NUMERICAL MODELING OF CHEMICAL EQULIBEUA BEm'EEN CHROMIAN SPINEL,

OLIVINE, AND BASALTIC MELT,

With Peîrologic Applications

by

Alexei Anatolievich Poustovetov

A thesis subrnitted to the Department of Geological Sciences

In confonnity with the requirements for

the degree of Doctor of Philosophy

Queen's University

Kingston, On'%io. Canada

January, 2000

Copyright 0 Aiexei Anatolievich Poustovetov. 2000

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ABSTRACT

A nurnber of equations connectuig the composition of coexisting chromian spinel, olivine, and

basaltic melt are developed, teste* and applied to natural samples. These equations include:

hproved version of olivine-spinel geothermometer,

A set of empiricai equations needed to calculate the composition of chrornian spinel given the

composition of basaitic meit fÎom which it crystallizes, and conditions of cqstallization;

Equation for calculating Cr content or O-xygen fbgacity (f03 of basaltic melts saturated with chromian

spinel.

The suggested olivine-spinel geothermometer can be used to determine the composition of

coexisting chromian spinel and olivine over a wide range of temperature, covering both magmatic and

metamorphic conditions. Although it is caiibrated mainly using the results of high-temperature

crystallization e.xperiments, it also reproduces the variation of olivine and chromian-spinel composition in

metamorphic rocks, such as alpine-type peridotites and metamorphoseci meteontes. Good agreement

between the suggested olivine-spinel and olivine-melt geothermometers has been demonstrated using a set

of primitive basaltic lavas that were rapidly quenched. The geothermometer has aiso k e n applied to

podifom chrornitites from the Kempersai Massif in northem Kazakhstan. indicatulg magmatic conditions

for theu origin.

The composition of chromian spinel has been calculated using the g l a s composition for a set of

primitive basaltic lavas using the suggested set of empirical equations. mie results, together with

petrographic data, indicate late aystallization of Al-rich chromian spinel that is obsewed in a number of

MORBs. The crystallization of the most Cr-rich chromian spinel fiom MORES likely occurred earlier, at

higher temperature and fi-om a more primitive rnelt.

The equation developed for calculaîing Cr content of basaitic melt, saturated with chromian spinel,

has been applied to a set of primitive naturai glasses with known Cr content to calculate fOz of quenching.

The f02 for the samples \vas also calculated fiom known ~ e ~ + / F e ~ + of the glasses. The good agreement

between the values obtained using these hvo methods confirms possible use of the suggested equation as a

new oxygen geobarometer.

The completion of the thesis wodd not be possible without ai i kùids of support, provided by my

thesis s ~ p e ~ s o r Dr. P. L. Roeder throughout my work on the project. This included sharhg ideas,

unpublished data, thoughtfiil discussions, patient guidance and encouragement, the value of that is

impossible to overestimate. I consider communicating with Dr. P, L. Roeder as a specialist and as a person

an invaluable lifetime lesson. Financial support \vas provided by a number of scholarships and gants fkom

Queen's University to myself, and NSERC research grants to Dr. P. L. Roeder.

1 am gratefid to many people tvho provided sampled or e.qertÏse: Dr. Henry Dick kindly let me

use his unpubkhed database of glas analyses. Dr- Heather Jamieson, Dr. Robbie Millard and Dr, Ron

Peterson s h e d theù e.xpertise in spinels in numerous discussions- Dr. James Man, Dr. Rosaiind Helz, Dr.

Susan Humphris and Dr. Hamidur Sigurdsson supplied çampies of basaltic Iavas. Dr, Grigory G.

Kravchenko and Natâlia Nikol'skaya were an invaluable source of help in organizing and conducting a

field trip to South Urals. Dave Kempson helped me with the microprobe on more than one occasion. IIsley

Colton is thanked for the unpublished data from his M Sc. thesis. Dr. Mïchael Ioffe helped nith PASCAL

progatnmhg and supplied some usehi code. 1 appreciate the exceilent work of J, Adwent and Roger Innes

in preparing microprobe samples. I'd like to thanli Rob Renaud, Mark Badharn, and Ela Rusak for their

help nith computing and pubIishing issues.

I would like to thank fa cul^ members, especiaily Dr. Herb Helmstaedt and Dr. Kurt Kyser, as

well as Dr. Tony Naldrett from the Universin of Toronto for iheir contiriuing interest and support 1 aIso

thank my felIow graduatc srudents. staffand Kingston residents including. among others. Beth and André

Tessier, Bi11 and A j a Kaninen, Andy Lee. Mike Peshko. Dave Love. Sarah Palmer. Aliona Valyashko,

Keny Klassen. and Donald Chipley who have made my Me at Queens' Universin so productive and

interesting.

1 thank my parents and sister for shaping rny interest in life and teaching man? inva1uabIe lessons.

1 am indebted to my tvife Aiiona n-ho ahvays ben-. through her love. u b t is needed in each particular

moment: a lvord of encouragement or an impromptu adventure. Witiiout l m influence. this project couid

never be completed. Finally. my daugliter Anya was a constant source of escitement and entliusiasm,

important attributes of any success.

STATEMENT OF ORlGlNALlTY

This thesis is my own original work, a i i data and results of previous work used in it are properly

acknowledged or refermi AU equations for Gibbs free energy of chromian spinel solid-solution and

foiiowing from them the nnal equations descxibing the chernid mass transfer in the reactions involving

chromian spinel are derived by myself. The equation describing total Cr content in basaltic melt as a

b c t i o n of temperature, oxygen fbgacity, and chromian spinel composition, and the resulting equation for a

new oxygen geobarometer are my own original contributions. Semiempirical equations describing major-

element distribution between basaitic melt and chromian spinel are also my own original equations. The

equations are calibrateci using pubiished experimentai data and comrnercially available cornputer programs

for regression analysis and for solving simultaneous equations. The application of the developed equations

involved using analytical data, some of which are published and the others are obtained by myseifwith an

electron microprobe at Queen's University. The interpretation of the application results is substantially my

own, but benefited fiom the discussions wïth my supervi~or~ Prof. P. L- Roeder. The geological samples

were obtained from severai depositones, except for the samples collected by myself d u k g field trips to

Kazakhstan.

TABLE OF CONTENT

.. ABSTRACT ........................... .................................................................................... ...................... ......LI

... ACKNOWDGMENTS .......-...O................ ........................................................................................ iu

. . ................................................................. LIST OF TABLES ...................~~~~.....H........~........................ WL

. . LET OF FIGURES ................. .... ................................................ .............................. VIL

CHAPTER 1 INTRODUCTlON ..... ........................................................................ 1

CHAPTER 2 THERMOOYNAMIC PROPERTIES OF CHROMIAN SPINELS ...... 8

Previous Work ........................................................................................................................................ 8 ...................................................................................... Thermodynamic ModeIs for Chromian Spineis 8

2+ ................................................................................... Cation Disnibution in (Mg Fe )MF4 Spinels 10 ..

Olivine-Spinel Geothennometry at High Temperatures ........................................................................ 13

Formulation of the Model ............ .... .................................................................................................. 15 . Calibration of the Model ............................................ ..................................................................... 19

WI5* .............................................................................................................................................. 19 AGrr. Wn ..................................................................................................................................... 21

Olivine-Spinel Geotbermometry ............................................................................. .... ................ 3 1 ....................................................................................................................... Theory and Calfiration 31

CHAPTER 3 OLIVINE-SPlNEL GEOTHERMOMETRY .......... .. ........ .. ........... 35

Olivine-Spinel Geothermometry at Metamorphic Conditions ............................................................. 36

................................................................. Alternative Thermodynamic Model for Chromian Spinels 38 Formulation ........................................................................................................................................ 38

................................................................................................................ Calibration and Application 42

CHAPTER 4 CHROMlAN SPINEL = MELT EQUILIBRIA ................................... 46

Existing models ..................... .. ......................................................................................................... 46

Formulation .................................................. ............... ....W..........H.......**.*..........*...H..*.......t...... 52

Data for Calibration and Calibration Results ...................... .............. .................................................. 54 Data Base ........................................................................................................................................... 54 Results of Regression Anaiysis ........................................................................................................ - 5 5

.............................. Preferred Empirical Model ...... .................. w 1- ....................................................... 58

CHAPTER 5 THE DISTRIBUTION OF CR BEWVEEN BASALTIC MELT AND

Previous Work ...................................................................................................................................... 65

Formulation of the Mode1 for Cr Distribution between Spinel and Melt .................. ............ ..... .... 67

........ ........ .......................................................................*...........*....... Results of Caiibration ,, ......... 70

CHAPTER PETROLOGICAL APPLICATIONS ............................................... 74

Introduction .......................................................................................................................................... 74

EIectron Microprobe Analysis ......................................................................................................... 75

Olivine-Spinel Geothermometry of Lavas .................. .. ....................................................................... 77

Olivïne-Spinel Geothermometry of Podiform Chromitites from the Kempersai Massif .................... 81

Composition of Chromian Spinel in Primitive Lavas ....................... ..... ....... ,.... ........................... 83

Cornparisou of Caiculated and AnaIyzed Spiael in Primitive Lavas ................................................... 83

Modeling Chromian-Spinel Composition in Primitive MORBs ........................................................ 93

The Cr Distribution between Cbromian Spinet and Basaltic Melt as an Oxygen Geobarometer ....... 96

CHAPTER 7 SUMMARY AND CONCLUSIONS .........-......... .... .................. 101

References ................ ,., ........................................................... 104

Appendir A. Electron microprobe analysis (wt . % oxide) of chromian spinel ..... .................... ............. 113 Appendir B . Electron microprobe analysis ( ~ v t . % oxide) of olivine .................. .......... ......................... 122

Appendir C . Electron microprobe analysis (wt . % oxide) of studied basaltic glasses .................... ., ....... 126

Appendix D . An example of rnuitiple linear regression (calibration of Equation 54) performed with Microsoft Excel 97 ......O.................. .... .......................................................................................... 129

Appendix E. Generai view and formuIas of the Corel Quattro Pm 8 spreadsheet used to caldate chromite composition fhm melt composition at a particuiar ternperature and pressure by solving a set of nonlinear equations (Equation 68 - EQuation 7 1). .,....., .....,,..., ....,,..,......,.,. ........ .... .......... .-..... .... ...,.. ........... 130

LIST OF TABLES

Table 1. Expressions of site mole fractions in terms of chosen independent variables ................-......resres..resres - 17 2+ 3-

Table 2. Molar free energy of Fe - Fe - Mg - Al - Cr spinel solid-solution in tenns of preferred themodynamic parameters .................................................................... 18

Z+ 3+ Table 3, Taylor e-upansion coefficients for Fe -Mg-Al-Cr-Fe spinels in terrns of preferred

thermodynamic parameters ........................................... ... ............................. 20 Table 4. Accepted values of thermodynamic parameters (independent parameters do not have

definition in îhe last column) ................................................................................. 3 1 Table 5 Values of thermodynwnic parameters (idependent parameters are rnarked as bold

symbols) corrected by regression ushg olivine-spinel e.upenments, .,..--.........-... - .-......... ....eseses......es.es. 34 Table 6. Expressions of site mole fractions in tenns of chosen independent variables .....,........-...- .. -.--....- 39

3+ Table 7. Taylor expansion coefficients for F ~ ~ ~ - M ~ - A ~ - c ~ - F ~ - Ti spinels in te- of

preferred thermodynamic parameters .............. - ........................................................................ . . 40 Table 8 Resutts of multiple regression ......................................... - . - - . - - . . . . . . - - - 44 Table 9. Results of multiple linear regression ...................................... .. . 55 Table 10. independent cornpositional variables and assumeci cation distribution in M~-F~'?A~-

c r - ~ i - ~ e ~ spinels ..... . . .. . . . . .. .. . . -. -. . -.. -- -. -.. .. . . . . . . . . . . . . . - . . . . . . . . . . . . . . . . .. - .. . . . .. . . . . . . . .. . . . . . . . . . . . -. . 59 Table Il. Taylor expansion coefficients for F~'~-M~-AL-C~-F~*-T~ spinels in tenns of

preferred themodynamic parameters ..................................... - ................................ 60 Table 12 Values of parameters in Equation 83 ....................................................................................... 71 Table 13. Summary of the studied samples ................................................... .... 75 Table 14. Average glas composition and calculated chrornian-spinel composition for studied

primitive lavas. ., .. . . ..., . .. . . . .. . . . . -. . ... -. -. . . - --. . . -. . . -. . .- -- -. . - .- -. . . -...... . . . . . . - . . . . . . - -. . - -. . . . . . . . - . . . . . .. . . . . - 86 Table 15. The composition of glas, chromian spinel and calculated loJf02 of selected

primitive lavas. ............................................................................................................... 97

LIST OF FIGURES

Figure 1. Typical occurrence of chromian spinel in quenched basaltic lava. Small chromïte crystais are hosted by olivine phenocrysts (white) or volcanic glass (gray). The sarnples are RS-774, Icelandic Iava (A), and IKI-21, Hawaiian pumice (B), Scale bars are 0.1 mm. Thin sections, transmitted Light. ...-............................+....-..~.............................................. 2

2+ Figure 2 Plot of the number of Fe and Mg cations in the teuahedral site versus temperature

for aluminate spinels. The X refers to number of cations assuming four oxygen atoms. Data for FeAL204 are from XRD stmctux-ai refïnements by Larsson et al. (1994) (L+,93), and by Harrison et al. (1998) (H+,98). Also shown is the curve describing ordering in FeAi204 calculated with the Sack and Ghiorso (1 99 1 b) model. Data for Mfl204 are £Yom Peterson et al. (199 1) (P+,9 l), and Redfern et aL(1999) (R+,99). ........,.............--...-..*......-.-. 12

Figure 3. The ciifference between temperature values calculated with Sack and Ghiorso's (199 lb) olivine-spinel geothermometer and real temperatures of expenments, plotted versus the temperature of expriment nie data are f3om Murck and Campbell (1986) and Thy et al. (1991). ..................... ... .................................................................................................. 14

Figure 4. The negative values of configurational Gibbs energy of the Mg-Ai exchange reaction in pure MgAI2O4 plotted versus a fiinction of the order parameter Sz using the chta of Peterson er al, (1991) for the temperature interval Born 700°C - 1000°C. The line is the

..................... least square regression line with AGrr = -19596 * 1200 and Wn = 1 1258f2900 J/& 23 Figure 5. Plot of Q* , dehed by Equation 24, versus S2-X3. Data are fiom Oka er al.

CF(AII-I ....................... (1984). Only 1250°C data were used for the calibration. ....-.....-.-.... 25

Figure 6. Plot of Q* , defïned by Equation 24, versus S2-X3. Data are from Petxic and Cr(AlH

Jacob (1982) and Oka et al. (1984). Values for Peiric and Jacob (1982) data were calculated as -RT[ ln( XJ(L-X3) - Y h ( auo3 / acno3 )]- The data points form hear trend

O

with a slope of %WIT and an intercept of YAG ~ e A l ) - , ................................................................... 26 Figure 7, Plot of QMfleFl VS. 1-2X2 for 1300°C da@ of Jamieson and Roeder (1984) and 650-

800°C data of Engi (1983). Note sub-parallei ciiaracter of the trends for these hvo data sets. .................... ,.. .......................... ,... .................................... 30

Figure 8- Plot of 1n.d (Kd = 0(Mg)0L(~F~rp/@~sp(xFef3"t) vs. X3 for olivine FOW and spinel with X9.05 (NFc3J2). Thin lines represent isothenns caicuIaîed with Equation 33 and values ffom Table 5. Values near the isotherms show temperature in OC. Evans and Frost (1975) 700°C "isothermn (E&F,75) is shown for cornparison .......................~.~..~..~.......~....~.............. 36

Figure 9. plot of ME^ M ci = (xMg)a'(xF~~.p/(x~sp(xF~~') VS. x3 for olivine Fos2 and spinel with Xfl.0. Thin lines represent isotherms calculated with Equation 33 and values f3om Table 5. Values near the isotherms show temperature in O C . Open squares are data for olivine-spinel assemblages from metamorphosed meteorite Gobabeb (Fudali and Noonan, 1975) .............................................................................................................................................. 37

Figure 10. Plot of lnKd (Kd = ( X & ~ ' ( X F C ~ ' P / O ( ~ ~ ~ ~ C F ~ ~ ~ " ' ) VS. X for olivine Fogo and spinel *th X d . 0 5 . Thin lines represent isotbenns calcdated with Equation 46 and values fiom Table 8. Values near the isotherms show temperatwe in OC. Evans and Frost (1975) 700°C "isothem" (E&F,75) is shown for cornparison ........................................................... 44

Figure II. plot of l a c i la ci = ~ O ' O ( F ~ ~ ~ / ( ~ M g ) v ( x F ~ ~ o ' ) VS- x3 for olivine FW and spinel with XS=O.OO. Thin lines represent isotherms caiculated with Equation 46 and values fiom. Values near the isotherms show temperature in OC. Open squares are data for olivine-spinel assemblages from metamorphosed meteorite Gobabeb (Fudali and Noonan 1975). ............................................................................................................................................. 45

Figure 12. The relationship of &O3 content in chromian spinei and meit in Roeder and Reynolds (199 1) 1300°C lbar experiments at logfOz -= -6 ( points) as compared to the relationship suggested by Maure1 and Maure1 (1982) (line, see Equation 49). ................................... 48

Figure 13. Plot of Cr content in melt saturated wiîh chromian spinel vs. logf;D2 as obtained by experirnent and by 'MELTS' program. Data are for Roeder and Reynolds 40 1

............................ experiments performed at 1 bar and -1300°C. See text for details of calculations. 50 Figure 14. Plot of CrD3 content in chromian spinel in experiments and calcuiated with the

'MELTS' program vs. 1ogfD2. Data are the same as on Figure 13 ...................................................... 5 1 Figure 15. Plot of Cr203 content in experimencal (selected Roeder and Reynolds (199 1) nuis)

....................................................................................................................... vs. calculatecl spinel. 56 Figure 16. Plot of Cr2(& content in e'iperimental vs. caiculated spinel, Spinel composition is

................... ...*........... . calculated uskg preferred ernpuical mode1 (Equation 68 Equation 7 1). ...,. 6 4 Figure 17. Plot of calcdated (Equation 83 and Table 12) total Cr content @pm) in melts versus

..................................... reported values fiom Roeder and Reynolds (i99 1) 1 bar experiments 7 1 Figure 18. Plot of the caldated versus analytically detennined values of ln(CrO/ GOl-> for

FAS and FAD melts (Schreikr, 1976). Both 1500 and 1550°C data are incIuded ............................. 72 Figure 19 Plot of caiculated vs. e.uperimentally determineci Cr content @pm) of synthetic rnelts . .

that were not used for the calibrahon- .............................................................................................. 73

Figure 20. Plot of Cr/(Cr+Ai) vs. ~e'+/@e'++~~) of anaiyzed chromian spinel as compared to the range of chromite composition fiom podiform chromitites and abyssal basalts outlined

......................... by Dick and Bullen (1984). .... .......................................................................... 76

Figure 21. Plot of the ciifference between temperatures caldateci with olivine-spinel geothetmometers fl (olsp)) and olivine-melt geothemometer of Beattie (1993) fl (olliq)) vs. M203 content in spinel for a senes of natural glassy basalts. ....................................................... 78

Figure 22. Fe-Mg partitioning data for olivine-spinel pairs from selected primitive MORBs formeci at a temperature close to 1200°C, shown as points, cornpared with the isotherms (OC) caiculated with Sack and Ghiorso (199 lb) calibration of olivine-spinel geothermometer. ...................................~..~.~..............................~.~-.................................................. 79

Figure 23. Fe-Mg partitioning data for olivine-spinel pairs fÏom selected primitive MORBs, shown as points, cornpared with the isotherms (OC) calculated with calibration, suggested in the present study (Equation 46)- .............................~..~....~~........................................................... 80

Figure 24. Plot of Cr/(Cr + Al) vs. F~*+/(F~~++M~) of chromite from ores and dunite fiom southeasteni part of Kempersai Massif (South Urals). Masive ores contain >75 vol..% of chromite, disseminatecl ores contain >lO vol.% of chromite, while dunites typically contain <5 vol.% of chromite, Lines represent calculateci composition of chromite with &=0.005 and Xp0.04 in equilibriwn with olivine F k 7 at diEerent temperatura. ...................... ,,.. 82

Figure 25. Plot of Cr/(Cr + Al) vs. F,"/(F?~) of chromian spinel h m MORBs (open symbols) and Hawaiian pumices (fïiied symbok). Outlùled fields indicate the range of spinel composition from MORBs and Hawaüan lavas (data are fiom Roeder, 1994). ........................ 84

Figure 26. Back-scatterai elecîron @SE) image of large spinel grain in glass fiom MORB-like lava F2-2 (Allan at al., 1988). Lighter areas of spinel have higher Cr content. Scale bar is 20 pn. ............................................................................................................................................ 85

Figure 27 A-E. Plot of Cr/(Cr + Al) vs. ~e~+/( 'e '++ Mg) of chromian spinel fiom Hawaiian pumices. Calculated values for each sample are indicated as 'A' (calculated according to Ariskin and Nikolaev, 1996b), and 'P' (calculated with the model described in Chapter 4)..,- ................................................................................................................................................ 88

Figure 28 A-H. Plot of Cr/(Cr + Al) vs. F~'+/(F-~*++M~) of chromian spinel fiom studied samples. Calculated values for each sampie are indicated as 'A' (calculated according to Anskin and Nikolaev, 1996b), and 'P' (calculated with the mode1 desctibed in Chapter 4) .................................................................................................................................................... 90

Figure 29. Plot of Cr/(Cr +AI) vs. ~e~+/(Fe?+~g) of chromian spinel fiom the MORB sample AI1 32-D 1 1-90. Calculated values for each sample are indicated as 'A' (calculated according to Mskin and Nikolaev, 1996b), and 'P' (calculated wvith the model described in Chapter 4). Circles and diamonds are for chromite hosted by g l a s and olivine respectively. Open markers are for small grains or rims of larger grains. Connected solid and open syrnbols show the values for the core and rim respectively of the sarne grain .................... 9 L

Figure 30. Plot of Cr/(Cr + Ai) vs. ~e '+/(Fe~++~g) of chromian spinel fiom the MORB sample F2-2. Calculated values for each sample are indicated as 'A' (catculated according to Ariskin and Nikolaw, 1996b), and 'P' (calcuiated with the model d e s c r i i in Chapter 4). Open markers are for small grains or rims of larger grains. Connected solid and open symbols show the values for the core and rim respectively of the same grah ......................... ... ....... 92

2+ t+ Figure 3 1. Plot of Cr/(Cr + Al) vs. Fe /(Fe +Mg) of spinel calculated using avaiiable data on

glass composition fiom primitive MORBs. Spinel composition calcuiated for average composition of liquid inclusions in Cr-rich spinel phenocrysts and composition of the spinel (see text) are also shown. Outlined field indiates range of chromian-spinel composition fiom MORBs (data are fiom Roeder, 1994). ................................................................ 95

Figure 32. Plot of calculated values of 10- vs. tem rature for selected naiural lavas. F Squares show the f02 calculated from known Fe /Fe2+ using equation by Sack et al. (1980). Circles refer to fD2 calculated using the method involving Cr described in the present study (Equation 84). Temperature is calculated using Beame (1993) equation for olivine-melt equilibna Solid symbols correspond to the samples where total Cr of the glass was analyzed at Queen's University, and open symbols are used for the samples studied by Gaetani et al., 1995. Error bars show the error of f02 calculation due to an analytical error of Cr analysis of +/- 20pprn ................................... .,. .......................................... 99

To my parents,

Tambme kfz~arnbee~e u Awrnonum Muxalinosuvy nycmosemosbclrr

Chapter 1 Introduction

Chromian spinel or c h m i t e (Mg, ~e'')(Al, ~ e ~ , Crh04-(Mg, F ~ ~ ' ) z T ~ o ~ is often one of the first

minerals to crystallize fiom basaltic melts and is commonly found as very small crystals in basaltic lavas,

frequently accompanied by olivine phenocrysts (Figure 1). The concentration of chromite by natural

magmatic processees bas led to the principal economic deposits of Cr as found in layered intrusions such as

the Bushveld Complex in South Afiica or in the ophiolitic ultramafic bodies such as Kempersai Massif in

south Urals. The principal purpose of the present study is to relate the composition of chromite to the melt

fiom which it crystallized and Uius use the composition of a chromite as a petrogenetic indicator.

