Multicomponent diffusion in basaltic melts

Pergamon Geochimlca et Cosmochimica Acta, Vol. S9, No. 2, pp. 3 13-324, 1995

Copyright 0 1995 Elsevier Science Ltd Printed in the USA. All rights reserved

C016-7037/95 $9.50 + .I0

0016-7037(94)00286-X

Multicomponent diffusion in basaltic melts

VICTOR C. KRESS* and MARK S. GHIORSO

Department of Geological Sciences, University of Washington, Seattle, WA 98195. USA

(Received Februury 16, 1994; accepted September 8, 1994)

Abstract-Experimental results are presented for eighteen experiments exploring multicomponent chemical-diffusion in basaltic liquids. Experiments were performed in Columbia River Basalt (CRB) composition doped with about 5 wt% SiOZ, Ti02, AlzOl, FeO, MgO, and CaO, under reducing conditions at 1 atm., at 1473 K, 1573 K, and 1723 K. Results indicate that diffusion behavior in CRB compositions is consistent with a simple Fick’s law formulation. This Fickian behavior in CRB compositions contrasts with more complicated diffusion behavior observed in MgO-AlZ03-Si02 and CaO-MgO-Al@-Si02 melts. Results of CRB experiments are combined to calibrate a diffusion matrix (D) in CRB liquids at 1473 K, 1573 K, and 1723 K. Our D estimates indicate negative coupling between CaO and both Fe0 and Al?O? components, though diagonal elements still dominate. These general features persist across the temperature range considered.

Self- and tracer-diffusion data from the literature are used with the predictive model of Richter ( 1993) to estimate a D matrix at 1573 K. The resulting matrix does not compare well with our measured 1573 K diffusion matrix. It cannot be established if this discrepancy indicates a failure of the Richter ( 1993) model, or merely reflects deficiencies in the available tracer- and self-diffusion data, or limitations in the melt activity model.

Effective Binary Diffusion Coefficients (EBDC) were also estimated for TiO?, A1203, FeO, MgO, and CaO at 1473 K, 1573 K, and 1723 K. These EBDC estimates are used to constrain a polythermal Arrhenian model for the prediction of EBDC values at super-liquidus temperatures. Results of experiments on alkali diffusion in CRB compositions are included to add NazO and K20 to the polythermal EBDC model.

INTRODUCTION

Three forms of diffusion are commonly distinguished: chemical, tracer, and self. Chemical diffusion refers to the diffusion of a component in the presence of a concentration gradient in chemical components. Chemical diffusion is driven by the gradient in the chemical potential of a given component, which is itself a function of the gradient in all chemical components in the system. Tracer diffusion refers to chemical diffusion at the limit of infinite dilution. In tracer diffusion, counter gradients in all other components in the material are negligible, so that one effectively considers the diffusive behavior of a single component in isolation. Self diffusion refers to diffusion of chemically indistinguishable isotopes of a component in the absence of chemical gradients in any component in the system. Experimental studies of tracer- and self- diffusion in natural silicate melts are relatively plentiful (e.g., Hofmann and Magaritz, 1977; Magaritz and Hofmann, 1978; Watson, 1981; Lowry et al., 1981, 1982; Henderson et al., 1985; Baker, 1992), due to the availability of comparatively simple and accurate experimental techniques and the relative simplicity of the theoretical treatment. Though tracer- and self-diffusion coefficients are useful in modeling isotopic contamination, assimilation, and mixing, virtually all natural applications will involve a significant aspect of chemical diffusion. Lesher (1990) demonstrated just how important chemical diffusion effects can be in his experimental study of isotopically important Sr and Nd components. Several models

* Present address: Geophysical Laboratory, Carnegie Institution of Washington, 5251 Broad Branch Road NW, Washington, DC 20015.1305. USA.

are available by which tracer- or self-diffusion coefficients can be combined with a valid multicomponent activity model to predict the chemical-diffusion matrix for a multicomponent material (Darken, 1948; Cooper, 1965; Lasaga, 1979; Richter, 1993). Baker (1992) demonstrates that the Lasaga (1979) model is only moderately successful at predicting chemical diffusion in natural silicate melt compositions. This is unfortunate, as it would be useful to be able to apply the relatively plentiful tracer- and self-diffusion data to chemical diffusion problems.

