Using Eq. (2.30) to calculate the contribution of the configurational entropy to the free energy of mixing is possible only for quite simple crystal structures in which the substituting atoms mix only on one crystallographic site (like garnet). Most of the common rock-forming minerals have crystallographically distinct sites over which cations of similar size are distributed non-randomly. A simple example is the much studied (Mg,Fe)2Si2O6 orthopyroxene binary (e.g. Mueller, 1962; Thompson, 1969, 1970; Ganguly, 1986; Shi et al., 1992; Strimpfl et al., 1999). The orthopyroxene crystal structure contains two geometrically distorted octahedral sites, conventionally labelled M1 and M2, over which Mg and Fe are distributed non-randomly, with Fe preferring the larger and more distorted M2 site. The degree of long-range ordering can be described using an order parameter, s, which must be defined relative to a standard state. In the (Mg,Fe)2Si2O6 orthopyroxene solution the two end-members are completely ordered (i.e. Mg2Si2O6 and Fe2Si2O6). For this type of solutions, the standard state is taken as complete disorder, that is, random mixing of the cations. It is then possible to define for the purposes of a reference an ideal entropy of mixing Sid-mix as that due to random mixing on the total number of sites per formula unit.
The free energy of the solid solution can then be formulated as a function of both s
and composition, xMg (where,
6 2 2 6 2 2 6 2 2 O Si Fe O Si Mg O Si Mg Mg n n n x + = and 6 2 2SiO Mg n and 6 2 2SiO Fe n are the number of moles of Mg2Si2O6 and Fe2Si2O6 in solution), by writing the configurational
entropy as in Eq. (2.34), with site occupancies defined as a function of s and xMg, and the total non-configurational free energy of the solid solution as a power series in s and xMg
(cf. Thompson, 1969): ex d o config mix T S G G G =− ⋅ + − + (2.32) where:
(
2 ....)
0 , 2 1 , 1 0 , 1 0 + + ⋅ + ⋅ + ⋅ = − Mg Mg d o s a a s a s x a s x G (2.33)For non-convergent ordering there is no restriction on the terms in s (e.g. Thompson, 1969; Carpenter, 1994).
The configurational entropy Sconfig is given in the usual way (e.g. Ganguly and Saxena, 1987): ⎟⎟ ⎠ ⎞ ⎜⎜ ⎝ ⎛ − =
∑∑
ζ ζ ζ ζ i i i config R n X X S ln (2.34)where, the summation is over all species i (‘ions’) on each crystallographically distinct site,ζ ; Xiζ is the fraction of i in site ζ and nζ the total number of sites ζ per formula unit. (Note that for crystals with only one type of site or for random mixing, Sconfig is of course identical to Sid-mix).
The excess free energy of mixing Gex then accounts for those contributions to lattice strain effects, etc., that are independent of s, and may also be given in the usual ways; for example, by applying the Margules formalism as in Helffrich and Wood (1989).
At internal equilibrium G is at a minimum with respect to the state of order-disorder (i.e. dG/ds = 0). If the parameters in the power series for Go-d are known, the value of s can be calculated at a given value of xMg. This value of s can then be substituted back into Eq. (2.33) to calculate the contribution to the free energy of mixing from the non- configurational entropy, Go-d (a mathematically more elegant but conceptually perhaps less
transparent way of doing this in one step has been given by Ghiorso, 1991, and Holland and Powell, 1996, their Appendix 1). The terms in Go-d can be constrained, in principle, by studying the order-disorder as a function of temperature and composition. However, it is important to note that there may be terms in G that do not contain s, which are written here as Gex; these terms obviously disappear in the differential dG/ds, hence the condition dG/ds
= 0 contains no information on these terms. Thus knowledge of the cation distribution as a function of temperature and composition cannot alone constrain the activity-composition relations of the solid solution.
Where the terms without s are large compared to the terms in s, the contribution from order-disorder may have only a minor effect on the thermodynamic mixing properties at high temperatures. This seems to be the case in many spinel solid solutions. The thermodynamic properties of several binary spinel solid solutions involving the mixing of 3+ cations (or charge-balanced 2+ + 4+ cations such as in the Fe2TiO4-Fe3O4 binary) have been measured, including those both with and without significant cation order-disorder, which allows an assessment of the relative importance of this factor compared to lattice strain energies. These data show that the size mismatch between the substituting cations is by far the most important factor contributing to the net deviations from an ideal mixing model, and this factor can be modelled independently of the cation order-disorder effect using a regular solution parameter (O’Neill and Navrotsky, 1984; see particularly their Fig. 14 in which the regular solution parameter WA-B is plotted against a volume mismatch term
for 3+ and (2++4+) cations).
