Thermodynamics of cation ordering in karrooite (MgTi 2 O 5 )

Transcription

1 American Mineralogit, Volume 8, page 37 37, 999 Thermodynamic of cation ordering in karrooite (Mg O 5 ) MARK S. GHIORSO,, * HEIONG YANG, AND ROBERT M. HAZEN Department of Geological Science, Box 353, Univerity of Wahington, Seattle, Wahington , U.S.A. Geophyical Laboratory Center for High Preure Reearch, Carnegie Intitution of Wahington, 55 Broad Branch Road, N.W., Wahington, D.C., 5-35, U.S.A. ABSTRACT A thermodynamic model of non-convergent cation-ordering in karrooite (Mg O 5 ) ha been calibrated from the ingle-crytal -ray tructure refinement of Yang Hazen (998) from etimate of the dependence of the bulk modulu on ordering tate (Hazen Yang 997). Derived value of the Gibb free energy, enthalpy, entropy, heat capacity of diordering of karrooite are reported a a function of temperature at bar at elevated preure. INTRODUCTION In thi paper we develop a thermodynamic model that decribe cation ordering a a function of temperature preure in the karrooite (Mg O 5 ) end-member of the armacolite olid-olution erie. The model i contructed utilizing technique firt developed by Thompon (969, 97) to decribe cation order-diorder phenomena in mineral in term of the principle of homogeneou equilibrium. The preent tudy wa prompted by the recent completion (Yang Hazen 998) of high quality, one-atmophere -ray tructure refinement of ingle crytal of ynthetic karrooite, which were equilibrated prior to analyi at temperature over the range 6 C. The reult of the tudy of Yang Hazen (998) are at odd with previou etimate of the equilibrium ordering tate of karrooite (epecially at elevated temperature), conequently, exiting formulation of the thermodynamic of cation ordering in thi phae (Brown Navrotky 989) require reevaluation in light of the recent reult. We take thi opportunity to develop a thermodynamic model that account for the Yang Hazen (998) meaurement, incorporate into the formulation a preure dependence of thi cation-ordering phenomena by accounting for the effect of ordering on the compreibility etimate of Hazen Yang (997). where M denote the mole fraction of Mg + on the M octahedral ite M denote the mole fraction of Mg + on the M octahedral ite. The experimental evidence demontrate that i a function of both temperature preure. Note that the order parameter defined in Equation i different from that adopted by Brown Navrotky (989) or by Yang Hazen (998). Our choice i motivated by a deire to implify the thermodynamic treatment by generating an order parameter that aume value over the range to in proceeding from the fully ordered ( M, M ; ) to the fully antiordered ( M, M /; ) tate. The algebraic implification with thi definition i coniderable. It hould alo be noted that a rom cation configuration i given by M M /3; /3. From Equation it follow that: M Mg + M Mg M (a) (b) (c) THERMODYNAMIC ANALYSIS A decription of the karrooite tructure it relationhip to other peudobrookite-type oxide olid olution i ummarized by Waychuna (99). For the preent purpoe, it i ufficient to undert that the Mg + + cation are ditributed over two type of octahedral ite, M M, which are preent in the ratio :, repectively. The cation ordering, which i conequently of non-convergent type, therefore can be quantified by inventorying the proportion of Mg + or + occupying either ite. We may define an ordering parameter,, to accomplih thi tak: M M () Mg Mg * ghioro@u.wahington.edu 3-/99/9 37$5. 37 M 3 +. (d) The configurational entropy (S conf ) aociated with the degree of cation order i: ( ) conf M M M M M M S R Mg ln + ln + ln + M ln M (3) which, via Equation a d may be written a a function olely of the order parameter, : conf S R ln + ln + ln + ln () We contruct a thermodynamic model for karrooite ordering by writing an expreion for the molar Gibb free energy (G ) of the phae in term of configurational ( TS conf ) lattice vibrational (G * ) contribution:

2 GHIORSO ET AL.: KARROOITE THERMODYNAMICS 37 G TS conf + G * (5) taking the vibrational component to be a econd-order truncated Taylor expanion in the order parameter: G * G * + G * + G *. (6) The Taylor coefficient in Equation 6, G *, G *, G *, are conveniently recat in term of more intuitive energetic quantitie. Adopting G Kar to denote the molar Gibb free energy of fully ordered karrooite (Mg M M O 5 ) at any T P, along with G E to decribe the free energy change of the ordering reaction: Mg M M O 5 M M (Mg,) O 5 (7) W G to refer to a regular olution type parameter defining the non-linearity of the Gibb energy along the Mg M M O 5 - M M (Mg,) O 5 join, Equation 6 may be tranformed into * G G + G Kar E( )+ WG ( ). (8) Equation, 5, 8 define our thermodynamic model. In general, the model parameter G E W G may be function of both temperature preure. Firt-order expanion in both T P are adopted for preliminary analyi of the experimental data et: G E H E T S E + V E (P ), (9) W G W H TW S + W V (P ). () Experimental meaurement of ordering tate in karrooite a a function of T P provide a mean of calibrating (T,P) under the aumption that the meaurement reflect an equilibrium tate of cation order. In thermodynamic equilibrium, the molar Gibb free energy i minimal with variation in at fixed T P: () which from Equation, 5, 8, 9,, reult in: ( + ) ( 3+ ) RT ln HE T S E +( P ) V E ( ) + [ WH TWS +( P ) WV. () The model parameter in Equation, e.g., H E, S E, V E, W H, W S, W V, mut be calibrated from experimental data. In Figure we plot value of ( + ) ( 3+ ) RT n ( ) a a function of, derived from the meaurement of Yang Hazen (998). The data point labeled 6 C i problematic. The karrooite ample Yang Hazen (998) utilized for their analyi were prepared from ynthetic oxide at high temperature ( C). Single crytal were then elected annealed at the temperature indicated in Figure. In the cae of the 6 C ample, an annealing time of 8 hour wa probably inufficient; repeated analyi of crytal equilibrated at thi temperature demontrated inconitent value of lattice contant. For thi reaon, in the ubequent calibration of our model from thee data we will utilize the 6 C meaurement for reference not contrain the model to fit thi datum. [ FIGURE. Analyi of Cation-ordering data for Karrooite according to Equation (ee text). Meaurement of Yang Hazen (998) are plotted with σ uncertaintie are labeled with the temperature ( C) of annealing. i given by Equation. The olid curve i the model repreentation of the data. A the Yang Hazen (998) data were collected at a preure of bar, the array of point hould be modeled according to (e.g., Eq. ): ( + ) ( 3+ ) RT ln H E T S E +[ W H TW S ] (') ( ) Extenive tudie of numerou oxide (Ghioro 99a; Sack Ghioro 99a, 99b) ilicate (Ghioro et al. 995; Hirchmann Ghioro 99; Sack Ghioro 989, 99a, 99b, 99c, 998) mineral olid olution have demontrated that exce vibrational entropie (i.e., non-zero value of S E or W S ) are never required to model cation order-diorder phenomena unle there i a trong diparity in the effect of temperature on the coordination number or geometry of one ite over the other (e.g., pigeonite high Ca-pyroxene, Sack Ghioro 99b). Such i the cae in karrooite (Brown Navrotky 989; Yang Hazen 998), the conequence can be een in the intrinic curvature of the array of point plotted in Figure. Even if the C point i removed from conideration, on the argument that the highet temperature equilibrium ordering tate may be unquenchable, the curvature i till apparent outide the σ uncertainty of the three point at, 8, 7 C. In the context of our model expreion (Eq. '), thi curvature can be accounted for by varying either the lope of the curve a a function