For the most part, chemical diffusion in natural silicate melts has been represented in terms of the Effective Binary Diffusion Coefficient (EBDC) treatment (Hougen and Wat- son, 1947; Cooper, 1968). It has been demonstrated experimentally, however, that cross-coupling of component fluxes can be significant in silicate melts, leading to diffusive effects which cannot be represented with the EBDC formalism (Su- gawara et al., 1977; Wakabayashi and Oishi, 1978; Oishi et al., 1982; Watson and Jurewicz, 1984; Koyaguchi, 1989; Kress and Ghiorso, 1993). It is important to determine how important such effects are in natural melt compositions.

Composition in an n component system can be expressed with a vector c of component concentrations. Only n - 1 elements of c are independent, the n th being determined by mass balance. In the volume-fixed reference frame, a diffusion matrix D may be defined relating the vector j of component fluxes to the gradient in composition:

j = -DVc, (1)

where V is the spatial gradient operator. Onsager ( 1945 ) proposed a representation based on nonequilibrium thermody-

313

314 V. C. Kress and M. S. Ghiorso

namics in a stable one-phase medium, where a symmetric matrix L relates component fluxes with gradients in chemical

potentials:

j = -LVp, (2)

where p is a vector of n chemical potentials. Based on ele- mentary kinetic theory and Eqn. 2, Kirkaldy and Young ( 1987) suggest that diffusion coefficients can be expected to be inversely proportional to concentration. Unfortunately, adding this effect makes the ensuing mathematical treatment of multicomponent diffusion dauntingly difficult. For this reason, we will consider the more traditional approach based on Fick’s Law (Eqn. 1).

D is generally asymmetric. Furthermore, the individual elements of D need not be positive (Kirkaldy and Young, 1987; Miller et al., 1986; Ghiorso, 1987). The only requirement of D in a stable one-phase system is that all eigenvalues of D be real and positive (Kirkaldy et al., 1963; Cullinan, 1965; Miller et al., 1986; Ghiorso, 1987).

For the remainder of this treatment, we will consider one- dimensional gradients in the z direction. If c is expressed in volume normalized units, such as moles per cc, Eqn. 1 can be combined with mass balance to form the more useful expression:

(3)

In this study, we explore multicomponent chemical diffusion in Columbia River Basalt liquid. Diffusion will be considered in terms of the simple oxide components, with SiOz as the dependent component. We will concentrate on the diffusion behavior of TiOZ, AlzOl, FeO, MgO, and CaO. The diffusive behavior of NazO and KzO are considered only to a limited extent.

EXPERIMENTAL TECHNIQUE

Diffusion experiments were performed using the method pioneered by Bowen ( 1921) Dissimilar melt compositions are loaded into half- capsules, and are joined so that the two compositions are juxtaposed across a planar interface. The resulting capsule assembly is then loaded into a furnace in a vertical orientation, with the denser liquid positioned at the bottom and allowed to diffuse for a controlled length of time.

Starting materials were synthesized in small batches by weighing powdered Columbia River Basalt (CRB) with about 5 wt% SiOZ, TiOz, A1203. FeO, MgO, and CaO. Resulting mixtures were then ground under ethanol in a agate mortar for 1 h. Each of the six resulting mixtures, along with an undoped CRB powder, was loaded into Pt crucibles and slowly heated, in air. to 1673 K. Starting materials were held at this temperature for 3-4 h, then quenched by dunking the entire crucible in distilled water. The resulting glass was then ground to a tine powder, and remelted for another hour before quenching to a glass.

Experimental capsules were constructed of 99.97% MO metal. 5 mm diameter rod stock was cut into 7 mm lengths and a 2.3 mm hole was drilled down the axis (Fig. la). A larger 4 mm diameter hole was then drilled 2 mm into one end to serve as a funnel for capsule filling. 200 mg glass fragments were positioned in the funnel end and both capsule and glass were inserted in a vertical furnace at 1723 K. As the sample reached furnace temperature, molten silicate flowed from the funnel end into the narrow capillary where it was held in place by surface tension (Fig. 1 b). The funnel end was then ground off.

(cl

Fttj. I. (a) Cross-section of experimental half-capsule with glass fragment in filling funnel. (b) Cross-section of experimental half capsule after filling and equilibration. Ends ground flat to dotted line. and one side polished to 1 pm smoothness. (c) Cross-section of fin- ished capsule assembly.