The great potential of chromian spinel as a tool in pettological studies \vas first recognized and

e'cplored by T. N. h i n e ( m e , 1965). He argued that, since chromian spinel shows a widc compositional

variation in rocks, it " must be relatively sensitive to the chernical or thermal conditions that accompanied

its formation". He also noted the potential use of chromian spinel as a pressure indicator. Irvine suggested a

set of chernical reactions bebveen spinel (sp), melt (m), pyroxene (px), and olivine (01)- that are useful in

fiiaher theoretical analysis. These reactions include:

Equation 2

X " MgAlp4sP + XF2+'Te~1204sP + Cr O = X %fgCrqO: f XFC2+'PFeCr O Mi3 2 3 Ms - 2 4

Equation 3

SP SP where Yi and X are parameters describing the chromian-spinel chernical composition

Figure 1. Typical occurrence of chromian spinel in quenched basaitic lava. S d chrornite crystais are hosted by olivine phenocrysts (white) or volcanic glas (gray). The samples are HS-774, Icelandic lava (A), and MI-21, Hawaüan pumice (B). Scale bars are 0.1 mm. Thin sections, transmitted light,

Assuming ideal mixing for spinels and coexisting phases, Irvuie used these reactions to constnict

thermodynamic equations. which relate the composition of chromian spinel and coexisting minerais or

melt Irvine used Equation 3 to show how oxygen fiigacity can be calculated when chromian spinel coexists

with olivine and pyroxene and stresseci the importance of chromian spinel as an indicator of oxygen

2+ figacity. Itvine aiso noted that Mg-Fe distribution between chromian spinel and coexïsting minerai(s)

(e-g., Equation 1) is temperature-dependent giving birth to the olivine-spinel geotherrnometer. It has also

been shown that, under certain conditions, chrornian-spinel composition alone can be used to compare the

conditions of the equilibrium or to deduce the changes during chromian-spinel crystallization. This is

achieved by comparing the range of chromian-spinel composition ftom basic igneous rocks ~4th dcuiated

"equipotentiaIW surfaces in the spinel compositional volume, representing the composition of spinel in

equilibrium with a phase of a fixecl composition a t specified conditions.

These equations aiiowed Irvine to interpret anaiytical data on chrornian-spinel composition fkom a

variety of geologicai environments (h ine , 1967). Specifically, he predicted the pentectic character of the

reaction involving chromian spinel, pyroxene and melf and suggested tlmt Werent reactions are caused by

the ciifferences in composition of crystallizing melt. This was Iater confirmai by a number e~~erimentai

studies in natural (e.g. HiIl and Roeder. 1974) and "simple" synthetic systerns (e.g. Onurna and Tohara.

1983). Irvine also conducted a comparative study of chrornianspinel composition fiom different geological

environments and connected them to the ciifferences in chernical and ptiysical parameters during chromian-

spinel crystallization

Irvine's work made it clear that more e.xperimenta1 and analytical data were needed in order to

interpret the composition of chromian spinel- For example. practically al1 coefficients in the equations

suggested by Irvine were undefmed, and, instead, rough "working" values were used for the demonshation

(Irvine, 1965). Also. no rnechanism existed to account for nonideality of chrodan-spinel solid-solution or

melt. That ied M e to suggest that further experimental work is necessary and he particularly noted the

need for expenmental investigations of reactions invoiving spinel. incIuding hi@-temperature experiments

and experiments on naniral matends (Iivine, 1967).

HU and Roeder were the first to conduct the experïments of this type (Hiil and Roeder, 1974).

Among the resuits, they found a dramatic dependence of chromian-spinel Cr samation on O-uygen fiigacity,

At the tirne, they interpreted this dependence as due to the changes of ferric iron content. As bewme clear

later, this dependence is largely due to the change in Cr oxidation state (Schreiber, 1976; Murck a d

Carnpbeii, 1986; Barnes, 1986; Roeder amd Reynolds, 199 1). The work of Irvine was followed by other

numerous experimental and theoretical -dies involving c h m i a n spinels. However, despite extensive

e'rperimental and geochemid snidies of chromian spinels. there has only been limited progress in

developing and caiibrating quantitative petrological tods (equations), invoiving chromian spinel. These

tools include:

olivine-spinel geothermometer ( Equation 1);

olivine-spinel-pyroxene oqgen geobarometer (Equation 3);

chromian spinel-melt equilibria (described by quations analogous to Equation 2);

oxidation state of Cr in melt

Many calibrations of the olivine-spinel geothermometer have been suggested (e-g- Jackson, 1969;

Evans and Frost, 1975; Roeder et al., 1979: O'Neill and Wall, 1987; Sack and Ghiorso, 1991b). using both

the original equations of Irvine. or more cornplex equations, which take into account cation disorder in the

spinel structure on M~F; partitionhg CSack and Ghiorso. 199 Lb). The cations in the spinel structure are

distributeci bettveen two distinct stmctural positions, a four-coordinated tetrahedral position and a six-

coordinated octahedral position* A spinel is described as normal if the +2 cations, Mg and ~ e - , are in the

tetrahedrd position The spinel is inverse if the +2 cations are in the octahedral position. The degree of

order of the spinel structure is temperame dependent and its accurate description is essential in formulating

a themodmc mode1 that involves spÏine1. Most of the suggested versions of the olivine-spinel

geoîhermometer are based on e~pr ïmenfa l data (e.g. Engi, 1983; Jarnieson and Roeder, 1984; etc.). The

composition of chromian spinel and oliviïne from natural lavas and rnetamorphic rocks was used by Evans

and Frost (1975) to caiibrate their version of the olivine-spinel geothermometer, and Allan et al. (1988)

used data on the composition of chromhm spinel and glas from oceanic basalts to calibrate a shiiar

equation describing partitionhg between chmmian spinel and basaltic melt Several calibrations

of the olivine-spinel-orthopyrosene oxygen geobarorneter have aiso been developed (O'Neill and Wall,

1987; Ballhaus et al., 199 1; Wood, 199 1). They were more d.if&cdt to achieve because of the difkulty in

defining and dculating the activity of the magnetite end-member in spinel sotid-solution Additionaily, an

olivine-pyroxene-Pt alioy oxygen geobarometer has been suggested (Jamieson et aL, 1992). The latest

olivine-spinel geothermometer, based on the most advanced thermodynamic mode1 of chrornian spinel

(Sack and Ghiorso, 199 lb), can d l be improved, as will be demonstrateci in subsequent parts of this study,

in order to better describe olivine-spinel equilibria, especially at magmatic temperatures.

The calibration of equations reiating the equilïbrium composition of chromian spinet and melt has

long been hampered by the absence of diable chromian spinel - melt experimental data The experimental

study of chromian-spinei crystallization in the Fe-free system forsterite - diopside - anorthite - picrochromite - silica, conducted by Irvine (I977), revealed major trends of chromian-spinel composition

during crysta1)ization of basaltic melt, uvine did not, however, report the complete analyses of the

experimentai phases, noting the esistence of zoning in chromian-spinel cxystals and suggesting that "special

experhents wiii be required to obtain data suitable for detailed considerations of C r M partitionkg"

(Irvine, 1977, p.467). Chrornian spinel u s intensively studied in many natural rocks, creating the

opportunity of empincal calibration of Xrvine's "equipotential surfaces" and thus e.uplaining other features

of chromian-spinel chemistry and occurrence. Dick and Bulien (1984) compiled a signifiant database on

chromian-spinel cornpositionai variations in different geological environments. They conducted a

comparative study of chromian-spinel composition depending on geologicai and tectonic sening, and ttaced

the observed trends to the conditions of chrornian-spinel crystaiiization using the Inine (1965) theoreticai

basis. A si& approach was later used by Roeder (1994). who colIected and organized much of the

pubtished chromian-spinel analyses into a database. He used the database to distinguish and decipher the

nature of sorne new features of chromian-spinei compositional variations. These features included the

influence of known peritectic reactions involving chromian spinel on distribution of chromian-spinels

analyses in the spinei compositionai volume, distinct composi tionai characteristics of magrnatic and

metamorphic magnetite, compositional variations of chrornite fiom kunberlites due to the oicidation during

late stages of evolution of kimberlitic melts, and the changes of chromian-spinet composition in chonciritic

meteontes caused by an unusual rnetamorphism, The continuhg expenmental studies of chromian spinel - melt equiiibria bas resulted in the creation of two cornputer propim* which can be used to calculate the

chem-cal composition ofchromian spinel given the composition of parent melt and conditions of

equili'brium (Sack and Ghiorso, 1995; Ariskin and Nikolaev, 1996b). However, both programs do not

adequately reproduce experimental data for a substantiai range of the equilibrium conditions. Particuiarly,

Ariskin and Nikolaev wamed against using their program to model crystallization of Al-rich chrornian

spinel. L t has been found in the present study, that their program works adequately for high-Cr chromite,

but gives signincant errors whiie modeling less Cr-rich chromian-spinel crystallization- The advanced

MELTS package (Sack and Ghiorso, 1995) produces good results, when trying to reproduce crystaIlization

of high-Cr chromian spinel under reducing conditions and using an arbitrary Cr content in melt At more

oxidiPng conditions, however, the MELTS program calculates chromian spinel wkch is systematidy too

rich in ferrite end-members as compared to an experimentally determined composition. The analysis of the

results obtained with these two programs is presented in a subsequent part of the present study.

A better understanding of Cr solubaity in silicate meIts was achieved when ex-perimentaI data on

Cr oxidation state in Fe-free synthetic rnefts became available (Schreiber and )ZasNàn, 1976). Additional

experimental data on natural and synthetic sampies were obtained by Murck and Campbell (1986), Barnes

(1986). Roeder and Reynolds (199 1)- Forsythe and Fisk (1994), and Hanson and Jones (1998). However, as

stated by the authors of the most recent study of Cr o'iidation state in synthetic and natural melts. "For now,

we must rely on experimentally derived Cr-saturation abundances to model Cr in basaltic systems" (Hanson

and Jones, 1998). There is not a general equation which can rnodel oxidation state of Cr in basalts over a

range of temperature and o.xygen fiigaciw- One of the challenges is the lack of an anaiytical method of

detennining Cr valence state in Fe-bearing silicate melts.

The present study has as an objective the development and caiibration of a set of equations which

allows the quantitative use of chromian spinel as a petrogenetic indicator. A simplified Sack and Ghiorso

(199 lb) thermodynamic mode1 of chromian spinel is formulated, and an olivine-spinel geothermometer is

calibrated in Chapter 2. The developed mode1 is applied in Chapter 5 to describe the distribution of Cr

between chromian spinel and basaltic melt, and to calibrate an equation for caldation of Cr oxiciation sate

in the melt. The rnodel has a iimitation of low Fez03 (<10w~O/o) and Ti02 (<5%%) content in spinel

bernase of the cation distribution scheme used for the spinel structure. However, since Cr distribution

between chromian spinel and basaltic melt is only considered at the conditions of low oxygen fugacity

(quartz-fayalite-magnetite b a e r or lower) and high (-1200°C) temperature, îhïs limitation is not important

for the purposes of the present study- A by-pduct of the mode1 calïi'bration is a new version of an olivine-

spinel geothermometer. As shown by Sack and Ghiorso ( 199 lb), the accurate description of cation ordering

in the spinel structure is essential for olivine-spinel geothermometry. in particular, accounting for the

structura1 ordering in spinel pennits the application of the geothermometer to the olivine-spinel

assemblages equihirated at very different temperatureS. sinœ the degree of disorder is a strong fiintion of

temperatute. The suggested geothermometer describes available experimental data on olivine-spinel

equilibria at magmatic temperatures better than the original Sack and Ghiorso (199 1b) calibration, but

contradicts available empirical data for metamorphic conditions. The alternative version of the olivine-

spinel geothermometer, based on a digerent chrornian-spinel cationsrdering scheme, is presented in

Chapter 3. This later version mn be applied to ail spinels, inciuding those nch in ~ e * and Ti and is in

better agreement with empiricai data on the compositions of olivine and spinel from metamorphic rocks. In

Chapter 4, a set of empincai equations is presented which ailows calculation of chornian-spinel

composition in equilibrium with a basaltic melt at known conditions. Chapter 6 presents electron-

microprobe analyses of olivine, chromite and glas from a set of naturai lavas and intrusive rocks and uses

these analyses to test the developed equations.

minerai as a Taylor expansion series in t e m of selected vanables describing these changes. The

configurational part of Gibbs fi-ee energy is calcuiated assuming that no short-range order exists in the

spinel structure. in other words, the crystal structure can be describecl as a whole, without paying much

attention to local cation arrangements. Sack and Ghiorso us& a second-degree Taylor expansion senes for

the coutïgurational part of Gibbs fkee energy of chromim-spinel solid-solutions, which produces a result

identical to that of the symmeuic-regular-solution formalisai, This form of equation for the Gibbs free

energy also reproduces the equation of O'Neill and Navrotsb (1983) to calculate cation distribution in

simple spinels and accounts for compositionai idluences on site occupancies in complex spinels.

The principal theoretical shortcomings of the model are discussed by the authors in some detail

(e-g., Sack and Ghiorso, 199 Lb) and are:

the model's inability to account for short-range cation ordering in spinel structure, which is particularly

important for Ti-rich spinels ;

3+ the failure to deal with nonstoichiometric spinels, such as Fe -rich spinels at hi& oqgen fbgacities

and hi& temperatures.

In addition to these theoretical diEculties, practical problems relating to the calibration of the

model are evident and important for many petrologically significant spinels. These problems arise fiom the

large number of thermodynamic parameters, ivhich are necessary to describe thermadynarnics of spinel. A 3+

total of 37 independent thermodynamic parameters is required for the Wg, F~~+)(AL Fe ,Cr)P4 - Wg, 2+

Fe I2TiO4 system (Sack and Ghiorso, 199 1b). As a resulc suaightfonuard and unambiguous caiibration of

the model requires an extensive database of diable experimentai data on cation-ordering and activity-

composition relationships for spinel compositional end-members and bïnary systerns, This database stiU has

rnany deficiencies, so that a less direct approach, together with some equivocal data, has to be used to

restrict îhe values of the thermodynamic parameters. In the following, a few of the problems in the Sack

and Ghiorso (199 1a,b) calibration are disnissed. One major problem is using e.uperimentai data involving

Ti-rich spinels for the purpose of the model calibration The authors of the model accepted that this can

lead to errors in some of the parameter values (Sack and Ghiorso, 199 Ib), since the energetic effects of

short-range ordering are not accounted for in the model. Another obvious di.8nculty is connected with the

ambiguity of published cation-ordering data for spinel end-members (see, for example, subsequent review

of cation ordering data for aluminate spinels). There is also no conkned experimental data on cation

distribution in compositionally complex spinel, These cation distribution data are necessary for calibration

of the themodynamic parameters describing interaction between compositionally and structurally different

spinel end-members. Finally, some of the data used for calibration may not represent equilibrium data and

thus do not place tight consûaints on the resulting values of the themodynamic parameters. For e.uample,

the Sad< and Ghiom (199 1b) calibration for Cr-bearing spinels is based. at l e m partiaüy, on M~-F~'+ -

exchange data between olivine and spinel, in this case ambiguity of the data used for calibration is evident

(e,g. see fig.5 in Sack and Ghiorso, 1991b). and is probably due to the problerns in achieving equilibrîum

especially in Iow temperature experirnents,

Given ai l these problerns, it is not surpnsing that sorne of the espenmental data published since

199 1 reveal inadequacies of the Sack and Ghiorso (199 lb) calibration, as will be s h o m later. Since the

major interest of the ptesent study is spinel found in magmas at high teinperature and relatively low o.qgen

fugacity, the subsequent discussion will deal rnostly with (Mg, F ~ ~ + ) ( A I . C ~ ) ~ ~ solid-solutions with only a - minor amount of (Mg, F$ ~e)+,q and (Mg, Fe2+) Ti04 cornponents (c 10 wt% of Fe203; c 5 M.% of .. 2

TiO?) at magmatic temperatures. These are the conditions at which tlic iiiodel seems to be adequate-

2+ Cation Distribution in IMq. Fe )Al O, Spinels

L - -

2+ The Fe -Mg duminate spinels are considered IargeIy normal at low temperame. with most of the

2+ Fe and Mg occupying tetrahedral positions. and Ai occupying the octahedrai positions (Sack and Ghiorso,

199 la) in the spinel structure. Increasing temperature l a d s to more random distribution of these cations in

the spinel structure. This is supported by numerous experimental data on cation distribution in qmthesized

spinels, most having a composition close to end-member spinels, and in natural samples. which are

typically mixtures of several end-rnernbers, The most studied is the MgA1204 end-member (Wood et al.,

1986; Peterson et al-, 199 1; Miiiard et al., 1992; Maekawa et al., 1997; Redfern et al-, 1999). The in-situ

data include those of Peterson et ai- (199 l), Maekawa et ai. (1997) and Redfern et al* (1999). W e r n et al.

(1999) pointed to the spinel nonstoichiometry to expiain the ciifference between their data and the results of

Peterson et al. (199 1). Hercynite, F e W 4 spinel, has also been the subject of several investigations (e.g-

Hill, 1984; Larsson et al., 1994; Hamson et al., 1998). The studying of temperature dependence of cation

distribution in F e q 0 4 is, however, more challenghg due to the difncuity in maintaining and controlling

3+ oxygen fbgacity during the experhent, the possïbie presence of Fe in the sample, and inabiiity to use

some methods iike NMR to study cation dimibution in the presence of ~ e ' + in the structure. Lanson (1994)

also cited faster kinetics of cation ordering in FeAl O4 in cornparison with MgA1304, m . g it more 2 ..

M c u l t to study high-temperature cation distribution in quenched samples. The results of Larsson have,

however, been confirmai by more rapnt in-situ study of synthetic FeA1204 by Hamson et al. (1998) as

shown in Figure 2.

Additional data on cation distribution in natual spinels close to the MgAlPd - FeA1204

join were published by Della Giusta et al. (1996). The naturai spinels in their e-xperiments tvere fiom a

slowly cooled pendotite body in the Italian Western Alps and contained only small amounts of Cr and Ti

and less than 3 M. % of Fe903. The spinel was anneaied at dinerent temperatures and the d o n -

distribution was determineci fiom XRD structural refinements and electron-microprobe analysis. Theu

3+ resuits indiate almost complete ordering of Fe2+ and Fe into teaahedral and octahedral positions

respectively, with temperaturedependent Mg-Ai distribution (Della Giusta et al., 1996). This suggests a

more ordered structure for FeAIPd compareci to M g q 4 , in agreement with the Sack and Ghiorso

2+ (199 lb) prediction, but rnay also be a result of faster redistribution of Fe during quenching. The most

recent in situ cation distribution study of synthetic spinel fiom the MgAI2O4 - FeA1204 join supports

2+ preferential ordering of Fe into the tetrahedral position (Pavese et al. L999a). On the other han& no

partitionkg of Mg relative to ~ e " between teûahedral and octahedrai sites is assumai by Harrison et al- 2+

(1999) for (Mg, Fe )(Al,~e)3 O4 spinels based on recent in situ studies of cation dimibution in FeA1204 2

and MgA1204, and previous smdies of Fe304 . MgFeP4, and (Mg ~e'+)Fe,0~. For example, a similar

Figure 2 Plot of the number of ~e '+ and Mg cations in the tevahedral site versus temperature for aluminate spinels. The X refers to number of cations assurning four oxygen atorns, Data for FeAi204 are fiom XRD structural rehernents by Larsson et ai- (1994) (L+,94), and by Hamison et al. (1998) (H+,98). Also shown is the m e describing ordering in FeA1204 caiculated with the Sack and Ghiorso (199 1b) model. Data for M g W 4 are fiom Peterson et al. (199 1) (Pi,91), and Redfern et aL(1999) (R+,99).

degree of order is reporteci for FeAl.>04 and MgA?04 by IEamson et al- (1998) and Redfern et al. (1999) ( - Figure 2). Thus, considerable controverjy exists on the relative degree of disorder of FeA1204 and

24 MgAlP4. and partitionhg of Mg and ~e'+ between teirahedrai and oaahedral positions in (Mg, Fe )Fe204

spinels. The two existing possibilities appear to be either q u a i degree of disorder of Fe+04 and M*O4

with no preference of Mg relative to ~e'+for a particuiar structural position, or a more ordered structure of

FeA1204 and preferentiai panitioning of ~e'+ into tetrahedra, dative to Mg, in the binary spuiels. Both of

these possibilities will be e.uplored in two separate models presented in this and the next chapter. The Sack

and Ghiorso (1991b) calibration implies a more normal structure for herqnite than indicated by Larsson et

al. (1994) (Figure 2).

