Densities of Na2O-K2O-CaO-MgO-FeO-Fe2O3-Al2O3-TiO2-SiO2 liquids: New measurements and derived partial molar properties

W16-7037/87/53.00 + .oO

Vol.51,pp.293I-2946 QPergamonJou~Ltd.1987.tintedinU.S.A.

Geochrmtca n Cosmoehrmica ~cta

Densities of Na20-K20-Ca0-Mg0-Fe0-Fe203-A1203-Ti02-Si02 liquids: New measurements and derived partial molar properties REBECCAA.LANGE and

IAN

S.E. CARMICHAEL

Department of Geology and Geophysics, University of California, Berkeley, CA 94720, U.S.A. (Received February 2, 1987; accepted in revisedform August 5, 1987) Abstract-The densities of 27 liquids in the system K&I-Na@-CaO-MgO-FeO-Fez03-AlzO~TiQ-SiOz have been measured using the double-bob Archimedean technique. These results indicate that multicomponent silicate liquid volumes have a linear dependence on composition with the exception ofthe TiO, component. The equation:

was used to derive values of the oxide partial molar volumes (v,) by the method of least squares. Regressions were made separatelyat 1573, 1673, 1773 and 1873 K (as liquid ferric-ferrousratios change with tempera+) with relative standard errors for each fit of 0.38%, 0.32%, 0.30% and 0.32%. respectively. Derived dV,/dT values by separate least squares regression for each oxide component reproduce the measured dV/dT of the experimental liquids by 20.2 I W on average. The effects of iron redox state on the density of a variety of natural liquids are demonstrated and at most amount to a variation of 1%. The-se.new data on silicate liquid volumes were used to x-derive oxide partial molar compressibilities, dti,;,ldP and &T, at 1 bar from ultrasonic velocity and calorimetric data from the literature. The fits for dV,,,/dP and & at 1673 K have relative standard errors of 3.9% and I .8%, respectively, which represent substautial improvements over previous fits. Two applications of these volume data are given: firstly, the fusion curve of diopside is calculated up to 95 kbr using an equation of state for liquid volume expressed as: V(T,P)= V(T)exp

s

-j3( T)( I - bP+ cP2)dP

(where @ = K-’ and therefore dK/dP = K’ = -j3-2d@/dp); secondly, using a range of K’ appropriate to a komatiite liquid. the pressure-temperature conditions where the density ofthe liquid equals that of its olivine phenocrysts are calculated.

INTRODUCTION

TINGA et al., 1982; STEBBINSet al., 1984) is a function of the techniques employed (maximum-pressure bub-

THE IMPORTANCEOF precise thermodynamic data on silicate liquids to igneous petrology is demonstrated through the successful applicatiod of a regular solution model to predicting phase relations in evolving magmatic systems (GHIORSO and CARMICHAEL,1985). In order to describe crystal-melt equilibria during fractional crystallization and assimilation, a complete description of the thermodynamic properties of the melt is necessary. Although accurate calorimetric measurements on multicomponent silicate liquids are available (RICHET and BOTTINGA, 1984. 1985; STEBBINSet al., I984), values of their volume and thermal expansion are less secure (BOTTINGA et al., 1984; GHIORSO and CARMICHAEL, 1984). primarily due to the effects of interlaboratory errors. The atthinment of precise liquid density data is of critical importance to petrogenetic models focused on crystal-melt density relations (HERzBEkG, 1984; NISBET and WALKER, 1982; OHTANI, 1984; STOLPERet al., 1981), especially since derivation of the pressure dependence of silicate liquid volumes (RIGDEN ef al., 1987a,b; RIVERS and CARMICHAEL, 1987) depends upon accurate 1 bar values as a function of temperature. The uncertainty within the density data set used to fit previous models of silicate liquid volumes (BOT-

ble method, counter-balance sphere densitometer, Xradiography, double-bob Archimedean method) which are of variable precision and accuracy. Our goal, therefore, is to provide a comprehensive data set for silicate liquid densities using the most precise technique available, the double-bob Archimedean method. In this paper, we present our results for 27 liquids in the K20Na20-CaO-Mg0-FeO-Fe203-A1203-Ti02-Si02 system and consider whether the volume of a liquid can be expressed as a simple linear function of composition. Our density measurements on iron-bearing liquids are combined witti those of MO el al. ( 1982) and allow an evaluation of the effect of oxidation state on natural melt densities, an effect rarely considered. Finally, the derivation of precise dri#rJdP values from ultrasonic velocity (RIVERS and CARMICHAEL,1987) and calorimetric data (RICHET and BOTTINGA, 1984, 1985; STEBBINSet al., 1984) combined with our v, and dv,,l dT values have been used in two petrologic applications: an analysis of the diopside fusion curve to derive values of the pressure derivative of the bulk modulus (K’) for diopside melt and a reexamination ofthe controversial proposal of STOLPERet ai. ( 198 1) that olivine will float in primary melts at sufficiently high pressures within the mantle.

2931

R. A. Lange and 1. S. E. Carmichael

2932 EXPERIMENTAL

PROCEDURES

Techniques for the double-bob Archimedean method are described in detail by NELSON and CARMICHAEL(1979). In this study, however, several modifications to their apparatus and procedure were made in order to improve the overall precision of the measurements. A MoSir furnace with an electronic temperature controller that operates up to 1873 K was used to heat the silicate liquid sample contained within a platinum crucible. An S-type thermocouple (Et-PtlORh), calibrated at the melting point of Au, measured the temperature at the top of the crucible. A correction was made for the thermal gradient between the top of the crucible and the surface of the silicate liquid. An electronic balance with a precision of kO.0001 g, mounted on a concrete platform above the furnace, measured the weights of the platinum bobs used in the experiment. The buoyancy of a platinum bob (weight in air corrected to vacuum less the weight in silicate liquid) is used to calculate the density of the liquid sample from the formula of Archimedes: B(T)+S(T) P(T)

=

f’(T)

where B(T) = buoyancy of the bob at temperature T, p(T) = density of the liquid at temperature T, S(T) = liquid surface tension on the emergent stem of the bob at temperature T, V(T) = submerged volume of the bob at temperature T. The data of WASEDAet al. ( 1975) and EDWARDS et al. (195 I) were used to compute the thermal expansion of pure platinum. The effect of surface tension on the emergent stem of the bob was eliminated by repeatmg the buoyancy measurements on each liquid with a second bob of different volume but identical stem diameter. The following relation allows density to be calculated without incorporation of surface tension effects:

p(T)=

IO x IOeJ K-i) this uncertainty in temperature contributes to an uncertainty in the buoyancy measurements of only 0.03% for the least expansive liquids and 0.07% for the most expansive. It should be noted that those liquids most sensitive to errors in temperature are the most fluid and hence, have the smallest scatter in actual buoyancy measurements. The error assigned to the values of the submerged bob volumes: I’, and V2(calculated from the measured weight of the bobs in air at 298 K and the density of pure platinum) is small, -0.06%. Therefore, the overall precision in our measurement of density varies between 0.14% and 0.33% depending upon the viscosity of the sample. Our assessment of the technique’s accuracy is based upon our measurement of the density and thermal expansion of liquid NaCl (Table 1) which are within 0.2% and 7.5%. respectively. of the values reported by the National Bureau of Standards (JANZ. 1980). However, the accuracy is expected to be worse for silicate liquids due to their higher VIScosities. Sample preparationand analyses Samples were prepared by mixing appropriate proportions of reagent grade Si02, Ti02, A1203, MgO. CaCOX, Na2COs and K2C03 powders. Each sample was initially decarbonated (if necessary) at - 1 I73 K and then fused at 1573 to I773 K. The samples were quenched to a glass, ground to a powder and re-fused. This procedure was repeated twice to ensure a homogeneous glass sample. The glass was then loaded into the crucibles used in the buoyancy measurements. After the measurements on a sample were complete, the liquid was heated in a box furnace to 1773 K and then poured onto a graphite slab and quenched to a glass. These quenched samples were analyzed by wet chemistry and their analyses are listed in Table 2. The uncertainty of a single oxide concentration is estimated to be less than 1.0% of the amount present. This error, combined with our estimated precision ofdensity, leads to an overall uncertainty in volume which varies between 0.2 I % and 0.39%. depending upon the viscosity of the liquid.

