GEOLOGICAL JOURNAL, VOL. 30, 297-305 (1995)

Diffusive equilibration between minerals during cooling: an analytical extension to Dodson‘s equation for closure

in one dimension

ROGER POWELL School of Earth Sciences, University of Melbourne, Parkville, Victoria 3052, Australia

AND

LEE WHITE School of Mathematics, University of Melbourne. Parkville, victoria 3052, Australia

Continuing element exchange between minerals during cooling from the metamorphic peak modifies mineral compositions from those existing at the peak. After closure, when the minerals stop changing composition, the compositional relationships hold important information regarding, for example, cooling rates. Whereas the classic result of Dodson (1973) regarding closure provides an important limit, the situation involving element exchange with a slow diffusing mineral equilibrating with aJinite faster diffusing mineral is more valuable in considering metamorphic systems. An analytical extension to Dodson’s equation for this situation in one dimension is presented here, and some of the consequences of departing from Dodson’s limiting case are illustrated.

KEY WORDS diffusion; re-equilibration; closure; Dodson’s equation; garnet-biotite

1. INTRODUCTION

There are different ‘controls of metamorphism’ that operate on different scales. On larger scales, the main controls relate to the tectonics which produce the P-T paths experienced by rocks, including how the increased temperatures in orogenic belts are achieved and how metamorphic rocks are returned to the Earth’s surface. On a thin section scale, the main controls relate to which mineral assemblages occur, the textural relationships between the minerals and the compositions of the minerals

Many processes may be involved in the development of the textural relationships, in particular the mineral compositional relationships observed in metamorphic rocks (see the reviews of Tracy 1982; Loomis 1983; Chakraborty and Ganguly 1991). Of these processes, diffusion is central and various workers have addressed the underlying partial differential equations of diffusion to further our understanding of metamorphism (e.g. Lasaga et al. 1977; Onorato et al. 1979; Lasaga 1983; Ozawa 1984; Muncill and Chamberlain 1988; Spear 1991; Chackraborty and Ganguly 1991; Ehlers et al. 1994).

In the so-called ‘equilibrium model of metamorphism’, equilibrium is continuously maintained during the prograde and retrograde history, at least on some scale. As the temperature increases during the prograde history, nucleation introduces new minerals to an assemblage and diffusion maintains the minerals at equilibrium compositions, causing changes in modal proportions, including the reacting out of minerals. Also, grain size increases, deformation notwithstanding. In this simplistic view, at the metamorphic peak the assemblage consists of a set of unzoned minerals constituting a system at equilibrium at these conditions.

If the minerals are unzoned, this implies that diffusion is sufficiently fast to overcome the tendency for growth zoning. ‘Sufficiently fast’ here should be understood in the context of grain size, given that volume

CCC 0072-1050/95/03029749 0 1995 by John Wiley & Sons, Ltd.

Received 15 December 1994 Revised 21 Jury 1995

298 R. POWELL AND L. WHITE

diffusion is generally much slower than grain boundary diffusion (e.g. Eiler et al. 1992, and references cited therein), temperature and time. Some minerals - garnet is the most common example ~ preserve growth zoning at the metamorphic peak. At the metamorphic peak, for a particular mineral in a particular assemblage (with a given mode), there will be a grain size below which the grains will be unzoned, and above which they will be growth zoned.

Superimposed on the composition-grain size relationships at the metamorphic peak are the effects of continuing diffusive equilibration of the minerals during cooling. The primary control on the extent of these composition changes is cooling rate, assuming the absence of infiltrating fluids and deformation causing a change in grain size. Of the unzoned (equilibrated) minerals at the metamorphic peak, the smaller grains will continue to be unzoned, completely equilibrating with their neighbours, until some temperature at which volume diffusion can no longer keep pace with the required changes in composition for equilibrium to be maintained. Finally, the mineral ‘closes’ with no more compositional changes occurring. The centre of the grain closes first, then closure moves outwards to the rim, producing retrograde zoning. In contrast with this behaviour for smaller grains, the central parts of larger grains may effectively close at the metamorphic peak, with diffusive equilibration only affecting their rims. For grains sufficiently large to be growth zoned, diffusive equilibration will also affect their rims, even though diffusion was not sufficiently effective at the metamorphic peak to smooth out the growth zoning.