Olivine-Spinel Geothermometrv at Hiah Tem~efatures

The composition of coe'iisting olivine and spinel, though insensitive to temperature changes at

magmatic conditions > 1000°C), does provide important information on the îherrnodynamics of

chornian spinel, in that sense. the temperatures which one might calculate fiom the analyses of olivine and

coexisting chromian spine1 should not reveal any sigrifkant non-random deviations fiom the values of r d

temperature of equiiibrium if properly calibratecl thermodynamic models of the phases are used. Figure 3

shows the resuits of the comparison of expenmental (Texp) and caiculated temperatures (Tolsp) for

olivine-spinel pairs produceci in some experimental stuclies, which report composition of coexisting olivine

and spinel in a wide temperature range. The values of calculated temperatures were obtained with the Sack

and Ghiorso's (199 1b) program. It \vas found that an error in calculated temperature caused by variation in

chromite composition as determineci by an electron microprobe can be higher than 100°C. Nevertheless, a

systernatic trend in the deviation of calcuiated and experirnental temperature (Tolsp - Texp) as a fiuiction

of the temperature of the experiments is apparent Thus, the disagreement between experimental data and

Sack and Ghiorso (199 lb) mode1 predictions suggest the need for improvhg the model. The problem, at

least as far as Ti-poor spinel at high temperatures is concerne4 resides not in the theoretical approach of

Sack and Ghiorso (199 la,b), but rather in the difficulties of calibrating the model. This is due to the large 13

O Murck & CampbeIl(1986)

Figure 3. The Merence between temperature values calculated with Sack and Gl~orso's (1991b) olivine- spinel geothermometer and real temperatures of experiments, plotted versus the temperature of experiment The data are from Murck and Campbell (1986) and Thy et al- (199 1).

number of independent ihermodynamic parameters required to describe energetics of compositionally

complex spinel solid-solution, as weii as a shortage of reiïable experimentai data, particularfy on cation

distribution in spinel end-members and binary joins. The two pnblerns result in the Sack and Ghiorso

(199 lb) calibration k i n g somewhat nontransparent to the outsider's examhaîion and analysis- Most

importanf however, it rnakes interpretation of other experimentai data used for the calibration more

diflïcult and ambiguous. For example, straighdonvard caiibraîïon requires the knowledge of cation

distribution, which is usually not available.

The next section is an attempt to obtain a new calibration of the Sack and Ghiorso (199lb)

thermodynamic mode1 that wouid be in better agreement with the hi&-temperature experimental data on

cation distribution and composition of coexisting olivine and spinel. This calibration is Iimited to

(Mg,~e?(Al ,~r ) l~ , spinels with only minor (Mg,~e'+)~e~-~ and ~ i ~ + - n c h components, making it

possible to signifïcantiy simpli@ the calibration procedure and hopefiilly obtain more reliable estimates of

the values of themodynamic parameters The restriction of the model to spinels low in ~ e ~ + and Ti is

teasonable since the Cr-Ai spinels that crystallize fiom basaltic melts at high temperatures are nonnaiiy Iow

in ~ e ~ ' and Ti (c 10 wt.% of Fe203; -= 5 wt.% of TQ).

Formulation of the Model

The formulation of the model requires making certain assumptions about cation distribution in the

spinels. AvailabIe e.upetunenta1 data on cation disuibution in Al-nch spinels has already k e n discussed.

This first model e.xplores the possibility of more ordered stnicture of FeA1204 compared with MgAi204 and

preferential o r d e ~ g of ~e~~ into teuahedtal position reIative to Mg (see above). Among other end-

mernbers, Cr-nch spiriels are accepted (e-g. O'Neill and Navrotsky, 1984) as completely normal, due to the

2+ large site-preference energy of c:* for the octahedral position. This Ieads to the first assumption of al1 Fe

residing in the tetrahedral position, for the spinels in question.

3+ The second major assumption deals with the Fe stmctural position For Cr and Al - nch spineis

3+ considered here, al1 Fe cations are assigneci to octahedral positions. This assumption is supporteci by

experimental studies of both synthetic (Robbh et al., 1971) and nahuai spinels (e-g. Osborne et al., 1981).

3+ Thus, by assuming Fe is octahedrally coordinated and F: tetrahedraily coordinated, the ordering is

restricted to Mg and Al. Thou& the assumptions about cation distribution are different fkom the mode1 of

Sack and Ghiorso, 1991). and, consequently, fewer independent variables are needed to describe

24 3+ compositional and ordering changes in Mg-Fe -&-Cr spinels with a minor Fe component, the needed

variables can be chosen analogous to Sack and Ghiorso (1991b):

Equation 4 '

X, = N,;

Equation 5

X, =No / 2 ;

Equation 6

where N. is the number of cations in a spinel formula based on 4 o.xygen atoms. The distribution of Ai t

between tetrahedral and octahedral positions is described by parameter Sz:

Equation 7

i where XM is the cation fraction of Al in sûucniral position i. so thac

Equation 8

and

Equation 9

i where NAl is the number of Ai cations in the siruchiral formula of spinel based on four oxygens atoms.

Defining cation hctions of other elernents in a simiIar rnanner, the following is me:

Equation 10

and

Equation I l

The additional condition of stoichiometry can be e.xpressed as:

Equation 12

The resulting expressions of site molar fractions in terms of these variables are derived fiorn Equation 4 - Equation 12 and shown in Table 1.

Table 1. E.upressions of site mole fractions in terms of chosen independent variables

Applying the same formalism as described in Thompson (1969), and Sack and G h i o ~ (199 lb), the molar

Gibbs fke energy G can be divïded into molar vibrational fk energy G* and a configurational km TS'~

a where S is the ideai moku conflgurational entropy. The molar vibrational fÎee energy G* can be

C+ 3+ Table 2. Molar £kee energy of Fe - Fe - Mg - Ai - Cr spinel solid-solution in terris of preferred thermodynamic parameters

represented as a second-degree Taylor expansion series in composition and ordering variables. The

coefficients of this series can be identified as more familiar thermodynarnic parameters by considering

cases of end-mernbers and binary joins in both the compositional and ordering sense. The definitions of

Taylor expansion coefficients in terms of preferred thermodynamic parameters are Summanzed in Table 3.

The resulting expression for molar fbx energy can be found in Table 2.

The chemicd potenriais of spinel end-members can be then dcuIated from the expression for G

and its partial derivatives as described in Sack and Ghiorso, (199 la,b). The site distribution of cations is

determineci by setting the denvative SG16S2 equal. to zero:

Caiibration of the Model

The step-by-step calibration for ali chosen linearly independent thermodynamic parameters is

described in this section, The calibration is obtained by minimuing errors of experimental data relative to

the respective theoretical mode1 with no additionai assumptions about their values. n i e parameters that are

obtaineci in eariier stages of calibration are used later to calcdate vaiues for other parameters.

The value of W, .y is obtained from thermodynamic analysis of the Petric and Jacob (1982)

experimental data on Cr-Al exchange between spinei (sp) and the oxide (A1203-Cr203) phase (ox):

Equation 15

FeAl,04 .P+ Cr O3 FeCr O 'P+ Ai2O3OX - t 2 4

The anaiysis of nonideality in the FeAltO4 - FeCr204 spine1 bin;irv is straightforward since it is assumed

that ali ~e'+ is restricted to the tetrahedrd position in the spinel structure. Though it seems as an

oversimplification, especially for FeAL204 - rich spinets, the final value for W1 is practicaliy the same as

19

in Sack and Ghiorso (199 lb), who considered FeAIzOs as partially disordered end-member. The resultant

expression for excess fkee energy of an Cr-Al exchange reaction uicludes the single symmetric reg&-

solution-type parameter, WIT. The value of the W13, can be directly obtained from the regression of Petrîc

and Jacob's (1982) data using the equation for fiee energy of the exchange reaction:

Equation 16

where and acm3 are activities of Al O3 and Cr20, in the oxide phase dcu la ted according to Z

Chatterjee at al. (1982) and AG^^^^^^ is the standard state free energy of the reaction, descnbed by

O

Equation 15. Simple Ihear regression of the e.primenta1 data produces values of A G FaNF, a.nci 'y,,

fkom Equation 16. Despite of the accepted assumptions made in the present study the value obtained for

the W13, parameter is 25535 J/@v and differs frorn the value accepted in Sack and Ghiorso (1991b) by less

These parameters are obtained from the cation distributions in MgAi Oq determineci by Peterson et 2

al. (199 1). The earlier NMR data by Wood sr al. (1986) on the same composition place tighter constraints

on the parameters, but were shown by subsequent NMR work by Millard ef al. (1992) to systematically

overestimate the degree of Mg-Al disorder. An even more ordered structure of M g K 0 4 is suggested by - the studies of Maekawa ef al. (1997) and Redfern et al. (1999). The data of Millard ef al. (1992) are in good

agreement with the resdts of Peterson et al. (199 1) in the 700-800°C interval, but show considerably more

ordered distribution at higher temperatures, This is attributed to a somewhat inadequate quenchhg

procedure of Millard et al. (1992) during which çamples were dropped into liquid nitrogen The resulting

gas coating developed around the samples preventing them fiom k i n g effectively quenched in hi&-

temperature runs (Milard, 1996, pers. CO-).

The cation distribution in pure MgA1204 is desrribed by the equatioa obtained nom the condition

of intemai equiiibrium:

Equation 17

BG/6S, =AG, + W,(l-2S2) +RTh Km = O

(ct ocL at fd

where Kw = @, Xa )/( X* XN 1-

The data of Peterson et aL (199 L) and the resuiiing least-squares regression Iine are shown in

Figure 4. Only the data f?om the runs in a temperature interval fiom 700- 10ûû°C, where samples with

initially contrasting cation dimibution showed statistically equivdent degree of disorder have been used

(total 10 mm) in the regression analysis. and two Iow-temperature nins have been left out,

In the absence of reliable data on cation distributions in &Cr spùiels, the value of W parameter 2

can be estimated £tom Mg-spinel-oxide Ai-Cr exchange data of Oka er al. (1 984) as described by Sack and

Ghiorso (199 lb). The reaction is analogous to one for the Fe-bearing system (Equation 15):

Equation 18

MgAlPd - 'P+ CrzO, OX= MgCr704 - " + Ai9OpX ..

The condition of chernical equilibrium for the exchange reaction is

Equation 19

'MgC1204 + 'A l203 - CLMgAKGl - 'C1203 = O*

The e.upressions for chernical potentiais p of the spinel end-members are obtained from the equation for

Gibbs molar fiee energy in Table 2 as described in Sack and Ghiorso (199 lab):

Figure 4. The negative values of configurational Gibbs energy of the Mg-Ai exchange reaction in pure MgAi204 plotted versus a function of the order parameter S2 using the data of Peterson et al. (1991) for the temperahue interval from 700°C - 1000°C- The line is the least square regression line with AG22 = -19596 *1200 and Wr;! = 1125&2900 J/gfw.

Equation 20

' kgCr204 = G + (1- X )6G/ôX2+ ( 1- XJ6G/6X3 - SSG/SS7 - X$G/5X5, and

2 - -

Equation 21

p ~ ~ m = G + (1- X 2 )6G/GX2 - X3GGBX3 + (1-S)6GBS7 - A - X56G/6X,.

The condition of interna1 equilibrium for the spinel phase is determineci by the equation;

Equation 22

These equations (Equation 19-Eqmtion 22), combineci nlth the e'ipressions for the activities of o.xide end-

rnernbers as defhed in Oka et al.(1984), produce:

Equation 23

The value of the standard-state volume effect of this exchange reaction. AVO WdAI>-1.

is 0.0358 Jhar (Oka et

al., 1984) and the ciifferences between partid molar excess volumes of m i - d g for Cr and aluminum end- USSP

rnernbers for spinel and oxide phase, 6V / 6X3 and GV? SX are calnilateci fkom the data of Oka C ~ Z O ~ '

et al. (1984) and Chatte jee et al. (1982) respectively. These values were not needed for the analysis of

Petric and Jacob (1982) data for the Fe-system, since their experirnents were performed at 1 bar. FoUowing O

Sack and Ghiorso (199 lb), S2 is assumed pressure-independent and AV .LfgcriAlbl'

SV""/ SX3, and 6vcrh1

6X are assumed to be pressure- and temperature-independent. Cr203

. . . Minunization of fiee energies of the exclunge reaction for the Oka et al. (1984) e-uperimental nuis

caidated according to Equation 23, whiie choosing the values of S sarisQing the condition of internai 2

1250°C I 1050°C A 7 w c

.-- Linear (1250°C)

Figure 5. Plot of QfwMbi , defined by Equation 24, versus SrX3. Data are fiom Oka et al. (1984). Only 1250°C data were used for the caiibration

[I Oka et-al (1 984) A Petric and Jacob (1 982) 1

Figure 6. Plot of Q* , defined by Equation 24, versus S2-X3. Data are fiom Petric and Jacob (1982) '=r(m-I

and Oka et al. (1984). Values for f etric and Jacob (1982) data were calculateci as -RT[ in( &/(1-X3) - %ln( O

amo3 / a~fl03 )]. The data points fonn linear trend with a dope of !4WlY and an intercept of YAG F ~ c ~ ( A I ) - ~ .

equilibrium (Equation 22), pmvides estimates for W and A G O ~ parameters. To demonstrate the redts of a'

calibration, the following fuoction Q*c<Nhl is introduced (se aiso Sack and Ghiorso, 199 lb, their equation

17):

Equation 24

From Equation 23 and Equation 24 it foElows:

Equation 25

A plot of Q* (Equation 24) ve- S2 - X3 (Figure 5) is Linear at 1250°C, in agreement with Equation Cr(MF1

25, but becomes progresively noniinear with respect to S2 - X3 r Lower temperatures. Due to this fact and

taking into account the difl[lculty experienced by Oka et al. (1984) in demonstrating equiiibrium in the

1050°C and 796°C e.uperimenta1 nuis, o d y data from the 1250°C experimentai nins were considered in the

f i d d b r a t i o n The value obtauied for W is - 4l kl'gfw, slightiy higher than the 40.585 kJ accepteci in =2'

O

Sack and Ghiom (199 Ib) . Assuming t h e same value of AG for these data as denved £iom Petric

and Jacobls (1982) data at 1 100°C. the value of A G O ~ is - 9 kJ. These values produce excellent agreement

between data from Oka et al. (1984) andl Peuic and Jacob (1982) as demonstrated on Figure 6 .

The value of the coefficient thaa modifies the X S tenn in the Taylor expansion series of 2 2

vibrational Gibbs free energy of spinei-solid-solution depends upon the degree of disorder in spinels with

2+ variable arnount of Fe . in this study the coefficient is represented as a combination of three independent

thermodynamic parameten: w~;w_-w~'' (see T'able 2). Cation distribution data of Della Giusîa et al. & -

(1996) for the nanual spinels with a composition close to the MgA1204 - F e q 0 4 join are used to assess the

3+ value of the coefficient, Since in a majority of their e.uperhents ail Fe is restricted to octahedral

3+ coordination, as determineci aom their experimental data, and because of Fe and Cr cation size similarity,

the values of WIT and W parametes are accepted to be equal to those vaiues of WIT and Wzr that have 22

already k e n obtauied. Thus Equation 14 can be rewritten as:

Equation 26

Data of Della Giusta et ai. (1996) give the average value of - 11 kl/gfiv for W -W--W (only the data 1 2 ,-

fiom the e.uperimental nrns performed in quenchable temperature interval of 650 - 920°C which produced

spinel with equilibrium cation distribution were used).

tet

The value of the parameter W can be obtained frorn compositions of coe.uisting Mg-Fe olivine

and spinel. Compositions of equilibrated Mg-Fe olivine and aluminate spinel wïth a %vide range of X t

produced in e.uperiments of Engi (1983) and Jarnieson and Roeder (1984) were used for calibration The

free energy change of the M~-F~'+ exchange reaction

can be calcdated for the case of aluminate spinel as ( see next section for more detailed e.uplanations):

Equation 28

OL OL 2+ where ( AH0* + 2WM ) is the parameter describing deviation ftom nonidedity in Mg-Fe olivine as

defined in Sack and Ghiorso (1989). The foliowing function is now introduced:

Equation 29

This function should be Iinear with respect to (1 - ZX,) for an assemblage of equilibrated olivine and - aluminate spinel at a particular temperature and have the slope of WC, as results fkom Equation 28.

Multiple regression of Engi (1983) and Iamieson and Roeder (1984) data produce a value of Wet equal to

5623 J/@k (Figure 7). This is compared to a value of 8368 J/@v as caiculated by Sack and Ghiorso

(199 lb) on the basis of their earlier analysis of Fe-Mg interactions betsveen olivine and spinel wïth a

nurnber of assumptions irnplying, among others, equai degree of disorder in spinel and hercpite (Sack and

Ghiorso, 1989). The derivation of wYL concludes the calibmtion procedure for (Mg,~e2+) (A.I,Cr) z Oa

3+ spinels. The e.utension of the mode1 for spinels containing small arnounts of Fe -rich component is

O

achieved by assignhg to W1 WZr, and AG 25 parameters the same values as for their respective Cr

a d o g s ( W13,, W ., and A G O ~ ) and asniming a vdue for Wyp equai to O. This is justified by assuming a a 3+ 3+

simiIar structural position of Cr and Fe in chornian spinels wiui low Fe content (see above) and by the

simiiar ionic radii for these cations. The other independent parameters representing vibrationai Gibbs

energies of completely nomial end-member spinels (G*,,, G*i G*3,. GL5) are not needed for the

description of nonideality, but make up a part of a standard state energy change of the reactions bebveen

spinel and another phase, The results of the compIete dibraiion are suinmarized in Table 4.

Engi (1 983)

A Jamieson and Roeder (1 984)

Figure 7, Plot of %idFe)-1

vs. 1 - 2 2 for 1300°C data of Jamieson and Roeder (1984) and 650-8W°C daia of

Engi (1983). Note sub-parailel character of the trends for these two data sets.

1 aole 4. Acceprea vaiues or uiermoaynamic parameters tinaepc;riut;riL parairiwxs uu I~UL IWG UGLLII~UULL UL

the iast column),

Parameter 1 Value, J 1 Definition for dependent parameters

-19596

Olivine-Spinel Geothermornetry

Theorv and CaIibration

The equilibnum composition of coexisting olivine and spinel is useful in construcîing

thermodynamic models for chromian spinels (O'Neill and Wall 1987; Sack and Ghiorso, 1991a,b; also see

above). Data on coexisting chromian spinel and olivine are used here to adjust the values of the

thermodynamic parameters Listed in Table 4.

The adjustment has been performed by m g the fkee energy of olivine-chroinian spinel

3 1

2+ Fe -Mg exchange ceaction using the compositions of coexisting phases from a set of crystallization

2t experixnents. The condition of chernical equilibrium for olivine-spinel Mg-Fe exchange (Equation 27) can

be presented in the following form:

Equation 30

j where p. is chexnical potentiai of component i in phase j and AG is the Gibbs fÎee energy change of the

oirp

zp sp reaction (Equation 27). Since phigA12@ - pFa4 = 6G6X , we can now use equations fiom Table 2 to

2

obtain:

The expression for the olivine part of Equation 30 is taken from Sack and Ghiorso (199 lb. their corrected

equation 19):

Equation 32

OL OL The t e m AHoX + 2WM descnbes nonideality in olivine solid-solution across the fomente-fwite join

(Sack and Ghiorso (199 lb)- Substituting Equation 3 1 and Equation 32 into Equation 30 and rearranging

produces:

Equation 33

O

where AG is the standard-sîate Gibbs fiee energy of the reaction (Equation 27) and is equal to G*= - w 01 01

G*, . + (GoFaiol - Ci0 ~ i a )R. The S2 ordering parameter is obtained by solving Eqyaiion 14. The

2+ 31 3+ compositional parameter X2 is calculated as Mgl(Mg + Fe ), and X5 as (Fe + 2TiY(A1+ Cr + Fe + 2Ti),

3+ since Ti has approximately the same effect on the exchange equilibria as Fe ( e g O'Neill and Wall, 1987;

Hill and Sa& 1987).

The experimental data for the minimization procedure were taken fiom Grove and Bryan (1983);

Tormey et al. (1 987); Roeder and Reynolds (1 99 1) (together with unpublished data on olivine); Sisson and

Grove (1993ab); Gaetani et aL (1994); Thy (1995a). Only data from the e-xperirnents which produceci a

sphel low in Fe O3 (clOwt%) and Ti0 (c5wt%) were used It is assumed that inadequacies arising fkom 2 2

the assumption about the equal effect of Cr. ~ e ~ + , and Ti (see above) on olivine-spinel equilibrium for

3+ spiriel containing these smaU amounts of Fe and Ti, as well as the presence of other trace eiements, does

not signîfïcantly change the values of free energy of the eschange reaction. During the minimization

tet procedure, the values of A G O ~ , W , and W were allowed to change, together with values of standard

IL

enthalpy (moo4 and enuopy (hsoOLp) change of the react io~ descxibed by Equation 27. Initiai

approximations for variable parameters were taken from Table 4. Initial values for AKO and ASO were olrp olrp

taken as -3672 J/@ and -8.129 J/K@ respectively, as determined from the data of Jamieson and Roeder

(1984) and Engi (1983). The results of the adjustment are iisted in Table 5. Equation 33 and JZquation 14,

together with the value of thermodynamic parameters Iisted in Table 5, constitute a potentially usefiil new

version of the olivine-spinel geothermometer. It turns out however, thaî AG+ h m Equation 33 is

relatively insensitive to temperature, resuiting in large errors in caldating temperature using composition

of equilibriurn olivine and spinel. The task of otivine-spinel geothermornetry is further explored in Chapter

3- Here it should be noted thaî the present formulation of the olivine-spinel geothermometer does not

Table 5 Values of thermodynamic parameters (independent parameters are rnarked as bold symbols) correcteci by regression using olivine-spinel experiments,

Parameter A G ,

A G O ~

w b'

w13*

*r; W

13' W

23'

wF w, wl.<l

Initial value*. .J 1 Final value, J -19596 -19596

9029 643 3

5623 6763

25535 25535

25535 25535

22129 25865 l

69665 65099

44054 44054

40 187 3873 1

O O

Initial value, .J 1 Final value, .J 9029 6433

11258 1 1258

27840 25870

2553 5 25535

25535 25535

22 129 25865

69665 65099

44054 M O 5 4

40 187 3873 1

* - same as in Table 4.

exhibit a systematic emor of calcdated temperature as it does with the Sack and Ghiorso (1991b) version

(Figure 3). This is important for ushg the spinei themodynamic mode1 in descnbing the spinel-melt

3+ equilibria, which is considered in Chapter 4 and Chapter 5, Though not e-xtending to Fe or Ti-nch spinets,

the suggested mode1 c m be used for rpinels with a low Fe 0, and Ti0 content such as chromian spinels f 2

found in most primitive lavas.

Chapter 3 OlivineSpinel Geothermometry

Introduction

Olivinespinel geothermometry is a ciassic petrologid application invoiving the equdibrium

between olivine and chromian spinei (Irvine. 1965; Jackson, L969; Roeder et al., 1979; O'Neill and Wall,

1987). Sack and Ghiorso (199 lb) were the first to point to the importance of cation distribution in spinel

stnicture for adequate description of olivine-spinei equiiibria. Shce cation distribution in the spinel

sûuctm is temperaturedependent, the accurate description of cation ordering in spinel can be crucial for

making olivine-spinel geothermometer applicable to rocks equilibrated at v e v different temperatures (e.g.

magmatic and metamorphic rocks), Altematively, it should be possible to use the avaiiable data on the

composition of coexisting olivine and spinel at different temperatures to check the vaiidity of the implied

cation distribution in spinel. Particulariy, this approach can be usehi to determine cation distribution in

( M g , ~ e ~ % L ~ 0 ~ and (Mg,~e~~[Al ,~r ]&, spinels, since available experimental data are inconclusive (see

cation distribution part from Chapter 2). Correct assumption about cation distribution is important in

constnictirig comprehensive thermodynamic mode1 for spinels.

