38 Acknowledgements

We wish to thank Peter Luffi for identifying the garnet solid solution error in the original

MELTS code, Ashley Nagle for pointing out the anomalously low spinel-garnet transition

pressures obtained when the corrected garnet model is used, and Aaron Wolf for helpful

discussions regarding statistical analysis. Comments by Associate Editor Jon Blundy are

greatly appreciated, as are the reviews of two anonymous reviewers. This work was

supported by the National Science Foundation and the American Recovery and

Reinvestment Act through award 0838244.

39

Tab le&1. !S um m ar y! tab le !o f!d ata! co ve rag e! in !c om po si tio n, ! P ,!an d! T Co m po si tio n To tal !E xp er im en ts Am bi en t< P ,! T Hi gh < P P# Ran ge !(G Pa) Hi gh < T T# Ran ge !(º C) End!m em be rs Mg Al

2

O

4

151 48 21 0. 6!<!29 82 132!<!1600 Fe Cr

2

O

4

1 1 0 < 0 < Fe

2

TiO

4

2 2 0 < 0 < Fe

3

O

4

55 9 46 0. 02!<!11. 11 0 < Fe Al

2

O

4

40 23 0 < 17 200!<!1150 Mg Cr

2

O

4

6 6 0 < 0 < Mg Fe

2

O

4

69 24 28 0. 11!<!34. 39 17 100!<!1200 Mg

2

TiO

4

53 2 0 < 51 70!<!1416 Bi nar ie s Mg Al

2

O

4

<Fe Al

2

O

4

14 14 0 < 0 < Fe Al

2

O

4

<Fe

3

O

4

1 1 0 < 0 < Fe Ti

2

O

4

<Fe

3

O

4

18 18 0 < 0 < Mg Al

2

O

4

<Mg Fe

2

O

4

68 15 4 1!<!4 49 200!<!1050 Mg Al

2

O

4

<Mg Cr

2

O

4

194 87 34 4. 4!<!8. 03 73 200!<!1000 Fe Cr

2

O

4

<Mg Cr

2

O

4

13 13 0 < 0 < Fe

3

O

4

<Mg Fe

2

O

4

1 1 0 < 0 < Mu lti <c om po ne nt 146 146 0 < 0 < To tal 832 410 133 289

TA BLES

40

Tab le&2.&& O pti m ize d) stan dar d) state )e nd .m em be r)m ol ar )v ol um e) fo r)g iv en )o rd er in g) state ) ) ) V

^{o )}

(J /b ar /m ol )) 1σ

*)

O rd er in g) State ) [ s

0

,%s

1

,) s

2

]) a )(Å )) ρ)(kg)m

.3

)) Mg Al

2

O

4)

3. 9722 ) 1. 7) ×) 10

.4)

no rm al ) [1,) 1,) 0] ) 8. 0808 ) 3582 ) Fe Cr

2

O

4)

4. 4243 ) 3. 9) ×) 10

.4)

no rm al ) [0,) 0,) 0] ) 8. 3765 ) 5059 ) Fe

2

TiO

4)

4. 6873 ) 3. 1) ×) 10

.4)

no rm al ) [0,) 0,) 0] ) 8. 5393 ) 4769 ) Fe

3

O

4)

4. 4553 ) 5. 5) ×) 10

.4)

inv er se ) [0,) 0,) 0] ) 8. 3960 ) 5197 ) Fe Al

2

O

4)

4. 0871 ) 2. 0) ×) 10

.4)

no rm al ) [0,) 1,) 0] ) 8. 1580 ) 4253 )

*

Bo ots tr ap )e sti m ati on )o f)s .d .)f or ) V

o

,)h ol di ng )o th er )p ar am ete rs )at) op tim al )v al ue s) ) ) ) Tab le&3.&& Stan dar d) state )e nd .m em be r)c oe ffi ci en t)o f)th er m al )e xp an si on ) ) α(1 0

.5

))(K

.1

)) 1σ

*)

) Mg Al

2

O

4)

2. 4413 ) 3. 8) ×) 10

.8)

) Fe Cr

2

O

4a)

2. 1691 ) .) ) Fe

2

TiO

4)

3. 3458 ) 4. 3) ×) 10

.8)

) Fe

3

O

4)

3. 3376 ) 2. 1) ×) 10

.7)

) Fe Al

2

O

4)

2. 6431 ) 2. 7) ×) 10

.8)

)

*

Bo ots tr ap )e sti m ati on )o f)s .d .)f or )α ,)h ol di ng )o th er )p ar am ete rs )at) op tim al ) val ue s) )

a

Val ue )o f)α )w as )h el d) co ns tan t)( se e) te xt) ) )

