Mg2+),(AI3+,Mg2+)’ D‘ral+,Ca2’],[Na*‘,Ca2+])Si4012 5. Clinopyroxene (Ca,Mg,Fe)m mg,Fe]Ml S&06. In all cases, it was demonstrated that the CEM could be used. Problems specific to the spine1 and the system Fe-Ti-0 are discussed in the text that follows.
1.1 Effect of Magnetic
Ordering
The following method will be used (1)
For one sublattice this reduces to
In line with opinions expressed by R-95, we recommend Inden’s model truncated to 3 terms. It has five model parameters: Tc B 1; m and n. It is recommended thatfand the
THERMODYNAMIC
MODELLING OF SOLUTIONS AND ALLOYS
exponents m and n be fixed for each phase, but may be optimised phase. Tc is described with Redlich-Kister,
(I/ (1 Lt.
’ “0 7-j +~~~y:y’.y; Tc = xxyiyj i jk ‘I i j
fl is more problematic
+ y: -y’.
L!..a+
J)Y.
-)
because it enters into an expression
249
for each new kind of
(3)
It@ + 1). G, is given for 1
mole of formula units. The effect of all atoms in a formula unit could be described
as xui
lncfli + 1) where Ui is the number of i atoms per formula unit. For practical reasons, a /3 value representing
the average effect of all i atoms could be used, yielding In(P(comp.)
+
1) 1-a;. In order to come closer to physical realities one may define CUi as the number of sites that may contain magnetic atoms. formula Fe304 and thus
CUi =
described with Redlich-Kister.
For spinels we may take magnetite with the real
3 as an example.
It is recommended
An alternative would be to replace
xU/
that p(comp.)
be
ln(pi + 1) by ln(Ji +
1) but those j? values could be much larger than individual /$. That does not matter for a compound but the composition
dependence
will be different.
1.2 Effect of pressure The problem of including the pressure effect in EC is closely linked to the pressure-volume relation for the end-member. At present, we use Murnaghan’s Equation of States (EoS) for expressing volume as a function of pressure. This EoS works well upto a pressure of about 10 to 15 GPa. It may be possible to extend this pressure range to 100 GPa. by using P-V-T data calculated from Birch-Mumaghan EoS which is recast in a form usable in ThermoCalc. Unfortunately, the Birch-Mumaghan EoS can not be used in deriving an expression for the Gibbs energy, Mumaghan’s EoS yields an expression for the Gibbs energy expressed as MG,=“G,,,(T=O)+G,P’eS(T,P)
(4)
where
A (l+nPK)‘-k G,p”(r
where
P) =
K(n - 1)
-1
1
+ PL
(9
M. HILLERT et al.
250 A = Jgo,o)-v,(o,oo)
(6)
n is sometimes
denoted K1.
1.3 Composition
dependencies
It would be helpful if n can be made constant for each phase.
A, L,q,
al and
K canbe
described with Redlich-Kister type of polynomial. This model is probably good up to 105 bar. To improve the model for P 1 105 bar, one could introduce a second region with new values of the model parameters but one should then try to avoid discontinuities between the two regions. This problem has, not yet been examined. In principle, there is an alternative MG,,, =
,+y;(
'G,#
P = O)+'G&"(Z:P))+EGr(T
P,comp.)
(7)
However, the composition dependence of EC is caused by the, variation of all model parameters with composition. It will probably be very difficult to make this alternative more attractive than the first one.
2 How to handle a large number of compound energies The compound energy model (CEM) was originally constructed to give the molar Gibbs energy for a phase with more than one sublattice under the assumption of random mixing of constituents within each sublattice. For the special case of two sublattices with the same number of sites on both and with all nearest neighbor bonds going between the two sublattices, CEM is identical to the classic bond energy model (BEM). However, it offers a possibility of writing a very general kind of program and in its programmed form it offers a convenient way of organizing the description of a phase with sublattices and defining the model parameters. It thus may be of value even for simple cases where the classical bond energy applies. A problem with both models, BEM and CEM, is that the number of model parameters increases rapidly with the number of constituents. In a system with two sublattices and m and n constituents on each one, there are rn. n bond energies in BEM and the same number of compound energies in CEM. For near stoichiometric phases, the number of mode1 parameters can be reduced by applying the Wagner-Schottky model, which is a dilute solution treatment whereas BEM and CEM in principle cover the whole composition range between the end-members of the solution. In very many applications the information is limited to a restricted range of composition and the energies of many end-members are not known from independent information. The number of significant parameters in BEM and CEM is then reduced and these models become equivalent to the Wagner-Schottky treatment but the model parameters may look different. From the very beginning BEM and
THERMODYNAMIC
MODELLING
OF SOLUTIONS
AND ALLOYS
251
CEM have parameters representing interactions between minor constituents and the questionis simply how can they be omitted when they would have a negligible effect. A practical way of doing this in CEM will now be discussed. In a simple, reciprocal phase (A,B),(D,E)t there are four compounds and all their energies are in principle independent.