The new version of olivine-spinel geothermometer developed in Chapter 2 is cali'brated using

hi&-temperature experimental da ta This chapter e-qlores the useiùiness of this calibration for

metamorphic assemblages, thus checking the asswnption about cation distribution in spinel. used in the

calibration. Another version of the olivine-spinel geothermometer is constructed, assuming the second

possible scheme for cation distribution in spinel, that of equal preference of Mg and ~ e ' + for tekiheàrai and

octahedrai positions (see Chapter 2 for more details). It is then calibrated using high-temperature

experimental data, and compared with available data from low-temperature metamorphic rocks in a simifar

way to the first version, This perrnits us to decide what cation distribution scheme is more probable from

the point of view of the M~-F~'+ exchange reaction between olivine and spinel, and what version of the

olivine-spinel geothermometer better describes the avaüable data on composition of coexisting olivine and

spinel fiom both magmatic and metamorphic rocks. 3 5

Olivine-Spinel Geothemometry at Metamorphic Conditions

The cornparison of avaiiable daia on olivine-spinel assemblages from metamorphic rocks with

results of calculations using the suggestd olivine-spinel geothermometer can provide an independent

check of the developed thermodynamic mode1 of chromian spinel. Figure 8 shows a set of isotherms

calculated d g Equation 33 for olivine F m and spinel with 5 mol.% of ~e*-rïch component. Aiso shown

is an "empincal isothem" mggested by Evans and Frost (1975) for the same conditions, based on thW

data on composition of coexisting olivine and chrotnian sphel from metamorphoseci peridotites. The

temperature of metamorphimi and mggested temperature of the "isotherm" is estimated as 700°C. Contrary

to this estimate, significantly lower temperature (- 400°C) can be calcuiated ushg Evans and Frost (1975)

data and Equation 33.

4 -

Figure 8. Plot of lnKd (Kd = ~ M S ) O ' ( X ~ ~ ~ ~ / ( X & ~ ( X F ? ~ " ' ) VS. Xj for olivine F m and spinel with X~0.05 (NF&&). Thin lines represent isotherms calculated with Equation 33 and values from Table 5. Values near the isotherms show temperature in OC. Evans and Frost (1975) 700°C "isothem" (EgtF,75) is show for comparison

Equation 33 was also used to calculate isotherms for olivine F k and duomian spinel with no ~ e *

or Ti. These conditions match the composition of olivine and cluomian spinel h m a metamorphosed

ordinary chondrite (Gobabeb) studied by Fudali and Noonan (1975). No estimate of the temperature of the

metamorphic event was made by the authors, but one would probably expect sirniiar quilibration

temperature for grains of chemically variable chromian spinel The grains are c k m i d y variable due to

much lower dinusion rates of Cr and Al in chromkm spinel in cornparison with the ciifhision rates of ~ e 2 f

and Mg (Fudali and Noonan, 1975). This results in the sarnple displaying M~-F~'+ equilîbrium, but not Cr-

Al equili'briurn. The calculated isotherms and analytical data are plotted on Figure 9. As in the case of

olivine and spinel fiom metamorphosed pendotites (Figure 8). the analyticai data deme a trend with a

steeper slope than predicted by suggested calibration of the olivine-spinel geotherrnometer. This

Figure 9. Plot of lnKd (Kd = ( ~ h d ~ ' ( ~ ~ ~ ~ ~ / ( x ~ g ) ' ( x ~ ~ ~ ~ ' ) VS- Xj for olivine F k and spinel with X ~ 0 . 0 . Thin lines represent isothem calculated with Equation 33 and values h m Table 5. Values near the isothem show temperature in OC. Open squares are data for olivine-spinel assemblages from metamorphosed meteorite Gobabeb (Fudali and Noonan, 1975).

inconsistency may indicate that the original asnunption about Fe2+and Mg cation distribution in spinel

structure is incorrect.

As wiu be shown in the next sec t io~ accepting equal degree of partitioning of Mg and ~e'+

between teûahedral and octahedral positions in spinel structure (îhe other alternative support& by crystal-

chernical data) results in a dramatic clifference in describing olivine-spinel equilibna in metamorphic rocks.

Alternative Thennodynamic Model for Chromian Spinels

Formulation

This section explores the second possible scenario of cation distribution in spinel structure,

assuming no partitionhg of Mg relative to ~ e ~ ' between tetrahedrd and octahedral sites:

Equation 34

This asnimption has k e n used in some themodyimic models and ~stal lographic studies (e-g.

Sack, 1982; O'Neill and Wall. 1987; Harrison er al., 1999). although it is not directiy supported by

independent crynal-chemical data, except for Fe304-MgFe0, sphels (Ne11 et aL, 1989). Since no

additional asnimptions about cation distribution are made at this stage. the mode1 can be applied for the

w hole range of (blg,Fe2'J(Al, ~ r , ~ e ~ ' ) r 0 ~ - ( M ~ F ~ ~ ~ ~ T ~ O ~ spinels- In order to describe composition and

order state of these spinels, tsvo variables should be added to X2, X3. XI. and S2 that were htroduced earlier

(Equation 4 - Equation 7):

Equation 35

Equation 36

Additional site-population constraints are:

Equation 37

Equation 38 O@. ou OQ oct oct x +x=2+== +XM +X 3- +XE +X = L ;

Ml3 Fe Cr

Equation 39

Expressions for cation fractions in spinel stnictural positions can noiv be obtained front Equation 4

- 7 and Equation 35 - 39. They are Bsted in Table 6.

Table 6. Expressions of site mole fractions in terms of chosen independent variables

Following the sarne procedure, as descnbed earlier, the motar ftee energy of solid-solution is

divided into vibrational and configurational parts (Equation 13), and molar b-ibrational free energy G* is

represented as a second-degree Taylor e.qansion senes in composition and ordering variables. The

resulting definitions of Taylor espansion coefficients in terms of preferred thermodynamic parameters are

summarized in Table 7. Table 7 contains two previously unused thennod'namic parameters, AGO~ and

AG°Fer which represent energ'. effects of the following exchange reactions benveen completely normal and

inverse spinel end-members (cations in octahedral position are in parenthesis):

Equation 4 1

~ e ~ ~ e ~ m ~ ~ + = ~ e ~ ' w g F ~ ~ T o ~ + ~e'+[ ~ e ~ ' ~ ' 1 0 4

2 t 3 t Table 7. Taylor expansion coefficients for Fe -Mg-Al-Cr-Fe - Ti spinels in tenns of preferred lhermodynamic paranieters

The other important ~ g - ~ e ' + exchange reactions involve completely normal spinet end-members:

Equation 42

~ e ' + [ ~ r ~ r ] 0 4 + Mg[AiAl]04 = Mg[CrCr]Oj + ~e'+[AI~1]04,

Equation 44

Fe2+lr;eX ~ e 7 0 , + Mg[ALAI]O, = M ~ F ~ ~ - F ~ ~ T O , + F~~+[AUI ]O~ .

The energy effects of these equations are referred to as AG0=, AG0=+ and AG"= respectively. Finaiiy, the

definitions for interaction parameten descnbing nonidulity of Mg and ~e ' - in teaahedral and octahedrai

position are identical; these parameters are named Wm.

An equation for molar free energy c m be formulateci using Table 6 and Table 7- It can then be

used to obtain expressions for condition of interna1 equilibrium and cliemical potentials of end-members as

described for the previous model.

Calibration and Ap~lication

Cornparing this mode1 with the previous one. fiom Chapter 2, it should be noted that the same

number of independent parameters are needed to describe the thermodynamics of (~g ,~e '+ ) (Cr,Al)fl~

spinels (see Table 3 and Table 7). Co~npositioiial and order variables are exaciiy the same in both cases.

Despite this, unambiguous calibration of tliis riiodel can not be achieved at the present time. One of the key

difficulties is the absence of data for two interaction parameters describing nonideality dong the Al-Cr join

(W1 3- and WI s3-). Available data on Al-Cr esclmge between spinel and oside phase (Petric and Jacob,

1982; Oka et al., 1984) could be sufficicnt if catiori distribution \vas known. The Merence with the

previous ( and Sack and Ghiorso, 199 lb) mode1 is iiiat, since equal degree of inversion is accepted for

MgAi204 and FeA1204, the two paranictcrs can iio longer be differentiated by comparing data from M g -

and Fe-bearing experiments. Thus, no atternpt will be made to comptetely calibrate the model. instead only

partial calibration necessary for an olivine-spinel geothermometer \vil1 be performed,

Equatîon 3 1 can be rewritten using Table 6 and Table 7:

Equation 45

'P Et O

%pu, -%-a = G*? - - G*,, + Wmf(l + & - 2X) - + X ~ A G O ~ + X~AGO: X ~ A G O ~ + XM AG +

X F= Y t ~ ~ O F e + RTln( X2 / 1 + X4 - X?

Equation 45, together with Equation 30 and Equation 32, produces the final equation for the fiee energy

effect of ~ g - ~ e ~ + exchange reaction between olivine and spinei:

Equation 46

kt tu

From Equation 46 it is clear. tint cation distribution in spinel plme (Xd and XFe ) must be

calcuIated to obtain a temperature estimates. in contrast with O'Neill and Waii (1987) model. Caldation

O O

of the cation distribution is not needed. hoivever. if values of AG and AG are all equal to zero, as

suggested by the structural studies of aluminate and femte spinels (O'Neill er al., 1992; Redfern er al..

1999). The vaiues of other parameters are obtained from published espentnental data by multiple hear

regession (values of parameters describing noriideality in olivine are taken from Sack and Ghiorso, 1989).

The composition of olivine and spinel. togetlier wîth the temperature of esperirnents from Ringivood

(1976); Akella et al. (1976): Waker et al- (1977); Delano (1980); Grove and B l a n (1983); Engi (1983);

Sack and Carmichael(1983); Murck and CampbeIl(1986); Mahood and Baker (1986); Takahashi (1986);

Tormey et al- (1987): Sack et al. (1987); Gee and Sack (1988); Grove and Juster (1989); Longhi and Pan

(1989); Usder and Glazner (1 989); Kennedy et al. (1990); Roeder and Rejnolds (199 1); Thy (199 1,

19950); Thy et al. (1991); Bartels and Grove (199 1); M e r and Grore (1992); Johnston and Draper

(1992); Draper and Johnston (1992); Sisson and Grove (1993a,b); Jurewicz et al. (1993); Snyder et al.

(1993); Gaetani et al., (1994); Wagner er al, (1995); Toplis and Carroll (1995) were used in the regression

analysis. The resulting values of parameters f?om Equation 46 are Listed in Table 8.

Table 8 Results of multiple regression

Figure 10. Plot of lnKd (Kd = o ( ~ ~ ' ( x ~ ~ ' ~ ~ / ( x ~ ~ ~ P ~ ~ ~ ~ ~ ~ ' ) vs- X for olivine Fow and spinel with X ~ 0 . 0 5 . Thin lines represent isotherms calculated with Equation 46 and values fiom Table 8. Values near the isotherms show temperature in O C . Evans and Frost (1975) 7û0°C "isotherm" (ElkF.75) is shown for cornparison.

Using Equation 46 and values from Table 8, isotherms were cakuiated for the same conditions as

StErr. 690

498

Parameter

AH0 - ASO

WFM

AGO

those on Figure 8 and Figure 9. They are plotted on Figure 10 and Figure 11.

Value. J -822

- 1

58 13

15713

Parameter

AGO 24

O

AG 25

AG"^ AGO

S t-Err. 790

0.6

285

3 73

Value, J 1653 9

19353

O

O

Figure 10 shows better agreement between the estimated rnetamorphic temperature of 700°C of

Evans and Frost and the isotherms caiculated with th& mode1 than the previous mode1 used in Figure 8. The

latter calibration aiso shows a consistent temperature for the range in composition of spinel in the Gobabeb

Figure 11. Plot of l a d (Kd = ( X ) " ' ( & ~ p / ~ ~ w ~ F C 2 ~ i ) vs. X for olivine Fos* and spinel with X~0.00. Thin Lines represent isotherms calculated wlth Equation 46 and values kom. Values near the isotherms show temperature in OC. Open squares are data for olivine-spinel assemblages frorn metamorphosecl meteonte Gobabeb (Fudali and Noonan. 1975).

chondritic meteonte. It is therefore recomrnended for use as olivine-spiriel geothermometxy. The similar

temperature for the metamorphoseci spinel in the terrestrial and meteontic spinel rnay reflect a temperature

below which diaision in the spinels W t s chernical reequdi'bration It has been reporte4 however, that

mechaniwlly induced recrystallization might lead to lower calculated temperatures (e.g. Jaroslow er al.,

1996). An e.xampIe of the application of the olivine-spinel geothermometer to magmatic rocks is describeci

in Chapter 6.

Chapter 4 Chromian Spinel - Melt Equilibria

The composition of chromian spinel is sensitive to the conditions of its crysîallization and is

recognized as a vaiuable petrologic indicator (Trvine, 1965, 1967; Dick and Builen, 1984; Aüan et al.,

1988). This chapter is devoted to the task of descnbing major-element distribution between basaltic (or

ultraInaîïc) melt and crystallizing chromian spinel. This task can be defined more precisely as a

construction of a set of equations to predict the composition of chromian spinel coezcisting in eqriilibrium

with a basaitic melt at conditions of interest, The existing models are discussed and the general approach to

deai with the probIem is formulateci, Then the set of equations is calibrated using preferred published

experimental data on coexisting chroxnian sphels and basaitic melts FinaIly, the duornian-spineL

composition is caiculated for the e~~erimental data uMg the suggested equations and the accuracy of the

mode1 is assessed by comparing calculated chrornian-spinel composition with the measured composition

Existing models

The first two experimental studies devoted specifically to the task of invedgating rile stability and

chemical variation of chromian spinel in silicate melts of basaltic composition were by Hill and Roeder

(1974) and Irvine (1977). They describeci the major trends of chromian-spinel compositional variation

during crystallization of a basaltic melt at atmospheric pressure but did not suggest any quantitative

relationship betiveen the composition of melt and chromian spinel.

The first equations comecthg chemical parameters of coe'iisting silicate melt and chrornian spinel

were based on the resuits of continuhg experimental work by P.L. Roeder in Canada and C. Maurel and P.

Maurel in France, These first equations were simple empirical relations not based on thermodpamic

principles or the crystal structure of chromian spinel. Nevertheles, they successfully described the limited

experimental data available at that t h e . As an e.uample, the following exnpirical equation with slightly

different constants A and B \vas independently suggested by Roeder (1982) and Maurel and Maurel (1982)

2+ 34-

in order to relate the Fe /Fe in basaltic melt and chromian spinel:

Equation 47

3+ 3+ Iog( ~ e % e ) = A + B 1ok(~e2+/~e )'9

As demonstrateci by Ariskin and NikoIayev (1 Wh), better agreement with currently availabIe eqerimentai

data can be achieved by including in this equation several additional parameters, However, the approach of

Anskin and Nikolayev is somewhat empind making it difficult to interpret the equation

One of the other, and probably the best of the equations suggested by C. Mamel and Iater used in

petrologic modeling (Auge, 1987) is:

Equation 48

At a constant temperature this equation has the form of the original olivine-spinel geothermometer ( Lrvine, 2+

1965). Allan et al. (1988) cornbined the olivine-spinel geothemometer and e.uperimenta1 data on Fe -Mg

distribution berneen olivine and spinel to consuuct a simiLat equation AlIan er al. (1988) assumed that

spinel behaves as an ideai reciprocal solution and diat silicate Iiquids are ideal with respect to mi.xing of

2+ Fe and Mg. The same result can also be acliicved using a more advanced ihermodynamic model for

chromian spinels (e-g. Sack and Ghiorso, 1989). but with several additional assumptions. In the case of the

Sack and Ghiorso (199 lb) rnodel, thesc asniinptions would be: F~'+/M~ is identical in both cation positions 2+

in the spinel structure. the excess Gibbs free energy of Fe and Mg mising in octahedral and tetrahedrai 3-

positions and parameters describing ordering in respective Mg- and ~ e - end-rnembers are the same, and

the effect of nonideaiity due to Mg - ~ e ' + subsiitution in rnelt and spinel cancels out.

Maurel and Maurel (1982) suggested. based on theü experiments. that the %O3 content of

c h m i a n spinel can be caiculated as a simple hinction of Al O3 content oicoexisting silicate melt: 2

Equation 49

242 AlZ03sp = 0.035 NO3" . -

Even through the authors denionsuritcd its good agreement witii some independently obtained

experimental data, it is now clear that tiie equation generally gives unsatisfactory resuits, On Figure 12,

data fiom the Roeder and Reynolds (1 99 1) expenments (1300°C and 1 bar) are compared with the values

calculated with this equation The equation suggested by Mawel and Maurel (1982) predicts significant

Figure 12, The reiationship of A1203 content in ciuomian spinei and rnelt in Roeder and Reynolds (1991) 1300°C lbar experhents at to@Oz < -6 ( points) as compared to the reiationship suggested by Maurel and Maurel (1982) (line, see Equation 49).

increase of A1 0 content of chromian spinel as AI O3 content of coexisting melt increases h m below 12 2 3 2

\VI.% to almost 17 W.% (he on Figure 12). Korvever, the actuai increase in A120, content of chromian

spinel from Roeder and Reynolds (199 1) e~pxiiiients is barely noticeable. Later improvement of the 3+

equation to account for the influence of Fe on the calculated values of Alq03 content of chromian spinel - (Maurel and Maurel, 1983) does not irnprove the results. In facc since cluornian spinel contains not only

3+ Al, Cr, and Fe , but also divalent cations, the suggested equation couid not be reasonably successful if

concentration of Mg) or Fe0 in melt varies independently of %O3 content.

The first computer program developed to caiculate chromian-spinel composition in equilibnum

with silicate melt (EQUIL) was introduced by Nielsen and Dungan (1983). The equations used in the

program were calibrated using a iimited number of published experimentai data on the composition of

coexisting chromian spinel and melî. and their own experiments for synthetic murtureSmurtures Most of the

equations employed in EQUIL are standard temperature-dependent expressions for simple distribution

coefficients of a cation or oxide (Nielsen and Dungan, 1983) and are not suitable for the task for the same

reasons why the Mawel's y03 equation is ineffective. The authors of EQUlL obtained 2 different

3- 3+ caiibrations of some of their equations for Fe -free and Fe -karing spinels and attributed the ciifference to

nonideal cation miàng in spinel solid-solution (Nielsen and Dungan, 1983). This is probably one of the

reasons for the inability of another, more recent program, SPINMELT (Arkkin and NikoIaev, 1996b), to

deal with chromian spinels relaiively ennched in Al or Fe. Some of the SPINMELT equations are not

thermodynarnicaiiy based and the others use an assumption of ideality of chromian-spinel-solid-solution

and fixed cation disiribution between tiie structurai positions. This results in the equations king too

shplistic to rnodel the crystallization of compositionaiiy diverse chromian spinel fiom relatively uniform

b d t i c melt 3+

The development of a comprehensive thermodynamic rnodel for ('g,FeZ+) (Cr.Al,Fe )204 - (Mg

~e'+) Ti0 spinels (Sack and Ghiorso. 199 Lb) \vas a necessary step toward the development of models free 2 4

of the limitations of the EQUIL and SPINMELT programs. The Sack and Ghiorso model made it possible

to account for the energetic effect of nonideal cation miuing, temperature and compositionally dependent

ordering in spinel structure on the calculated values of chernical potentials of chromian-spinel end-

members.

The Sack and Ghiorso model has been used in at Ieast two programs, including their own MELTS

program, which calculate the composition of cluoinite crystallizing frorri silicate melt at h o w n

temperature, pressure and o.qgen fhgacity (Neilsen el al., 1994; Ghiorso and Sack, 1995). A set of

equations describing reactions, where a ciuorruan-spinel end-member is a cornponent, is used in rhe

programs. Consequently, the amount of ~ 3 ' in melt is required for the calculations. Both programs assume

that C? content is qua1 to the total Cr concentration in melt It has been shown, however. that both C?

and &+are present in basaitic melt at petrologically important conditions ( e g Murck and Campbell, 1986;

Barnes, 1986; Roeder and Reynolds, 199 1). The inconsistency between esperimentai data and calculated

should therefore be evident for the condition of varying eOz. This is supportecl by comparing some

49

by 'M ELTS'

O Experimental

Figure 13. Plot of Cr content in melt saturateci wilh chrornian spinel vs. 10gR3~as obtained by expenment and by 'MELTS' program. Data are for Roeder and Reynolds 40 1 experiments performed at 1 bar and -1300°C. See text for details of calculations.

by 'MELTS'

fl Experimental

Figure 14. Plot of Crfi content in chromian spinel in experirnents and calculateci with the 'MELTS' program vs. logflDz. Data are the same as on Figure 13.

of the expeximentai data on the compositions of coesisting melt and chromian spinel reported by Roeder

and Reynolds (199 1) and the chromian-spinel composition caiculated wi th 'MELTS' (v. l.O.O.aIpha, Java

Version 1.1.5). The results of the calculations and the expnmental values are shown on Figure 13 and

Figure 14. Good agreement between calculated and observeci values of Cr content in both melt and

chmmian spinel is achieved at log£Q = -8 to -10. At these conditions, both C? and C? are present in the

melf contrary to the assumption used by MELTS. At higher fDz the Cr content of the melt decreases, due

to decreasing amount of c?. and ody partiaily due to increase of ~ e ' + . The f m cause is not accounted for

by MELTS, thus resuieùtg in progressive underestimating of Cr content in chromian spinel (Figure 14), a1

the expense of higher than expected FC~' content. On the other hand. at Io\ver f02, only a very limited

increase of Cr content in the melt can be aciiieved as a result of lowering of ~ e ~ + . and the increase is mainly

due to increasing C? (Figure 13).

The following two major requirements should be satisfied to successfiiIly describe major element

distribution between equilibrated chroinian spinel and rnelt:

1). The equations used to describe the clements partitioning sliodd be thermodynamically based

and account for nonideai mi'ring in spinel solid-solution.

2). Provision should be made to account for varying o'udation state of Cr in the melt and different

behavior of various Cr species in the melt relative to crystdlizing chromian spinel, if the Cr content in melt

is used.

As will be shcrwn in the following section, using Cr content in riieit to calculate chromian-spinel

composition is not an absolute requireineiit. if ai assumption about stability of chromian spinel at specified

conditions is made. This may be useîÙI, since reliable analytical data on die low Cr content (< 5 0 0 ppm) in

melt equilibraîed with chromian spinel is often not available.

Formulation

The task of calculating the composition of a phase given the coniposition of another equilibriurn

phase is dtim=ltely achieved by solving a system of simultaneous equations describing element distribution

between the phases (e.g Ghiorso and Sack. 1995; A r i s k i n and Nikolaev, 1996b). For the chromian spinel T+ 3+ 3+

the foUowing six components are usually considered: Mg, Fe , Al, Fe , Ti and Cr . However, due to

stoichiometry restrictions, only four compositionai parameters are linearly independent (e-g Sack 1982).