41

Tab le&4.& Bu lk )m od ul i)an d) pr es su re )d er iv ati ve s) of )e nd ) memb ers ) ) K

oT

)(G Pa) ) K ') Mg Al

2

O

4)

190. 8

a)

6. 77

a)

Fe Cr

2

O

4)

203. 3

b)

6. 5

c)

Fe

2

TiO

4)

181

d)

5. 5

d)

Fe

3

O

4)

181

e)

5. 5

e)

Fe Al

2

O

4)

210. 3

f)

5. 5

d) a

)F ro m )L ev y) et) al .)( 20 03 )) )

b

)F ro m )D or ai sw am i)( 19 47 )) )

c

)C on str ai ne d) us in g) data) of )F an )e t)al .)( 20 08 ))( se e) ap pe nd ix ))

d

)A ss um ed )to )b e) eq ual )to )th at) of )F e

3

O

4)

(s ee )te xt) )

e

) Fr om )N ak ag iri )e t)al .)( 19 86 )) )

f

)F ro m )W an g) an d) Si m m on s) (1 97 2) )

42

Tab le&5. )Mo de l)p ar am ete rs ) & ) 1σ

*)

) W

hc.ch)

0. 0108 ) 0. 018 ) (J /b ar /m ol )) W

ch.mt)

0. 0715 ) 0. 19 ) ) W

sp.ch)

0. 0578 ) 0. 014 ) ) dW

sp.ch)

0. 0489 ) 0. 063 ) ) W

mt.hc)

0. 0833 ) 0. 025 ) ) W

mt.sp)

0. 1020 ) 0. 025 ) ) W

s0)

.0. 0692 ) 0. 00049 ) ) W

s1)

0. 0264 ) 0. 00049 ) ) W

s2)

0. 1035 ) 0. 0019 ) )

*

Bo ots tr ap )e sti m ati on )o f)s .d .)f or )e ac h) par am ete r,) ) ho ld in g) al l)o th er )p ar am ete rs )at) op tim al )v al ue s)

43

43

FIGURE CAPTIONS

Fig. 1 Compositional coverage of the data used in model calibration. Diamonds are experiments measured at ambient conditions, circles are experiments measured at high-T, squares are data measured at high-P. Triangles are data not included in calibration due to lack of measured site occupancies, and are used only for comparison to model results below. a Mg# (Mg/[Mg + Fe

²⁺

]) vs. Ti

⁴⁺

/(Ti

⁴⁺

+ Cr

³⁺

+ Fe

³⁺

+ Al

³⁺

); b Mg# vs.

Cr

³⁺

/(Ti

⁴⁺

+ Cr

³⁺

+ Fe

³⁺

+ Al

³⁺

); c Mg# vs. Fe

³⁺

/(Ti

⁴⁺

+ Cr

³⁺

+ Fe

³⁺

+ Al

³⁺

).

Fig. 2 Compositional coverage of the data used in model calibration. Symbols the same as in Fig. 1. a Ti

⁴⁺

- Fe

³⁺

- Al

³⁺

ternary; b Cr

³⁺

- Fe

³⁺

- Al

³⁺

ternary.

Fig. 3 Order-dependence of observed volume vs. ordering term for a s

0

, b s

1

, and c s

2

. The order-dependence for s

0

, for example, is calculated by subtracting the model V

ideal

and all other model terms not dependent on s

0

from the observed volume, V

obs

. The linearity of the data shows the simple form of the order-dependence of the data and the slope of the linear trend is proportional to the W

s0

parameter (see Eq. 7). The

corresponding calculations were done for s

1

and s

2

. The solid lines are the model order- dependences, plotted over the relevant range of ordering states.

Fig. 4 Model volume vs. measured volume. a Model results using spinel volume model

currently implemented in MELTS (shown in box); b Ideal model volume results after

44

44

fitting of end members. Model results from a shown grayed out; c Final model results.

Model results from a shown grayed out.

Fig. 5 Model residuals vs. measured volume. Gray diamonds are samples measured at ambient-P, T, open circles are samples measured at high-T in situ, and black squares are samples measured at high-P in situ.

Fig. 6 Model volumes and data in the MgAl

2

O

4

-MgCr

2

O

4

-FeCr

2

O

4

-FeAl

2

O

4

reciprocal square. End members are labeled at the corners. a Gray surface is the model of Brey et al.

(1999) with end-member contributions subtracted. Diamonds are the data of Brey et al.

(1999) and Doroshev et al. (1997); open circles are the rest of the room-T, P calibration data set with <5% magnetite or ulvöspinel component; b Gray surface is the model presented in this work, with end-member and ordering contributions subtracted. Symbols are the same as in a.