According to CEM, the energy of the system is
and it is illustrated in Fig. 1.
The colon is used to separate constituents of different sublattices. Whenever not necessary for clarity, the colon may be omitted. The quantities yi and &” are site fractions. The composition components,
can be varied within a square and the phase thus has three independent just as a ternary system where the composition can be varied within a triangle.
Only three of the compound energies the components
s?i;j are needed to define references for the energy of
because one of the components
can be formed from the other three by the
reciprocal reaction AD + BE + AE + BD. That reaction has the following standard Gibbs energy of reaction,
In any actual case, the quantity A ‘G,,~B;DEwill have a value independent
of the choice of
references and it is thus convenient to choose it as the fourth model parameter rather than the fourth compound energy. It describes the shape of the surface in Fig. 1. One may know its value or may evaluate it from information through an assessment or one may even guess its value.
If two of the constituents
are minor, e.g. B and E, then the value of A@AB:DE
has a small effect and it could perhaps be put to zero, which means that the surface in Fig. 1 is assumed to be planar and that the plane is a tangent plane to the real surface as illustrated in the BE-AD section in Fig. 2. This diagram illustrates that, having decided to use ‘GB:D, ~GA:Dand @B:E as references, one can give any value to ~?A:E or A LTI;AB;DE. The system will be very close to the tangent plane at the comer of the A:D end-member if B and E are both minor constituents, and that the tangent plane is not affected by the value of ~A:E
~~A~AB:DE
In a system with two sublattices with m and n constituents pound energies,
each, there will be rn. n com-
‘i;ii:kl, but m + n - 1 of them will be needed as references
The term -1 comes from the stoichiometric
for components.
constraint caused by a fixed number of sites of
the two sublattices. The number of parameters remaining will be rn. n - m - n + 1 and that is equal to (m - 1) . (n - 1). However, the number of independent reciprocal subsystems is also equal to (m - 1) (n - 1) All the remaining model parameters may thus be defined as standard
Gibbs energies
of independent
defining the model parameters
reciprocal
reactions
is strongly recommended.
A‘GV;u. That method
It may be emphasized
of
that there
M. HILLERT et a/.
252
are many ways of defining
the set of independent reciprocal subsystem and it is an important question how this would be best done. If there is only one major constituent on each sublattice, then one would still need m + n - 1 of the compound energies as references and the (m - 1). (n - 1) that remain may be defined through (m - 1) (n - 1) independent
reciprocal subsystems.
put to zero or any other arbitrary value. In a system with k sublattices and many constituents
Their A ‘lC;iitk,can be
there will be IIni compound
energies if ni represents the number of constituents on the i-th sublattice. The number of compound energies needed for defining the references is Z “i - k + 1 and one of them will be the reference for charges if there are ions. As already explained, it is proposed that the values of the remaining compound energies be defined through the standard Gibbs energy of independent reciprocal reactions. There may be a practical problem of finding a set of independent reciprocal reactions. The following procedure is recommended: Define the most important compound, say ABCD if there are four sublattices. Then arrange the remaining constituents in the order of decreasing importance, e.g. judged by the content. Replace one constituent at a time with a new one, obtaining e.g. AECD. Thus, the Z ni - k + 1 compounds have been defined that will be used as references and they will be the main model parameters. Then replace a second constituent with a new one and describe the corresponding compound energy through reciprocal reactions involving compounds obtained with one or no replacement e.g.