Filled half-capsules were inserted in the furnace under the desired conditions for the diffusion experiments, and allowed to equilibrate their redox state. Equilibration times ranged from 24 h for the 1723 K experiments to 48-72 h for the 1473 K experiments. The furnace atmosphere consisted of a controlled mixture of CO and CO*, defin- ing a furnace oxygen fugacity roughly I/? log unit below that defined by the MO-MOO assemblage (Chase et al., 1985). Estimated FezO?/ (Fe0 + Fe,O,) for endmember compositions under these reducing conditions range from 0.017 to 0.022 (Kress and Carmichael, 199 1 ) We ignore this minor FezOl component and assume all iron in our experiments to be in its ferrous state.

Both ends of each “equilibrated” half-capsule were ground flat and one end of each half-capsule was polished to I pm smoothness. Two half-capsules were then machined to press-tit into a larger Mo sleeve (Fig. lc). The resulting capsule assembly was then quickly inserted into a hot furnace under conditions nearly identical to equilibration conditions, and allowed to diffuse for a measured length of time. Temperature was measured by an S-type thermocouple positioned immediately adjacent to the capsule. The thermocouple was enclosed in a high-purity alumina sheath in order to prevent contact with the reducing furnace atmosphere. Thermocouples were calibrated against the melting temperature of Au at irregular intervals. Measured Au melting temperatures range from 1335.2 to 1337.8 K, which compares with the accepted Au melting temperature of 1337.6 K (IPTS 1969). Temperatures reported in Table 1 are uncorrected. The temperature gradient measured along the length of the capsule was less than 1 K. In order to further inhibit convection. the capsule was positioned in the thermal gradient so that the hottest portion was towards the top. The 60- 120 s required for the sample to attain equilibrium temperature is considered insignificant relative to the total diffusion time ( l-2% of our shortest-duration experiment), and was ignored. The experimental durations in Table 1 represent the interval between the time at which the sample attained within 5- 10 K of the final temperature and the time of the final quench.

Experiments were quenched by rapidly pulling the sample from the furnace. More rapid quenching was found to induce large stresses in the experimental glasses. which usually lead to fracturing during sample preparation. Quenching at an excessively slow rate resulted in the formation of bubbles in the melt. Similar bubbles were observed when samples were quickly taken to more reducing conditions at high temperatures. These bubbles cannot be a consequence of the reduction of ferric iron, as a bubble resulting from iron reduction alone would be composed of pure 02, and would have an oxygen fugacity of one. The mechanism for bubble formation in these experiments remains a mystery. The optimum quench rate was achieved by quickly pull-quenching into a region of the furnace which is near room temperature. The quench was performed entirely under the rem ducing furnace atmosphere.

Quenched experimental capsules were mounted in epoxy, sec- tioned, polished, and coated for microprobe analysis. Samples were rejected if bubbles were present which were larger than about 5% of the capillary diameter.

Diffusion in basalt magma 315

Table I. Conditions for diffusion experiments.

Experiment T(K)

DN17 1723

DN18 1724

DN21 1723

DN23 1723

DN25 1726

DN27 I725

DN28 1473

DN29 1474

DN30 1473

DN33 1572

DN34 1573

DN35 1573

DN36 1573

DN37 1473

DN38 1473

DN39 1573

DN40 1724

DN41 1724

r(seconds) Top

4680 CRB

5160 CRBS

5730 CRBS

5479 CRBS

5580 CRB

6180 CRB

28179 CRB

27490 CRB

21990 CRBA

31004 CRB

21629 CRBA

36411 CRB

31413 CRBT

32551 CRBT

34574 CRBA

26585 CRBA

9780 CRBA

7200 CRBA

Bottom

CRBF

CRBC

CRBF

CRBT

CRBT

CRBM

CRBM

CRBF

CRBC

CRBF

CRBM

CRBM

CRBF

CRBF

CRBF

CRBC

CRBM

CRBC

Solution of melt components into the MO metal could not be re- solved on the electron microprobe with up to 100 s counting times. Solution of MO in the silicate melt was also found to be negligible, however several experiments showed dispersed trails of very fine (2 pm) metallic droplets near the capsule walls. These droplets were too small to permit accurate determination of composition, but qualitative probing suggests that they are primarily MO. These metal trails ap- peared at first glance to indicate lines of convection. Electron microprobe examination revealed, however, that metal trails crosscut un- disturbed compositional interface features, suggesting that they formed in a static liquid. These metallic droplets are not associated with bubbles. The nature of these metal trails remains unclear, but one hypothesis is that minor alloying along the metal-melt interface created a small amount of immiscible metallic liquid, which seems to have remained in suspension in the liquid. Using liquid density and viscosity calculated for CRB composition at 1723 K from Lange and Carmichael ( 1987) and Shaw ( 1972) respectively and liquid MO density from Weast ( 1986), the estimated Stokes settling velocity of a 2 pm drop is estimated to be on the order of 1O-8-1O-9 m/s. Thus, settling of these droplets during the diffusion experiment can be considered negligible. Regions rich in these metal trails were easily avoided during sample analysis.