Fig. 7 a Excess model volume in the MgAl

2

O

4

-MgFe

2

O

4

-Fe

3

O

4

-FeAl

2

O

4

reciprocal

square. End members are labeled at the corners. Gray surface is the model presented in

this work, with end-member and ordering contributions subtracted. Open circles are the

room-T, P calibration data with < 5% chromite or ulvöspinel component; diamonds are

the data of Mattioli et al. (1987), only the most stoichiometric data (samples synthesized

at 1400 °C and oxygen fugacity of 10

^-4

atm) are plotted; triangles are the data of Golla-

Schindler et al. (2005) for an equilibration temperature of 1100 ºC (varying this value

45

45

within the stated range of annealing temperatures has little effect); squares are the data of Nakatsuka et al. (2004); data from Zhao et al. (1998) (MgFe

2

O

4

-Fe

3

O

4

) are not shown due to minor amounts of Ca in the analyses; b Excess model volumes along the MgAl

2

O

4

– Fe

3

O

4

binary. The upper solid line is the model excess volume along this join, including ordering states calculated with the MELTS ordering routine for the Mattioli et al.

experimental temperature of 1400 °C. The lower solid line is the excess volume model plotted without additional ordering effects, i.e., the order is perfectly normal at spinel and perfectly inverse at magnetite. The dashed line is the excess volume model of Mattioli et al. (1987); gray circles are the data of Mattioli et al. (1987); open diamonds are

calibration data for the magnetite end member, measured at ambient conditions; the optimized end-member contributions from this work have been subtracted from models and data.

Fig. 8 Excess model volume (black curve) along the Fe

3

O

4

– Fe

2

TiO

4

binary. The

ordering state is calculated using the MELTS ordering routine at 1100 ºC; in other

respects the volume is calculated for ambient conditions. Open circles are the data of

Wechsler et al. (1984) and gray triangles are the data of Bosi et al. (2009). Optimized

end-member contributions from this work have been subtracted from the data. The

volume along this binary is treated as ideal in the present model; asymmetry along this

join is captured in the model by the effect of ordering alone.

46

46

Fig. 9 Model volume (black curve) along the FeCr

2

O

4

– Fe

3

O

4

binary, plotted with the data of Robbins et al. (1971) (open circles) and Woodland et al. (2009) (gray circles). The model curve is calculated using the MELTS ordering routine and the volume model from this work at ambient conditions. This binary is modeled with a symmetric excess term.

The asymmetry of the data, which were not included in the calibration, is captured by the s

2

ordering term.

Fig. 10 Order variable s

0

vs. temperature for pure MgFe

2

O

4

data. The Levy et al. (2004) and Antao et al. (2005a, 2005b) data were measured at high-T, in situ. O’Neill et al.

(1992) samples were annealed over a range of temperatures; the annealing T is plotted

here. The solid lines are calculated using the updated MELTS ordering model at 1 bar,

3 GPa, 5 GPa, and 6 GPa for pure MgFe

2

O

4

.

47

47

0 0.2 0.4 0.6 0.8 1

0 0.2 0.4 0.6 0.8 1

Mg#

(a)

0 0.2 0.4 0.6 0.8 1

0 0.2 0.4 0.6 0.8 1

Mg#

(b)

0 0.2 0.4 0.6 0.8 1

0 0.2 0.4 0.6 0.8 1

Mg#

(c)

Ti

⁴⁺

Ti

⁴⁺

+ Cr

³⁺

+ Fe

³⁺

+ Al

³⁺

( )

Cr

³⁺

Ti

⁴⁺

+ Cr

³⁺

+ Fe

³⁺

+ Al

³⁺

( )

Fe

³⁺

Ti

⁴⁺

+ Cr

³⁺

+ Fe

³⁺

+ Al

³⁺

( )

FIGURES

Fig. 1

48

48

Fig. 2

(a) ^Ti

4+

Al ³⁺

Fe ³⁺

(b)

Fe ³⁺ Al ³⁺

Cr ³⁺

49

49

-1 -0.8 -0.6 -0.4 -0.2 0

-0.02 0 0.02 0.04 0.06 0.08

s V olume Dependence (J/bar)

0

(a)

-1 -0.8 -0.6 -0.4 -0.2 0

-0.05 -0.04 -0.03 -0.02 -0.01 0 0.01

s V olume Dependence (J/bar)

1

(b)

-0.2 0 0.2 0.4 0.6

-0.04 -0.02 0 0.02 0.04 0.06

s V olume Dependence (J/bar)

2

(c)

12

(

s₀ 1

)

^Xsp

2

(

s₁ 1

) (

^Xsp+X_hc

)

2s

2

X

mt

Fig. 3

50

50

3.6 3.8 4 4.2 4.4 4.6 4.8

Measured Volume (J/bar/mol)