c AECF
= 4931
+ GABCF
+ CAECD
- %ABCD
All those compounds are already defined. AAer having treated all such alternatives, introduce a third constituent and describe the corresponding energy through a reciprocal reaction involving compounds obtained with two or one replacement, etc. If the constituents are introduced in the order of decreasing importance, then one can pay less and less attention to the corresponding A”Gi quantity in the assessment of the properties of the phase. The experimental information: often will be limited and somewhere down the line one may start to introduce arbitrary values for the corresponding A”Gi ,e.g. zero, which do not need to be adjusted during the assessment. It should not be necessary to reassess a subsystem which has been assessed previously using a reciprocal subsystem not appearing accord to the above procedure. Instead, one could rearrange the order of the constituents in a such a way that it does appear. However, if a compound appears in two different subsystems, that have both been assessed previously, it may be necessary to make a reassessment
3 Symmetry Constraints on Compound Energy Models The compound energy model (CEM) is often applied to complex systems in which ordering may, or may not, result in symmetry breaking. In the geological literature these two types of ordering phenomena are typically called convergent- and nonconvergent ordering, respectively [69Tho, 7OTho). Only the symmetry breaking (convergent) ordering
THERMODYNAMIC
MODELLING OF SOLUTIONS AND ALLOYS
253
process is associated with a phase transition. Some recommendations are made on the notation to be used for representing CEM models in a way that, at least partially, specifies the relevant crystallographic features (symmetry constraints) of the system to be described. Specific examples are used to illustrate problems that can arise when symmetry constraints are ignored. In particular, it is shown that if order parameter coupling [SSSal, 86&l, 87Dav] is not treated correctly the model may predict two transitions even though space group symmetry only permits one.
3.1 Sublattice Notation To promote understanding of sublattice/CEM models we recommend some conventions in notation that clarify symmetry constraints on CEM models. Consider for example a system with one sublattice at high-T and two at low-T, the usual representations are: high-T: (A,B,C ,.... )t low-T: (A,B,C ,... )I/,(A,B,C ,.... )I/~ If sublattices 1 and 2 become identical at high-T via an order-disorder the low-T system we suggest:
transition
then for
([A,B,C ,...I[A,B,C ,...I) to emphasize that the low-T sublattices are derived from the high-T sublattice through a symmetry breaking transition. The essential recommendation is that parentheses be used for sublattices which will never be symmetrically equivalent, and that square brackets, braces, etc. . .. [. . .], { . . . }. . . be used for sublattices that can become equivalent through an order-disorder transition. For example, consider ordering between tetrahedrally and octahedrally coordinated cations in spinel: (A,B,C ,... )~(A,B,C
,... )y’
Now, if we consider ordering on the octahedral sites of an inverse spinel, (B)(A,B), such as Mn2Ti04:
With this notation one can immediately see that ordering of Mn 2+ and Ti4+ between (..) and ([..],[..I) sublattices does not break the symmetry of the high-T phase, but ordering Note that there are Mn2+ and Ti4+ on the octahedral and octahedral’ sites does. symmetries in the interaction parameters that describe interactions between sublattices that
M. HILLERTet a/.
254 may become equivalent.
For example, pair interactions between the act and act’ sites must
be invariant with respect to interchange of species, so for A = Mn*+ and B = Ti4+, we have: ocr-ocf’
act-act’ =&BA
&AB
tet-act EAB
(10)
f p-act
(11)
Therefore, the symmetry information in the notation is sufficient to know which ordering processes may or may not be associated with a phase tradition.
4 Examples 4.1 Ti20J - FeTi03 - FezOJ A recent calculation of the Ti,O, - FeTiO, phase diagram by Eriksson evaluated the sublattice model:
et. al. [96Eri],
(Ti ‘+.Ti”,Fe*‘) (Ti”,Ti4’,Fe2’)
which treats borh the ilmenite (FeTiO,) and (Ti,O,) solid solutions as though they have the R3 ilmenite structure. In reality however, T&O, has the R3c corundum structure, from which an ilmenite structure phase may form via a second-order order-disorder tradition. In fact, the system FeTiO, - Fe,O, exhibits just such a transition which suggests that one can model the entire Ti,O, - FqO, binary with a sublattice model of the type: ([Ti”,Ti4’,Fe2’,Fe3’J[Ti3’,Ti4*,Fe2’,Fe3’])
4.2 Order Parameter Coupling When ordering breaks the symmetry on more than one sublattice, then more than one long range order (LRO) parameter will be required to, properly describe the system. In the simplest case of two LRO parameters (51 and 52) they may be symmebicdy coupled or
symmetrically decoupled: 1. If ?,I and 52 are symmetrically
coupled: the system will have one phase transition
and
one T,, 2. If 51 and 52 are symmetrically two z-c’s,
decoupled the system will have two phase transitions and
THERMODYNAMIC
MODELLING OF SOLUTIONS AND ALLOYS
Note, that E,t and 52 are always energefically Consider
the two ordered
considered
structures
as ordered I:1 compounds
coupled through the interaction
represented
in Figures
in the system a4-bB.