Consequently, on& four equations are necessaxy to calculate chromian-spinel composition fiom melt

composition at known temperature, pressure. and logf02. To satim the first major requirement for

constructing the model, a thennodynamic model of chromian spinel should be applied to calculate

properties of the spinel phase. The mode1 developed in Chapter 2 is used for that purpose. This model does

not apply to spinels containing signifiant amount of Ti. Consequently, the following equauons do not

include thermodynamic parameters of Ti-bearing end-members. and c m not be used to calculate Ti content

of the spinel. Thus the resulting rnodel is only intended for Ti-poor spinel and it is assumed that a minor

amount of Ti does not change its thermodynamic properties significantly, In the absence of Ti as a

compositional parameter, only 3 equations need to be solved. Since published experirnental data on

equifibrated chromian spinel and melt usually contain unreliable values of low Cr content in melk it was

decided not to use Cr content in melt for the msk. The following three reactions are used in calibrahg the

Equation 50

FeAl 2 O 4 'p=Fe0m+2A10~5m

Equation 5 1

Fe O q + 2 A 1 0 , ~ 5 =FeAl O '+2FeO15* 3 4 2 4

Equation 52

It is assuma for the purposes of this study. tliat Ihe chemicai potential of a spgies in melt pi, can be

O

expressed via the chernical potential of pure phase. F- , and simple molar fraction of the component X. I

using the foiiowing semiempirical rehtion:

where c is a constant If the activity of a component in the melt is a linear h c t i o n of its molar fiaction, this

constant c shouid approach 1.

The values of the diEerence of chernical potentiai of chromian-spinel end-member and the

vibrational Gibbs energy of the end-meinber wïth a completeiy ordered cation distribution p. - G , can be t t

calculated using the model formulated in Chapter 2. The equations describing chernical equilibrium for

crystallization reactions (Equation 50- Equation 52) be written using Equation 53 as follo~vs:

Equation 54

%- Gy ' = AH0 hc - TASO h~ + c FC R m F d + 2cURTlnX aol-5

Equation 55

Equation 56

In these equations G. 's are defmed according to the scheme of Sack and Ghiorso (199 lb). and sp, I

hc. and mt represent spinel (MgA?04). hercynite (FeAI 04), and magnetire (Fe304) end-members 2

Solution of these simultaneous equations (Equation 54- Equation 56, together with the Quation

14 describing Al ordering in spinei structure), provides the composition of cluomian spinel coe.xisting with

a silicate melt of known composition at a particular temperature, pressure. and logf02.

Data for Calibration and Calibration Results

Data Base

An attempt to calibrate the suggested equations has been performed using Roeder and Reynolds

54

(199 1) data (lbar) with low Fez03 (c 10 W.%) content in the spinel phase. These data are considered as

both reiiable and spannhg a significant range of reciprocal Al-Cr variations of chromian spinel. At the

same tirne, the range of melt composition is relatively narrow. Roeder and Reynolds (L99 1) data are

therefore ideal for testing the constnicted equations. which are b d on a very simple mode1 for the melt

components (Equation 53). The spinels produced in Roeder and Reynolds experhents are also Iow in Ti0- - ( C 3.5 wt% of Ti02 ). Low TiOz and Fe 0, content is required in order to use the thermodynamic model

2

formulated in the Chapter 2 to calculate the values of p- - G . fiom Equation 54- Equation 56. The I t

calibration is also limited to 1 bar- data since the model applies oniy to these conditions (e-g. no mechanisrn

is included to account for pressure influence on cation distribution in the !spi.net structure).

Results of Remession Analvsis

Multiple linear regression of the selected experimental data (see the previous section for the

database description) was performed to obtain coefficients in Equation 51 - Equation 56. The values of p. - t

G. were calculated using the thennodynarnic mode1 for chrornian spinel formuiated in Chapter 2, and z

Fe 03, and Fe0 content in melt was caiculated aiter Kress and Camiichael(199 1). n i e renilts of the 2

regression analysis are listed in Table 9- The regression can be performed with most available spreadsheet

programs; an e.uample of the calibration procedure for Equation 5-1 using Regression Data Anaiysis TooI of

Microsoft Exce197 is given in Appendiv D.

Table 9. Results of multiple Iinear regression

Parameter 1 valuew 1 Std. Error 1 Parameter 1 Value 1 Std. Error

AH0 988 13 6550 AS" 33 5 hc hc

0.953 0.05 0.999 0.05 '=Al

AH0 13638 35087 A s 0 31.335 23 hant hmt

=AiFe L.1535 0.11 AH0 7925 2065

h=p

AS",, 3 -3 1.3 c MeFe 0.8 16 0.0 1

-- values of enthalpies are in J/mol; for entropies - UmoK

The fit of the experimental data using these coefficients and equations gives a squared value of the

correlation coefficient higher than 0.9. Regression residuals do not e.uhibit any obvious systematic

variations with composition of meIt or conditions of experiments (there is. however, a systematic variation

with the composition (Cr/(Cr+Al)) of chmrnian spinel as will be discussed later). The vaIues for R, cu,

c and c in Table 9 are ail close to 1, suggesting that the activity of a cornponent in the melt is a F = A ~ ~ g ~ c

30 40 50

Cr203 in experimental chromite

Figure 15. Plot of Cr O3 content in esperimentai (selected Roeder and Re-ynolds (199 1) runs) vs. calculated 2

spinel.

linear function of its molar fraction However. large residuals for the F2- 1 runs of Roeder and Reynolds

(199 1) suggest potential problems in calculathg the composition of spinels relatively e ~ c h e d in Ai. To

fiirther explore this and check the accuracy of the calibration, the spinel composition was caiculated using

the melt composition from Roeder and Reynolds (1991) for the conditions of their experiments by solving

Equation 14, Equation 54 - Equation 56 and using values from Table 5 and Table 9. The e-uplicit analytical

solution of any of these equations in terms of wt.% of oxides is not available, due to the complex

dependence of chernical potentials of spinel end-members on composition; instead, a set of nonlinear

algebraic equations should be solved using a numeric method. Some of the program used to perform this

and other simiiar tasks include Optunizer from Corel Quattro Pro, or SoIver nom Microsoft Excel. An

example of using Optimizer to solve a similar set of equations (Equation 68 - Equation 71) is given in

Appendix E. Figure 15 compares the resui ts of the calculation nith the composition of spinel reported in

Roeder and Reynolds (199 1). From this corn parison it is clear. that while good agreement between

experimentai and cainilated values is observeci for high-Cr spinels. calnilateci Crz03 content is

systematically higher than e.uperimentally determined for spinels relatively poor in Cr. This dependence of

regression residuals on Cr203 content in chromite (Figure 15) is possible. since chemicai potentials of

chromïm-spinel end-members used in die regression anaiysis are not Iinear hc t i ons of an oxide content in

chromite. The fact that the Merence between calculated and e.\perimentnlly determined values of Cr-O3 - content in spinels does not reveai obvious correlation with any particular element in the coe.uisting melt

suggests that the problem Iies in the tliermodynainic mode1 for spinels. A Iugher degree of nonideality in

chrornian spinels dong Cr-Ai join is required to correct the problem. Increased nonideality codd be the

result of an assumption that disorder in the spinels is not Iimited to Al and Mg, but dso includes ~e'*. The

quai degree of partitionhg of Mg and ~e '+ between positions in spinel structure leads to a more plausible

explanation of ~ g - ~ e " distribution data beliveen olivine and chromian spinel at dinerent temperatures

Additionally, the Oka er al. (1984) data. used for the calibration were collected at high pressure. Accordhg

to Pavese et al. (1 999b), increasing pressure makes MgAllOj more 'normal', which, in tum, shodd

decrease nonideality dong the MgAI2O4- MgCr204 join, resulting in underestimating of the respective

parameter wz), if the pressure effect is negiected. Thus, more data. especially on cation ordering (or

degree of inversion of spinel structure) dong die AI-Cr spinel join, are rcquired to better restrict the values

of parameters in the thermodynamic mode1 for cluomian spinel. Therefore a better calibration of the

thennodynamic mode1 for chromian-spinel solid-solutions and equations analogous to Equation 53 - Equation 56 used to predict the composition of clirornian spinet crystallizing fiom the melt of known

composition has to wait for the e-upen~nental detemination of cation 0 r d e ~ g dong the MgA120a-

MgCïz04 join.

Preferred Empirical Model

Based on the results described in the previous section, it is concluded that better calibration of an

advanced thermodynamic mudel of chornian spinel such as Sack and Ghiorso (199 1b) model, should be

achieved before it can be successfiilly applied to relate the composition of chrornian spinel and coexisting

basaltic melt Therefore, a more empincal approach will be used where the spinel is arbitrarily assumed to

be normal at aii temperatures and a regression of experimental results will give a series of constants which

have little to do with actuai themodynamic parameters.

The following arguments can be considered before formdating such an approach. Shce it is

chornian spinel, and not basaltic melt whose composition exhibits great variabiiiiy in e.xperimenta1 runs

and naîurai lavas, accounting for nonideality of chromian-spinel solid-solution appears to be much more

important than t r C î ~ g nonideality of species mising in the melt. Thennod-vnamic properties of chromian-

spinel solid-solution are greatly affected by cation distribution between structural positions. However, as

has k e n shown in a nurnber of cases, a simple model can be used to successfully describe the behavior of

3+ I+ systems where the structure is not very weIl understood. Some esamples are calcuiating of Fe /Fe in

natural silicate melts (Sack et ai., 1980) md tiie description of minerai-iiielt equilibna (Ghiorso and Sack,

1995). In both cases a wetric-regular-solution mode1 with no constxaints for the values of interaction

parameters is used to account for nonidcality of cornpositionaiiy variabte silicate melt with a Iargely

unknown structure,

The approach adopted here therefore is to use regular-solution formalism for chrornian spinels

with Exed cation distribution to relate its composition to the composition of coexisting basaitic melt. The

chernical potentials of melt components, on tiie other hand. are caiculated by assuming direct

proportionality of their activity to concentxation. The resultuig espressioris describing chernical equilibria

will be nodinear algebraic equations. wliicli include several tenns modieing chromian-spinel

compositional parameters. This is in conuast witii the equations of Ariskin and Nikolaev (1996b). which

include several melt, rather than chromian-spinel compositional parameters- The approach adopted in the

present study is considered more prornising because of the sensitivity of chromian-spinel composition to

variations of melt composition. Another important merence is in using Cr content in melt to calculate the

composition of chromian spinel. The Cr content in melt is used in equations by ArisIan and Niolaev

(1996b), thus limiting experirnentai data used for the model calibration, since the Cr content of giasses is

often not determineci or the rneasured vaIues are unreliable.

The nec- eicpressions for chernical po tentials of spinei end-inembers can be obtained

following the p r d u r e described in Chapter 2. The independent variables needed to describe

3+ cornpositional changes in M ~ - F ~ ~ * - A ~ - c ~ - T ~ - F ~ spinels (chosen analogous to Saçk and Ghiorso, 1991b)

and the assumeci cation distribution are surnrnarized in TabIe 10. In con trast to the model describeci in

Chapter 2, no ordenng variables are needed to describe cation dismbution in spinel structure (fked

distribution is assumeci). However. an additional compositional variable. X4. is introduced to extend the

model to Ti - beaxing spinels. Following Tliompson (1969)- molarvibra~ional fiee energy G* is represented

as a seconddegree Taylor espansion series in selccted independent composition variables.

TabIe 10, Independent compositional variables and assumeci cation distribution in M~-F~-AI -c~ -T~-F~* spinels

. N. is the number of cation i in spinel formula based on 4 oxygen atoms

These Taylor expansion coefficients c m be espressed via more familiar qmmetrical regular solution

model interaction parameters and reciprocal eschange reaction energies (e.g. O'Neill and Navrotsky, 1984)

as shown in TabIe 1 1. Baseci on TabIe 10, the expression for configurational entropy of spinel solid-

ie solution, S , is defineci as:

Equation 57

s ' = - R [ ~ x ~ I o ~ ( x ~ ) + x 4 log(X4/2)+2X510g(X5)+2( l -X3-X4-X5) log ( 1 - X 3 - X 4 - X 5 )

-t-(~+x,-X2)1og(( 1+&-X2)!( 1+JC))+X*log(Xd(I +&)II

Note, that the coefficient, modifying log / 2) term, is X4, rather then 2X4, reflecting the coupled nature

Table I l . Taylor expansion coefficients for F~''-M~-A~-c~-F~*-T~ spinels in temis of preferred thermodynamic parameters.

3-4- of 2(Cr,ALTï,Fe ) = (M~,F;)T~ substitution. The molar Gibbs free energy of spinel solid-solution, G, is

obtained h m its vibrational and configurational romponents. G = G* - T S ~ ~ , and is used to derive the

expressions of chernical potentials of spinel end-rnernbers. The finai e.upressions of chernical potentials of

selected spinel end-members are:

Equation 58

% = G + (1- XJ6G/6X7 - X36GBX3 - X$GK4 - X56G/6X5 .. -

Equation 59

II mt = G - X2G/6X7 - X 3 6 G 5 - X46G/X4 + ( i - X5)6G/6X5 -

Equation 60

% = G - X?G/SX - X3SGfiX3 + (1 - X4)GGK4 - X56G/6Xs 2

To constmct the equations reIating composition of chromian spinel and coe'ùsting basaltic liquid,

the following crystallization reactions are considered:

Equation 6 1

M g Y O P = MgOrn + 2AIO ' - 1.5

Equation 62

Fe O ' = F a m + 2Fe01.," 3 J

Equation 63

Ushg Equation 53 (and accepting c=l), the condition of chernicd equilibnwn for the these crystallization

mictions can be written as:

Equation 64

- G: = AH0 - TASO -+ (p - L)AVO + RTW( + 2RTLnX;Uol~S % - SP P =P h W

Equation 65

Equation 66

4 - G~,- = AH0 - TASO + @ - l)AvO + 2RTlnXFc0 + RTLnXri02 P rp =P

Combining Equation 58 - 60 and Equation 61 - 67 produces three equations used to fit selected

experimental data on compositions of coexisting chrornian spinel and basaltic melt

The fourth equation needed to calculate spinel composition, given the composition of coexisting

melt is similar to what is referred to as 'spinelmetex'. used to relate Mg/(Mg +FA of coexisting spinel and

basaltic melt (Man et al., 1988. Man. 1992):

Equation 67

~ ~ ~ X ~ I X ~ ~ ~ = ~ ~ X ~ I ~ L + X ~ - X ~ ~ ~ - A S ~ - ~ ~ ~ - A ~ ~ - A ~ ~ - A ~ ~ ~ ~ R T -

The unknown coefficients in the four equations were obtained by multiple linear regession of

most of the published experimental data. The procedure was simiIar to tint used to calibrate Equation 54.

Data used for calibration cover the maxiinuiri coinpositional range of spiiiel coe'risting with what is broady

defined as "basaltic" melt. Some expcninental studies were excluded from the resulting database.

Particuiarly, data Born Murck and Campbell (1986) and some nins fiom Barnes (1986) were not included

since the experimental technique used in these esperïments. when the initial mixtures were seeded with

chromian-spinei crystals, has proven unsuitable for studying chromian spinei - melt equilibrium (h ine,

1977; Roeder and Reynolds, 199 1). Some other eqerimental data, indicating Little change in composition

of chromian spinel coexisting with melt at contrasting temperature my et ai-, 199 1; Thy, 1995b) were also

excluded Thus, experimental data from the following studies were used: Roeder and Reynolds (199 1),

Akella et al, (1976), Grove and Bryan, (1983), Barnes, (1986). Fisk and Bence (1980), Sack et al- (1987).

Toplis and Carroll, (1995), Bartels et al. (199 l), Mukhopadhyay (199 1). Draper and Johnston (1992),

Capobianco and Drake (1990). Thy (1995a)- No additional constraints separating data within seIected

sources were applied. The calibration (regression analysis) \vas performed to achieve closest agreement of

each equation with experinental data, rather ttun to obtain a set of consistent thermodynamic parameters,

As a resuit, each equation contains its unique coefficients, not comected to coefficients fiom other

equations. The final equations with caicdated coefficients are (some statisucaiiy insignincant terms are

ornitteci) :

Equation 68 2 7

in ( Xw XMoi-5 ) = in ( X$l+X4) (1-x3-X;XI)) - 0.390867 + [ - 6615.07251 - O. 064161 (pl)

Equation 69 2

h(X FCO X ~c01.s )=h((l+Xd-X,)/(i+X4)~5Z)+6.905697-[-23780.78-0.11618/(p-l) -

+ (7105.519586X3 -9188.804319X4+6282.248784( 1 - X 3 - X -X5)) ( 1 - X 5 )

+ 805.2- 43 7161 X X - ( 3544.856034 X, + M28.04993 X4 ) ( 1 - X3 - X4 - X5 ) - ( 8455-38893 3 4

X3 + 8841.717738 X - 4818.62107( 1 - XS)) X ] IT -

Equation 70 2

in (XF4 XTio2)=h((( 1 + X - X ? ) / ( 1 + X4)?x4) + 3.325468 i [ - 13410-39- 0.115429 (p-1) -

Equation 71

in ( Xw I XFcO ) = 1" ( X2 1 (1 + X4 - XZ)) - 0.404074 - ( 537.987252 - 2207.833813 X3 - 2903.433014 X4

-2914.212902 X5 ) /T

These equations relate melt compositional parameters from the left side to chromite compositional

parameters fiom the right side of the equations. Many more digits are sliown for the coefficients than are

statistically signifïcant but are included to avoid round-off errors. To calculate chromite composition given

melt composition, pressure, and temperature ( ~ e ~ / ~ e ' ~ can be calcdated using equation of Krw and

Carmichad, 1991), this set of equations should be solved in tenus of chromite compositional variables X2,

X3, JC4, and X5- A possible procedure and an example (Corel WordPerfectS Suite Quattro Pro spreadsheet)

is given in Appendiv E. The solution of the simultaneous equations successNly predicts the composition

of spinel coexisting with the meit of known composition for most e-xperimental data used for dibration.

Figure 16 illustrates this by using data of Roeder and Reynolds (nins perfonned at oxidized conditions are

not included to aiiow direct cornparison with Figure 15).

3 0 4 0 50

Cr,O, in experimental chromite

Figure 16. Hot of Cr& content in experimental vs. calcuiated spinel. Spinel composition is catculated using preferred empirical mode1 (Equation 68 - Equation 7 1).

64

Chapter 5 The Distribution of Cr between Basalt ic Melt and

Chromian Spinef

Previous Work

The solubility of Cr in silicate melts sarurated with chromian spinel has b e n the subject of several

studies (Hill and Roeder, 1974; f i e . 1977; Maure1 and Maurel, 1982; Barnes, 19%; Murck and

Caf~~pbeiI, 1986; Roeder and Reynolds. 199 1) that show an obvious positive correlation behveen Cr 2 O3

content in spinel and melt phase. The same strong correlation can be seen in the results of other

experirnental studies, tvhere the Cr in quenched giass has been carefuily analyzed (e.g, Schreiber, 1976;

Delano, 1980; Sack et al., 1987).

The fïrst quantitative e . ~ p l e s of Cr distribution between basaltic melt and chromian spinel were

published by Barnes (1986), and Murck and Campbell (1986). These studies adopted a relatively simple

approach to describe the influence of chromian-spinel composition on the equilibna, implying that aU

spineis present in theü nrns have completely normal structure and that rni.xing within each c~ystafIographic

position is ideal. The formulation of Bames (1986) made no provision for the influence of divalent cations

on the equilibria, so that spinel t a s treated as an ideai oxide phase. One of the major probIems is the -)i 3+ 3+

di8idty in aaounting for the presence of both CF and Cr in the melt, with only Cr present in

chromian spinel (Schreiber, 1976). Barnes (1986) trîed to caicuiate the valence state of Cr species in 34-

basaitic melt as a funciion of temperature and oq-gen fugacity (fD-). Banies assurned that Cr is the only - Cr species in the meIt at oxygen fiigacities of the e~mpolated quartz-fayalite-magnetite (QFM) buffer or

3+ higher, and, he used a theoreticai coefficient of 0.25 to describe the depetidence of log (Cr KrZ) on

logf0-. Roeder and Reynolds (199 i) showed îhat avaihble experimentai data could be better fit accepting .. 2+ 3+

that both Cr and Cr are present at oxygen fugaci ties of the QFM buEcr. They calibrated an equation 2+ 3+

which represented Cr /Cr in the melt at l30O0C for one basaitic composition as a function of foi

Poustovetov and Roeder (1994) used a similar approach to describe the Cr distribution between chromian

spinel and silicate melt for a range of temperatures and meit compositions. This study did not account,

however, for the nonidealiîy in chromian-spinel solid-solution Forsythe and Fisk (1994) were the first to

include thermodynamically based vaiues of spinel end-member activities, caicuiated according to the Sack

3+ and Ghiorso (1991b) expressions in order to model Cr -Al chromian spinel-melt equilibria. However

Forsythe and Fisk ignored the influence of varying disorder on the value of the chernical potential of the

pure spinei end-member, They applied the equation of Roeder and Re_vnoIds (199 1) designed for 13ûû°C to 3+

calailate c,"/c~ in melt overlooking the important effect of temperature on the valence state of Cr in the

melt The creation of the MELTS computer program has been a major advance in modeling element

distribution between silicate melt and coexisting mineral phases (Ghiorso and Sack 1995). The program

uses thermodynamicaiiy based models for solid plmes calibratecl usîng a variety of available experimental

data such as data on cation ordering, ixmiscibility, and element distribution bebveen two different phases.

This program utilizes a set of eaergetic parameters for a chosen species in the melt phase rnaking it possible

to account for nonideality in siIicate meI& over a wide range of composition and conditions. However, 3+

Ghiorso and Sack (1995) choose to assume tlut al1 Cr in the meIt is present in the Cr form, even at low

oxygen fùgacities, thus rnaking it hadequate for the task of modeling cluomian spinel - melt equilibria. An

empirical model by Ariskin and Nikolacv (1996b) was successfid in describing the Cr distribution behveen

hi&-Cr chromian spine1 and meft. Howver. it was not caiibrated for duomian spineIs with lower Cr

contenf or for melts with a Cr content, hrpical for naturai lavas (< 500ppin)- In addition, the assumption

about the ideality of chrocnian-spinel solid-solution can Iimit the appiicability of the model to lavas

containing only high-Cr chromian spinei.

This chapter de& with the development of a quantitative mode1 for Cr distribution between chrornian

spinel and basaltic meit, based on a thcrmodynamic model for chromian spinel developed in Chapter 2. an

adequate description of Cr oxidation state. and published experirnental data on Cr solubifity in basaltic

melts saturated with chromian spinel. This model is used in Chapter 6 to calcdate the o.xygen figacity of a

melt in equilibrium with chromian spinel. given a temperature and the careful determination of the Cr

content of the melt

Formulation of the Modal for Cr Distribution between Spinel and Melt

Any successful mode1 describing Cr distribution between chromian spinel and silicate melt of

variable composition in the range of peuologicaily important conditions should:

1) account for the presence of Cr species of varying valence in the silicate melt since distribution

coefficients for these different species between chromian spinel and melt are quite Meren t ( e,g.