In practice, one must combine the results of a number of diffusion experiments performed in a variety of compositional directions in order to reliably estimate the elements of D. D estimates are best constrained if the experimental joins are close to orthogonal in composition space (Trial and Spera, 1994). Our experiments were performed in a variety of compositional directions with a vertex at the original CRB composition.

Experimental capsules were designed to maximize the contact between the liquid and the furnace atmosphere. This was done in order to minimize gradients in oxygen fugacity, and thus minimize potential com- plications due to inhomogeneities in ferric-ferrous distribution. Unfor- tunately, the very feature. which allows accurate consideration of iron diffusion in silicate liquids, also limits our ability to consider diffusion of alkalis. Alkali cations have significant vapor pressure at experimental

temperatures, and alkali-loss at the liquid-vapor interface can be quite rapid. This alkali-loss invariably leads to the formation of concentration gradients in alkali components in the vicinity of these interfaces. Thus, one cannot assume a homogeneous alkali distribution in each half-capsule at the start of the experiment. Without this initial boundaty condition, it becomes impossible to consider alkali diffusion using the methods employed in this study. With this in mind, an experimental strategy was devised to minimize the effects of alkalis in our diffusion experiments. An unfortunate consequence of this strategy is that complete character ization of alkali diffusion in basaltic liquids must await future experimental study.

The effects of alkali inhomogeneity were minimized by allowing the sample to devolatilize a substantial portion of its original content during ferric-ferrous equilibration. As much as 50% of the original alkalis were lost at this stage. Both ends of the filled capsule were then ground off in order to eliminate the most severe alkali gradients near the original melt-vapor interface. The remaining alkali gradients, though significant relative to the alkali contents, were minor relative to the other melt components considered.

SAMPLE ANALYSIS

Microprobe analyses were performed on a JEOL 744 Su-

perprobe using a spot size of about 10 pm, a 15 nA beam current and 10 seconds counting time. The large spot size and

short counting time were chosen to minimize alkali loss during analysis. A wet-chemically analyzed Columbia river Ba- salt was used as a standard. Standardization was performed both before and after the analytical traverse. Results of 15 30 analyses were averaged for each standardization. Back- ground-corrected counts were output directly to a PC running PRTASK software (Donovan and Kress, 1991; Donovan et al., 1992). PRTASK interpolates between standardizations so


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0 1 2 3 4 5 6 millimeters

FIG. 2. Compositional profiles across experimental charges. Lines indicate profiles predicted for individual experiments on the basis of the D matrices in Table 3.

that illusory compositional discontinuities are avoided when standards are reanalyzed. Samples were probed for eight ma- jor elements and totals ranged from 98.0- 100.2%.

A typical analytical traverse consists of 500 points spaced between 10 and 40 pm along the length of the capillary. The effects of distortions due to sample contraction during quench were minimized by performing probe traverses within 100 pm of the capillary wall. The effect of sample contraction on the probe profiles is estimated to be on the order of 0.5% or less. Two-dimensional composition maps of several experiments

showed no evidence for convection. Compositional profiles for diffusion experiments are displayed in Fig. 2.

Because of variations in volatilization history, and differ- ences between individual batches of starting materials, common starting-material compositions could not be assumed for endmember compositions in the individual experiments. End- member compositions for each individual experiment were determined by averaging 20-50 analyses from both ends of each capsule. Care was taken to insure that these analyses were performed as far as possible from the region where dif-


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0.02 p Ti02

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millimeters

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(f-f-0 DN36 (n) DN37 0.17 0.17

0.15 0.15

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0.13 E

‘Z 0.11 ‘S 0.11

t” ~ 0.09 JI E 0.09

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millimeters millimeters

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FIG. 2. (Continued)

fusion effects had penetrated. Compositions listed in Table 2 represent averages of these endmember compositions.