255
parameters.
3a and 3b which
can be
If the @,*)-sites and (Cl,W)-
sites always remain symmetrically distinct then both ordered phases (Figs. 3a and 3 b) can be represented by the four-sublattice model:
For simplicity,
assume that species a and b are confined to the (o,*) sublattices, and species
A and B are confined to the (Cl,=) sublattices; this approximation does not have any qualitative effect upon order-parameter coupling Components of the full binary are aA and bB, but we will only consider the transition behavior of the pure 1:l compound. As shown in Figure 3c, the ordered phase represented in Fig. 3a requires two LRO parameters (51 and j-J for a satisfactory description, but because 51 and 52 are symmetrically coupled there is only one T,.‘, The phase represented
in Figure 3b also requires two E,‘sbut in this
case 5, and 52 are symmetrically decoupled so there are two Tc’s. (decoupling)
of E,t and 52 can be understood qualitatively
The coupling
by examining the near neighbor
(nn) environments of (0) and (*)-sites with respect to (Q-site coordination. In the structure depicted in Fig. 3a (o)-sites have two empty (0) nn’s plus zero (W) nn’s, and the (*)-sites have zero (Cl) no’s plus two (m) no’s. Therefore ordering of A and B on the (O,W)-sites breaks the symmetry of the (0, *)-sites and vice versa which implies: 51’0-52’0
51=Oo~2=O~oneT, Note that crystal structure analysis of the mineral omphacite [68Cla, 87Dav]
indicates that it exhibits cation ordering that is topologically equivalent to the schematic shown in Fig. 3a. In the structure depicted in Fig. 3b both (o)-sites and (*)-sites have two empty (Cl) nn’s plus (W) nn’s. Therefore, the environments of (o)- and (.)-sites remain identical when ordering occurs on the (Cl,= )- sites, so in general 51 # c2, and the system exhibits two rc’s.
This is similar to the experimental
transformation
situation
in Heusler alloys that exhibit
the
sequence: L2t +B2 + BCC, except that in the Heusler alloy all sublattices
at high-T so that the sublattice description should be become equivalent ([a,b,..],[a,b,..]‘,[a,b,..]“, [a,b,..]“‘). An example that would strictly correspond to the sublattice structure depicted in Fig. 3b is ordering on the A- and B-sites of an (A)(B)03 perovskite [(A)-sites are 6coordinated coordinated by oxide ions].
by oxide ions, and (B)-sites
are (ideally)
l2-
Assuming equal contents of a, b, A, and B, site occupancies are as given in Table 2. Analytically, order parameter coupling/decoupling in the structures represented in Fig. 3 can be understood as follows in the next two sections.