of T (non-zero W S ) or the intercept (non-zero S E ) or both. Leat-quare analyi of thee three poibilitie, utilizing maximal F-value a an optimality criteria (Daniel Wood 98), indicate a parameterization with S E et to zero i the bet choice. The reulting value for H E, W H, W S are provided in Table. The model curve i plotted in Figure. In Figure, we plot a a function of T howing our model prediction indicating the meaurement of Yang Hazen (998) the inference of Brown Navrotky (989). Ordering tate etimate derived from the latter tudy were not obtained from ingle crytal tructure refinement conequently are not to be regarded with the ame weight a the data

3 37 GHIORSO ET AL.: KARROOITE THERMODYNAMICS TABLE. Suggeted model parameter Model parameter Value H E 7.77 (kj) S E V E W H (kj) W S.6 (J/K) W V.7367 (J/bar) V Kar, (J/bar) V Kar,T (J/bar-K) V Kar,P (J/bar ) V Kar,T, P. (J/bar-K ) Note: V Kar V Kar, + V Kar,T T + V Kar,P(P ) + V Kar,T, P(P ) V V + V Kar E( )+ WV ( ). (6) The compreibility (β) i the invere of the bulk modulu (K) i defined in term of the iothermal preure derivative of the volume: β V V P. (7) To obtain a model expreion for the compreibility, we take the derivative of Equation with repect to preure at contant temperature: V d G d G G + P P P + d d d P + P. (8) The firt term on the right h ide of Equation 8 vanihe in the pecial cae of equilibrium cation order, but the next two term do not! They repreent the change of volume with repect to preure induced change in the ordering tate. The partial derivative of G are eaily evaluated from our model expreion: G RT( 5 + 3) ( + )( ) ( 3+ ) W G (9) FIGURE. Ordering parameter () plotted a a function of temperature (T) for Karrooite. Heavy vertical black line repreent data of Yang Hazen (998) with two igma uncertaintie in. Data plotted a thin olid dahed vertical line repreent inference of ordering tate calculated by Brown Navrotky (989) from high-temperature lattice parameter data. The olid curve i the model repreentation of the data at bar preure. Dahed curve are model iobar at preure of,, 6, 8 GPa (only the 8 GPa curve i labeled). of Yang Hazen (998). The greatet diparity between the two et of meaurement arie at elevated temperature mot likely reflect difficultie aociated with quenching of ample. Having etablihed a model for the variation of the order parameter with temperature at bar preure, we next turn to meaurement of molar volume compreibility in order to calibrate the remaining model parameter. From the definition of the total derivative of G : G dg G + T T + d d d P P (3) we obtain a general expreion for V ( d G at contant T), d V + () P dg which for the pecial cae of equilibrium cation order ( ), d implifie to: G V P. (5) Differentiating our model expreion for G according to Equation 5 yield: G V P G P E WV () V Kar. () P The variation of the order parameter with preure in an equilibrium tate of cation order can be obtained utilizing the method of Ghioro (99b). A G in a tate of equilib- rium order, the total derivative of G mut alo be zero: G G d d d d T T G + + P P () from which at contant temperature: d G G P. (3) Subtituting the definition in Equation 3 into Equation 8, give the preure dependence of the volume in an equilibrium tate of cation order: V G G G +. () P P P Subtitution of Equation 9,, relate thi expreion back to our adopted model parameter. In Figure 3 we plot meaured molar volume of karrooite from Yang Hazen (998) a a function of ordering tate (a) temperature (b). In addition to thee data, iothermal bulk moduli for thi phae are plotted in Figure a a function of temperature. Thee etimate are from the analyi of Hazen Yang (997) are baed on diamond-anvil cell experi-