DATA ANALYSIS

Substituting the Boltzmann variable 6 = z/h (Boltzmann, 1894) into Eqn. 3, and integrating both sides yields the expression of Matano ( 1933) as extended to multicomponent systems by Kirkaldy ( 1957):

(1) DN35

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‘ii . Fe0 m 0.10 - t

CaO _

0.04 MO

* “‘.‘.I. ‘. ‘. ‘. 0 1 2345678

millimeters

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0.02 -

0.00 . ’ . ’ . ’ . ’ . ’ . ’ . ’ . ’ . 0123456789

millimeters

The origin for 5 in the above expression is the Matano interface z,, defined by the solution of:

s

Cl,_, (dc’ = 0, (5)

c,,mmt

which is given by:

1 s

c, ,:,,,,,i Zm = zdc’. (6)

CtC:,“,x) - clGm) (lC&,,“i

It is important to note that, in the derivation of Eqn. 4, it is

assumed that ac, = 0 at the ends of the profile. The Boltz- 36


Table 2. Approximate end member compositions for diffusion experiments. Averaged, normalized, post-experiment compositions in wt%. Compositions in individual experiments vary somewhat, due to variations in initial mix and differing alkali-loss

I CRB CRBS CRBT CRBA CRBF CRBM CRBC

SiO2 58.03 60.30 54.54 53.84 54.69 55.23 55.28

TiOa 2.40 2.55 6.96 2.3 1 2.28 2.31 2.37

A’203 13.61 14.03 13.09 17.42 13.10 13.51 13.39

Fe0 12.17 10.37 11.80 11.97 15.99 12.02 II.25

MgO 3.45 3.51 3.38 3.41 3.49 6.94 3.44

CaO 7.25 7.43 7.06 6.95 7.13 7.03 11.69

Na;?O 1.83 0.65 1.88 2.64 2.00 1.74 1.42

K20 1.26 1.17 1.30 1.47 1.34 1.22 1.15

mann-Matano-Kirkaldy method can, therefore, only be applied in experiments where the effects of diffusion have not reached the ends of the capsule. This condition was satisfied with a large margin in all experiments from this study. Meth- ods for fitting profiles in which diffusion has reached the capsule ends are presented by Ghiorso ( 1987) and by Trial and Spera ( 1994). Methods for treating diffusion experiments in which diffusion effects have reached the capsule ends require strict control of the geometry of the ends of the diffusion capsule, and thus, can not be employed with our experimental design.

Numerically integrating the left-hand side of Eqn. 4, and assuming constant D yields n - 1 linear expressions for each analyzed point, corresponding to the n - 1 rows of D. Thus, D can be estimated from a series of experiments performed at a given temperature using standard linear regression methods. Techniques for making this large regression problem more tractable are outlined in V. C. Kress (unpubl. data). Errors in estimates of the Matano interface, and analytical drift can lead to significant bias in estimating D using the BMK method. For this reason, experiments were fitted with the expression:

_ e;, i(k)* - 5(l)’ c,(k) - c,(l) 4

~ eL 2 ’ (7)

for component i at point k, where e i,, is the drift-error rate in component i, and e2 is the error in the Matano interface estimate (V. C. Kress, unpubl. data). In all regressions, cases are weighted according to the inverse of the propagated uncer- tainty in the left-hand side of Eqn. 7. High-frequency analytical noise in the composition profiles was filtered before fitting using the Fourier smoothing method described in Kress and Ghiorso 1993). Filter widths corresponding to approxi- mately 2 J- Dr were employed.

Alkali components were not considered in the fitting process. By omitting alkalis from consideration, we are effec-

tively lumping alkalis with Si02 and other minor components in the solvent. Alkali contents generally contrast across the couple to a small extent, due to the effects of oxide doping of the original CRB compositions, along with dif- ferences in volatilization history. According to Fick’s Law (Eqn. I), the flux of a melt component i will couple with the alkali gradient through off-diagonal elements of D. To a first approximation, the gradient in alkali concentration is proportional to the initial contrast in alkali concentration across the couple. Alkali compositional contrasts range between 1 and 20% (mole fraction basis) of the contrasts in the primary doped components. Thus, D,.alkal, must exceed D r,pr,marsr by a factor of 5- 100 if the alkali gradient is to influence the flux of i to the same extent as the gradient of the primary doped component. If there is a substantial effect due to diffusion of alkalis, it is likely to be on the least abundant components considered in our diffusion analysis. Even then, the effect due to alkali gradients is likely to be small, relative to the effect due to the primary gradient. Fur- thermore, the direction and intensity of these incidental alkali gradients are likely to be random. Thus, the effects of these alkali gradients will, for the most part, average out in the fitting process.