M. HILLERT et al.
256 4.2.1
Coupled Ordering
AI-I= =AHM~ +AHM~ +~MIM~ where:
hy42 AHM~=T ‘Z 1,2
[(l+s2X~+e2)+(1-52Xl-42)]=~(1+~:)
03)
mMlM2-7 - 1~(~+5,)[2,,3(l-SZ)]+hZ2,4~[(1+12X1-51)] 4
=wofl+ w, [&,3 + z1.4 +(z1,4 4
%22[2
(14)
- Z1,3j5152]
- Q&52)]
AHMIM~ =
(15)
4
where WMIM c Was + WbA and AZ= 2, ,3 - Z1,4. Note that the bilinear term _ WMIM2=&52) 4
. only present because Z,,3 # ZIP. The excess Gibbs energy is: IS
Minimizing with respect to 51and 52 yields:
a@ =
o =
as2
wMIzMMl -<2__-
wMIM2;MIM2 5, _&-&I
2
a2
respectively. Consider the first of the above equations, if it is solved for 52 (with 51s 0):
and a similar analysis of the second equation indicates:
THERMODYNAMIC tJ=ozt,
MODELLING OF SOLUTIONS AND ALLOYS
257
=o
.‘.<, = 0 c3 62 = 0
This implies that there is only one T,. even though there are two LRO parameters that take independent values in the range 0 < T < T,. The essential point here is that Ht,,t~~ is bilinear in
4.2.2 Decoupled Ordering Now consider the structure represented in Fig. 3b. In this case, the bilinear term is not present because: Z1,J = z,,4 = &,) = z&t = 2 * AZ = 0 The derivation of hH~t* for the structure depicted in Fig. 3b is exactly the same as for the structure shown in Fig. 3a, but because AZ = 21,~ - Zt,4 = 0, the AHhrltM2term is a constant:
AH MIM2
h4,,2(q3 =
+ z1.4) 4
= 2WMI
M2
Clearly, there is no coupling of 51and 52 because A&t,, independent of 52 and there may be two Tc’s.
4.3
f f& or 52). Therefore, 5) is
The Problem associated with Pseudo End-Member (PEM)
a) The number of PEM
For solutions with more than two sublattices and/or with mixing of more than two cations on each crystallographic site, the number of PEM grows with each addition. Fig. 4 shows the crystal structure of an amphibole. If now, we were to count all the possible endmembers, we will have 1536 of them! We note that in counting the numbers, we have already exercised the crystal-chemical constraints; for example Na is not allowed in Ml, M2 and M3 sites or Al not permitted on A and M4 site etc.. It is quite clear that we need to reduce the number of independent PEM to a manageable few. b) Cation Site Preference Data
It will be important to tabulate data on cation site preference for oxides. This is already shown above and in Fig. 4 for amphibole. We may adopt the scheme of writing chemical
258
M. HILLERT et a/.
formula such that the cations are listed in the preference order. For example orthopyroxene, we write (Mg,Fe),, [Fe,Mg]Mz to indicate that Fe prefers the M2 site.
for
c) Calculation of the Enthalpy of PEM It would be quite desirable to calculate the gibbs energy of PEM either from first principles or by some other empirical method. One possibility is that we use the polyhedral approach as discussed below. We may consider that each crystal is composed of several polyhedra and a certain characteristic enthalpy can be assigned to such coordination unit. Thus one may consider that an orthopyroxene (MgMtMg&Si20,j consists of an octahedron of the MgMt type, a second octahedron of the Mgm
type and the third one of a Si type. We add
a second mineral to this data set e.g. an amphibole called anthophyllite
Mg$Si8022(OH)2
by comparing the structure shown in Fig. 5 with an orthopyroxene. We find that the Ml, M2 and M3 octahedra in amphibole are very similar to the Ml site of orthopyroxene; the M4 octahedron in amphibole is similar to M2 in orthopyroxene etc. Using such arguments, we build a data set with a series of compounds (e.g. silicates or spinels) for which we have enthalpy data. We convert the enthalpy data to lattice enthalpy, e.g. for enstatite, where AHL refers to the reaction: Mg*+ (gas) + Si 4+ + 30*- (gas) = MgSi03 (crystal) Such lattice enthalpy data can be regressed for finding the polyhedral contributions; enstatite the equation is a (Mg-Ml) + b (MpM2)
for
+ 2 e (Si) = AH, (enstatite)
We write similar equations for all solids and obtain the polyhedral energies for several silicates. Table 4 shows such energies for the polyhedra which may be used to calculate AHL for any silicate PEM of suitable composition. Let us use the data in Table 1 to calculate the PEM’s for ortbopyroxene (Mg,Fe)Mt [Fe,MglM2Si206. We need the enthalpies for the ordered, PEM MgMtFe&i206
and for the antiordered MgMZFeMtSi20,j.
From the table we have (KJ/mol): MgMt = -3911.4; FeMt - 4032.3; 2*Si -26282.0 MgM2 -3857.9; FeM2 -4015.6. Converting back to Ah’_‘_ (6*0-2 = 5400,2*Si = 20856.6, Fez+ 2842 and Mg*+ = 2349), we have mf for MgMlFe&i206 as -2762.0 and for FeMtMg&i20,j as - 2725.0. These values lie between that of MgMtMgM2Si206(-3095.45)
and FeMlFe&i,O,(-2389.35).