Such closure - the focus of this paper - is one of the important ‘controls of metamorphism’, in that the observed compositional relationships in minerals in metamorphic rocks are caused, at least in part, by the processes associated with closure. Thus if geothermometry and geobarometry are to be undertaken successfully - for example, to determine peak metamorphic conditions - the effect of diffusive equilibra- tion on the peak metamorphic mineral compositions have to be understood (e.g. Spear 1991). On the other hand, given that this equilibration depends on the cooling rate, study of the consequences of closure may allow the determination of cooling rate (e.g. Ehlers et al. 1994).

2. CLOSURE AND DODSON’S EQUATION

Although the consideration of closure for real systems is difficult, involving multicomponent diffusion, grain size and the geometrical disposition of the grains in three dimensions, modal proportions and so on, some progress has been made through numerical solution of the underlying partial differential equations, at least for simple interdiffusion. This has been done for closure during cooling for systems with simple exchange: in oxygen isotope systems by, for example, Eiler et al. (1992) and Jenkin et al. (1 994), and for element exchange by, for example, Spear (1991) and Ehlers and Powell (1994). In many situations, such numerical solutions are more easily and efficiently obtained, for example, by the explicit method (Spear 1991) or the Crank- Nicholson method (Ehlers and Powell 1994), than by substitution into infinite sum solutions (Lasaga 1983; Chackraborty and Ganguly 1991), even when these are available.

Although such numerical modelling is the only way to make progress in most situations, analytical solutions where they have been found (e.g. Dodson 1973) are particularly valuable in allowing the depend- ence of results on parameters to be clearly seen, without a prohibitive amount of numerical experimentation being required (cf. Ehlers and Powell 1994). In an important paper, Dodson (1973) presented an analytical solution for closure during cooling involving diffusion of an element at low concentration in a system consisting of a slower diffusing mineral in a faster diffusing infinite reservoir (mineral), with particular reference to isotopic systems. Dodson also averred that closure of element exchange between minerals during cooling will be formally identical to his results.

As presented in Dodson (1986), using the notation in Table 1, his equation relating closure temperature to diffusion parameters, grain size and cooling rate is

DIFFUSIVE EQUILIBRATION DURING COOLING 299

Table 1 . Notation

Faster diffusing phase Slower diffusing phase Diffusion coefficient of B, D = &exp(-(Q/RT)) Pre-exponential diffusion coefficient of B Closure function as a function of x ( P = 0 implied) Closure function as a function of P and x k = X,(l - XA)/XB(I - X B ) , the slope of K D contours on a X A - X B diagram k at To Equilibrium constant, KD = x A ( 1 - XB)/( 1 - &)Xe I = lB , grain size of B Grain size of A Modal proportion of B, PB = l / ( r + 1) Activation energy of B r = I A / I B , the grain size ratio Gas constant Cooling rate Time Temperature Temperature at start of cooling Reference temperature Closure temperature (x = 0 implied) Closure temperature as function of x Position in slower diffusing grain (0 = centre, 1 = edge) Mole fraction of 1 in B; mole fraction of 2 in B is 1 - X B Mole fraction of 1 in A; mole fraction of 2 in B is 1 - X , XB at To X , at To P = l / rko

The closure function, G(x) , as a function of position in the slower diffusing grain, x, for the one-dimensional (or plain sheet) case is

cos(n - l).rrxln(n - i).rr (n - :)T

in which y is Euler's constant (y = 0.577216). The cooling time constant, T, the time for D to decrease by a factor, e, is given by

R

This is usually written in terms of the cooling rate, s = -dT/dt, so that

R T ~ Qs

7 = - .

The following assumptions are involved in the derivation:

at the start of cooling, the minerals are in equilibrium (i.e. there is no growth zoning in the slower diffusing mineral); the minerals continue to remain in equilibrium at the start of cooling, so that the closure temperature is not dependent on the temperature at the start of cooling, T,; diffusion of the faster diffusing mineral is sufficiently faster than the slower one that the closure temperature depends only on the diffusion properties of the slower diffusing mineral;

300 R. POWELL AND L. WHITE

4. the modal abundance of the faster diffusing mineral is large, so that the faster diffusing mineral does not change composition with re-equilibrium during cooling;

5 . the diffusing element is present at low concentration; 6. cooling occurs approximately linearly in l / T during closure; 7. mineral compositions change only by diffusion (so that, for example, net transfer reactions are not

involved).