3.6 3.8 4 4.2 4.4 4.6 4.8

Model V olume (J/bar/mol)

(b)

3.6 3.8 4 4.2 4.4 4.6 4.8

Measured Volume (J/bar/mol)

3.6 3.8 4 4.2 4.4 4.6 4.8

Model V olume (J/bar/mol)

(a)

V

(

X,T,P,s

)

^{=Videal+XmtXuv}

(

^{0.1250Xmt+0.1018Xuv}

)

V ( X, T, P, s ) ^{= X}

ⁱ

^V

ⁱ

i

(T, P)

3.6 3.8 4 4.2 4.4 4.6 4.8

Measured Volume (J/bar/mol)

3.6 3.8 4 4.2 4.4 4.6 4.8

Model V o lume (J/bar/mol)

(c)

Fig. 4

51

51

Fig. 5

3.6 3.8 4 4.2 4.4 4.6

Measured Volume (J/bar/mol)

-0.02 -0.01 0 0.01 0.02

Model Residuals (J/bar/mol)

52

52

Fig. 6

E xce ss V o lu m e ( J/ b a r/ m o l) E xce ss V o lu m e ( J/ b a r/ m o l)

53

53 0 0.2 0.4 0.6 0.8 1

X

0 0.01 0.02 0.03 0.04

Excess V olume (J/bar/mol)

mt

(b)

Fig. 7

E xce ss V o lu m e ( J/ b a r/ m o l)

54

54

Fig. 8

0 0.2 0.4 0.6 0.8 1

-0.015 -0.01 -0.005 0 0.005 0.01 0.015

Excess V olume (J/bar/mol)

Fe ₃ O ₄ Fe ₂ TiO ₄

55

55

0 0.2 0.4 0.6 0.8 1

4.41 4.42 4.43 4.44 4.45 4.46 4.47 4.48

Molar V olume (J/bar/mol)

FeCr ₂ O ₄ Fe ₃ O ₄

Fig. 9

56

56

Fig. 10

0 200 400 600 800 1000 1200 1400

Temperature (ºC)

-0.9 -0.8 -0.7 -0.6 -0.5 -0.4

s

O'Neill et al. 1992 Levy et al. 2004 Antao et al. 2005a

Antao et al. 2005b (3 GPa) Antao et al. 2005b (5 GPa) Antao et al. 2005b (6 GPa)

0

1 bar

3 GPa

5 GPa 6 GPa

57

57

Supplement)1.!Data!sources!used!in!model!calibration Reference

1 Andreozzi,!G.B.!and!Lucchesi,!S.!2002 2 Andreozzi,!G.B.!and!Princivalle,!F.!2002 3 Andreozzi,!G.B.!et!al.!2000

4 Andreozzi,!G.B.!et!al.!2001 5 Bosi,!F.!et!al.!2007 6 Bosi,!F.!et!al.!2009 7 Brey,!G.P.!et!al.!1999 8 Carbonin,!S.!et!al.!1996 9 Carbonin,!S.!et!al.!2002 10 Carraro,!A.!2003

11 Della!Giusta,!A.!et!al.!1996 12 Doroshev,!A.M.!et!al.!1997 13 Finger,!L.W.!et!al.!1986 14 Fleet,!M.E.!1981 15 Fleet,!M.E.!1984 16 Gatta,!G.D.!et!al.!2007 17 Girnis,!A.V.!et!al.!2003 18 Harrison,!R.J.!et!al.!1998 19 Hill,!R.J.!1984

20 Ishii,!M.!et!al.!1982 21 Larsson,!L.!et!al.!1994 22 Lavina,!B.!et!al.!2003 23 Lavina,!B.!et!al.!2005 24 Lavina,!B.!et!al.!2009

25 Lenaz,!D.!and!Princivalle,!F.!2005 26 Lenaz,!D.!et!al.!2004

27 Lenaz,!D.!et!al.!2007 28 Lenaz,!D.!et!al.!2009 29 Levy,!D.!et!al.!2003 30 Levy,!D.!et!al.!2004 31 Lucchesi,!S.!et!al.!1998 32 Martignago,!F.!et!al.!2003 33 Martignago,!F.!et!al.!2006

34 Menegazzo,!G.!and!Carbonin,!S.!1998 35 Millard,!R.L.!et!al.!1995

36 Nakagiri,!N.!et!al.!1986 37 Nakatsuka,!A.!et!al.!2004 38 Nestola,!F.!et!al.!2007

39 O'Neill,!H.St.C.!and!Dollase,!W.A.!1994 40 O'Neill,!H.St.C.!et!al.!1992

41 O'Neill,!H.St.C.!et!al.!2003