D estimates for experimental data at 1473 K, 1573 K, and 1723 K are presented in Table 3. All eigenvalues for the three D estimates are positive and real (Table 4), confirming that stability criteria are satisfied for all three matrices. Predicted profiles from D estimates are compared with the experimental data in Fig. 2. The predicted curves generally match the experimental data well, though some observed features are not well reproduced in the predicted profiles. These deviations may reflect inadequacy in our fits, inconsistency between the results of these experiments and those of the rest of our experiments, or diffusion effects which are not accounted for in our six-component Fickian formulation.

Uncertainties in the individual values of D, are estimated to be on the order of IO-20%. It is difficult to obtain a more precise estimate of uncertainties, due to the problem of accurately propagating the effect of compositional un-


Table 3. Full D matrix fits atl473K. 15733 and 17233 (lo-ttn

1473K

TiOl

-4’203

Fe0

MgO

CaO

1573K

TiO,

*@‘3

Fe0

MgO

CaO

1723K

Ti02

*‘2O3

Fe0

MgO

CaO

Ti02 A1207 Fe0 MgO CaO

2.6 -0.076 -0.41 0.38 0.28

0.99 2.8 0.96 1.1 -2.5

-2.9 -0.67 3.0 0.73 -4.1

0.7 1 0.30 -0.27 5.4 0.21

-1.1 -3.9 -2.6 -0.44 6.4

Ti02 *‘Z03 Fe0 MgO CaO

18. 1.4 -0.45 -1.6 1.1

0.55 5.3 2.4 0.91 -9.4

2.2 0.46 19. -9.2 -11.

0.12 14. -7.5 23. 10.

-2.0 -14. -9.2 -4.6 40.

TiO2 *‘Z03 Fe0 WO CaO

1.5. 0.62 2.1 -0.56 0.12

0.95 26. 9.1 -4.9 -15.

15. 12. 65. -5.5 -7.8

-7.2 -14 -1.2 44. -7.7

-8.8 -32 0.19 -8.5 78.

certainties through the Fourier smoothing and differentia- tion processes.

DISCUSSION

The values in Table 3 represent the first estimates of five- component D matrices for basaltic liquids. The structure of these D estimates can be more clearly displayed by contouring the D matrix at each temperature. D estimates at 1473 K, 1573 K, and 1723 K are contoured in Fig. 3. In Fig. 3, D, values are represented as “altitudes” in an [n - l] x [n - l] grid, contoured by interpolation. It is evident from Fig. 3 that the detailed structure of D varies with temperature; however, significant features span the temperature range considered. All three matrices are dominated by a ridge along the matrix diagonal, with a peak value at Dcaoo.c.o. All three matrices have a broad symmetry across the diagonal with a tendency towards negative coupling between the flux of CaO and the gradients of A120,, FeO, and to a lesser extent, TiOZ and MgO. There is a reciprocal negative coupling between the fluxes of A1,OX and Fe0 and the gradient of CaO, but inter- estingly, the coupling between the fluxes of TiO, and MgO

with the gradient of CaO are positive to slightly negative. Trends in the orientations of the diffusion matrix eigenvectors with changes in temperature are not obvious.

Experiments considering multicomponent-diffusion in simple analog systems such as MgO-Al,O,-Si02 and CaO- A1203-MgO-Si02 show dramatic uphill diffusion effects which cannot be reconciled with the traditional Fickian treatment of diffusion (Kress and Ghiorso, 1993). This contrasts with equally dramatic uphill diffusion effects observed by Varshneya and Cooper (1972) in K20-SrC-Si02 melts, which can be reproduced with a simple Fickian treatment. The effects observed by Kress and Ghiorso ( 1993) are not observed in the CRB compositions considered in this study. It is possible that the phenomenon responsible for the features observed in the three- and four-component silicate melts are active in the basaltic compositions considered in this study, but their effect is diluted. It is also possible that diffusion in complex natural silicate melts is fundamentally different than diffusion in MgO-Al&-Si02 or CaO-A1207-MgO-Si02 melts.

Though the diffusion matrices presented in this study are important building blocks in our understanding of multicomponent diffusion in basaltic systems, they are of more limited use to the average igneous petrologist, who must consider diffusion in arbitrary compositions at arbitrary temperature. Some formalism must be derived by which results of this study can be extrapolated to arbitrary compositions and temperatures. It is likely that such a formulation will not be based on Fick’s Law, but rather on a chemical potential driven form, requiring a detailed mixing model for the basaltic melt, more diffusion experiments, and perhaps, further structural and spe- ciation information on the silicate melt phase.