S,(T) - B2( T) V,(T)- V2(T)’

An electronic temperature controller enabled the temperature in the furnace to be reproduced within f 1.5K at a given setpoint which allowed the buoyancy of both bobs to be measured at the same specified temperatures. For each liquid sample, buoyancy measurements were made at three to four temperatures with the lowest temperature approximately 50 K above the liquidus and the highest temperature between I823 and I873 K. This procedure differs from that previously followed (BOCKRISer al., 1955; NELSONand CARMICHAEL, 1979) and eliminates the necessity of fitting a least squares line to the buoyancy-temperature data on the large bob and solving for density at the measured temperature of the small bob, and then inverting the procedure by fitting the small bob buoyancy data and solving for density at the measured temperature of the larger bob. Instead, four to six buoyancy measurements were made on each bob at each temperature. Experrmental precrslon and accuracy The replicate buoyancy measurements for each bob allowed an average buoyancy to be calculated at each temperature as well as a standard deviation based upon the scatter for a given set of measurements. The most viscous liquids have the largest standard deviations corresponding to a precision in buoyancy of 0.282, whereas the most fluid liquids have the smallest standard deviations corresponding to a precision in buoyancy of 0.09%. Considering the precision of S-type thermocouples at these elevated temperatures and additional uncertainties regarding thermal gradient corrections, we report an uncertainty in temperature of +5 K. Since the thermal expansions of silicate liquids are small (typical values range between 3-

RESULTS

K,O-Na20-CaO-MgO-A1203-Si02 Density

measurements

on

liquids I9 liquids

(chemical

analyses in Table 2) in the KzO-NazO-CaO-MgOA1203-Si02 system are presented in Appendix 1. Within the resolution of our measurements, the volumes of the melts have a linear relationship with temperature: V = a + bT. Table 3 lists the equations for molar volume as a function of temperature for each experimental liquid. Density measurements were initially made on six liquids confined to the CaO-A1z0&02 system. The results of our measurements were analyzed using four different regression equations from which the partial molar volume of each oxide component, r, at 1773 K, were obtained. The first equation assumes that the partial molar volume of each oxide component is inTable

I

Denmy

of mollen

NaCI (B/cc)

T (K)

P-

PC&’

1209 1281 1354

I.409 I .a9 1404

1.483 I.444 1.404

Th

study:

‘JANZ

(1980):

p = 2.1985

- 5863xW’

T

P - 2.1389

- 5.426xlLP

T

Densities of silicate liquids Tabk 2. 0emicrl

sample

1

3 53.23 _

53.40 _

Sio, siq

23.07

z_-_-_-pea Mp

5.53

_-

-

4

5

36.70 _

6.33 _

27.09 _-_-__

6 45.21 _

maMe

of cxpmmmul liquids (mole %).

1 36.12 _

2933

10

11

12

13

14

15

61.20 -

56.11 -

49.02 _

51.14 -

46.74 _

49.83 -

47.59 -

13.07

17.28

15.27

-

-

9

8 60.74 _

2969

11.64

8.25

9.46

9.51

10.48 -

-

-

-

-

-

-

-

33.27 0.09 _ _

37.82 17.45 _ -

14.12 2209 _ -

24.55 25.63 -

15.90 35.29 55.233

---

N-0 &O

19.06 4.47 _

41.u _ _

36.21 _ _

64.18 _ _

43.15 _

55.63 _

29.80 _

0.66 _ -

28.64 0.14 -

efw

68.961

60.734

69.906

69.863

63.200

61.312

62.027

58.269

57.758

57.923

63.734

62.360

54.133

16

17

18

19

20

21

22

23

24

25’

26’

27’

28’

2

56.13 _

55.02 _

65.84 -

67.47 -

58.23 15.47

40.54 8.44

62.96 9.79

31.92 36.73

59.52 -

65.30 -

56.53 0.24

47.56 2.20

49.30 3.44

22 pea MIO (A0 N-0 LO

_ _ 25.11 18.79 -

5.25 _ _ 17.20 17.27 5.26

5.60 _ 2856

_ 16.03 16.50

-_ _ 26.30 -

8.27 _ 34.75 -

11.74 _ 15.51 -

_ 31.35 -

_ 30.51 9.97

13.67 4.61 _ 16.43 -

9.79 2.71 2.39 12.31 13.15 2A6 0.42

10.12 2.50 3.97 15.41 14.07 3.18 1.00

4.13 6.77 4.54 15.07 12.37 2.26 2.14

Efw

54.307

59.907

72.101

65.828

63.391

63.8%

67.051

65.154

62.244

74.554

64.443

64.500

67.066

C80

Sample

’ Compositim at 1773 K

dependent of composition and hence, includes no excess volume terms. The other three equations each include an excess volume term between Al203 and Si02, CaO and AlzO3, and CaO and Si02, respectively: V,,,(T)= CXE

Table 3

SpccYc volrmea of exprimentd liqulac. V = .,nlr

Sunpie 2 3 4 5 6 7 8 9 10 11 12 13 14

+ b (T b x ld

0.398% 0.38320 0.37392 0.32306 0.37759 0.37203

15 16 17 18

0.39531 0.40433 0.39658 0.38803 0.38815 0.38428 0.38182 0.37569 0.39234 0.40420 0.45957

19 20 21 22 23 24 25 26 27 28

0.46639 0.43729 0.37642 0.42760 0.34994 0.41464 0.39696 0.38733 0.38064 0.36684

2.516 2.121 2295 2.262 2.397 2.626 1.821 1.814 1.700 1.847 1.888 2.C03 2.834 2.395 1.981 2.514 4.047 5.010 5.037 3.982 2.267 4.566 3.425 3.634 1.832 3.101 2.UO

1773 K) cc/g Temp. range (K) 1795 1692 1727 1696 1646 1677 1527 1674 1695 1778 1696 1671 1695 1682 1682 1652 1622

-

1851 1866 18% 1876 1847 1854 1827 1823 1855 1840 1854 1823 1833 1830 1832 1840 1826

1421 1351 1% 1592 1713 1644 1635 1580 1590 1650

-

1828 1743 1795 1793 1835 I745 1770 1760 1780 1800

For each equation we have listed the total F statistic, its probability level, the equation’s multiple correlation coefficient and the relative standard error on the fit (Table 4). Based on these criteria, the fourth model equation provides the best fit. Density measurements on I3 additional liquids with the added components K30, Na20, and MgO were combined with those above and with density measurments made in our laboratory on eight liquids in the Na20-A1203-Si02 system (STEIN et al., 1986). This extended data base was used to regress values of the partial molar volumes ofthe oxide components using two different model equations. The first equation, based on the results from the CaO-A1203-Si02 system, includes a single excess volume term between CaO and Si02. The average deviation between the measured and calculated volumes (r) is 0.17%. However, by eliminating two liquids which each have a mole fraction of CaO greater than 0.50 (#5 and #7; Table 2), a second model equation with no excess volume terms provides a comparable fit with an average value of r equal to

Table 4a.

l4odel : 3 4

Ruults

F 24890.0 10S18.1 46080.1 4049816.5

of n@wiona

on 6 CaO-Al@,-Sio,

Rob level

R

.ooO Ml0

.999!a :Z .5?xw9

:Z

Table 4b. Results 00 K20-N8PCa0-Mg0-A120,-SiOl

Modal 1 4

F 1356409 725694

Rob kvel .ooO .cmo

R I multiple caelmion ax5cknl s.R.=r&tiveslMddamrcmcbc6t * I number of compositions in tbc remon

melu. SE.

0.78 1.35 % 9b 0.57 9b 0.06 % ID&R

*

SE.

25 27

0.21% 039 %

R. A. Lange and I. S. E. Carmichael

2934

0.19%. The total F statistic, its probability level and the relative standard error on each of the two fits are shown in Table 4b. These criteria indicate that the linear model equation is valid within the constraint that the liquids have a mole fraction of CaO not greater than 0.50. Titanium-bearing liquids Four additional density measurements were made on silicate liquids containing titanium (Tables 2 and 3). When the volume data on the four titanium-bearing liquids are regressed with the data on the previous 25 liquids (X,o less than 0.50) using a linear model equation, the fit is poor. The average deviation between the measured volumes of the titanium-bearing liquids and those computed from the regressed parameters is equal to 0.71%. However, when the measured volumes of the two sodium-titanium-silicate liquids are regressed separately from the two calcium-titanium-silicate liquids, two excellent fits are obtained with an average deviation between measured and calculated volumes equal to 0.18%. Regressed values of the partial molar volumes of the oxide components ( r,) at 1773 K are presented in Table 5 for both data sets. It is intriguing that the derived partial molar volumes of the oxide components are virtually identical between the two regressions, except for r~,q. The change in rTi% is substantial as it has a value of 28.32 cc/mole in the sodium-titanium silicate liquids and only 23.87 cc/ mole in the calcium-titanium silicate liquids, a difference of 18%. This is especially significant considering that neither pa0 nor v,+,@ change between the two fits. It should be noted that one of the two sodiumtitanium-silicate liquids contains alumina, while the other does not (although they are both peralkaline), and the same is true for the two calcium-titaniumsilicate liquids. Therefore, the difference in PTis between the two fits is not a reflection of effects caused by alumina. Based on these results, the form of the linear model equation was changed to include an excess volume term between NazO and TiOz:

Table 5.