Point 2 may be stated alternatively in terms of the rate of cooling: if the rate is ‘slow’, which, in terms of the derivation of Equation (4), means that

then Dodson’s equation is applicable. For ‘fast’ cooling, the equation is not applicable and this implies that closure takes place at, or close to, T, . It is important to note that the grain size of the slower diffusing phase appears in Equation (3). Thus Dodson’s equation only applies to sufficiently small grains of the slower diffusing phase. Other aspects of these assumptions are discussed in Ehlers and Powell (1994) and Ehlers et a/. (1 994).

The difference between the equation as presented in Dodson (1986) and Dodson (1973) is that in the latter the closure function is put inside the log term in Equation (4) (and called A )

- Q = In( +) A r D R TC (4)

Also the values of A given were for the closure temperature for the whole grain, rather than as a function of position in the grain. Thus the relationship of A to C(x) using Equation (2) is

and, for example, substituting into this gives A = 8.66, as given in Dodson (1973) for the one-dimensional case.

3. EXTENSION TO DODSON’S EQUATION

In the element exchange case in rocks, Dodson’s equation is inappropriate because the requirement of an infinite reservoir and diffusing elements at low concentration (points 4 and 5) are not normally met (see also Eiler et al. 1992). We have solved the underlying partial differential equations (e.g. Ehlers and Powell 1994: p 242) for the element exchange case, and for a finite reservoir of a faster diffusing phase, using Laplace transforms (details in Powell and White unpublished data). The results presented here are for the one- dimensional (plane sheet) case, so that comparison with numerical experiments of Ehlers and Powell ( I 994) may be made.

Dodson’s equation is extended by making the closure function, G, a function of position in the grain and a non-dimensional parameter, p, which relates to the concentrations of the diffusing elements and the relative proportions of the two minerals

DIFFUSIVE EQUILIBRATION DURING COOLING

Table 2. Values of the closure function, G(p, x)

30 1

p x : o 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 0.95 0.99

0 0.989 1.008 1.064 1 2.222 2.234 2.272 2 2.965 2.974 3.003 3 3.501 3.509 3.532 4 3.922 3.929 3.948 5 4.269 4.274 4.291 6 4.564 4.568 4.583 7 4.820 4.824 4.838 8 5.047 5.051 5.063 9 5.251 5.255 5.266

10 5.436 5.439 5.449 11 5.605 5.608 5.618 12 5.761 5.764 5.773 13 5.905 5.908 5.916 14 6.040 6.043 6.051 15 6.167 6.169 6.176 16 6.285 6.287 6.294 17 6.397 6.399 6.406 18 6.503 6.505 6.512 19 6.604 6.606 6.612 20 6.700 6.702 6.708

1.163 1.311 1.522 2.338 2.438 2.583 3.053 3.131 3.244 3.573 3.637 3.730 3.983 4.037 4.117 4.321 4.368 4.438 4.610 4.652 4.714 4.862 4.899 4.955 5.085 5.119 5.170 5.286 5.317 5.364 5.468 5.496 5.540 5.635 5.661 5.702 5.789 5.813 5.851 5.931 5.955 5.990 6.065 6.087 6.120 6.189 6.21 6.242 6.307 6.327 6.357 6.418 6.437 6.465 6.523 6.541 6.568 6.623 6.640 6.666 6.718 6.734 6.759

1.820 2.794 3.410 3.869 4.236 4.543 4.808 5.040 5.247 5.435 5.606 5.763 5.909 6.044 6.171 6.290 6.403 6.509 6.610 6.706 6.798

2.521 2.920 3.108 3.618 3.663 4.086 4.083 4,447 4.422 4.744 4.708 4.997 4.956 5.219 5.175 5.417 5.372 5.595 5.550 5.758 5.713 5.908 5.863 6.047 6.003 6.176 6.133 6.297 6.255 6.411 6.370 6.519 6.479 6.621 6.581 6.717 6.679 6.809 6.772 6.897 6.861 6.982

4.166 5.483 4.632 5.778 4.962 5.997 5.226 6.180 5.450 5.338 5.645 6.478 5.819 6.606 5.976 6.723 6.120 6.831 6.252 6.932 6.375 7.027 6.491 7.117 6.599 7.202 6.701 7.282 6.798 7.359 6.889 7.433 6.977 7.503 7.060 7.571 7.140 7.636 7.217 7.699 7.290 7.759

8.642 8.732 8.804 8.868 8.926 8.980 9.030 9.077 9.122 9.165 9.206 9.246 9.284 9.320 9.356 9.390 9.424 9.456 9.488 9.519 9.549

with A and B being the faster and slower diffusing minerals, respectively, then /3 is defined as

/B x!(1 - xi) /A xi(1 - xi) p = - .

in which lis the grain size and xis the concentration. Note that points 4 and 5 imply p = 0. In the notation of Ehlers and Powell (1994), Equation (6) would be written as p = l / rko , with r and ko defined in Table 1.