Predicting Diffusion Matrix from Self and Tracer Diffusion Data

Richter ( 1993) proposes a promising model by which diffusion matrices may be estimated. This model is an extension of the model of Cooper ( 1965), which is itself an extension of the Darken ( 1948) mobility model. According to the Rich- ter ( 1993) model, the flux of a neutral oxide melt component i is given by the equation:

j, = - (1 - c,U,)E;+ c, i “$(l - 6,) ci 2 [ Ir=l 1

- ci ,r, lix$l - 6s) i (1 - &,)(I - 6,k,~, (8) ,=I where vi and 9, are the partial molar volume and self-diffusion coefficient of component i, respectively, R is the gas constant, and 6,, = 1 if i = j, or 0, otherwise. Combining Eqn. 8 with Eqn. 2 yields:

4 = ci i ]6,,&(1 - c,u,) + 1 h,(l - 4k) k=l

+ (1 - 6,,)(’ - &,)(l - 4i))C,Uklj$ (9)

Combining L from Eqn. 9 with the Hessian of the Gibb’s


Table 4. Eigensystem of D fits at1473K. 1573K and 1723K. Eigenvalues are arranged in order of increasing value at each temperature. The highest magnitude element is underlined for each eigenvalue.

Eigenvalues Eigenvectors

14733 Ti02 403 Fe0 MgO CaO

S.O~~X~O-~~ 0.115 0.073 O&&j.!. 0.008 0.453

1.362~10-‘~ 0.213 -0.304 Q.&Y? 0.030 0.267

3.060x10-‘* -0.329 -0.227 0.843 0.195 0.302

5.734~10-‘~ 0.165 0.090 -0.300 0.907 0.228

9.548x10-‘* 0.068 -0.346 -0.486 0.058 eaes

1573K Ti02 Alz03 Fe0 MgO CaO

1.933x10-‘2 .O. 142 0.720 -0.249 -0.620 0.127

1.157x10-” 0.116 -0.073 m 0.361 0.315

1.781x10-” ila55! -0.036 0.221 0.052 0.167

2.747x10-1’ -0.110 0.032 -0.626 n -0.171

4.638x10-” 0.005 -0.197 -0.45 1 0.379 QJQ

1723K Ti02 Al+J1 Fe0 M8O CaO

1.387x10-” .0.432 QJJj -0.2 12 0.309 0.356

1.534x10-” QA.z -0.002 0.322 0.276 0.160

4.487x10-11 0.010 -0.138 0.372 0.912 0.100

6.595x10-” 0.048 0.148 0.924 -0.245 0.249

8.817x10-” 0.001 -0.267 -0.425 -0.058 Q&jJ

function with respect to composition, H, one can estimate D

from:

D = L’A’H’ (10)

(Cussler, 1984, p. 200), where A is the matrix given by A,,

= b,, + *, s denotes the solvent component, and L’, A’,

and H’ are n - 1 dimensional matrices formed by deleting the sth row and column of L, A, and H, respectively.

Equation 10 was used to estimate the 1573 K D matrix in Table 5. Partial molar volumes are from Lange and Carmi- chael ( 1987). The H matrix in Eqn. 10 is estimated from the liquid solution model of Ghiorso and Sack ( 1994). Values for 9, in basaltic composition at 1573 K for Fe0 and NazO are from Lowry et al. (1982), while MgO and CaO values are from Sheng et al. (1982) and Hofmann and Magaritz ( 1977) respectively. Self- or tracer-diffusion data for SiO*, TiOZ, AlzOl, or K20 in basalts at 1573 K could not be located. 2, values for these components were estimated from the relation: log,, (9) = -1.47 log,, (2’~) ~ 5.56 (Hofmann, 1980)) where Z is the cation valence and r is the cation radius (Whittaker and Muntus ( 1970) The Hofmann ( 1980) predictive model was calibrated on diffusion data in basalts at

1573 K. These self-diffusion estimates are a poor substitute for experimentally determined values. Nevertheless, provi- sional use of these values allows us to make a preliminary assessment of the Richter ( 1993) model. The structure of the matrix predicted from the Richter ( 1993 ) model is quite different from the 1573 K D in Table 3. The most conspicuous difference is that all of the values in the predicted matrix in Table 5 are positive. This reflects the nature of the input data, rather than a priori limitations of the model. Nevertheless, the predicted matrix is inconsistent with the measured D values from this study. Though eigenvalues of the matrix in Table 5 are all positive and real, they are one to two orders of magnitude lower than the 1573 K eigenvalues in Table 4. Similar calculations were performed using a gs,o, value from the Baker ( 1992) 1 atm dacite experiment extrapolated to CRB composition using the Stokes-Einstein relation. The resulting ZI estimate was very similar in both magnitude and structure to the matrix in Table 5.