Fit parameters 6erkd

fivm 3 scpamte ren~rions:

2) ~,.,7?3, maok)

3) b73, ~cc/mo*)

SiQ

26.85 (0.06)

26.86 (0.06)

26.92 (0.07)

TiOl

28.32 (0.28)

23.87 (0.10)

23.92 (0.11)

WA

37.29 (0.09)

37.28 (0.09)

37.31 (0.11)

Me0 cao

I 1.92 (0.09)

11.92 (0.10)

11.83 (0.11)

16.94 (0.07)

16.93 (0.07)

16.85 (0.08)

NM W

29.96 (0.16) 46.91 (0.17)

29.96 (0.17)

29.80 (0.19)

46.90 (0.17)

46.78 (0.19)

NW-TQ

18.33 (129)

V,,,(T)= C-UT)

E.nllr+$(T-

1773 K) I + &.@&,0~ Vt+oJ,q.

Volume data on all four titanium-bearing liquids plus the previous 25 liquids were used in this regression. The significance of the large Na*O-TiOz interaction parameter and the implications for either a change in titanium coordination or distortion of TiO, groups is explored further in the discussion section. Comparison wifh previous densit_v measurements A critical evaluation of density measurements reported in the literature is necessary as previous model equations of silicate liquid volumes (calibrated on these data) have errors of l-2% (B~TTINGA and WEILL. 1970: NELSON and CARMICHAEL. 1979; MO et al., 1982; BOTTINGA et al., 1982: STEBBINSet al.. 1984). As the different techniques employed to measure densities (maximum bubble pressure method, counter-balanced sphere densitometer. X-radiography. double-bob Archimedean method) are of variable precision and accuracy, substantial interlaboratory errors are incorporated into the derived models. It is necessary, therefore, to establish some criteria by which previous measurements can be evaluated. The following discussion is an assessment of the various techniques described in the literature and outlines the criteria by which previous measurements were either incorporated into or excluded from our final data set. The maximum bubble pressure method is a technique with an estimated precision of +-2.00/oin density (BARRETT and THOMAS, 1959). The two most recently proposed models of silicate liquid volumes (BO-ITINGA et al.. 1982; STEBBINSet al., 1984) do not include measurements obtained by this method in their data base. We have followed suit and exclude data on the CaOA1203-Si02 system from BARRETTand THOMAS ( 1959). the FeO-SiOz and CaO-FeO-Si02 systems from HENDERSONet al. ( 196 1) and HENDERSON ( 1964), and the FeO-SiOz , CaO-FeO-SiO* and MO-FeO-SiOz (M = Ca, Mn, Co, Ni) systems from GASKELL and WARD ( 1967), LEE and GASKELL ( 1974) and GASKELL et al. ( 1969), respectively. A second technique uses a counter-balanced sphere densitometer (RIEBLING, 1964, 1966) or similar ap paratus (SHARTSIS et al.. 1952; LICKO and DANEK, 1982). This method measures the velocity of a falling bob (attached to an analytical balance) in a silicate liquid and is also used to derive the liquid’s viscosity. As the change in weight of the bob versus velocity is a straight line, the weight of the bob in the liquid sample is determined by extrapolation to zero velocity. The buoyancy of the bob is calculated from the difference between the weight of the bob in air and that (extrap elated) in the liquid sample. However, an important correction involves subtraction of the liquid’s surface force on the bob’s stem. RIEBLING ( 1964, 1966) followed the correction of SHARTSISand SPINNER( 195 1)

Densities of silicate hquids

and assumed that the surface force is independent of temperature and is a linear function of some mean melt surface tension. The equation S = 0.46~

2 was 0 g used to calculate the liquid’s surface force where d = diameter the suspension wire, g = acceleration of gravity, y = surface tension of the melt, 0.46 = empirical correction for thin wires, and the contact angle is assumed to be zero. STEINet al. ( 1986) used a doublebob Archimedean method to measure densities of the same liquid compositions as RIEBLING (1964) and found their density measurements to be systematically lower by 0.4%. Furthermore, STEIN et al. (1986) used their double-bob data to calculate the surface force (S = p V - B) for each melt composition and found that S decreases strongly with temperature. The inadequacy of a simple correction for surface force is apparent (not only for silicate liquids but for molten NaCl as well: STEIN et al., 1986) and may explain the deviation between the two sets of measurements. We have chosen, therefore, to incorporate the measurements of STEIN et al. (1986) into our data set and to exclude those of RIEBLING ( 1964). To be consistent we have excluded the data of RIEBLING(1966) on liquids in the system MgO-A1203-SiOz and the data OfSHARTslSet al. ( 1952) on alkali-silicate liquids. The density measurements of !%KOLOVet al. (1971) and LICKOand DANEK( 1982) were also obtained using a falling sphere method. SOKOLOVet al.‘s (197 1) density measurements on CaO-AlzOs-SiOz liquids are an average of 1.8% lower than our calculated values while LICKO and DANEK’S (1982) measurements on CaOMgO-SiOz liquids are both higher and lower than our calculated values by an average of 2.8%. For the reasons outlined above, we exclude these data from our data set. In addition, LICKO and DANEK (1982) fitted their measured volumes to equations which include excess volume terms of the form: V= P,X, + && + &( 1 - &)(A + BX2 + CA-:). Their reported values of the oxide partial molar volumes at 1873 K ( rslol = 25.96 cc/mole, vMp = 13.54 cc/mole, v-o = 18.28 cc/mole) are similar to those derived by BOI-TINGAet al. (1982) ( psro, = 26.86 cc/ mole, PMp = 12.12 cc/mole, Tao = 17.10 cc/mole). Yet, when we regress their volume data using a simplified model equation with no excess terms:

the fit is excellent with an average deviation between measured and calculated volumes equal to 0.1796, well within their experimental precision. However, the derived values of the oxide partial molar volumes (~s,s = 8.14 cc/mole, vm = 13.31 cc/mole, V,, = 35.80 cc/mole) have unrealistic values for SiOz and CaO. This result is probably due to a narrow representation in Si02-CaO compositional space. An innovative technique developed by RASMUSSEN and NEL..SON( I97 1) measures the densities of refractory liquids from their X-radiographic images. The samples

2935

are sealed in molybdenum containers and heated to temperatures up to 2273 K (monitored with a twocolor pyrometer to + 10 K). The volume of each melt is calculated from the measured height and crucible diameter (corrected for thermal expansion). AK~AY et al. ( 1979) used this technique to measure the densities of melts along the A1209-Si0z binary with a reproducibility of -+0.004 g/cc. However. the accuracy of their measurements is difficult to assess due to the formation of bubbles and the presence of cores in the melts. Extrapolation of our fitted model equation (calibrated on measurements made between 1373 and 1873 K) to the elevated temperatures of the A1z03-Si02 liquids studied by AKSAYet al. (1979), indicates volumes which are systematically higher than those measured. The deviation is 2.4% for a liquid with XAIf13= 0.30 and 6.6% for one with X,,,,o, = 0.47. These discrepancies are not surprising considering the large extrapolation in temperature coupled with substantial errors in the fitted dr,,ldT terms. More fundamentally, these discrepancies might also be attributed to a breakdown in compositional linearity in volume due to changes in Al coordination at higher mole fractions of A1203. We recommend, therefore, that our fitted model equation not be used in temperature and composition regimes which are very different from those of the measurements used to calibrate the model. Other experiments using the double-bob Archimedean technique have been reported in the literature, such as those by TOMLINSONet al. (1958) using MO and W crucibles and bobs. We do not incorporate their measurements on MgO-SiOz, CaO-SiOz, BaO-Si02 and SrO-Si02 liquids as some replicate poorly (cu. 1o/o); the MO/W thermocouples used by them have a large uncertainty; some of the liquids would undoubtedly react with the metal apparatus between 1873 and 2223 K; and the thermal expansions up to these temperatures for both MO and W were not well known in 1958 (TOULEKIANet al.. 1975). However, in a comprehensive regression of volume measurements on iron-free liquids, we include the data of BOCKRIS et al. (1955) on binary alkali silicates and those of STEIN ef al. (1986), as mentioned earlier, on sodium aluminosilicates as both were obtained by the platinum doublebob technique. The derived partial molar volumes and their temperature derivatives for each oxide component are presented in Table 6. The average deviation between measured and calculated volumes is 0.17%, while the relative standard error in V (1773 K) and dV/dT for the data set are 0.19% and 22.6%, respectively. We do not present a value of dFNa+Tiq/dT, since its standard error is large and incorporation of this term into the regression changed neither the quality of the fit nor the derived values of v, and dVJdT sufficiently to be considered statistically significant. Iron-bearing liquids In order to derive a volume equation applicable to magmatic systems, density measurements on four liq-

2936

R. A. Lange and 1. S. E. Carmichael

dii/dT

%77,r. SiO, TiO,

26.88 (0.02) 23.98 (0.10)

fiuQ3 Me0 Cl0 Na20 K20 Liz0

3752 11.85 16.84 2953 47.10 17.42

Na,O-TiOz

20.10 (1.12)

One mndud deviation is in

(0.08) (0.06) (0.05) (0.04) (0.07) (0.06)

pumthe~~. Dua

ments of chir study. Stein cI al.

m in cc/k&e

x 10”

-0.33 8.76 0.74 2.45

(0.28) (1.05) (0.76) (0.87)

4.22 7.90 12.68 5.82

(0.51) (0.46) (0.78) (0.73)

ICI includes

Ihe measure-

(1986) and Bockris et 0. (1955). Units

md cc/mole K

uids containing ferric and ferrous iron were made (Table 2). These were combined with the measurements of MO et al. (1982) on eight FezOr-Fe0 muhicomponent liquids. Although these measurements on ironbearing liquids in air were made using the double-bob method, the errors on their measured densities are notably higherihan on measurements on iron-free liquids. The principal cause for these increased errors arises from uncertainties in the ferric-ferrous ratios in the liquid samples at the temperatures of buoyancy measurements. After a density measurement was complete, a Pt loop tied to a ceramic rod was lowered into the furnace, dipped into the melt and rapidly pulled out and quenched in a cup of distilled water. The process was timed to take approximately five seconds. The analyzed ferric-ferrous ratio of the glass sample was assigned to a quench temperature equivalent to the temperature of the melt in the furnace (i.e. temperature of the last buoyancy measurements). This procedure differs from that followed by MO ef al. (1982) where the crucible was removed from the furnace and the sample was poured onto a graphite slab and quenched to a glass. As this does not allow an adequate assessment of the quench temperature, we loaded MO et al.‘s (1982) samples onto Pt loops and equilibrated them in air over six hours at the temperatures listed in Table 7. The samples were drop-quenched into distilled water and analyzed for their ferric-ferrous ratios. These determinations of ferric and ferrous iron at the respective quench temperatures differ from those reported by MO