4 GHIORSO ET AL.: KARROOITE THERMODYNAMICS 373 DISCUSSION The volume change aociated with diordering the cation ditribution in karrooite i given by V V ( ) : di V V E( )+ Wv ( ). (5) Subtitution of parameter value from Table demontrate that the right-h ide of Equation 5 i alway poitive, thereby implying that the Gibb free energy of the diordered tructure i alway more poitive than the ordered tructure at a given temperature a preure i increaed. Conequently, there i a thermodynamic driving force for karrooite to become more ordered a preure i increaed at contant temperature. Thi effect can be quantified by calculating the variation in ordering tate a a function of temperature at variou preure. Curve depicting thi variation at a total preure of,, 6, 8 GPa are plotted in Figure. One intereting apect to thee calculation i that the effect of P on appear to be maximized at intermediate temperature. The maximum repone correpond to the point of inflection in the bar T- curve of Figure. We examine thi apect of the dependence of on P in more detail in Figure 5. Here the derivative, FIGURE 3. Volume (V) plotted againt ordering parameter () in a againt temperature (T) in b. Data are from Yang Hazen (998) hown with σ uncertaintie. The olid curve are the model repreentation of the data. FIGURE. Bulk modulu (K) plotted againt temperature (T). Etimated value are from Hazen Yang (997) hown with σ uncertaintie. The olid curve i the model repreentation of the data. ment on ynthetic ingle crytal annealed at the indicated temperature quenched to room temperature. Both of thee dataet are fitted to the propoed model imultaneouly; for reaon outlined above, the 6 C volume determination were afforded zero weight. The reulting model curve are plotted on both figure the optimal model parameter value are provided in Table. d V E + WV RT( 5 + 3) W TW + ( P ) W + 3 ( )( )( + ) [ H S V ] (6) (ee above) i evaluated at a preure of bar i plotted a a function of temperature. The maximum in d/, een at a temperature of around K, confirm the inference drawn from the pacing of the iobar in Figure. The calculation diplayed in Figure 5 alo illutrate a general feature about the thermodynamic repone to non-convergent cation ordering phenomena: maximal temperature-variation in thermodynamic propertie i oberved at a temperature that correpond to the maximal rate of change in. Thi principle i bet illutrated for karrooite by calculating a a function of temperature the Gibb free energy, entropy, enthalpy, heat capacity of diorder: G di, S di, H di, C P di repectively. Thee quantitie are defined analogouly to Equation 5 are given by: di conf G TS + GE( )+ WG ( ) (7) di conf S S + S E( )+ WS ( ) (8) di H HE( )+ WH ( ) (9) di G G CP T T (3) where the later follow from an expreion for the molar heat capacity, G G G CP T T (3) T T which may be derived in a fahion identical to Equation. Curve depicting the temperature dependence of G di, S di,

5 37 GHIORSO ET AL.: KARROOITE THERMODYNAMICS H di, C P di at bar G di at 8 GPa are plotted in Figure 6. Note that the maximum in C P di of ~55 J/K-mol at ~95 K i roughly 3% of the total heat capacity of karrooite at thi temperature (Berman Brown 985). Note alo the ympathetic variation of S di H di how thi variation i to a certain extent, moderated by cancellation in G di. The thermodynamic analyi of cation-ordering in karrooite developed in thi paper could be improved if in itu high-temperature cation-ordering data were available. The aumption that the meaurement of Yang Hazen (998) repreent quenched equilibrium tate of order need verification a the kinetic of ordering may be rapid enough, epecially at high-t, to introduce ytematic bia in the calibration. Neverthele, the model formulation developed in thi paper can be readily ap- FIGURE 5. The rate of change of ordering parameter () with repect to preure (P) plotted againt temperature (T). The modeled curve (Eq. 6) diplay a maximum at ~7 