The closure function for the one-dimensional case is

in which y is Euler’s constant ( g = 0.577216) and a, is provided by the solution of the non-linear equation

Note that, for /3 = 0, a, is given by the solution of cos a, = 0, which is a, = (n - 1/2)7r. Then equation (7), for /3 = 0, i.e. C(x, 0), is identical to Dodson’s closure function, Equation (2). Values of G(x, /3) are shown in Table 2 and Figure 1, obtained from Equations (7 and 8).

In solving Equation (8) to substitute into Equation (7), any standard non-linear equation solver can be used. From the form of the equation written as

sin a, f = c o s a n + / 3 - an

as shown in Figure 2, it can be seen that a, always lies within the bracket (n - 1/2)7r H (n+1/2)~, for finite /3, and as n increases, a, slowly approaches (n- 1/2)7r. An approximate solution, at least for small /3 or large n, is given by: (1) writingfas a quartic in a,,f’, by series expansion around (n- 1/2)7r; (2) solving f ’ = 0 for a,; and ( 3 ) then, with a, = (n - 1/2)7r + Aa,, writing Aa, as a quintic in l/(n- 112)~.

302 R. POWELL AND L. WHITE

4

2 Pcontours 1

I

0.2 0.4 0.6 0.8 1

x contours

20 P

5 10 15

Figure 1. Dependence of the closure function, G(x, p), on /3 and x. Note that C(x,P) has a strong dependence on P, particularly at small x

n in (n-l/2)p

Figure 2. Form o f f = cos a,+,O(sinan/a,) as a function of n, for two ranges of n, and for /3 = 0, 10, and 20. Note that for /3 = 0 (Le. Dodson's assumption), the curve goes through (n - 1/2)x Aa%:i ::I,**L 0.8 ~ : ~ : * * * * * ~

0.6 0.6

0.05 0.4 0.4

0.1

p = 1 0.2 p - 10 0.2 p = 2 0

5 10 15 20 5 10 15 20 0 5 10 15 20

Figure 3. The first 20 values of a, as a difference from (n- 1/2)a, AcY,, for /3 = 1, 10 and 20, showing that the approximation for cy, Equation (9), is good for large n, and/or small

This gives

Figure 3 shows how well this equation performs compares with the numerical solution of Equation (8).

4. COMPARISON WITH NUMERICAL EXPERIMENTS

The extensive numerical study of Ehlers and Powell (1994), relating to the same set of conditions as addressed by Equation (5 ) , can be compared with this analytical result. Interestingly, the empirical term they found, which allowed a representation of their data, is in terms of a parameter that is closely related to /7. However, that parameter, which will be denoted by p' here, is in terms of the compositions of the phases after closure. In contrast, the PA and x"! in Equation (6) refer to To, the temperature around which the thermodynamic and diffusion properties are linearized in derivation Equation of (5 ) , a temperature at which the minerals are still at equilibrium. For the linearization to be appropriate, it should be at a temperature

R. POWELL AND L. WHITE 303

1 In(1 +P) 0 0.5

Figure 4 Comparison of Equation (5) (the curve), with the numerical experiments of Ehlers and Powell (1994, Figure 5) In this, A is the discrepancy in Q/RTc from Equation (4) as p increases Note that A is not quite linear in In(l+p)

close to the closure of the system. As the core of B closes over a small temperature interval (Ehlers and Powell 1994: Figure 2), the observed core composition of B can be used in Equation (6), and so TO can be taken to be T,(O). For A of finite size, the composition of A will change while the remainder of B closes, so that the observed composition of A will depart from PA progressively as lB/lA increases. The effect of this can be seen from

0 k = (XA + SxA)(l - XA - SxA) xB(1 - XB)

in which XB is the observed composition of B, XA is the observed composition of A , and PA = XA + SXA. Expanding in S X A

in which k is in terms of observed compositions

k = x A ( 1 - XA) xB(1 - XB)