Table 7. Composition

of MO el &‘a

ef al. (1982) and are therefore listed in Table 7. We eliminated their sample 003 as our analysis of its total iron differs from that reported by MO et al. (1982). Given the analyzed ferric-ferrous ratio at a known temperature, the appropriate value could be calculated for the liquid at any other temperature using the equation of ULINC et al. ( 1983). Unlike iron-free liquids, &,/dTterms cannot be regressed directly in our model equation as the ferricferrous ratios of iron-bearing liquids change in air as a function of temperature. Instead, we mgmssed partial molar volume values for each of the oxide components separately at four temperatures: 1573, 1673, 1773, and 1873 K (Table 8) using the model equation: b#q(T) = 2 x, ( 7) c(T) + XNagTiOI

Despite the care taken to adequately assess the ferricferrous ratios in these liquids, the incorporation of ironbearing liquids into the data set increases the standard errors on the fit for p,.r and enlarges the average deviation between calculated and measured volumes from 0.17% to 0.2 1% at 1773 K. The iron-bearing liquids as a separate group have calculated volumes which deviate from their measured volumes by 0.49%, 0.40%, 0.39% and 0.41% on average at 1573, 1673, 1773, and 1873 K, respectively. The cause for these increased deviations may reside in structural changes in the liquid related to changes in ferric iron coordination from tetrahedral to octahedral. However, no systematic compositional effect could be detected (i.e., tetrahedral ferric iron in peralkaline melts and octahedral ferric iron in calcic melts). Furthermore, Miissbauer spectroscopy on MO et al.‘s samples (MYSEN et al., 1985) indicate that ferric iron is predominantly in tetrahedral coordination. The dV,/dT terms (Table 8) were derived by fitting a least squares linear regression to the partial molar volumes at each temperature using the equation: v,(T)=

v,.r7rlK+s(T-

(1982) experimental hquids (wt.%).

Oxide

002

004

cul5

006

009

010

52.26

29.05

49.61

44.77

40.08

41.51

32.41

TiOl AI,O,

23.93 7.36 13.46 0.02

-

3.26 13.40 9.66 4.04 ::;

2.97 12.20 14.78 7.02 5.36 8.90 2.25 0.61

2.68 10.77 26.18 7.76 4.75 8.06 2.03 0.55

0.11 2.44 27.26 4.33 17.81 0.27 4.60 0.01

23.35 9.23 35.27 -

k20,

Fe0 Mgo cao N@ W

1732

32.81 5.76 31.63 -

1788

2.63 0.67

1744

1773 K).

Again, we do not present a value of dVNae&dT as the standard error on each value of vr.+o_r,02at a specific temperature (-2.50 cc/mole) far exceeds its vari-

SiO,

003b

pNNp+TiOz.

1788

1788

1744

1788

2931

Densities of silicate liquids

W02 TiO, &Ox Fe,O, Fe0 JMJ cno Ns,O K,O Ll*O NP~O-TIO~

26.92 22.43 36.80 41.44 13.35 II.24 16.27 28.02 44.61 16.19 20.33

(0.07) (0.32) (0.21) (0.31) (0.18) (0.15) (0.11) (0.12) (0.20) (0.18) (2.71)

26.90 23.16 37.11 42.13 13.65 11.45 16.57 28.78 45.84 JO.85 20.28

(0.06) (0.26) (0.18) (0.28) (0.15) (0.13) (0.09) (0.10) (0.17) (0.15) (2.25)

26.91 23.89 37.37 42.97 13.97 11.73 16.85 29.51 47.01 17.36 20.21

(0.06) 10.25) (0.17) (0.29) (0.14) (0.12) (0.09) (0.10) (0.16) (0.14) (2.14)

26.90 24.60 37.63 43.9) 14.23 11.98 17.15 30.26 48.22 17.90 19.99

(0.06) (0.27) (0.18) (0.36) (0.16) (0.13) (0.10) (0. JO) (0.17) (0.15) (2.32)

0.00 7.24 2.62 9.09 2.92 2.62 2.92 7.41 11.91 5.25

(0.50) (0.46) (0.17) 0.49) ( 1.62) (0.61) (0.58) (0.58) (0.89) (0.81)

One standard deviulon LS III pucnlheses. Data xl ineludes the meLPuremems of lbls study. Stern ec A (1986). Bockns ct ti. (1955). md MO et A. (1982). Umts ore UI cclmoJc =d cc/mole K.

ante over 300 K (0.34 cc/mole). The fitted parameters, dv,,ldT, for each oxide allow the dVfdT of each experimental liquid to be reproduced within 20.21% on average. This represents a substantial improvement over previous fits with errors in dV/dT closer to 28.0% (BOTTINGA et al., 1982: STEBBINS et al., 1984). The most direct assessment of the ability of our model equation to predict the densities of natural melts is to compare measurements of their volumes to those calculated by the model. Three of the four iron-bearing liquids we measured are natural melts: an andesite from Manam Island, New Guinea (#26), a basanite from the Korath Range, East African Rift (#27) and a ugandite spiked with 10 wt.% Fe203 (#28). These liquids were removed from the data base and values for vi.r were re-derived and used to calculate the volumes of these natural melts. These calculated volumes at the temperatures of measurement (Appendix 1) reproduce the measured volumes by 0.47% on average. We conclude, therefore, that our model can predict the densities of anhydrous and carbonate-free natural melts over a large range of iron redox states. DI!XXJ.SSlON Volumes of the liquid oxide components Values of the liquid oxide partial molar volumes can be compared on many different bases, such as per mole, per gram atom or per oxygen atom. STOLPER ef

Table 9. solid

SIO, ~ghw

CryshJJmc

Volumes of cryAtiiac

solid

liquid

liquid

1673 K

1673 K

1673 K

WclmoJ)

(Cc(mOl)

(CclhDl)

27.27 25.79 22.69 20.53 25.57 30.31 11.98 II.25 16.78 25.88 40.38

27.32 27.44 23.49 21.61 26.57 31.98 12.67 11.95 17.76 -

298K

SIO, (cmmJMte) SiO2 @wW TiO; (A-) ALO. Fe-$; Ho w-0 00 NM KzD

al. (1986) suggest that the volumes per oxygen atom of various silicate melts (anorthite, diopside, An(0.36)Di(0.64) eutectic, and fayalite) are all similar due to similar packing of oxygen atoms in these liquids at atmospheric pressure. It is instructive, therefore, to compare the regressed values of the oxide partial molar volumes on a per oxygen atom basis (listed in Table 9). It is’also informative to compare the liquid partial molar volumes of the oxide components with their crystalline molar volumes (Table 9). Al203 and Fe203 each show a large increase in their liquid partial molar volume values compared to their isothermal crystahine volumes in which the respective cations are in octahedral coordination. The implication is that the regressed values of l&, and rr& at 1673 K are representative of each cation in tetrahedral coordination. Numerous spectroscopic studies on aluminosilicate glasses (i.e. MCKEOWN et al., 1984; WHITE and MINSER, 1984) support the evidence for tetrahedral aluminum in non-peraluminous liquids. In addition, Mbssbauer spectroscopy (MYSENet al., 1985) on the samples studied by MO et al. ( 1982) indicate that ferric iron is also predominantly in tetrahedral coordination. Elimination of the two calcium-titanium silicate liquids in a regression without a Na20-Ti02 interaction parameter results in a partial molar volume for TiOz equal to 28.32 cc/mole at 1773 K (Table 6). This liquid volume is much larger than the volume of anatase which contains Ti4+ in octahedral coordination. By analogy with the liquid/solid ratios at 1673 K for Al203

volume mhi tJwmrl

exparion

26.90 26.90 26.90 23.16 37.11 42.13 13.65 11.45 16.57 28.78 45.84

ud J#luid oxldes. Jiqhnl

solid

1673 K

dv,ldT (JO-’ cc/m01

(@) 13.45 13.45 13.45 11.58 12.37 14.04 13.65 11.45 16.57 28.78 45.84

drtr we from TAYLOR

0.99 0.98 1.15 1.07 1.40 1.32 1.08 0.96 0.93 (1984b.

c. d. 1985).