Schfeiber, 1976; Roeder and Reynolds, 199 L);

2) include terms describing the effect of the phase composition and associated nonideal

interactions on the equili'brium, 3+

The following AlCr exchange reaction is used to describe Cr distribution between chromian

spinel (sp) and melt (m):

Equation 72

FeCr2)4v+2A10,5rn= FeAl 2 049>+2Cr0 1.5 "

The choice of melt components is completely arbitrary and reflects the simpiicity of the adopted

themodynamic mode1 of basaltic melt (see below). At equilibrium:

Equation 73 m rn

P F ~ ~ O ~ ' - ~ F ~ C ~ , O ~ ~ + ~ .. (PCrOi5 - FAIOI.~ )=O;

j where p. is chernical potential of component i in phase j at the conditions of interest Using the definition

1

of activity, Equation 73 can be rewitten as:

Equation 74

Enthaipy and entropy effects of the reaction, described by Equation 72. at a standard state of 1 bar and

O O

temperature of equilibrium, âH c~M(-~) and AS cr~(-,) , can be defined as:

Equaîïon 75 m m

G*10+2 PO croi~ - G*Y - 2 CLa~ior.5 =m CA(-II - T AS0 C~AIC-I),

where GeI, and GIY are molar vibrational free energies of cornpletely n o d Fe%04 and FeCr201 (se

Chapter 2 for more details) and are chernical potentials of component i in it's pure state. It is accepted

here that the effects of nonideality in the melt on the activity of chromian and aiumina species cancel out in

Equation 74, such that

where X is the simple moiar fmction of an oside compooent in rnelt i based on one cation ( equals to

cation fraction ), Combining Equation 74, Equation 75, and Equation 76 produces:

Equation 77 zp sp nt m

('F4204 - G-14 - (3.3 ) + Mo CIM-I) - T AS0 C r ~ c - l ) = - 2RTïn ( Xcai /X Al01.5

1-

3+ St The reaction involving Cr and Cr in the melt c m be wvritten as:

Equation 78

Equation 79

where and AS0 =&are enthalpy and enuopy effects of Cr ollidation reaction, dewribed by the

Equation 78, at a standard state of 1 bar and temperature of equiiibriuin. It has k e n previously

demonmaied, îhat a better agreement wvith esperiinentally d e t e n - e d or implied values of XF2+ / XFe3+,

and Xc2+ I Xcr* may be achieved by changing the theoretical coefficient of -0.25 that modifies the hKIZ

value in equationç similar to Equation 79 (FudaIi. 1965; Roeder and Reynolds, 1991). This equation can

thus be transformed into a more empirical equation making it anaIogous to the equation used by Roeder

and Repoids for 1300°C:

Equation 80

where a, b, and c are empirical fit parameters-

Fuially, it is assume4 that the only Cr species. prcsent in the melt at olygen fugacities below that of

Q-3, are CF (CrO) and c:~ (CrOl

Equation 8 1

Equation 80 and Equation 8 1 can be used to define the concentration of C? s w e s in rnelt as a firnction of

total Cr concentration. temperature, and fOz:

Equation 82 m - m

X cr01.5 -XcrOl-pw U l +e.q[fl+blnfOl+cl)

m Substituthg this expression for Xcai .5 in Equation 77 results in the final equation, describing total Cr

concentration in the melt as a function of temperature. oxygen fiigacity. cluomian-spinel composition, and

Ai content in melt:

m The calNJated vdue of XcalS- cari be converted to wt% or ppm of Cr, if the major-element

composition of the meIt is known ( the value of the norrnalizing factor is required).

Results of Calibration

Equation 83 can be calibxated with experimental data on coexisting chromian spinel and melt

using available constraints on Cr oxicla~on state. Aü values in Equation 82, descnbing Cr oxidation state in

basaltic melt at Merent temperature and f;Dz are obtained fiom the analogous equation of Roeder and

Reynolds (199 1) caüiraied for 1300°C, and ternperature dependence of the reaction as detemiined by

Schreiber and Hashkin (1976)- Thus, thie value of coefficient b (- 0.28) is taken directly from Roeder and

Reynolds (199 1). Coefficient a is obtaimed fiom the value of the enthaipy change of Cr reduction reaction

in basaltic melts reported by Schreiber and Hasldcin (1976) (their value of 38 kcaVmol gives -19 122 for the

parameter a.) The other parameter fkom Equation 82, c, is dculated frorn the adopted value of a. and the

value of the fiee member fiom 1300°C equation of Roeder and Reynolds (199 1).

The remaining parameters f iom Equation 83. AH0 and AS0, describe energy change of c?- Al

exchange reaction (Equation 72). They are obtained by linear regression of experimental data on coexistùig

chromian spinel and basaltic rnelt. It is important that the Cr content in glas should be carefully analyzed

since its content is quite often just a f ew hundred pprn. Data used for the regression are from B m e s (1986),

Murck and Campbell (1986). Roeder and Reynolds (199 1). and Forsytlie and Fisk (1991). in these studies

the Cr content in g las from the e.xperinienta1 runs was carefully anaiyzed. satiwng one of the main

requirements for the data selection Simce values of pFeCr2-- Gfr and pFeA120"- Gfl, were caiculated using

the thermodynamic mode1 derived in Chaptcr 2. only the data on 1 bar especirnents, whicb produced

3+ chrornian spinel low in Ti (Ti0 < 3.5 ri?.%) and Fe (Fe-O3 c 10 wt.%) were med.

2 - The resulting values of the coefficients from Equation 83 are listed in Table 12. The caiculated

total Cr in the melt for the e-xperiments used in the calibration are cornpared to the rneasured Cr in Figure

17.

Table 12 Values of parameters in Equation 83.

Parameter 1 Value 1 Parameter 1 Value, J* 1 St, Error, J CI -19122 AH0 CA(-1 293091 1 1373

c 6.6 1 * not al1 digits are statistically significant and are given to exclude round-off errors.

Equation 83 and vdues fiom Table 12 can be wed to predict the Cr content in meIt of known

composition sanirated with chromian spinel at the temperature and o.xygen fugacity of interest

Alteniatively, it can be apptied to çaiculate the i02 if the tom1 Cr content of the melt is laio\vn, or it can be

used together with Equation 80 to calcuIate the Cr oxidation state in a melt

O 2000 4000 6000 8000 12000

Cr observed (ppm)

Figure 17. Plot of calcuiated (Equation 83 and Table 12) total Cr content (ppm) in melts versus reported values from Roeder and Reynolds (199 1) 1 bar experirnents.

FAD FAS SiOz 51.6 54.2 &O3 2.3 9.9 Mg0 26.9 30.3 Ca0 19.2 5.6

-4 -3 -2 -1 O 1 2 3 4

In(CrOlCr01.5) analysed

Figure 18. Plot of the calculated versus analyticaily detennined values of in(CrO/ Cr0 1.5 ) for FAS and FAD

rnelts (Schreiber, 1976). Both 1500 and 1550°C data are included.

An independent check of the validity of this approach and calibration was performed by

cdcuiating the Cr content and osidation state for a number of e.xperimentally produced runs that were not

used in the caiibration The cornparison of the eqerimental data of Sclireiber (1976) on Cr oxidation state

in the synthetic Fe-free FAS and FAD melts approaching basalt with cdculated values are shown on Figure

18. Though the agreement is good. Equation 83 does not cornpensate for the influence of total melt

composition on Cr oxidation m t e as s h o m by the Merence in FAD and FAS on Figure 18.

The experirnentai data of Hanson and Jones (1998) and Roeder (unpublished data) on synthetic

Fe-free melts saturated with chromian spinel were used to check calcularions of the total Cr content in

rnelts. Hanson and Jones (1998) e'lperirnents were performed at very dflerent oxygen fugacities (fiorn air

2+ 3+ to well below IW buffer), but only the data for IogfO-= -5.9 to -10.5 wvere used. where the Cr and Cr - melt species of Cr predominate. The calculated Cr content is cornpared to the reported values on Figure 19.

6000-

5000- LI

E a L40'30- O

I Q 3000- O - I O

2000 -

I Roeder (unpubl,)

O 1000 Zoo0 3000 4000 5000 6000 7000

observeci Cr (ppm)

Figure 19 Plot of caicuiated vs. e.upenrnentaUy determined Cr content (ppm) of synthetic melts that were not used for the wlibration

The agreement between caicuiated and reported Cr content is good, especialiy at low Cr content Calculated

values are significantly higher for two of the four of Hanson and Jones (1 998) experhnents performed at

low oxygen fugacity (logfDz= -10.5). Hanson and Jones (1998) found substantial loss of Cr kt0 Pt rWe in

their Fe-free e?cperiments performed under reducing conditions. and tried to miniMze this effect by using

the wire fiom their previous experiments. However. a cornparison of dieir "reversal" esqerimental run with

a normal nrn for l o m 2 = -10.5 (connected points on Figure 19) indicates that their anempts to achieve

equiiibrium at these reducing conditions might not have k e n dways successful. Thus. kigher then

e.upected calculated Cr content in Fe-fiee synthetic giasses, produced in some of reducing Hanson and

Jones (1998) e.uperiments may be at least partiaily explained by the Cr loss into Pt nlre during the

experiments,

Chapter 6 Petrological Applications

Introduction

The equations described in previous chapten are used in this chapter to extract petrological

information on the conditions of chromite crystailization for a number of geological environments.

Specifically, the temperature of crystai1Uation of podiform chromitites from the K e r n p e d Massif (South

Urals) is dadateci, probable crystallization conditions (rnelt composition temperature. and pressure) of

chrornite fiom primitive MORBs and Hawaiian lavas are determined and the oxidation state of Hawaiian

lavas is estimated. The suitabifity of the equations \vas checked by performing the analogous calculations

for a set of primitive lavas and e'cperhnental data, for which the values in question were knoum or could be

estimateci by an independent method. The e'rperimental data include recently published data of Hanson and

Jones (1 998) and unpublished data of P. L. Roeder (personal communication, 1998).

A set of primitive nahuai lavas and piutonic rocks has been studied in some detail to provide

necessary input information for the calculations and as an independent check for the caiculation reçulu.

The lava sarnples were carefully chosen so that they represent lavas tliat were rapidly quenched and thus

presewed the melt as glass and the spinel had little opporninity to change composition upon cooling. It was

hoped that the spinel in the glas wouid thus represent the equiiibrium composition and be a good test of

how well the equations couid predict the spinel-melt equiiibrium. The samples include 5 Kilauea Iki

pumices from Hawaii (R. Helz), 5 primitive glassy MORB samples froin the Woods Hole Oceanographic

Institution collection, 3 primitive MORB sampIes supplied by H. Sigurdsson and used by Colton (1986,

unpublished M. Sc. thesis), and a primitive MORB Iava, provideci by J. AlIan (sample F2-2, Allan et. al,

1988). Plutonic sarnples inciude a set of dunites and chornitites fiom the ophiolite of the Kempersai Massif

(South Urals) that were collected during field work organized by the Insti tute of Geology of Ore Deposits,

VemadsSf ïnstitute of Geochemistry and Analytical Chemistry in Moscow, Russia, and Queen's University

in Kingston, Ontario. A briefdescription and other reIated information about the samples is given in Table

13.

Electron Microprobe Analysis

Sample # 234 -... ... 183/2 3588 360 1 16D 2D AIL3 2-D 11-90

The sarnples were anaiyzed with an ARL SEMQ elecaon microprobe fitted with an energy-

dispersive specuorneter (EDS). The analysis of glass, chromian spineI. and oIivine \vas performed using

EDS at an accelerathg voltage of 15kV using the Bence and Albee (1965) and Albee and Ray (1970)

correction scheme. The initial calibration was achieved by collecting die spectra of a g l a s (US National

Bureau of Standards NBS 470 K12) for 300 sec. and utilizing sofivare supplied by Tracor Northern and

witten at Queen's University, Chromian spinel and olivine were analyzed with a focused beam and glas

was anaiyzed over a scanned area (25.~25 pm) to avoid Na loss d m g die anaiysis. The composition

obtained using spot andysis and area andysis \vas statistically indistinguishabte, except for the Iower Na

content in spot analyses. A series of O ther standards were analyzed duriiig each microprobe session

enabling the analyses to be corrected at a later date (uiis \vas hoivever riot necessary since there $vas good

agreement between the obtained composition and listed values for the secondaq standards). The number of

cations in chramite has been calculated assuming a stoichiometric spinel structure so that both ~ e ~ ' and

~ e ~ + can be caiculated. A plot of Cr# (Cr/Cr +Al) versus ~ e ~ + # (F~~+/(F~'"+M~) has k e n found to be one of

the most usefùl ways to compare analyses of chrornite that have forrned under a range of geological

Location Kempersai Massif, South Urals Kempersai Massif, South Urds Kempersai MassX South Urals Kempersai Massif. South Urals Mid-Oceanic Ridge Mid-Oceanic Ridge Mid-Atlantic Ridge

Descnp tion Chromitites Chrornitite Chromitite Dunite Pillow rirn Pillow rim Piiiow rim

ALV-5 19-2- 1 1 Mid-Atlantic Ridge

Km22 Kilauea Iki Lava Lake Pumice Scowen et al. (199 1) IKI3 3 Kilauea Iki Lava Lake Pumice Scowen et ai. (199 1 )

Reference Poustovetov (unpubl,) Poustovetov (unpubl.) Poustovetov (unpubl,) Poustovetov (unpubl,) Colton (1986) Colton (1986) SIubata et al, (1979)

Piliow rim Piilow rim Pillow rim P a o w rim Pïiiow rim Pumice .

Pumice Pumice

ALV-520-1- 1 ALV-528-2-1 ALV-529-3-2 F2-2 KI7 IKIl 1 IKI21

Bqan and Moore (1977) Bryan and Moore (1977) Bryan and Moore (1977) Bryan and Moore (1977) Man et ai. (1988) Scowen et ai. (199 1) Scowen et al. (199 1 ) Scowen et al. (199 1)

Mid-Atlantic Ridge Mid-Ahtic Ridge Mid-AtIantic Ridge East-Pacifie Rise Kilauea Iki Lava Lake Kilauea Iki Lava Lake Kilauea Iki Lava Lake

+ *,> ,. *----. '. <+.$:6* .- '*$ . . . Podifom Chromitites

(Dick & Bullen, 1984)

Abyssal Basalts (Dick & Bullen, 1984)

Kempersai chromite MORB chromite

2+ Figure 20. Plot of Cr/(Cr+Al) vs. ~e?(Fe +Mg) ofanalyzed in the present study chromian spinel as compareci to the range of chrornite composition fiom podiform chromitites and abyssal basaits outiîned by Dick and Bullen (1984).

conditions. Al1 the chromite anaiyzed in the present study are shown in Figure 20. These chromite analyses

are used in this chapter to demonstrate the efficacy of the equations that Iiave been developed in previous

chapters. The electron microprobe anaiyses of chromian spinel, olivine, and glass are Iisted in Appendices

A, B, and C.

Olivine-Spinel Geothennornetry of Lavas

Since the first calibration of the 01ivi.e-spinel geothemometer by Jackson (1969) and his attempt

to calculate magmatic temperature of olivine and spinel qstallization during the formation of the

Stillwater Cornpiex, this geothermometer has been rarely used for magmatic ternperatures. Olivine-liquid

geothermometry (Roeder and Emslie. 1970) provides a rnuch more accurate and reliable method of

calculating temperatures for volcanic rocks (e-g. Ford et al.. 1984; Beattie, 1993)- Consequently, using

compositions of coexisting spinel and olivine may be of some interest only when no glass is available for

the analysis. Two possible e-uamples would be altered or recrystallized volcanic rocks (e.g- kimberiites) or

plutonic rocks where olivine and spinel preserved their original composition. The later situation is not

typical due to the process of subsolidus reequilibration but can be achieved when magmatic olivine and

spinel formed separate layers such as in the Stillwater Cornplex (Irvine, 1967). Later in this chapter the

composition of olivine and c hromian spinel from the Kempersai podiform chromitites and surrounding

rocks is used to estimate the tempexature of the chromite precipitation.

The oiivine-spinel geothermometer has been applied to a set of glassy naturai lavas, for which

temperature of quenchuig can be estimated with one of the olivine-melt geothermometers (Beattie. 1993).

This provides an independent check for the suggested geothemometer and is shown on Figure 2 1. The

open squares on Figure 2 1 are the resuir of using the Sack and Ghiorso (199 lb) formdation for cdcuiatiag

temperature and the calculated temperature deviation varies with A.i2O3 in the spinel. The solid squares on

Figure 2 1 are the result of using the olivine-spinel formulation developed in the present study and the

calculated temperature deviation shows no obvious systernatic variation with A1203 in the spinel. In order

to ensure equilibnum between spinel and olivine found in the samples, the following sample selection \vas

used for the temperature calculation: srnail (<30 pu) spinels and the average composition of olivine

microphenocrysts, spinel fiom vermiform rims of larger gains using the same olivine composition, and

spinel completely included in large olivine phenocrysts in combination wïth olivine close to the spinel

inclusion

1 1 ~ h i s study a S a c k a n d Ghiorso , 1991 ]

Figure 2 1. Plot of the difference between temperatures calculated with olivine-spinel geothermorneters (T (olsp)) and olivine-melt geothermometer of Beattie (1993) (T (olliq)) vs. A1203 content in spinel for a series of natural glassy basalts.

The resuits of the calculation (Figure 2 1) support earlier conclusion about the inadequacy of the

Sack and Ghiorso (199 lb) calibration based on the analysis of eqerimental data (Figure 3). Sack and

Ghiorso ( 199 1 c) reported good agreement behveen the temperatures calculated using their formulation of

the olivine-spinel geothermometer and actual temperatures. Ho wever, ùie isotherms from S ack and Ghiorso

(1991~) could not be duplicated ushg the equations and data fiom Sack and Ghiorso (199 lb). It was found

that the disagreement may be due to an error, which is also present in tlieir earlier version of the olivine -

spinel geothermometer cornputer program. Specifidy, the X3 (or S3) term \vas missing in one of the

equations describing the condition of intenial eqwlibrium (f3st equation frorn Table 5, Sack and Ghiorso,

t 99 lb). The calculated isotherms and the composition of the olivine and coexisting spinel for the primitive

lavas are s h o w in Figure 22, where in Kd = Ln ( Xdw pF:'/ ~ ~ ~ x 0 ' ~ c 2 t ) , YCr= Cr/(Cr + Al + ~ e ? , and

YF=== ~ e * / ( ~ r + AI + m. This method of representing coexisUng olivine and spinel was introduced by

Irvine (1965) and is often used (e.g, Evans and Frost, 1975) to represent Uiis equiiibrium, The Ln Kd for the

primitive MORB lavas shows a consistent variation with respect to Cr content in the spinel (Ycr)-

Superimposeci upon Figure 22 are isothenns caiculated with the Sack and Ghiorso (1991b) calibration of

Figure 22. Fe-Mg partitionhg data for olivine-spinel pairs from seIected primitive MORBs formed at a temperature close to 1200°C, sho~vn as points. coinpared wïth the isothenns (OC) calculated with Sack and Ghiorso (199 lb) çalibration of olivine-spinel geothermometer.

the olivine-spinel geothermometer. The same data with i so them calculated using Equation 46 and values

fiom Table 8 are shown on Figure 23. The points shown on Figure 22 - Figure 23 represent o d y those

samples with olivine and spinel that closeiy matches the composition of olivine and spinel used for the

3+ isotherms calcuiation ((Fe,Mg)([Al.Crlo,9s[Fe ]0.0J204 and (Mgo9Feo,)- Si04 ). Thus, olivine in selected

samples is more Mg-rich than Fog8, wlule chrornian spinel has Y- < 0.06. The temperature for the lavas as

calcdated using the olivine-melt equilibrium with the Beattie (1993) fonndation is 1190-1243OC (Table

14). As seen fiom Figure 2 1 - Figure 23, the formulation of the olivine-spinel geothermometer as

developed in the present study is a definite irnprovement However, the temperature resolution for lavas is

still not nearly as precise as those calculated using the olivine-melt equilibnum The average deviation of

Figure 23. Fe-Mg partitionhg data for olivine-spinel pairs fiom selected primitive MORBs, shown as points. cornpared \vit6 the isothem (OC) calnilated with caliblation suggested in the present study (Equation 46).

temperature using the olivine-spinel geothermometer as shoun in Figure 21 is about 70°C whereas the

average deviation using an of ivine-rnelt geothennometer is Iess than 20°C. As e.xplauled eariier the r d

usefiilness of the olivine-spinel geothermometer is for situations where the composition of the melt phase is

Olivine-Spinel Geothermometry of Podiform Chromitites from the

Kempersai Massif

The temperature of chromite crystailiration in slowiy cooled plutonic rocks can only be

determined if the composition of chromite fiom concentrateci chromitite and olivine fiom a geneticaUy

relafed dunite are present (Irvine, 1965; Dick and Bullen, 1984). in particular, the folIotMng requirements

should be met:

the chromitite body should be large enough and composed 1argeIy of chrornian spinel, in order for

chromite not to change its composition during subsolidus reequilibration wiîh silicate minerds;

the olivine shouid be fiom dunite where it preserved its original composition;

the olivine and chromian spinel which are anaIyzed, should be genetically related (e-g. crystallized

h m the same magma).

S a t i w n g these criteria can be dficul t. Podiform chromitites often consin of disseminated ore. a

significant part of which is silicate mauix, or the chromitite f o m thin lenses or veins. Second, the dunite

envelope immediately adjacent to chromitite is commonly narrow and the olivine may have been changed

by an Fe-Mg exchange reaction with ore cluomite during subsolidu recquilibration. Finaliy, dunite,

surrounding chromitite, may be the product of a reaction b e m n media lransporting Cr (e.g. basaltic

magma) and surrounding country rock (e.g. Kelemen, 1990) and may have never been in equilibrium wïth

ore chromite.

The Kempersai Massif includes iarge bodies oichromite ore, often of very hi& grade ( e g

Kravchenko and Grigoryeva, 1986). where the chromite had a better cliance of preserving its original

composition The ore bodies are always surrounded by a dunite envelope of variable thickness. The dunite

typically grades into harzburgite furtiier away from the contact with thc ore,

The olivine chosen for olivine-spinel equilibrium temperature calculation is fiom dunite sampled

near the contact with the ore body of the Diamond-Pearl deposit in the southeastern part of the massif. This

dunite contains large (- 1 cm.) euhedrai grains of chromite and abundant relicts of olivine surrounded by

low-temperature alteration produm. The chromite h m the dunite does not differ in composition fiom

chromite nom chromite-pwr ores. mggesting that they are genetically related. The average composition of

the olivine (3 miaopmbe analyses) corresponds to F k 7 - A set of isothenns for olivine of this composition

was dculated using Equation 46 and vaiues from Table 8. The composition of chromite and the results of

caldations are shown on Figure 24. The variation in duomite chemistry is mainly limited to ~ e ~ / @ f ~ +

~e '+ ) and reflects dinerent de- of subsoiidus reequilibration. This is proven by the dependence of the

chmmite composition on the chromite content of the rocks (the chromite fiom massive ores has the lowest

[* Massive ore D Disseminated ore X Disseminated in dunite/

Figure 24. Plot of Cr/(Cr + Al) vs. ~e"/(Fe'-+~g) of chromite from ores and dunite fiom southeastern part of Kempersai Massif (South Urals). Massive ores contain >75 vol..% of chromite, disseminated ores contain >10 vol.% of chromite, while dunites typically contain ( 5 vol.% of chromite. Lines represent calnilateci composition of chromite with &=0.005 and X~0.04 in equilibrium with olivine Fogr7 at different temperatures.