Historically, the view that the order-independent mixing terms may exist has not been universally accepted, and several early studies have in fact sought to constrain macroscopic thermodynamic mixing properties from cation order-disorder measurements, particularly in (Mg,Fe)2Si2O6 orthopyroxenes (Mueller, 1962; Saxena, 1973). This may be because it is difficult to justify the order-independent terms if one views the excess mixing in solid solutions as due to A-B type ‘interactions’ of the type described in section 2.5.4. But this is not a problem if the excess energy of mixing is viewed instead mainly due to lattice strain, as it is here argued to be more appropriate for ionic crystals.
As emphasised by Thompson (1969, 1970) and Sack (1980), the contribution to the free energy of mixing from order-disorder comes in two parts, that is, from both the
config
S
T⋅ and Go-d terms. An important general point is that the two quantities
(
Sconfig Sid mix)
T⋅ − − and Go-d are opposite in sign and thus tend to cancel each other out, increasingly so with increasing temperature. For solid solutions in which the order-disorder does not occur in the end-members, as in (Mg,Fe)2Si2O6 orthopyroxene, Sconfig< Sid-mix, hence the contribution of this term to Gss (where, here, Gss = Gmix – Gex) is such as to produce positive deviations from ideality, while the ordering energy Go-d produces negative deviations. If Go-d has a simple form in the order parameter, s, (i.e. only the a0
term in Eq. 2.33 is important), then Go−d >−T⋅
(
Sconfig −Sid−mix)
and the net result is that the solution has negative deviations from ideality (NB In the absence of a contribution fromGex). For solutions in which the end-members also show order-disorder phenomena (like spinels, see O’Neill and Navrotsky, 1984), this simple relationship does not hold, but nevertheless the tendency for Go-d to compensate for the difference between T⋅Sconfig and
mix id
S
T⋅ − remains a general conclusion. Hence a better approximation, at least at high
temperatures, is achieved by simply ignoring long-range ordering between sites, than by taking the T⋅Sconfig alone, without the compensating Go-d term.
Secondly, the appearance of rigour conferred by explicitly considering long-range order wears thin when the problems of extrapolating such models into multicomponent solid solutions have to be confronted. In the case of (Mg,Fe)2Si2O6 orthopyroxenes, for example, nothing is known about the effects of other components, such as Al2O3, on ordering. Neutron diffraction measurements on Mg2Si2O6-MgAl2SiO6 orthopyroxenes show that Al is actually considerably disordered between the two octahedral sites M1 and M2 and not confined to the smaller M1 site (S.A.T. Redfern and H.St.C. O’Neill, personal communication), but unlike Mg-Fe2+ order-disorder, this has never been taken into account in thermodynamic modeling. Indeed, what is modelled in solid solution formalisms that include effects of order-disorder is always incomplete and is usually quite arbitrary.
In summary, the explicit modeling of order-disorder in multicomponent solid solutions greatly increases the complexity level, for little gain in accuracy to which the mixing properties can be represented. Moreover, only a few of the important bounding binary joins in the common-rock-forming solid solutions have been studied for order- disorder. For example, in orthopyroxenes the join Mg2Si2O6-Fe2Si2O6 has been studied to
exhaustion (e.g. Strimpfl et al., 1999), but the exact relationship between order-disorder and the macroscopic free energy of mixing still remains contested (Aranovich, 2004).
Therefore, in this work it is proposed to model solid solutions with long-range order in one of two ways:
1) where disordering is extensive, as in (Mg,Fe)2Si2O6 orthopyroxenes or in the various spinel solid solutions reviewed in O’Neill and Navrotsky (1984), it is essentially ignored, modeling an ideal entropy of mixing using the appropriate number of sites per formula unit. The difference is then accounted for entirely empirically in the Gex term.
2) where ordering is almost complete, as in (Ca,Mg)2Si2O6 pyroxenes, (Ca,Mg)2SiO4 olivines, or low-temperature (Ca,Mg)CO3 carbonates, end-members are defined at the perfectly ordered composition, in these examples at CaMgSi2O6, CaMgSiO4 and CaMg(CO3)2 respectively.
Accordingly, when deriving G expressions for complex solutions, the simplified assumption that Sconfig ≡Sid (i.e. Go-d = 0) has been made and to express Gid-mix Eq. (2.70)
has been used. This implies that cations of a like charge and size (e.g. Mg and Fe2+, or Al, Cr and Fe3+) are assumed to partition randomly between the sites they occupy. Thus, the site occupancies Xik’s that appear in Eq. (2.70) (and also in Eq. 2.71) should more properly
be called ‘virtual’ site occupancies, since they are defined without taking into account order-disorder phenomena. Similarly sites they occupy should be called ‘virtual’ sites (see also section 2.8 and section 3.4).
Results presented in chapter 4 and chapter 5 of this thesis attest the validity of this approach.