0.04 0.08 -1.18 0.79 0.97 1.26 0.54 0.55 0.74

Jlquid dv,ldT

K)

(lo-‘cc,koJ 0.00 0.00 0.00 7.24 2.62 9.09 2.92 2.62 2.92 7.41 11.91

K)

2938

R. A.

Langeand 1. S. E. Carmichael

and Fe203, it can be inferred that Ti4’ is ~~ornih~~y in tetrahedral coordination in our sodium-bearing experimental liquids. The partial molar volume of 23.87 derived for TiOz in calcium-bearing melts is strikingly similar to that for anatase and suggests that ~tanium is in octahedral coordination in these melts. Although density measurements are not the most direct probe of melt structure, our results are suggestive regarding the coordination of ti~nium. Ti4’ is known to be four, five and six-fold coordinated by oxygen atoms in various crystal structures. A number of recent spectroscopic, X-ray emission and phase equilibria investigations (YARKER et al.. 1986; DICKINSON and HESS, 1985; GREEGOR et al., 1983; IWAMOTO et al., 1982; HANADA and SOGA, 1980) indicate that similar Ti4+ coordinations exist in liquids and glasses. However, the conclusions drawn by these studies arc not always in agreement and the factors controlling the coordination of titanium (i.e. titanium concentration, silica concentration, etc.) are not unambiguously established. A program to measure the densities of several titanium-bearing liquids in order to examine the effect of MgO and K20 on derived values of Prier as well as the effect of mixed cations (i.e. K&4ZaO-TiO~SiOr), is currently underway (JOHNSON and CARMICHAEL, 1987).

Thermal dependence of liquid volumes The precise estimation of the thermal coefficient of expansion for natural silicate liquids is a primary aim of this study. Our measurements of liquid volume with an experimental precision of -0.2% over an average span in temperature of 200 K allows the property, dV/ d7’, to be derived with an estimated average precision of 28%. As described earlier, the partial molar quantities, dF,/dT (Tables 6 and 8), were regressed assuming a linear dependence on com~sition. Calculated values of dV/dT for the experimental liquids are listed in Table IO and compared to their measured values. The average deviation between calculated and measured dV/dT is 13.1% for the liquids measured in this study which is well within their estimated experimental precision. As our determination of dV/dT for each liquid is not direct but derived from measurements of the integral property as a function of temperature, the errors of dpJdT are large. It is thus important to examine our derived dpt/ dT values and consider whether these values represent reasonable estimates. In general, the therma expansion of Iiquids results from an increase in the average amplitude of thermal molecular vibrations. However, covalent bonds such as Si-0 undergo little, if any, expansion and the principal mode of expansion in silica-rich melts involves changes in the Si-0-Si bridging angle. As silica-rich melts are expanded structures with regard to Si-04 bond angles, we anticipate limited expansion through this mechanism. This is consistent with our derived value for dvsloJdT is small and negative in our regres-

SMlpk

dV/dT_ (IO-’ cc/&v K)

dV/UT,, (IO-’ cc/@ K)

r%

1.74 1.29 1.60 1.51

1.49 1.3s I 76 1.56

144 4.6 10.0 3.3

1.12 1.02 1.u 1.33 1.33 I.4b 1.39 1.48 1.20 1.72 3.54 3.14 3.06

:~;I 17:3 24.3 10 8 16.8 9.1 12.1 11.1 13.9 21.2 48

:::

42. 2% I

3.23 2.08 1.91 1.35 1.82 1.99

i:: 29.5 16.1 6.7 12.4

2 3 4 6 8 9 IO 11

12 13 14 I.5 16 17 18 19 20 i: 23

:‘: 26 27 28

1.14 1.06 0.98 1.07 1.20 1.25 1.53 1.32 1.08 1.51 2.92 3.30 3.19 2.54 1.52 2.98 2.13 2.71 1.18 1.95 1.77

13.1

h’jj.

r % - [(dV/dT_.

- dV/dT,,,$dV/dT_JxlOO

sion of iron-free liquid volumes (Table 6) and essentially zero in our regression, which includes iron-bearing liquids (Table 8). In contrast, the Al-O-B bond angle is apparently more responsive to thermal effects as indicated by the derived values for dpa+,/dT listed in Tables 6 and 8 (although their values differ by a factor of 3). For those cations which are octahedrally coordinated, we can compare their respective oxide values of dp,/dT as a function of cation field strength (Fig. l), defined as z/( 1.35 + r)2 where z is the cation charge, r is the cation radius for octahedral coordination and 1.35 is the radius (in ~n~tr~rns) of the dl- anion (SHANNON and PREWITT, 1969). The trend in Fig 1 indicates a pronounced correlation between decreasing thermal expansion and increasing field strength which is analogous to that found for the constant volume partial molar heat capacities (STEBBINSet al.. 1984) as well as for the derived molar sound speeds (RIVERS and CARMICHAEL, 1987) of the oxide components in silicate melts. Volumes qffusion A fundamental application of our volume and thermal expansion data is to calculate liquid volumes and volumes of fusion (AVf ) for a variety of minerals for which the crystalline volumes are known as a function of temperature. Table 11 lists the liquid volumes of several mineral compositions. These values can, as for the oxide partial molar volumes, be compared either on a per mole or on a per oxygen atom basis. We have tabulated the liquid volumes on both bases for each minerai and again note the similarity in volumes per oxygen atom. Of significance. is the small volume of

Densities of silicate liquids .I(

FIG. 1. Fitted values of dv,,ldT and dv,,ldP plotted as a function of cation field strength. This is defined as z/Q + 1.35)* where z is the cation charge, r is the cation radius and 1.35 is the radius of the O-* anion.

fusion for pseudowollastonite relative to diopside and enstatite. This result is primarily an effect caused by the large volume of crystalline Casio3 relative to the

two pyroxenes. In support of this observation, the “Si NMR spectra of crystalline and glassy CaSi03, MgSiOs and CaMgSizOd samples (MURDOCHetal., 1985) suggest that crystalline CaSi03, with its silicate chain repeat length of three tetrahedral units, relaxes in a glass to a chain structure more like that of the pyroxenes CaMgS&O, and MgSiO, . Our calculation of liquid and solid volumes for these metasilicate minerals is in agreement with this interpretation. The volume of fusion for calcic feldspar is also significantly smaller than that for either albite or sanidine, though the crystalline molar volume of anorthite and albite are close at their respective fusion temperatures. This is consistent with TAYLOR and BROWN’S(1979) interpretation of X-ray radial distribution functions (RDF) derived from diffraction data on NaAISisOs, KAlSi-,Os and CaA1&Os glasses. Their experimental RDFs indicate that the alkali feldspar glasses have a tridymite-like structure based on six-membered rings of tetrahedra, whereas the RDF for the calcic feldspar glass indicates a four-membered ring structure of the type found in crystalline feldspars. The lower volume of anorthite liquid is, therefore. a reflection of the tighter T-O-T bond angles and the small A V, reflects the similarity in structure between crystalline and molten anorthite. In contrast, alkali feldspars with fourmembered rings of tetrahedra undergo a significant volume change during fusion to a tridymite-like structure. An important application of these calculations relates to volumetric changes within crystallizing magmas. Ubiquitous quartz and aplite/pegmatite veins

’ The value for vH, is estimated to be 17.1 cc/mole after SILVERand

STOLPER

(1985).

2939

within granitic intrusions exemplify the contraction of silicic magmas during solidification. Approximate calculations, based upon our liquid volume and dVldT data, combined with data for quartz. albite and orthoclase (SKINNER, 1966; TAYLOR, 1984d), indicate that a typical anhydrous granitic melt will contract by 8.2% during crystallization between I 173 and 873 K. Hydrous silicic melts. on the other hand, with l-3 wt.% Hz0 (assumed to completely exsolve during crystallization), undergo a more significant decrease in volume of 9.6-l 1.4% over the same temperature range.’ Likewise, the devitrification of an obsidian glass also leads to a contraction, although the volume of devitrification is smaller than the volume of fusion. To estimate the magnitude of A C>ev,,nhcnuon, the density of an obsidian glass from Mono Craters, CA was calculated at 673 K. Under the assumption that 50% of the glass devitrifies to form spherulites (with the relative proportions of the Qtz, Ab and Or components derived from a CIPW norm calculation), the calculated volume decrease is 2.9%. If the remaining glass is assumed to hydrate with 1 wt.% of water, resulting in expansion, the overall contraction approaches 1.4%. As noted in Table 11, the pyroxenes, enstatite and diopside, are characterized by larger AV, than either quartz or the feldspars. Thus an intrusive tholeiitic liquid is expected to undergo a significant contraction during crystallization comparable to a hydrous silicic melt. Based upon the calculated phase proportions crystallizing from a Thingmuli olivine tholeiite (GHIORSO and CARMICHAEL,1985) the volume data in Tables 8 and 1 I indicate that such a magma’s volume will decrease by 9.4% during cooling from 1473 to 1073 K. It has been postulated by GHIORSO and CARMICHAEL(1987) that the consequent decrease in hydrostatic head during crystallization may induce fracturing of the surrounding country rock and thereby provide for repeated cycles of isochoric crystallization followed by pressure drops until some cracking threshold is reached.

Table II.