2+ Z+ Fe /(Mg + Fe ). The values of the calculated temperame increases steadiiy rvith increasing chromite

contenc reaching more than l30O0C for samples of massive ore. This suggests magmatic conditions for the

chromite precipitation

composition of Chromian Spinel in Primitive Lavas

The composition of spinel fiom selected primitive lavas (total of 2 11 microprobe analyses) are

presented in Appendix A and are shown on Figure 25. The individuai MORB and Hawaiian spinel analyses

shown on Figure 25 f d in the more-primitive (low ~e'+#) part of the respective MORB and Hawaiian

fields described by Roeder (1994). One of the characteristic features of the compositional variation of the

spinei fiom some MORB samples, or even a single grain. is its wide range in Cr#. An example of a grain

composeci of chemically very dinerent chromian spinel is showm on Figure 26. The most Cr-rich part of the

&cain (Crf13 = 40 wt.%) is surrounded by Al-rich spinel (Cr203 = 23 WL%). which, in turri, is nmmed by

verrniform chromian spinel of intermediate composition

The wide range of composition of chromian spinel from one salnple has been weli documented in

a number of MORB lavas (e.g. Dick and Bryan, 1978; Dick and Builen, 1984; Allan et al., 1988; Gaetani,

1990). It has been explaineci by either varying pressure of chrornian-spinel crystallization (Sigurdsson and

Schilling, 1976), by miuuig of magma of contrasting chernical composition (Allan et al., 1988. Roeder and

Reynolds. 199 1) or by di8tiision-controlled cvstaf Iization (Roeder et a/. . 1999).

Cornpanson of Caiculated and Analyzed Spinel in Primitive Lavas

The composition of spinel in equiiibrium wïrh basaitic melt from a series of rapidiy quenched lavas tvas

calcuiated ushg Equation 68 - Equation 7 1 from Chapter 4- The input parameters include the composition

of the meit (glas), temperature and Iogf02. TIie tcmperature \,.as obtained from the condition of olivine

saturation using the glass composition and tlie equation of Beattie (1993). Christie et al. (1986) determined

thai fO2 for an extensive set of MORB lavas is 1-2 log unis below the conditions of the QFM buffer, based

on ~etC/Fe~' of the glasses. O.xygen fugaciw of Kilauea Iki pumices as calcdated from the limited data on

Fe2'/Fe3' of Murata and Richter (1966) corresponds to the conditions of the QFM buffer.

Figure 25. Plot of Cr/(Cr + Ai) vs. ~e=>(J?e'+ + M ~ ) of chrornian spinel fiom MORBs (open symbols) and Hawaiian purnices (filled symbols). Outlined fields indicite the range of spinel composition fiom MORBs (dotted line) and Hawaiian lavas (solid iine) as indicated by Roeder (1991).

Figure 26. Back-scattered elecuon @SE) image of large spinel grain in glas fiom MORB-like lava F2-2 (Man at al., 1988). Lighter areas of spinel have higher Cr content, Scale bar is 20 p n

Oxygen fbgacity does not affect the spinel composition (Cr/Cr + Al) significantiy (Roeder and

Reynolds, 199 l), unless conditions are very osidking. Thus. for the purposes of these calculations, the

10- values for al1 samples were chosen co~~espondulg to the QFM buffer, as defined by Myers and

Eugster (1983). The input data and the calculated chromian-spinei composition for the L3 samples are

shoivn in Table 14. The calculated spinel composition (P) is cornpared to the range of analyzed spinels on

Figure 27 and Figure 28. Also shown on the diagrans is the chromian-spinel composition (A) calcuiated

with the 'SPINMELT' computer program of Ariskin and Nikolaev (1996b). Good agreement between the

two calcdated compositions (A and P) is observed for Kilauea Lki pumices (Figure 27) containhg Cr-rich

chrornian spinel. Both calculated compositions (A and P) plot near the low-Cr end of the observed range

for aii Hawaiian sarnples, with the exception of KI-22 pumice (Figure 27 E). That particula. sample m a s

found to contain recxystallized portions of lava, with distinctly different relatively Cr-poor, chrornian

spinel, and more 'evolved' g l a s The Cr/(Cr + Ai) of chromite in the more aluminous MORB samples

Table 14. Average glass composition and calculatcd chromian-spinel composition for studied primitive lavas,

Sampk 1 2 3 4 5 6 7 8 9 10 11 12 13

Rock MORB MORB MORB MORB MORB MORB MORB MORB HAW HAW HAW HAW HAW

L0gflD2 -8.2 1 -8.29 -7.97 -8.02 -8,ll -8.4 1 -8,36 -8.06 -7.83 -8.15 -8.25 -8.20 -8.21 Glass (average electron microprobe analyses)

Suin 99-34 98.33 100.05 99,99 100.54 99.48 99.61 98.36 98,33 98.83 98.87 98,81 98,99 Chromian spinel (calculated)

M g 0 16 16.6 17.5 18 18.1 14.6 16 19,9 12,7 12 11.8 11.9 12

Cr203 JO. 1 33.2 30.8 25.1 23.8 313.4 3 1.3 2 O 41.1 40.1 38,l 38.7 39.8 Samples: 1 : AI1 32-D 1 1-90 (Shibaia er al,, 1979); 2: ALV-529-3-2,3: ALV-528-2- 1,J: ALV-520- I- 1,5: ALV-5 19-2- 1 (Bryan and Moore, 1977); 6: 2D, 7: 16D (Collon, 1986); 8: F2-2 (Allan et al., 1988); 9: IK122, 10: IKI2 1, 1 1: IKI 1 1, 12: IK17, 13: KI33 (Scowcn et al., 199 1). Rocks: HAW - pumices erupted by Hawaiian volcano Kilrtuca Iki in 1959. TOC: calculated with Beattie (1993) olivine equations. Logfü2: calciilaicd corresponding to QFM biiffcr as dcfincd by Mycrs and Eiigster (1983).

Figure 27 A-E. Plot of Crf(Cr + Ai) vs. ~e*+/(~e'*+ Mg) of chromian spinel nom Hawaiian pumices, Cdcuiated values for each sample are indicaîed as 'A' (caiculated accordiug to Ariskin and Nkolaev, 1996b), and 'P' (calculateci with the mode1 describeci in Chapter 4).

4

1

0.8 -.

0 7 .

0 6 -

0 5 .

Cr I) 0 4 .

0 3 .

O 2 -

0.1

0

FdW

D [ALVS2932I

-%-y-

U A

& '- -.- .-

-. _ -- 5. p

.

- O 0.2 O 4 0 8 0.8

I P.'

Figure 28 A-K Plot of Cr/(Cr + Al) vs. ~e'- /@e'-+~g) of chmmian spiilel fkom studied samples. Calcuiatd vaiues for each sample are indi~ted as 'A' (calculateci according to Ariskùi and Nikolaev, 1996b), and 'P' (dculated with the mode1 described in Chapter 4).

Figure 29. Plot of Cr/(Cr +Al) vs. F~'*/(F~'-+M@ of chrornian spinel froin the MORB sample AI1 3 2-D 1 l- 90. Calculateci values for each sample are indicated as 'A' (calculated accordiiig to Ariskin and Nikolaev, 1996b), and 'P' (calculated with the mode1 described in Chapter 4). Circles and diamonds are for chromite hosted by glass and olivine respectively. Open markers are for small grains or rims of larger grains. Connecteci soiid and open qmbols show die values for the core and nm respectively of the same grain-

Figure 3 0. Plot of Cr/(Cr + Al) vs. ~ e " / (F~''+M~) of chromian spinel from the MORB sample F2-2- Calculated values for each sample are indicated as 'A' (calculated according to Anskin and Niolaev, 1996b), and 'P' (calculated with the mode1 describeci in Chapter 4). Open markers are for s m d grallis or rims of larger grains. Connected soiid and open sjmbols show the values for the core and rim respectively of the same grain.

(Figure 28) shows a much larger variation than thai for the H a d a n samples (Figure 27). The composition

caldated with the mode1 suggested in the present study (P) is near the low-Cr end of the observed range,

while the Ariskin and Niolaev (1996b) model (A) calculates a more Cr-nch chromian spinel. This

Merence appears to increase with increasing Al content in glas, For exampIe, the biggest dinerence in

caldateci compositions is observed in samples ALV-520-1-1 and F2-2, which contain the most Al-rich

giass uable 14). Since the Anskin and Niolaev (1996b) model was calibrateci mostly for high-Cr

chromian spinel and is not recomrnended by the authors for the purposes of modehg high-Al spinel

crystallization, it is suggested that the model developed in the present study rnay give more realistic resuits.

This suggestion is supported by the detaüed study of chromian-spinel cherni* in selected MORB

samples. It turneci out that the most Cr-rich chromian spinels in each particu1a.r sample e-xhibiting large

variations of chromian-spinel composition are usually found in the cores of large crystals, whiIe the more

Ai-rich chromian spinel compose rims or smaller grains. Two e.uamples of the composition of the rim of

chromite king closer in composition to the chromite calculateci using die equations of the present study (P)

are shown in Figure 29 and Figure 30. Therefore, the mode1 suggested in the present study seerns to be

successfül in predicting chromian-spinel composition crystallizing from selected primitive magmas. The

most Cr-rich chromian spinel apparentiy crystallized earfier, fiom a more primitive magma Tt shouId be

stressai here, that this conclusion is only valid for the studied set of primitive samples wvith no significant

plagioclase crystallization frorn the g l a s pnor to eruption PlagiocIase cnstdlization in more evoived

basalts l ad s to AI depletion of basaltic melt and coeiùsting chromian spinel (e-g. Irvine, 1977), so rhat late

chromian spinel becomes more Cr-rich. The increase of Cr/(Cr+A.l) of chromite during crystaiiization has

been documented in many plagioclase-bearing basalts (e.g. Sigurdsson 1977. Dick and Bryan, 1978).

Other mechanisms c m dso significantiy a e c t composition of crystallizing chromian spinel. For e.xample,

complex zoning found in chromian spinel from some MORBs (F2-l) is probably best explained by magma

mixing (Allan et al, 1 988).

Modeling Chrornian-Spinel Composition in Primitive MORBs

In this section the spinel composition is calculateci using Equation 68 - Equation 7 1 and an

assemblai database of published analyses of Mg-rich gIasses fiom MORBs, and these are compared with

the known range of spinel composition for this type of rock This permis us to decide whether the

conditions of the spine1 cxystaiiïzaîion were close to the conditions of the magma quenchhg.

For the calcuIaiions, the temperature was determineci fiom the major-element composition of the

glasses with the olivine-melt geothermorneter of Beattie (1993) and the o.ygen fiigacity of the QFM buffer-

The data on giass composition from Melson et al. (1977), DSDP party (1 977 , and Dick (personal

communication, 1995) were usxi The giass analyses were screened, so that the calculated temperature was

close to or higher then 1200°C. where the suggested mode1 is considered most reliable. The resuits of the

calculations are depicted on Figure 3 1.

The calcuiated chromite on Figure 3 1, as contrasteci with the andyzed chromite in Figure 25, does

not extend into the MORB spinel field with the highest Cr/(Cr + AI) and Mg/(Mg + ~e'+) vahes. Th-

chromite that has lower values of~e'+/(M~ + ~ e ~ + ) than those calcuiated in Figure 3 1 rnust have

crystallized from magmas that are more primitive than the glasses used for caiculation in Figure 3 1. It has

been shown by Roeder et al. (1999) that often chromite included in olivine crystals cxystallized from a

2+ 2+ more primitive, lower Fe /(Mg + Fe ), melt than those e.uposed to g l a s This indicates the existence of a

MORB melt with lower Al and higher Mg contents than the most primitive gIasses recovered from MORS.

Donaidson and Brown (1977) report the composition of glas included iri a spinel crystai from a MORB

sample which might be an e.uample of a melt composition close to that parental to MO=. The composition

of the host spinel is rich in Cr (Cr203 = 44.37 W.%. Cr / (Cr + Al) = 0.55) as s h o w by the open square

(Figure 3 1) and is more Cr-rich than the calcuiated spinel compositions on Figure 3 1. The g la s from the

inclusions was interpreted as magma trapped in the spinel host just before the emption, The spinel

composition calcuhted using the average composition of these g l a s inclusions is shown by the filled

square on Figure 3 1 and lies near the Cr- and Mg- nch boundaxy of the outlined MORB spinel field not far

away fiom the composition of the host spinel. The caldations and analyticd data support the conclusion

that the most Cr-rich spinel found in MORBs must have crystaliized from the magmas, which are more

primitive than @ses usually found in MORBs. At the same t h e , no special conditions (e-g. high

pressure) seem to be necessary to expiain the low Cr/(Cr + Al) values of some spinels. as previously

suggested (cg. Sigurdsson and Schilling, 1976; Fisk and Bence, 1980). This is because the composition of

-

:/---.. i - helt inclusim 'i / [7 host spinei'----. --...

... -.. - _i

.+-4 -/-= I

-

-

-

f

1 1 I

O 0.2 0.4 0.6 0.8

1 1

~ e ~ + #

Figure 3 1. Plot of Cr/(Cr + Ai) vs. F~~)(F~'&+M~) of spinel calculated using amilable data on g las composition from primitive MORBs. Spinei composition calculated for average composition of liqpid inclusions in Cr-rich spinel phenocrysts and composition of the spinel (sce text) are aiso s h o w . Outlined field indicaîes range of chromian-spinel composition from MORBs (data are fiom Roeder, 1994).

spinel caiculated with the equations suggested in the present study using the composition of MORB glasses

can be as Al-rich as the most Al-rich spinel found in MORBs.

The Cr Distribution between Chromian Spinel and Basaltic Melt as an

Oxygen Geobarometer

Equation 83 and Table 12 provide a method to caicdate the Cr content in melt sanirateci with

spinel at known temperature and oqgen iùgacity. Altematively, the equation can be rearranged for the

purposes of fOt calcuiation using the Cr content in the rnelt:

A set of samples (primitive. chromian spiriel-saturated natural lavas) wi th rneasured Cr content of g las and

FeO/Fez03 \vas compiled to test the technique. The samples include 4 basaits fiom the Mid Atlantic Ridge

studied by Colton (1986), 2 basaits fiom the East Pacific Rise (EPR) described by AlIan et al. (1988),

pumice fiom KiIauea iki volcano on Hawaii (Murata and Richter. 1966). and 8 MORB-like lavas fiom the

Blanco Fracture Zone (Eastern Pacific. Gaerani el al., 1995). The Cr content in glass was analyzed at

Queen's University (Poustovetov and Roeder. 1999) using the technique described in Roeder and Reynolds

(1 99 1). The Cr content in gIass £tom Blanco Fracture Zone samples l a s taken fiom Ciaetani er al, (1 995).

The fOz calculation tvas performed for each sample using the following steps:

1. Glass and the chrornian spinel most likely to have k e n in equilibrium with the melt represented by the

glass are analyzed for major elements with the electron microprobe:

2. Tempemture of quenching is estimated or calculated fiom the composition of glas in equilibriurn

with olivine using the Beattie (1993) equation;

3. The values of chemical potentials of MgA1204 and MgCia~ are dculated using the thermodynamic

mode1 for chromian spinels developed in Chapter 2:

4. The fQ value is calculateci using Equation 81, Table 12, and rneasured Cr content in glas.

The choice of chromian-spinel composition for the calcuiations is often compiicated by the presenœ of

chromian spinei having varying chernical composition (e.g Allan et aL, 1988 and Figure 27 - Figure 30).

The composition of the rùn of chronüan spinel beside glas was usually chosen. Data necessary for the

calculations and the results are listed in Table 15.

Table 15. The composition of glas, chromian spinel and catculated of selected primitive lavas.

1 Setring 1 Mid Atlantic Ridge 1 A a d 1 Eanem Pacifie 1

Source 2 2 2 2 3 6

Cr, ppm 3 90 3 80 402 354 5-10 289

A 1 2 0 3 27.82

TiOz 0.67

Cr203 35.83

Fe0 12.88

Fe203 5.53

M@ 15.30

Mn0 0.32

Source 4

Table 15 (continuai).

Seaing Blanco Trough

--

Source

Cr, ppm

1 - temperature caIculated after Beattie (1993); from FeO/Fefi3 using Sack et al. (1980); 3- 1ogf02

ushg Equation 84; 'ND- not detemllned; ' - ~ d b i = ( ~ F K ~ ~ O , - G*3;) - (pFoU,04- GII*) (J, Chapter 2)- Sources: 1 - this study; 2 - Colton (1986); 3 - Murata and Richter (1966): i - Sigurdsson (unpublished); 5 - Gaetani et al. (1995); 6 - AIlan et al- (1988).

The caiculated vaiues of the ternperanire of lava quenching and Iogf;Dz are plotted on Figure 32.

There seems to be good agreement between the logtio2 caiculated by the method suggested in the present

midy (circles) and the IogfQ calculateci fkom the fe*/J?eZf analyzed from the bulk sample (squares). The

dinerence between logfû2 values, calcuiated ushg the F ~ ~ + / F ~ Z + of the glas. and that using Equation S4

may in part be expiained by the diniculty of m e a s u ~ g the low levels of totai Cr content in the giass, The

Figure 32. Plot of calculated vaiues of IogfiO, vs. temperature for selected natural Iavas. Squares show the Q calculated from known ~e~+/Fe*+ using equation by Sack et al. (1980). Circles refer to KI-, calculated using the method invoiving Cr describeci in the present study (Equation 84). Temperature is calculated using Beattie (1993) equation for olivine-rnelt equilibria. Solid symbols correspond to the samples where total Cr of the glass was analyzed at Queen's University, and open symbols are used for the samples studied by Gaetani et al., 1995. Error bars show the error of fO2 calculation due to an anaiytical error of Cr analysis of +/- 20ppm

values of logfOz calculated using Cr content in glas reported by Gaetani et al. (1995) appear somewhat

higher than the values caiculated from the ~e~+/Fe"+ of the glas. This is in contrast with Poustovetov and

Roeder (1999) data and might indicate interlaboratory merence in analyzing giass for Cr. One advantage

of the suggested method is that it can be used for the quenched melt phase directly whereas it is very

difncuit to directly masure the ~e~+/Fe'+ in the melt if there are crystalline phases present in the bulk

sample. The suggested method is dependent on carefiil analysis of Cr in the glass and will not work weii at

a t02 much higher than QFM because the arnount of Cr in the melt becomes very low- The mettiod should

however work mucb bener at lower £Clt because of the large increase in Cr solubility as fOz is decreased

below QFM (Roeder and ReynoIds, 199 1).

Another method of caiculating the iQusing the electron microprobe analysis of an element of

variable oxidation state in glass is that using the distribution of sulfur species described by Matthews er al.

(1999). The method invotving sulfur works best above QFM whereas the technique using Cr works best

for fiO2 below QFM and thus the two methods cornpiement each other.

Chapter 7 Summary and Conclusions

Thermodynamic analysis has b e n used in an attempt to better describe published experimental data

involving spinel and to understand the conditions of chromian spinel a y s t a i f on in natural magmas.

The first logical result of îhis thermodynamic andysis is an improved version of the olivine-spinel

geothermometer, which is a b-product of the spinel thermodynamic model calibration. The suggested

version of the olivine-spinel geothermorneter appears valid for both magmatic and memmorphic

assemblages and gives reasonable magmatic temperatures of olivine-spinel equili'bria in podiforrn

chromitites from the Kempersai Massif in South Urais. This may be one of the few examples of direct

evidence supporthg magmatic origin of podiform chromitites.

Olivine-spinel equilibria can also be used to predict some features of cation ordering important in

formulaihg and calibrahg thermodynamic models for spinels. The existhg experimentai data on cation

distribution in spinels are stiit inconclusive or, as in the case of many binary spinel solid-solutions, lacking.

For example, various crystal-chemical studies support either a more ordered structure for FeA1=04 relative

to MgAlZ04 and a preferred partition of ~e '+ relative to Mg into the tetrahedral position or an equal degree

of inversion of FeAi204 and MgA120,. The Sack and Ghiorso (1 99 1 b) thermodynamic model, and one of

the models, suggested in the present study, asnime the fus possibility, but can not reproduce ~ e ' + - ~ g

partitioning data between chromian spinei and olivine over a temperature intetval covering both magmatic

and metamorphic conditions. Much better results can be achieved by assurning the second possibility of

equal partitioning of ~ e ' + and Mg behveen tetrahedral and octahedmi positions in the spinel structure.

Calculateci dependence of I?e2+-~g distribution coeflicient between olivine and chromian spinel on the Cr

content of spinel closely matches observed variations of olivine and chornian-spinel composition in

metamorphic rocks (e-g. Evans and Frost, 1975). The complete dibration of the later model is, however,

impossible, due to the lack of e.uperimenta1 data on cation ordering in binary joins, especially in ALCr

spinels. There is also practically no data on the influence of pressure on cation distribution in petrologically

important spinels.

A correctly formulated and calibrated thermodynamic model for chromian spinel is essential in

describing chramian spinel - melt equilibria Since this mode1 is still unavailable, an enpirical solution for

the probkm has been suggested. It includes dibraiion of a number of empirical parameters a c c o u n ~ g for

nonideaiity of spinel solid-solution On the other hanci, the activities of melt species are assumed

proportional to their cation fractions. This approach (relatively compiex spinel model and simple melt

madel) is different h m some other published empirical equations. An e.uample is the model of Ariskin and

Nikolaev (1996b), which assumes ideal mixing of cations in the spinel structure, but a more complex model

for calculating activities of the melt components. One way to look at this problem is to consider the

sensitivity of chromian-spinel composition to changes of composition of basaitic melt fiom which spinel

crystallizes. The large change of chromian-spinel composition associated rvith a small change of melt

composition implies that accounting for nonideality in the spinel phase is probably more important than

treathg the nonideality of basaltic mek

The suggested empirical equations seem adequate to d e s c n i the large variability of chromian-spinel

composition associated with relatively small changes of melt composition documented in a number of

experixnental studies. One should be cautious, however, in using these equations outside the range of spiniel

or melt compositions of the e.uperimental data used for caiibration. The füture development of a more

"theoreticai" model, based on a better knowledge of structure and thermodynamics of both spinel and melt

phases should ease these restrictions. The empirical equations that have been developed were used to

calcdate the composition of chrornian spinel for a set of glassy primitive naîurai Iavas sanirated rvith

olivine. The dculaîed chromian-spinel composition is close to the composition of the most aluminous

chromian spinel found in each particdar sampfe. Some of the MOEü3 samples contain chromite with a

more Cr-rich core, which suggests crystaliization at a higher temperature. Arnong the calculated chrornian-

spinel compositions for ihe MORB samples, some f a in the e-weme Al-rich category of the MORB

chromites, the origin of which is disputeci- One hypothesis explaineci this enrichment in Al by

crystallization at high pressure. The resuits of the present calnilations suggest no need for this. Thus,

surprisingly, it is the most Cr-rich; not Al-nch spinel from MORB lavas that apparently crystallized under

conditions, d i f f e ~ g fkom those just prior to eruption (this does not apply, however, to fiactionated basalts

where plagioclase crystallization can produce high-Cr chromite). The calcuiating of the chromian-spinel

composition using a larger database of primitive MORB glass analyses supports this conclusion, since no

calculated composition fall close to the rnost Cr-rich chromian spinel found in MORBs. However, if the

composition of a high-Mg, low-Ai glas inclusion found in one of such Cr-rich chromian spinel crystals is

use& the resuiting calculateci composition of chromian spinel is simiiar to the composition of the most Cr-

rich chromian spinels fkom MORBs.