This shows that using k for ko is an increasingly poor approximation as XA+O or XA-, 1 .

of p, at least for small departures from Dudson’s equation, using Equation (10) Ehlers and Powell (1994) represented their data using In( 1 +p’); this expression can be represented in terms

Thus for small p, In( 1 + p’) M In( 1 + p). It turns out that Equation (5) is not quite linear in this; Figure 4. The agreement between the numerical experiments and Equation (5) is good at lower /3, but starts to diverge to higher p as In( 1 + p’) departs from In( 1 + p). The scatter in the numerical experiments is a consequence of this departure varying with XA, and all of the variables through SXA .

5. DISCUSSION

Although the formulation only applies to closure in the one-dimensional (plane sheet) case, nevertheless it is of interest to observe some of the consequences of varying /3, starting with the effect on calculated zoning profiles for the slower diffusing phase; Figure 5. In this ‘forward’ calculation, with specified cooling rate, the flattening of G(x, p) with x as P increases means that the zoning profiles also become flatter as P increases.

304 DIFFUSIVE EQUILIBRATION DURING COOLING

0.22. p contours

0.19-

0.18-

0.1 6.

0 0.2 0.4 0.6 0.8 1 X

Figure 5. Calculated zoning profiles across a grain of the slower diffusing .phase, B, for p = 0, 2, 5, 10 and 20. Note that the profiles become flatter as p increases. The curves were calculated with A = biotite, B = garnet, I , = 0.05 cm, s = IO"C/Ma and the diffusion and thermodynamic parameters used in Ehlers et al. (1994): Do = 9.81 x m2/s and Q = 239 kJ/mol in D = Do exp [ -(Q/RT)] (Cygan and Lasaga 1985) and AHR = 15.00 kJ and ASR =0.00485 kJ/K in Rln[(l - X&g)Xkg/Xhg(l - X k J ] = AHR - TASR, using the

data of Holland and Powell (1990)

T1look 1050 T

1050 A A

Figure 6. Dependence of closure temperature, T, (in K), on p. In (a), T, is plotted against fl; in (b) T, is plotted against modal proportion of the slower diffusing phase, p B , for ko = 0.04, 0.14,0.4,1, 2.5, 6.25 and 25. (The curves were calculated with 1, = 0.5 cm,

s = IO"C/Ma and the diffusion parameters referred to in the caption to Figure 5)

8.

In s In s

20 0.2 0.4 0.6 0.8 1 lo P 1 5 Pbi

0 5

Figure 7. Dependence of cooling rate, Ins, on p. In (a), Ins is plotted against p; in (b), Ins is plotted against modal proportion of the slower diffusing phase, p ~ , for ko = 0.04, 0.14, 0.4, 1,2.5, 6.25 and 25. (The curves were calculated with lB = 0.5 cm, Tc = 1123 K,

diffusion and thermodynamic parameters referred to in the caption to Figure 5)

The forward calculation of closure temperature using the centre of the slower diffusing phase, with specified cooling rate, gives Figure 6 . Although (a) is the summary diagram, (b) allows the effect of changing the proportions of the minerals to be observed, for different k values. Such diagrams can be related to, for example, Jenkin et al. (1994, their Figure 2), with in this instance k z 1. For large k, for which XB 4 1 and X B > X A , or X B 4 0 and XB < XA , the effect of p B is not dramatic until p~ -+ 1. In the converse case, even small values of pB will dramatically affect T,.

The 'backward' calculation of cooling rate is possible with element exchange because T, is provided by application of the element exchange thermometer itself (e.g. Ehlers et al. 1994). The effect on the cooling rate

DIFFUSIVE EQUILIBRATION DURING COOLING 305

calculated using Equation ( 5 ) of varying p, with a specified T,, is shown in Figure 7. Because of the form of Equation (9, the cooling rate is sensitive to changes in T,, for example, but also G(x, p), and so the dramatic effects in Figure 7 are to be expected. This means that evaluation of ,B will be particularly important in calculating the cooling rates of rocks as, for example, by the approach of Ehlers et al. (1994).

ACKNOWLEDGEMENTS

We thank Karin Ehlers for discussions over several years, Frank Spear and two anonymous reviewers for helpful comments on the manuscript and Alan Boyle for organizing COM94 and editing this issue.

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