Vohmcs

@..i&

(CC%,)

N&USt,O,

1373

112.83

14.10

KAISi,O,

121.24

CA&O,

1473 1830

15.15 13.55

MgSiO, caste,

1830 1817

38.85 43.89

caMgsi,o,

1665

81.80

14.63 13.63

f+fgsto, Fqsto,

2163

52.37

1490

53.15

SiOz

1999

‘SKINNER

108.41

103.72 111.40 103.42

e: al. (1983)

‘SUZUKI

et al. (1981)

6/3-cnstobahte;

1.09 1.09 1.05 1.18

69.10

14.08 11.52

1.04 1.18

13.09

47.742

1.10

4.63

13.28

48.ad

1.09

4.55

26.91’

13.46

27.816

0.97

-0.90

28.15’

14.08

27.816

1.01

0.34

%x~rapokted from fitted vm fim

12.97 13.93 12.93 10.97

12.95

32.92 42.25

(1966) uttkss othenvtse specified

amJxu

‘eXtl~kted

of fttsttm.

t” Tabk 8

M -II

TAYLoR(19844)

of BACON et al. (1960)

R. A. Lange and 1. S. E. Carmichael

2940

Oxidation state and melt densities Although the volumes of iron-bearing liquids are known less precisely than iron-free liquids, the incorporation of ferric and ferrous iron components into our volume equation allows the effect of oxidation state on the densities of melts to be evaluated. The variation in a natural liquid’s ferric-ferrous ratio as a function of temperature, bulk composition, and oxygen fugacity can be calculated from the equations of SACK et al. (1980) and KILINC et al. (1983). This relation, combined with our liquid volume equation, can be used to calculate the variation in a liquid’s density as it equilibrates along four oxygen buffers: iron-wustite (IW), fayalite-magnetite-quartz (FMQ), nickel-nickel oxide (NNO) and magnetite-hematite (MH). These variations in liquid density are presented graphically in Fig. 2 for six natural liquids: a komatiite, ugandite, mid-ocean ridge basalt (MORB), high-alumina basalt, olivine andesite and peralkaline rhyolite. In the case of the komatiite, the liquid is well in the metastable region (i.e. below the liquidus), but nevertheless correctly shows the relative contrast between this and other lava types. An obvious deduction is that liquids with the highest concentration of iron have densities which vary most as a function of oxidation state. Ugandites are ironrich, potassic, undersaturated lavas which, as shown in Fig. 2, are extremely sensitive to redox conditions. Although rare, it is often these deep-seated, alkaline

2.9 Ml4

2.8

Ra 2.6 MM

2.7

lavas which entrain mantle-derived nodules and hence, the transport of these lavas to the Earth’s surface (which depends in part on their density) is of interest. In recent years, there has been considerable debate regarding the oxidation state of the mantle. Intrinsic oxygen fugacity measurements on mantle minerals (SATO. 1972. 1978: ARCULUSand DELANO, 198 I: ARCULUS et al.. 1984) indicate equilibration along the IW buffer, although one type of lherzolite and kimberlitic minerals (ARCULUS et al., 1984) apparently equilibrated between FMQ and NNO. This much variation in oxidation state will affect the density of a ugandite liquid at 1 bar by 0.81%. Mid-ocean ridge basalts are also iron-rich and as a consequence, have densities sensitive to oxygen fugacity. Analyses of ferrous and total iron from 78 handpicked MORB glasses (CHRISTIEef al., 1986) indicate that these liquids (at the point of eruption) have relative oxygen fugacities l-2 log,, units below the FMQ buffer. Previous assumptions regarding the oxidation state of MORBS were based on whole-rock analyses which indicate equilibration close to FMQ. Our density calculations show that equilibration at IW rather than at FMQ produces a difference in density at 1 bar of the order of 0.42%. For comparison, these liquids require a 65 K change in temperature to produce a similar variation in density. Pressure dependence qf liquid volumes Directly derived from our measurements is a model equation which allows the volume of multicomponent silicate liquids to be calculated as a function of temperature and composition at 1 bar. However, magmatic phase relations and fluid dynamic mechanisms relevant to petrogenetic processes cannot be modelled without knowledge of silicate liquid volumes at elevated pressures. Therefore. two essential parameters are the first and second derivatives of volume with respect to pressure. RIVERS and CARMICHAEL(1987) measured ultrasonic sound velocities (c) through silicate liquids to derive isothermal compressibilities (&) using the relation: _

I

1673

I 1673

I 1473

TIKI

FIG. 2. The density of six natural liquids as a function of temperature,with each equilibrated along four oxygen buffers: IW (iron-wustite), FMQ (fayalite-magnetitequartz), NNO (nickel-nickel oxide), and MH (magnetite-hematite). For comparison, the density of crystalline Ana is plottad as it is a common liquidus phase of mid-ocean ridge basalts (MORBs). Note that the komatiite liquid is in a metastable temperature region.

V

TV&

At the time their data were published, the model equations of silicate liquid volumes representing the best fits were those of BorrlNGA et al. ( 1982) and STEBBINS et al. (1984) with standard errors of 1%. As our measurements have produced a substantial improvement in the precision of calculated liquid volumes, it is important to propagate this improvement into the derived isothermal compressibility terms. We have, therefore, recalculated values of (dV/df), and & for the liquid compositions studied by RIVERS and CARMICHAEL (1987) using their velocity data (c). the heat capacity data (C,) of RICHET and BOTTINGA (1984, 1985),

Densities of silicate liquids

2941

STEBBINSet a/. (1984) and ZEIGLER and NAVROTSKY (1986), and volume (VT) and thermal expansion coefficient (a*) values calculated from our model equation. From the recalculated (dl/ldP)r values for each liquid and their analyzed compositions (RIVERS and CARMICHAEL, 1987), values of the partial molar properties, (dv,,ld& can be regressed assuming a linear compositional model, i.e.:

~ompre~ibility, however, for an ideal solution will be a linear function of volume fraction (RIVERSand CARMICHAEL, 1987):

Our fit for (dpJdP), at 1673 K has a relative standard error of 3.9% which is a substantial improvement over the fit based on previous volume data which had a standard error of 13.4% (RIVERS and CARMICHAEL, 1987). As mentioned above, our fit for @,,r Iisted in This is a less convenient formulation than a simple Table 12 assumes a linear dependence on compositi@n algorithm relating compressibility to the mole fractions rather than on volume fraction. However. both regresof the constituent oxides: sions were made for comparative purposes resulting in relative standard errors of 1.8% and 3.8%, respectively. PT= 2 x,@,,T The latter value can be compared to the relative standard error of I 1.5%on the fit reported by RIVERSand where fi,,= is not to be mistaken for a strict partial molar CARMICHAEL( 1987) for &,,=which is based on previous quantity. We have chosen to use the latter formulation volume data. Prior to the exclusion of any liquid comin our regression of compressibility as it provides a positions included in the regression of RIVERS and superior fit to the data. CARMICHAEL(1987) the standard error on our fit for In addition to the 22 synthetic and three natural (~~~~~~ was 5.3%. The reduction in the standard error compositions studied by RIVERS and CARMICHAEL from 13.4% to 5.3% is thus entirely due to the precision ( 1987), their regression of (&,/d&and /?,,r terms also of our volume equation. This is not surprising since a included sound speed data on five NatO-Si02 and six detailed error analysis (RIVERS and CARMICHAEL, K20-SiQ liquids measured by BOCIGUSand KOJONEN 1987) demonstrates that 60% of the un~nty in (dV/ ( 1960), seven K@-SiO* , nine CaO-SiOz and ten l&OdP), arises from errors in aT. SiOz liquids measured by BADOV and KUNIN (1968) The derived values of dV,/dP for each oxide comand three CaO-A120s-SiOz liquids measured by Soponent can be evaluated in a similar manner as those KOLOV etal.(197 I). We also include these data into of dv&dT. Again, the covaient nature of the Si-0 bond our regression with the exception of three CaO-SiO, is such that its principal mode of low-pressure and three Li@-SiOz liquids measured by BAILIOVand compression does not involve shortening of the bond KUNIN (1968) which contain mole fractions of CaO but rather changes in the Si-0-Si bond angle (JORGENand L&O greater than 0.50 (as our volume equation SON, 1978; LEVIEN et af.. 1980). As a high-silica melt is calibrated on liquids which exclude this composirepresents an expanded structure with regard to Si-Otional range). We also exclude one synthetic compoSi bond angles, we expect that (at low pressures) it can sition (K2Si03) studied by RIVERS and CARMICHAEL accommodate a substantial degree of compression ( 1987) which during the course ofthe experiment took through this mechanism. Thus, we anticipate a value 4.7 mole percent of MoO from the crucible and buffer for d&,~2/dF derived from t bar sound speeds which rod into solution. Ail of the other liquids studied by RIVERS~~~CARMICHAEL( 1987)containlessthan 1.0 is not exceptionally small relative to the other oxide component values as it is for dt7s,,/dT. This is, in fact, mole percent MOO. Finally. as our volume equation observed in Table 12 with dp&%/dP more negative does not include R&O, C%O, SrO or BaO, four of than derived values for FeO, MgO and CaO. Like valRIVERSand CARMICHAEL’S(1987) liquids which conues of dV,/dT. those of dp,/dP (for the octahedrally tain these components were eliminated from the fit. coordinated cations) are correlated to cation field Hence, our final regression of (dp,,ldflTand fii,r values strength (Fig. 1) with dF*/dP, in general, becoming is based on 54 liquids relative to the 65 of RIVERSand more negative with decreasing field strength. CARMICHAEL(1987). Table I2 presents these partial molar values and their associated standard errors. Also Fusion curve of diopside shown are the derived values of the temperature dependence of (dV,/dP), and &T with standard errors of Before applying our recalculated (dV/d~~ terms to the order of 100%. predict high pressure melt densities, an appropriate