C. FIGURE 6. Calculated variation of the Gibb free energy ( G di ), entropy ( S di ), enthalpy ( H di ), heat capacity ( C P di ) of diorder plotted a a function of temperature (T). The olid curve correpond to a preure of bar. The dahed curve how variation in G di at a preure of 8 GPa. plied to the iotructural Fe O 5 (known a ferro-peudobrookite), in principle, can be extended to armacolite (Fe + -Mg) olid olution, more oxidized varietie of peudobrookite (in the ytem Mg O 5 -Fe O 5 -Fe O 5 ) found in ome alkalic volcanic rock. What i required to extend the preent analyi to thee compoition i data on cation-ordering of a quality comparable to that of Yang Hazen (998). A demontrated in thi paper, thee data provide all the eential information to define an equation of tate for the phae that include the important effect of cation order-diorder. ACKNOWLEDGMENTS Material upport for thi invetigation wa provided to MSG by the National Science Foundation (OCE-95979). REFERENCES CITED Berman, R.G. Brown, T.H. (985) Heat capacity of mineral in the ytem Na O- K O-CaO-MgO-FeO-Fe O 3-Al O 3-SiO -O -H O-CO : repreentation, etimation, high temperature extrapolation. Contribution to Mineralogy Petrology, 89, Brown, N.E. Navrotky, A. (989) Structural, thermodynamic, kinetic apect of diordering in the peudobrookite-type compound karrooite, Mg O 5. American Mineralogit, 7, 9 9. Daniel, C. Wood, F.S. (98) Fitting Equation to Data, nd ed. Wiley, New York, 58 p. Ghioro, M.S. (99a) Thermodynamic propertie of hematite-ilmenite-geikielite olid olution. Contribution to Mineralogy Petrology,, (99b) Application of the Darken equation to mineral olid-olution with variable degree of order-diorder. American Mineralogit, 75, Ghioro, M.S., Evan, B.W., Hirchmann, M.M., Yang, H. (995) Thermodynamic of the Amphibole: I. (Fe +,Mg) Cummingtonite Solid Solution. American Mineralogit, 8, Hazen, R.M. Yang, H. (997) Increaed compreibility of peudobrookite-type Mg O 5 caued by cation diorder. Science, 77, Hirchmann, M. Ghioro, M.S. (99) Chemical potential of NiSi.5 O, CoSi.5O, MnSi.5O in magmatic liquid application to olivine-liquid partitioning. Geochimica et Comochimica Acta 58, 9 6. Sack, R.O. Ghioro, M.S. (989) Importance of conideration of mixing propertie in etablihing an internally conitent databae: Thermochemitry of mineral in the ytem Mg SiO -Fe SiO -F-SiO. Contribution to Mineralogy Petrology, 68. (99a) An internally conitent model for the thermodynamic propertie of Fe-Mg-titanomagnetite-aluminate pinel Contribution to Mineralogy Petrology, 6, (99b) Chromian pinel a petrogenetic indicator: thermodynamic petrological application. American Mineralogit, 76, (99a) Thermodynamic of multicomponent pyroxene: I. Formulation of a general model. Contribution to Mineralogy Petrology, 6, (99b) Thermodynamic of multicomponent pyroxene: II. Phae relation in the quadrilateral. Contribution to Mineralogy Petrology, 6, (99c) Thermodynamic of multicomponent pyroxene: III. Calibration of Fe + (Mg), Al(MgSi), Fe 3+ (MgSi), AlFe 3+ (MgSi), NaAl(CaMg), Al (MgSi) Ca(Mg) exchange reaction between pyroxene ilicate melt. Contribution to Mineralogy Petrology, 8, (998) Thermodynamic of feldpathoid olution. Contribution to Mineralogy Petrology, 3, Thompon, J.B. Jr. (969) Chemical reaction in crytal. American Mineralogit, 5, (97) Chemical reaction in crytal: Correction clarification. American Mineralogit, 55, Waychuna, G.A. (99) Crytal chemitry of oxide oxyhydroxide, In Lindley, D.H., Ed., Oxide Mineral: Petrologic Magnetic Significance. Mineralogical Society of America, Review in Mineralogy, 5, 68. Yang, H. Hazen, R.M. (998) Crytal chemitry of cation order-diorder in peudobrookite-type Mg O 5. Journal of Solid State Chemitry, 38, 38. MANUSCRIPT RECEIVED APRIL 3, 998 MANUSCRIPT ACCEPTED APRIL, 999 PAPER HANDLED BY REBECCA LANGE