The data presented above are illustrative of how we can approach the solution to the PEM energy problem. We expect that such databases, one each for different types of compounds (silicates, spinels, sulfides etc.), can be used with advantage in future for assessing solid solution data from phase equilibrium experiments. It is necessary that we build such databases by determining the enthalpy of, as many different types of polyhedra as occur in oxides and silicates.
THERMODYNAMIC
MODELLING OF SOLUTIONS AND ALLOYS
References [69Tho] J.B. Thompson, Am. Mineral. 54,341 (1969) [7OTho] J.B. Thompson, Am. Mineral. 55,628 (1970) [85&l] E. Salje Phys. Chem. Minerals 12,93 (1985) [86Sal] E. Salje and V. Devarajan, Phase Transitions 6,235 (1986) [87Dav] P.M. Davidson and B.P. Burton Am. Mineral. 72,337 (1987) [96Eri] G. Eriksson, A. Pelton, E. Woermann and A. Ender, Ber. Bunsenges. Chem,to appear, 1996 [68CIa] J.R. Clark and J.J. Papike, Am. Mineral. 53,840
259
Phys.
Text for figures: Fig. 1. Gibbs energy diagram for a reciprocal system at constant T and P. The Gibbs energy surface is curved but sections parallel to the sides are straight. Fig. 2. Diagonal section through Fig. 1. Alloys close to the AD comer fall very close to the tangent, which is independent
of the value of ‘t?,,
Fig.3. Two ordered structures are shown in Figures 3a and 3b. Figure 3c shows that the ordered phase represented in Fig. 3a requires two LRO parameters (51 and 52) for a satisfactory
description,
but because
T,‘. The phase represented symmetrically
coupled there is only one
in Figure 3b also requires two 5’s but in this case ??,tand 52 are
decoupled so there are two Tc’s.
Fig.4. Structure of an amphibole. Double chains of Si04 tetrahedra are viewed along the c axis. The M 1, M2 and M3 cations form chains of edge-sharing octahedra between the apices of the tetrahedra while the larger M4 site does so between the bases. Site occupancy: A (Na‘,Ca*‘,Va), Ml, M2 or M3 (Fe2’,Mg2’,Fe’*,A13’), M4 (Ca2’, Na’, FeZ*,Mg2’), T (Si”, Al”) (OH, F) O,, .
Table 1: Nearest Neighbor Coordination
Table 2: Site Occupancies
Numbers for structure 1
for Structures 1 and 2
M. HILLERT et al.
260
Table 3: Nearest Neighbor Coordination Numbers for Structure 2
Table 4: Regressed lattice enthalpy of the polyhedral unite (W/mole of formula unit) Polyhedral
Ca+2
Tw nsmrt
_?CK, d”“,
.,..&I.“.
c ..,
Olivine Ml Olivine M2
-3525.4 -3531.9
Pyroxene Ml Pyroxene M2
-3545.3
Amphibole Feldspar Talc 4-coord 6-coord Kaol (03H3) Zoisite
.,,
.
A *._
I
I
_A,,?n
A1+3
Fe+2
II
Na+’
.lAh7 C
I”
,
K+’
1
._,.”
I
I
I
-387 -3911.4 -3857.0
-3889.
(coU7\
.....L......-
-3011
-3555.8 -3595.2
.All~lf-im.=
Fe+3
Mgf2
, -‘I”,‘.>
,
-,4>,.‘l I
I
Si+4
H20
THERMODYNAMIC
AE
MODELLING OF SOLUTIONS AND ALLOYS
AD
Fig. 1. Gibbs energy diagram for a reciprocal system at constant T and P. The Gibbs energy surface is curved but sections parallel to the sides are straight.
BE
AD
Fig. 2. Diagonal section through Fig. 1. Alloys close to the AD corner fall very close to the tangent, which is independent of the value of ‘GBE.
261
262
M. HILLERT et a/.
a
1
b
‘5 d T
Figure 3
-
T,
T,'
THERMODYNAMIC
Figure 4
MODELLING OF SOLUTIONS AND ALLOYS
263