R. A. Lange and 1. S. E. Carmichael

2942

DIOPSIDE FUSION CURVE I

TIKI

P373-

---V=V*.xF/-fil-bP+cP'1dP 2273-

-Birch-Mumghan

2t73-

P lkbl 10

20

50

10

50

60

10

80

90

100

110

(20

FIG. 3. Equilibrium between diopside crystals and liquid as a function of pressure (in kilobars). The solid line is the calculated curve using a Birch-Mumaghan equation for the liquid. and the dashed line is the calculated curve derived from the integration of the exponential equation. This involves an expression for the compressibility of the liquid at zero pressure, and its first and second derivatives with respect to pressure. as explained in the text. The sources of experimental data are given. of state for silicate melt volumes, V(T, P) must be formulated. Our volume measurements indicate that over the experimental temperature range (I 373-l 873 K) silicate liquid volumes have a linear dependence on temperature: V = a + bT. The functional dependence of compressibilities on pressure, d&/dP, can be determined from measurements of sound velocity through silicate liquids at pressures greater than 1 bar. Although these experiments have yet to be done, they are nonetheless feasible. In lieu of experimental results, however, we can predict that the isothermal compressibility will generally decrease with pressure, more rapidly at low pressures and ultimately will reach an essentially constant value at some elevated pressure. A simple polynomial of the form: equation

,&B(T,P)=&T)[I -bP+cP2] could be used to fit the behavior of B over an experimental range in pressure.2 Volume, which is a thermodynamic state function, would then be expressed as V( T, P) = V(T) exp (s

B(T)[ 1 - bP+ cP’]dP

1

(1)

derived from the relation: tb=-

1 W’) yijjg.

’ Note that for SiOl glass, which shows an increase in compressibility at room temperature up to 25 kbrs KOND~et al. ( I98 I ), the form for @may be more complicated. 3 Note that a correction must be applied to the isentropic bulk modulus to obtain a value for the isothermal bulk modulus. However, a Gruneisen parameter of approximately 0.3 for silicate melts results in an isothermal bulk modulus which is roughly 1%lower than the isentropic bulk modulus. This correction is negligible considering the uncertainties in the derived K’ values.

The advantage of this formulation for the volume of a phase is that it ensures exact differentials, a thermodynamic requirement for any state function (LEWIS and RANDALL, 1961). The form most commonly used to relate pressuredensity relations is the third-order Birch-Mumaghan equation of state: P=;K(R7”-R5’3)[

1 - $(4-K’)(R2”

- l)]

(2)

where R = V,o/V,p and K is the zero pressure bulk modulus and K’ is the derivative of K with pressure. This equation is a truncated series expansion derived from finite strain theory and models the compressions of solids under hydrostatic pressure extremely well. Its applicability to liquids is less well documented, although the use of Eqn. (2) to fit values ofthe isentropic bulk modulus (KS) and its pressure derivative (K’s) for diopside melt from shock-wave data (RIGDEN ef al., 1987b) predicts a value for KS (224 kbr) which compares favorably with that determined from 1 bar ultrasonic sound speeds (242 kbr: RIVERS and CARMICHAEL, 1987). In order to compare results derived from each equation of state (Eqns. I and 2), it is necessary to choose a mineral such as diopside whose thermodynamic properties and fusion curve are well known (up to 95 kbrs; Fig. 3) and to derive the functional dependence of the melt’s compressibility with pressure using Eqn. (1) and to compare the resultant rr values (where K; = -&2d&/dP) with the Kk value obtained by shock-wave experiments (RIGDEN et al., 1987b).3 The results of this exercise are illustrated in Fig. 3 in which the calculated fusion curve is plotted together with the experimental brackets on the melting reaction. The thermodynamic data are taken from BERMAN and BROWN( 1985), RICHET and BOTTINGA( 1984),S~~~

2943

Densities of silicate liquids BINSet al. (1984) ZEIGLERand NAVROTSKY(1986) and Tables 8 and 11. The volume of diopside crystals is represented as:

v ~eclmd)

I’(T)=65.566+

do/dT (IO-’

Fou

wo

44.04

43.90

43.76

2.74

2.75

2.76

1.02

1.02

1.03

K (Id blrs)

12.92

12.87

12.82

WdT (ld dK/dp

-2.35 5.30

-2.32 5.35

-2.30 5.40

a (IO-’

1.7591 X 1O-3 K + 3.1454 X lo-’ KZ cc/mole

which is taken from ROBIE et al. ( 1967) and FINGER and OHASHI ( 1976). The compressibility at 298 K was derived as p = 8.7638 X lo-’ - 2.9864 X lo-‘*P (bars-‘) from the data Of LEVIENet ai. (1979). with the number of decimal places in both equations being included to avoid roundoff errors. The temperature dependence of the solid compressibility was assumed to be equivalent to that for the liquid, namely 2% of its value every 100 K (RIVERS and CARMICHAEL, 1987). The values for 8, p’ and K’ derived from Eqn. (1) at various T, P points along the fusion curve are listed in Table 13. Also shown in Fig. 3 is the fusion curve derived using the Birch-Mumaghan equation of state with a K’ value of 6.9 as measured by RIGDEN ef al. (1987b). It is interesting to note that a K’ value of 6.9 in Eqn. (2) and K’values ranging between 6.2 to 5.7 in Eqn. (1) (derived from the relation: K’ = +F’d&/dP) model essentially the same fusion curve despite the fact that the 1 bar compressibility, volume and calorimetric data used in each equation are identical. The coefficients derived for b and c in Eqn. (I) (b = 8.42 X low6 + 2.49 X 10e9T (bars-‘) and c = 8.98 X lo-” bars-*) were used to calculate the density of diopside melt at elevated pressures. At 1913 K, the calculated density of diopside melt is 2.72 g/cm3 at 12.5 kbrs and 2.75 g/cm3 at 15 kbrs. If & is assumed not to decrease with pressure, then the respective calculated densities are 2.75 and 2.78 g/cm3. Neither of the two sets of calculations match the measured densities of diopside melt obtained by the falling-sphere technique (SCARFEet al.. 1979) of 2.96 g/cm’ at 12.5 kbr and 3.00 g/cm3 at 15 kbr, respectively. This indicates that the falling-sphere method is inaccurate for determining diopside melt densities.

Table13.

PUMClerS for diopside melt derived from

fUnOn CIUW wilb V(T.P) = VotXpj-pcT)[ r

P

p x 104

(K)

(kbr)

(bus-”

1746 1812 1871 1925 1972 2014 2CM5 2075 2100 2146 2191 2255 2311

4.4 9.6 14.9 20.2 25.5 30.8 35.2 39.8 43.9 52.1 61.2 74.9 91.2

4.12 4.19 4.27 4.35 4.42 4.50 4.55 4.61 4.65 4.14 4.82 4.91 4.99

the I - bP + c@]dP

p;x_l_y

K

-1.05 -1.08 -1.11 -1.14 -1.17 -1.20 -1.23 -1.25 -1.27 -1.30 -1.34 -1.37 -1.41

6.2 6.1 6.1 6.0 6.0 6.0 5.9 5.9 5.9 :I 5.7 5.7

K-l) K-*) bars K-l)

Fo9J

V,aandK~eat298Kandlbar. DuauehmSumioo (1979) md Swakn cl al. (1981. 1983).

Crystal-melt density relatrons at high P

It is unfortunate that derivation of a liquids K’ value using Eqn. (1) is restricted to fusion curve analyses. The ultimate goal, however, is to model the densitypressure relations of natural multicomponent melts and for this task the Birch-Mumaghan equation of state is most promising. As demonstrated in the previous section, our 1 bar volume and compressibility data used in the Birch-Mumaghan equation together with the measured K’ value of 6.9 (RIGDEN et al., 1987b) models the experimental fusion curve of diopside extremely well. It is worthwhile, therefore, to use our liquid volume data to re-examine the controversial proposal (HERZBERG, 1984; NISBET and WALKER, 1982; OHTANI, 1984; STOLPERet al., 1981) initially suggested by STOLPERef al. ( 198 I ) that olivine may float in primary melts at sufficiently high pressures within the mantle. The melt composition chosen for our calculations is an average low-MgO spinifex-textured komatiite from the Munro Township, Northeast Ontario as reported by ARNDT et al. (1977). Based on shock-wave experiments on diopside, anorthite and the Di(64)An(36) eutectic composition and the derived K’ values of 6.9, 5.3 and 4.85, respectively (RIGDEN et al., 1987a,b), K’ values for the liquid komatiite were chosen to vary between 5 and 7 in our calculations. The experimental data used to derive VT, q, KT and K; values for Foss, Fh and FOBS(Table 14) are from SUMINO ( 1979) and SUZUKI et al. ( I98 1, 1983). Calculated regions in temperature-pressure space where Ap = 0 for Fog5, Few and Fog5 relative to the melt composition are shown in Fig. 4. According to ARNDT et al. (1977), the average olivine composition in the komatiites of the Munro Township is Fo,o which experimental evidence (BICKLE et al., 1977) indicates is the liquidus composition of olivine in komatiite melts. Our calculations indicate that Fh will begin to float between 100 and 120 kbrs if the K’ for the melt is assumed to be 5. For K’ values approaching 7, Fw will never float as it converts to the spine1 structure well before the projected density cross-over at 180-200 kbars. A K’ value of 5 or less is consistent with the prediction of AGEE and WALKER ( 1987), that phenocrystic olivine in komatiites will float between 80 and 100 kbrs. CONCLUSIONS Volume measurements on 25 multicomponent silicate liquids combined with data from the literature