Calculating the chromite composition with the suggested ernpjricai equaîions requires values of the

majorelement composition of the melt, temperature and f02. Temperature can often be calculated £tom the

major-element composition of the melt and the condition of olivine (or other mineral) saturation. The fOz is

usually more difncuit to assess, However. if the Cr content of the melt is known, the KIz can also be

calculated using another equation suggested in the present study (Equation 84)- This equation is the first

quantitative theoreticai mode1 describing Cr solubility of basaitic rneit saturated with chromian spinel in

petrologically important conditions. It is based on the thermodynamic anaiysis of Cr oxidaîion/reduction

reaction and Cr-AI exchange equilibria between chromian spinel and basaltic melt It aiso uses expressions

of chemîcal po t enas of Al- and Cr-rich spinel end-rnembers fiom the suggested thermodynamic mode1 of

chromian spinel, Thus, by knowing the composition of the meic including carefiil Cr analysis, and the

ternperature, both the fû2 and the chromite composition can be calculated The usefiilness of the developed

equation as an oxygen geobarometer was tested using the data on Cr content and FezOfieO of the gIass for

a set of primitive lavas. Calculated KI2 for primitive mural lavas using the suggested equation agrees well

wïth the values, obtained by the traditional technique h m using the Fe203 andFe0 of the melt

References

Akella, J., Williams, RJ., and Mullins. 0. 1976. Solubility of Cr, Ti, and Al in CO-existing olivine, spinel, and Liquid at 1 atm. Proc. Lunar Sci-Conf. 7th, 2, 1179-1 194.

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Wood, B.J., Kirkpatrick, RJ., and Montez B- 1986. Orderdisorder phenornena in MgAiz04 spinel. Am. Miner., 71,999-106.

Appendix A. Electron microprobe anaiysis (W. % oxide) of chromian spinel.

' ~ o c k Analysis # '~ ize '~hape 4 ~ o c S i 9 Ti02 Cr203 Fe0 Mn0 Mg0 Ca0 Na20 K20 Tolal '~e'+# 'cr# CHR+ 75.881144.55 >IO00 S 0.14 0.15 8.58 62.40 13.45 0.00 14.79 0.10 0.00 0.00 99.61 0.29 0.83 CHRt 91.371183.15 >IO00 S 0.09 0.16 8.71 62.45 13.49 0.00 14.83 0.14 0.00 0.06 99.93 0,291 0.828 CHRt 91.37/183,15 >IO00 S '0.09 0.18 8.67 62.76 13.37 0.00 14.86 0.11 0.00 0.04 100.08 0,291 0,829 CHRt 36,85/162,05 >1000 S 0.16 0,07 8.20 62.72 13.21 0.00 14,71 0.08 0.00 0.00 99.15 0.288 0,837 CHRt 36.851162.05 >IO00 S 0.12 0,lS 8.04 63.25 13.36 0,OO 14,86 0,lO 0,08 0.02 99.98 0.287 0.841 CHRt 36,851162.05 >IO00 S 0.19 0.14 8.36 62.50 13.18 0.00 15.02 0.08 0.08 0.00 99.55 0.279 0.834 CHRt 49,45186 >IO00 S 0.13 0,16 8.45 62,43 13,15 0,OO 15.11 0.17 0.00 0.00 99.60 0.276 0.832 CHR+ 49,45/86 >LOO0 S 0,11 0.21 8,31 60.80 12,69 0,00 15,14 0,lO 0.10 0.05 97.51 0.262 0.831 CHR 53.51217.1 >IO00 S 0,27 0.15 8,45 62.62 13.69 0.00 14.75 0.08 0.00 0.04 100.05 0.293 0.833 CHR 69.12/226,7 >IO00 S 0,00 0.18 8.30 62.59 13.86 0.00 14.68 0.15 0.10 0.00 99,86 0,296 0,835 CHR 69,121226.7 >IO00 S 0.25 0,11 8.43 62.29 13,42 0.00 15.01 0.10 0.00 0.03 99.64 0.279 0.832 CHR 18.131185,8 >IO00 S 0,18 0.16 8,69 61.32 1537 0,OO 13.89 0,lO 0,12 0.00 100,03 0,332 0.826 CHR 28.71196.75 >IO00 S 0.12 0.17 9.18 61.56 14.86 0,00 13.96 0.15 0,10 0.00 100,lO 0,33 0,818 CHR 36.141188.12 >IO00 S 0.22 0.16 8,77 61.52 14.91 0.00 13.61 0.11 0.00 0.00 99.30 0.34 0.825 CHR 3,515 >IO00 S 0,13 0.25 8.60 61.62 15.25 0.00 13,99 O,11 0,00 0.03 99.98 0,329 0,828 CHR 3.515 >IO00 S 0.13 0.18 8.56 62.02 14.81 0.00 13.98 0.13 0.00 0,OO 9931 0,327 0,829 CHR 7/124,2 >IO00 S 0.1 1 0,09 8.55 61.84 15,21 0,OO 13,87 0,lO 0,08 0,OO 99.85 0.33 0,829 CHR 290.18t85.72 >IO00 S 0.22 0.14 8.77 62.00 13.40 0.00 1430 0,11 0,00 0,00 99.44 0,289 0,826 CHR 292,24193.22 >IO00 S 0.24 0.13 8.97 62.27 12.77 0.00 14.95 0.10 0.00 0.00 99,43 0,282 0,823

C

CHR 299.24/95,8 >IO00 S 0.13 0.13 9.08 61,43 13,18 0.00 14.86 0.10 0,OO 0,04 98,95 0,284 0,819 CHR 307t91.76 >IO00 S 0.14 0.10 8.72 62.29 13.33 0.00 14.84 0,10 0,OO 0.00 99S2 0,287 0,827 CHR 125.341140.2 >IO00 S 0.14 0.19 8.47 62.90 14.80 0.00 14.13 0.11 0.00 0.00 100,74 0.326 0.833 CHR 152,651139.7 >IO00 S 0.08 0.13 8,48 62.10 15,02 0,OO l4,18 0,lO 0.00 0.04 100.13 0,32 0,831 CHR 152,651139.7 >1000 S 0.24 0.13 8.47 61.68 14.52 0.00 14.09 0.07 0.00 0.04 99,24 0,317 0.83 CHU- 58.85t54.15 >IO00 S 0.13 0.25 8.88 61.86 14.05 0.00 14.83 0,12 0.14 0.00 100.26 0.294 0,824

CHRt 58.85/54,15 >IO00 S 0.16 0.15 8,62 62,34 13.66 0.00 1437 0.16 0,OO 0.00 99.66 0.3 0.829 CHRt 74.15167.1 >IO00 S 0.17 0.13 8.73 61.84 13.69 0.00 14.65 0,07 0.16 0,04 99,48 0,295 0,826 CHRt 1 11,081227.04 >IO00 S 0.17 0.10 8.89 61.93 13.17 0.00 15.24 0.12 0.00 0.06 9968 0.271 0,824

m * m m I I I

~ 1 + 1 N N N

Analysis # Size Shape Loc SiOz TiOa Alz03 Cr203 lllA 10 E 1.16 0.59 22.26 39.98

Ca0 Na20 K20 Total Cr# 0.63 0.00 0.00 97.98 0,328 0.546 0.33 0,OO 0.07 98.61 0,329 0,543

1 1 1 1 1 1 1

III A 1 l O I E I 10.33 10,57 122.95 140,62 1

IXA 100 E OC 0.33 0.53 20.72 47.20 IXB 100 E OC 0.23 0.69 20.86 47,19 IXC 100 E OR 0.31 0.51 21.46 45.72 VIA E 0.52 0.35 22,19 45,42 WC E 0.61 0.53 24.32 42,44

.V-5 J 9-22 1 IMOR ,V-5 1 9-2-1 MOR

IlIA 5 E 0.62 0.36 31.39 33,lO IVA 5 E 0.78 0.32 30.50 33.17

LV-520-1-1 (MOR

Analysis # Size Shapc Loc Si02 7A 50 E OC 0.20

MOR - MOR - MOR - MOR - MOR - MOR

MOR F2-2 MOR

f\ 00 m crl 9 ? 3 0

9 00 -t m s-? 3 0

LI

2% z g 0 0 9 9 0 0

0 0 9 9 0 0

2 5 0- O

CA \D 0. -? z? z fl b a. C? 0 0

\O O 09 =? Vi CI N c-4

m 00 z- 2 -P e

OI - p. -? 2 11

= 5 N' t?4

2 $ 0- O

d

W U

g o

@ s 4 t?4

3 3

f

Appendix C. Eleciron microprobe nnnlysis (W. % oxidc) of studied basaltic glasses.

Sample # ' ~ock Analysis # Si02 Ti02 16D MORE 161.58A12.22 50.77 1.17

1

AI2O3 Cr2O3 Fe0 Mn0 Mg0 Ca0 NazO K20 Total 1

115,95 0.12 8.99 0,25 8.25 12.04 2.56 0.08 100.18

16D MORB 64.651172.92 50.28 1,27 1 6 ~ MORB IXC 50,68 1.19

2D MORB 5411392.84 51.07 1.22 20

pp

ALV-519-2-1 MORB lllC 49.75 0.70 ALV-519-2-1 MORB VC 49.42 0.73 ALV-519-2-1 MORB VID 49.91 0.77 ALV-520-1-1 MORB I D 49.44 0.73 ALV-520-1-1 MORB I F 49.05 0.75 ALV-520-1-1 MORB 2E 49.14 0.64 ALV-520-1-1 MORE 30 48.90 0,65

1

F2-2 MORB 2C F2-2 MORE 3B

1

F2-2 'MORB 40 F2-2 MORB 5C F2-2 M O R B - ~ B ~ - - F2-2 MORB 9G

IKI11 HAW 318 !KI1 1 HAW 4/C

IK121 HAW 4lB IK121 HAW 518 IK121 HAW 6lD IK122 HAW 1lC IK122 HAW 1 g IK122 HAW 2a

Appendix D. ~n example of multiple iinear regression (calibration of Equation 54) performed with

linput Range" SUMMARY OUTPUT

Regression Statistics Multiple R 0.985080053 R Square 0,9703827 1 Adjusted R Square 0.9683 163 87 Standard Errcx 1246.636686 Observations 47

ANOVA Df SS

Regression 3 2 189505264 Residual 43 66826430.11 Total 46 225633 1694

X 1 32.99794 193 4.908067583 X Variable 2 0.953326697 0.052760522 X Variable 3 0.998920786 0.0582159 14

Appendix E. ûenerai view and formulas of the Corel Quaüro Pro 8 spreadsheet used to calculate chromite composition fiom melt composition at a particuiar temperature and pressure by solving a set of nonlinear equations (Equation 68 - Equation 71)-

The actual spreadsheet is available upon request from the author (alexei(iù..neol.quee~lsu.ca) or Prof. P.L. Roeder (mederO,,neol.aueensuca),

Sheet "Input" (al1 cells are protected, e m p t shaded ones, some celis are hidden)

Sheet "Output" (ali cells are protected, some celIs are hidden)

I 1 raal I I I I I I I I I I i

Sheet ‘‘Macre" (dl œlls are protected)

Input:B2: Input:C2: Input:E2: Input:F2: Input:H2: 1nput:U: 1nput:BQ: Input:C4: Input:D4: InputE4: Input:F4: Input:G4: Input:H4: 1nput:Ia: Input:J4: Enput:K4: 1nput:LQ: Input:M4: Input:N4: 1nput:AS: 1nput:BS: Input:CS: 1nput:DS: InputSS: 1nput:FS: Input: G5 : 1nput:HS: 1nput:U: Input:J5: 1nput:KS: 1nput:LS: Lnput:M5: input:NS: Input:A6: Input:B6:

mizer.Reset} 1 1 mizer.Solution-cell Output:EG)

TC= O 'Pbar = O 'TK =

+C2+273.15 'Si02 'A1203 Ti02 'Cr203 Fe0 'Fe203 'Mg0 'Ca0 'Mn0 'Na20 'K20 'P205 'H20 'OxWt 60.0848 101.9612 79.8988 151.9902 7 1,8464 159.6922 40.3 114 56.0794 70.9374 61.979 94.2034 14 1.9446 18.01534 ' AtNum 1

I

rnizer.Solutionlgoal Min) 1 mizer.Variable cells lnput:B1 3..lnput:D13) mizer.Add 1 ,"lnput:B13.,lnput:D13",~=,"0.0000001~ mizzr.AdG 2,"0utput:F3",<=,"1") 1 mizer.Add 3,"0~tput:C3",~=,"0.0000001") mizer.Max-Time 1000) 'mizer.Max-lters 1000) 'mizer-Precision 1 E-06) imizer.Auto-scale 1 ) imizer.Linear 0) 1 imizer.Show-lters 0) imizer-Estimates Quadratic

1 1 1 !

imizer-Derivatives Central) imizer-Search Newton) imizerSoIve) ]

f

l 1

Input:C6: 2 Input:D6: 1 KnputE6: 2 Input:F6: 1 Input:G6: 2 Input:H6: 1 InputI6: 1 Input:J6: 1 Input:K6: 2 Input:L6: 2 Input:M6: 2 InputN6: 2 Input:A7: 'W.% Input:B7: O Input:C7: O Input:D7: O InputZ7: O Input:F7: O 1nput:Gï: O Input:H7: O Input:L7: O Input:J7: O Input:K7: O rnput:L7: O Input:M7: O Input:N7: O InputAg: 'CatFr 1nput:BS: @ARRAY(B9..N9/$Input:$OS9) InputAg: 'Cat Input:B9: @ARRAY(B7..N7/B5..N5*B6..N6) Input:-: @SUM(89..N9) 1nput:B 1 1: 'Initial approximations 1nput:B 12: 'X2 1nput:C 12: 'X4 Input:Dl2: 'X5 1nput:B 13: 0.8 Input:C13: 0.01 Input:D13: 0.0 1 1nput:B 15: @IF(Output:E6<1E-lO,"","Chromite not calculateci yet- Complete your input and play rnacro

in Uacro:Al..A17") 1nput:B 16: @IF(Output:E6clE-lO,"Chromite calculateci, see it in the output","") 1nput:B 17: @IF(@ISERR(Output:E6)=1," Wrong or incomplete input data","") InputB20: 'Instnictions: Input:A2 1: '1). 1nput:BZI: 'Enter melt composition (wt.%), temperature (OC), and pressure (bar) in shaded areas. Input:B22: '(calculate FeO and Fe203 with Kress and Carmichael(199 1) if necessary)

InputA23 : '2.) Input:B23: 'Leave initiai approximations unchanged or enter defadt values of X2=0.8, X4=0.0 1, X5=0.0 1 Input:A24: '3 .) Input:B24: 'Calculate chromite corn psi tion by playing macro in Macro:Al. .Al7 Input:B25: '&ou might want to create a quickbutton to do it wvith "(BRANCH Macro:Al..A17}" rnacro

property) Input:A26: '4.) InputB26: 'If it says: "Chromite calcuiated", you are done! InputA27: '5.)

Input:B27:

Input:B28: Input:B29: Input:B30: Input:B3 1: Input:B3 2: Input:B34: OutputB2: OutputC2: OutputD2: OutputE2: OutputB3 : OutputC3 :

Output:D3 : OutputE3: OutputF3 : OutputBS : Output CS: OutputDS: 0utput:ES: Output:B6:

Output:C6:

Output:D6:

UutputE6:

'Kit says: "Chromite not ..." after you run the macro a few times, change initial approximation(s) and nui the macro agah

'Try choosing initial approximations close to expected chrornite: 'X-parameters are connecteci to N(number of cations in chrornite formula based on 40): 'X2=N@fg); X4=N(3); X5=N(Fe3 +)/2 'Note: program is designeci to work for primitive or miidly evolved broadly basaltic rneIts 'at temperatures higher than 1200.C or a Little les, 'Aiexei Poustovetov January, 2000 'X2 'X3 'X4 'XS +InputB 13 (@LN($Inpu~$~8/$hput:$F$8/$Output:$B$3 *(l+$Output:$D$3- $Output$BS3))+0.404074+(537-987252-2903 -4330 14*$Output:$DS3- 2914.212902*$Output:$E$3)/$Inpu~$I$2)/2207.8338 13 *$Inp~r$l$2 +InputC 13 +Input:D 13 @SUM(C3. E3) 'Eq68 'Eq69 'Eq70 ' S m @LN(SOutput:$B$3/(1+SOutputSD$3)*(l-$Output:$C$3-$Oritp~t$D$3-$Output$E$3)~2)- @LN($Input$H$8*$Inp~t:SC$8~2)-0.390867-66 15.0725 1/%Inpu~$I$2- 0.06416 l/$lnput$I$2*(Shput:$F$2- 1)+(50 17.8673 13/$Input:SIS2*$Output:SC$3+12 13.5 16 174/$Inpuc$I$2*$Output:$D$3+717 9.32608/$Input.61$2*$0utput:$~3)*($0utp~t:$C$3+$0~tp~t:$D$3+$0utp~t$E$3)- (1724.29439 l/$hput:SI$2*SOutput:$D$3+1486.533648/$hput:SIS2*$Output:$E$3)*$Outpu t:$C$3+3 104.5 14209/$Input:$I$2*$Output:$DS3*$Output:%E$3+(1- $Output$B$3)*(1413.575609/$hput:$1$2*$0~~:$~3+10188.56975~1n~ut~~$2*$0utpu t:$D$3 -2829.436475/$Input:$I$2*$Output:$E$3) @LN(SInput$FS8*$hp~t:âG$8~2)+@LN((l+$OutputSD$3 - $Outpu~SB$3)/(lt$O~tput:$D$3)*$0utput:$E$3~2)+6.905697-23780.78/$hput:$I$2- 0.1 16 18 l/$Input$I$2*($Input:$F$2- 1)+(7 105.5 19586/$Input:$IS2*SOiitput:$~3+ 9 188.8043 19/$Inpuc$I$2*$Output:$D$3+628 2.248784/$Inp~t$I$2*(l-$Output:$~3-$~~~t$D$3-$0~~~:~$3))*(1- $Output$E$3)+8052.43716 l/$Input:$1$2*$Output:$C$3 *$Output:SD$3- (35QQ.856034/$Input:$I$2*$0utput:$~3+16428.04993/$1np~t$1$2*$0utput:$D$3)*(1- $Output%C$3-$Output$D$3-$Output:$E$3)- (8455.38893/SInput:$I$2*$0utput:$C$3+884 1717738/$Input:$IS2*$Uutput:$D$3 - 4848.62 107/$Input:$I$2*(lâOutput:$E$3)) *$Output$B$3 -@LN($Inp~t:$F$8~2*$Input:SD$8)+@LN(((l+$Output:$DS3 - $Output$BS3)/(l+$Outp~t:$D$3))~2*$Output:SD$3)+3 -325468-1 54 10.3 9/$hput:$I$2- 0.115429/$Input$I$2*($Input:$F$2-l)+(564.652751/$Input:$IS2*$Output:$C$3- 4376.293677/$Input:SIS2*$0utput:$E$3+139 1.840597/$Input:$I$2*(l-SOutput:$C$3- $oUtput:$DS3-$oUtput:$E$3))*(1- $Outpu~SD$3)t8084.928786/$Input:$IS2*$Output:$C$3 *$0utput:$E$3+(23.823SS l/$Input: $IS2*SOutputS~3-13005.248984/$Input:S~2*SOutput:$ES3) *(l-$Output:$C$3 - $Output$D$3-$Output:m3> (9275.364957/$Input:$IS2*SOutput:SC§3+456 1.214%4/$Input:SI$2*$Output:$E$3- 10592.609753/$Inp~tSI$2*(l-$~tp~tSDS3))*SOutp~t:$BS3 +$Ou~utSB$6~2+$Output:$C$6~2+$Output:SD$6~2

OutputB 13 : @IF(E6<lE-lO,"Final calcuIated cluomite composition MI.%", "This is only an approximation, and NOT the fimi clmmite composition")

OutputB 14: ' M g 0

Outputc14: 'Fe0 OutputD 14: 'A1203 OutputE14: 'Fe203 OutputF14: Ti02 OutpucG 14: 'Cr203 OutpuCH14: 'Fe& OutputAlS: 'a% OutputB 151 +B 16/$0utpuc$EB 16 OutputC15: +Cl6/$Output$H$l6 OutputD 15 : +D 16/$Output:$H$ld OutputE15: +El6/$0utput:$H$16 OutputF15: +F16/$Output:$H$16 OutputGlS: +G16/$0utput:$H$16 OutputHlS: +C15#.9*El5 OutpucB 16: +$Output$B$3*$hput:$H$S OutputC16: (l+$Outpin:$D$3-$Output:$B$3)*$Input:SF$S Outpu~D16: (l-SOutput:$~3-$Output:$D%3-$Output-S~3)*$Input:$C%5 OutputE 16: +SOutput:$E$3 *$Input:%G$S OutputF 16: +$Output:$D$3 *$Input:%DSS OutpucG16: +C3 *$hpuc$E$S OutputH16: @S~(B16.,G16)/100 Mam:A 1 : '{Optimizer.Reset) Macro:A2: '{Optimizer.Solution-ce11 OutputE6) Macro:A3: '{Optimizer.Solution~Min) Macro:A4: '{Optimizer,V~able-ceUs I.nput:B 13 .. input:D 13 ) Macro :AS: '{Optimizer.Add 1,"InputB 13 ..Input:D 13",~=,"0.000000 1") Macro:A6: '{Optimizer.Add 2,"0utputT3 ",<=," 1 ") Macro:A7: '(Optimizer.Add 3,"0utput:C3 ".>=,"OO00OO0O 1" ) Macro:A8: '{Optimizer.Max-The 1000) Macro:A9: '{Optimizer.Max_Iters 1000) Macro: AlO: '(0ptimizer.Precision 1E-06) Macro: A 1 1: '{Optimizer.Auto-scale 1 } Macro:A12: '(0ptimizer.Linea.r O ) Macro:A13 : '(Optimizer.Sbow_Iters O) Macro:A 14: '{OptimizerEstimates Quadratic) Macro:AlS: '{Optirnizer.Denvatives Central) Macro:Al6: '{Optimizer-Search Newton) Macro:A17: '(Optimizer.Solve)