R. A. Lange and I. S. E. Carmichael

2944 Contours

of Ap-0

d-5

In ‘ KomWft8

I’-7

Meit

by comments from M. Ghiorso, P. Richet, C. Scarle and J. Stebbins. Editorial handling: P. C. Hess

REFERENCES AGEE C. B. and WALKERD. (1987) Static compression and olivine flotation in ultrabasic silicate liquids. J. Geophjs. Rex (submitted). AKSAYI. A.. PASK J. A. and DAVIS, R. F. (1979) Densities of SiO,-AbO, melts. J. Amer. Cerum. Sot. 62. 332-336. ARCULU~R.-J.-and DELANOJ. W. (1981) fn&nsic oxygen fugacity measurements: techniques and results for spinels from upper mantle peridotites and megacryst assemblages. Geochim. Cosm~him, Acta 45.899-9 13. RG. 4. The caktdated pressure-temperaturecoordinates for ARCULUS R. J., DAWSONJ. B., MITCHELL R. H., GUST D. A. and HOLMESR. D. (I 984) Oxidation states of the the density of a komatiite liquid matching exactly the density upper mantle recorded by megacryst ilmenite in kimberlite of olivine phenoctysts of the labelled composition. Two sets and type A and B spine1 Iherzohtes. Contrib.~i~erai. Petrol. of curves are given: one for a liquid with 1y’(pressure derivative 85.85-94. of the buik modulus) = 5 and the other with R = 7. Note that the projected density cross-overfor Fh at 180-200 kbars ARNDTN. T.. NALDRETTA. J. and PYKE D. R. (1977) Komatiitic and iron-rich tholeiitic lavas of Munro township, is fictive as Fh will convert to tbe spine1structure. northeast Ontario. J. Petrol. 18.3 19-369. BATONJ. F., HASAPISA. A. and WHOLLEYJ. W. JR. (1960) Viscosity and denstty of molten silica and high silica content glasses. Phys. Chem. Glasses 1.90-98. obtained by the double-bob Archimedean technique BAIDOVV. V. and KUNIN L. L. (1968) Speed of ultrasound allow a precise model equation of silicate liquid voland compressibility of molten silica. Sov. Phys. Dok. 13, umes to be derived. Although a previous fit of volume 64-65. BARRETTL. R. and THOMASA. G. (1959) Surface tension data (BGTTINGAm al., 1982) postulates a compositionat and density measurements on molten glasses in the CaDdependence to the partial molar volume of alumina, A&O3 system. Sot. Glass. Tech J. 43, 179T-l90T. our measurements do not provide such evidence. BERMANR. G. and BROWNT. H. (1985) Heat capacity of However, our data do indicate a compositional deminerals in the system Na20-KPCao-MBO_FeeFe*O~AlrOs-SQ-TiO~H@-COz: Representation, estimation and pendence to the partial molar volume of TiOz sughigh temperature extrapolation. Contrib. Mineral. Petrol. gesting either a change in the coordination of titanium 89, 168-183, or distinct populations of distorted and undistorted BI~KLEM. J., FORD C. E. and NISBETE. G. (1977) The petTiO, groups between sodium-bearing and calciumrogenesis of peridotitic komatiites: Evidence from highbearing titanium silicate melts. pressure melting experiments. Earth Planer. Sci. Left. 37, 97-106. Our density measurements on iron-bearing liquids, combined with those of MO et al. (1982), enable a BOCKRISJ. O’M. and KOJONENE. (I 960) The compressibilities of certain molten silicates and borates. J. Amer. Ceram. volume equation to be derived which allows the effect Sot. 82,4493-4497. of oxidation state on the densities of natural liquids to BOCICRIS J. O’M., TOMLINSONJ. W. and WHITEJ. L. (1955) The structure of the liquid silicates: partial molar volumes be evaluated. Molar compressibility values for the oxide and expansivities. Furuday Sot. Trans. 52,299-310. component, calculated from the ultrasonic velocity BOETTCHER A. L., BURNHAMC. W., WINDOM K. E. and BOHdata of RIVERS and CARMICHAEL(1987) and the calLENS. R. ( 1982) Liquids, glasses, and the melting of silicates orimetric data of Rtcri!z’r and BOTI-INGA( 1984,1985), at high pressures. J. Geol 90, 127-138. STEBB~NSet al. ( 1984) and ZEIGLERand NAVROTSKY BOIXNGA Y. and WEILLD. ( 1970) Densities of liquid silicate systems calculated from partial molar volumes of oxide (1986), have been rederived using our new volume components. Amer J SCI.269, I69- 182. data, leading to a significant reduction in the errors on BOTI’INGAY., WEILL D. and RICHETP. (1982) Density calthe fit. These results are used to calculate the density culations for silicate liquids. 1. Revised method for alumiof diopside liquid at elevated pressures using a shocknosilicate compositions. Geochtm. Cosmochim. Acta 46, 909-919. wave determination of the pressure derivative of the BOTTINGAY., WEILL D. and RICHETP. (1984) Density calbulk modulus (K’); there is excellent concordance with culations for silicate liquids: Reply to a Critical Comment the fusion curve up to 95 kbr. The agreement with by Ghiotso and Carmichael. Geochtm. Cosm~him. Acta published fafling-sphere determinations of diopside 48,409-4 f 4. liquid density at two pressures is poor. The region in BOYDF. R. and ENGLANDJ. L. (1963) Effect of pressure on the melting of diopside, CaMgSi20h, and albite, NaAISilOs, temperature-pressure space where liquidus olivine in the range up to 50 kbars.j Geophys.Res. 68,3 11-323. neither fioats nor sinks in a komatiitic liquid broadly CHRISTIE D. M., CARMICHAEL I. S. E. and LANGMUIRC. H. matches the results of AGEE and WALKER (1987). (1986) Oxidation states of mid-ocean ridge basalt glasses.

Earth Planet. Sri. Lctt. 79, 391-4 I I. Acknowledgement+--The authors are grateful for the support DICKINXINJ. E. and HESSP. C. (1985) Rutile solubility and of the National Science Foundation grants EAR 81-03344 titanium coordination in silicate melts. Geochim. Cosmoand 85-00813. Some of this work was prepared for the Dichim. Acta 49,2289-2296. rector, G&e of Basic Energy Sciences, Division of Engineering EDWARDSJ. W., SPEISERR. and JOHNSONL. (1951) High and Geosciences of the US Department of Energy, under temperature structure and thermal expansion of some metcontract DE-ACU3-76SFOO98.This manuscript was improved als as determined by X-ray diffraction data. 1. Platinum,

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2945

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2946 SHARTSIS L., SPINNER R.

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Appendix1. Meuurcd volumesof expcrimenlPl liquids. SSlllpk NO. m

;

3 3 3 4 4 4 5 5 5 6 f 7 7 : 8 8 ; 9 10 10 10 11 :: 12 12 13 13 13 14 14 14 I5 15 IS 16 16

Temp. W

V0llWlle Wgfw)

1851.4 1795.3 1692.0 1777.6 1866.2 1727.2 1828.4 18%.l 1696.5 1797.0 1876.1 1645.5 1745.7 1847.2 1677.1 1749.9 1854.0 1527.3 1677.3 1827.4 1697.4 1770.5 1845.7 1695.5 1777.0 1855.2 1777.9 1839.8 1695.6

27.618 27.522 23.168 23.279 23.393 26.066 26.226 26.337 25.272 25.442 25.559 23.672 23.821 23.976 22.478 22.595 22.763 24.554 24.727 24.898 23.476 23.566 23.640 22.826 22.913 22.982 22.535 22.583 24.653 24.738 X812 23.836 23.931 24.027 20.591 20.701 20.764 20.656 21.315 20.854 21.233 21.315

1770.9

1835.6 1694.0 1769.7 1846.3 1722.0 1795.1 1833.4 1704.8 1780.4 1853.4 1705.5 1780.4

Smiple No.

Temp. W

VdlUllC wgfw

16 17 17 17 18 18 18 19 19 19 19 19 20

1855.2 1652.1 1753.0 1839.7 1622.3 1723.3 1825.6 1421.0 1526.3 1622.9 1725.5 1828.1 1350.5 1445.6 1543.8 1641.5 1743.3 1593.8 1692.7 1795.4 1592.2 1691.4 1792.8 1712.9 1753.4 1793.9 1834.6 1644.0 1694.1 1744.9 1639.2 1688.9 1737.4 1759.1 1680.6 1583.2 1591.2 1691.2 1777.6 1662.4 1727.2 1786.6

21.393 24.030 24.186 24.312 32.702 32.980 33.295 29.55 1 29.876 30.192 30.556 30.886 26.519 26.858 27.206 27.502 27.778 23.560 23.822

E 20 20 21 21 21 z ; 23 23 23 24 24 24 25 25 25 26 26 26 27 27 27 28

24.070

28.3% 28.548 28.701 22.618 22.744 22.864 22979 25.535 25.635 25.751 29.154 29.118 29.080 24.958 24946 24.916 24.432 24.552 24.547 24.625 24.608 24.600