Clearly, the Richter ( 1993 ) model has failed in this case. This should not be considered a definitive test, however. The structure of the predicted matrix is sensitive to input 9, values. Thus, the fact that we were forced to use estimated diffusivities for SiO?, TiO?, AlzOj, and KZO may have detri- mentally affected the predicted D. Furthermore, H used in the prediction of D may not have been accurate. Though the liquid model of Ghiorso and Sack ( 1994) is entirely self-consistent in the context of prediction of phase equilibria in magmatic systems, there is no way of knowing if properties of the predicted Gibbs surface are accurate in an absolute sense.

1573K

1723K

TlO, Al,O, Fe0 MS0 CaO

FIG. 3. Contour plots comparing D estimates at 1473 K, 1573 K. and 1723 K. Values of matrix elements are represented as “altitudes,” contoured by interpolation.


Table 5

1573K

Ti02

*l2O3

Fe0

MgO

CaO

Na20

K20

D matrix at 15733 estimated from the model of Richter (1993) (lO-llm2/sec).

TiO? A1701 Fe0 M80 CaO Na*O &O

3.11 0.775 0.592 1.03 0.694 2.46 4.98

13.3 3.37 2.57 4.49 3.02 10.7 21.7

16.5 4.16 3.18 5.54 3.74 13.2 26.9

8.48 2.15 1.64 2.88 1.93 6.8 1 13.8

13.2 3.36 2.57 4.47 3.04 10.7 21.7

8.41 2.17 1.66 2.83 1.97 7.78 13.1

3.39 0.901 0.705 1.18 0.820 2.71 7.14

Some distortion of the D matrix can also be attributed to com- parison of a measured four-component D with a predicted six- component D. Such distortion should be relatively minor, however. In any case, definitive testing of the Richter ( 1993) model will require experiments constraining multicomponent chemical diffusion of alkalis, along with tracer diffusion data for SiO*, TiOz, A1203, and K20.

Temperature Dependence and the EBDC Form

The temperature dependence of diffusion coefficients is usually expressed according to the Arrhenius relationship:

D,,, = Dy,, exp (11)

(e.g., Hofmann, 1980). Unfortunately, Eqn. 11 cannot be applied to our fits, as many of the coefficients change sign over the temperature range considered. Alternatively, one might express the temperature dependence of D by assuming that the eigenvalues follow the Arrhenius relation, while the corresponding eigenvectors remain relatively constant. Unfortu- nately, inspection of the eigenvectors in Table 4 reveals no obvious basis for the choice of common eigenvector directions. Any choice of common eigenvector directions for the matrices in Table 4 would be essentially arbitrary.

Given the fact that common polythermal eigenvector directions must be arbitrarily chosen anyway, it is most convenient to chose vectors which lie along the five oxide component

5.9 6.3 6.7

v l-lo _--t

??AI,o,

X Fe0 -_-_

+ MgO -----

axes. This gives a diagonal D, whose elements are the EBDC values for each component. Most of the more significant features in Fig. 2 can be represented with this form. In adopting an EBDC form, however, we lose the ability to predict many potentially important details in the composition profiles. An- other serious limitation of the EBDC form is that accuracy may be significantly dependent on the trajectory of the couple in composition space. Though accuracy and generality are lost in this treatment, we gain simplicity, as well as the ability to derive a preliminary polythermal model for diffusion in CRB composition.

1473 K, 1573 K, and 1723 K datasets were each used to estimate a D which is constrained to be diagonal. Resulting parameter estimates are reported in Table 6. Figure 4 repre- sents a plot of the natural logarithm of these EBDC estimates plotted against inverse temperature. Also plotted in this figure are best-fit lines for each component. Slopes and intercepts of these best-fit lines correspond to the A, and 0: values listed in Table 7. Activation energies for the five nonalkali components deviate from the mean value by, at most, 20%. It is evident from Fig. 4 that EBDC fits follow a broadly Arrhenian trend. In detail, however, the EBDC values diverge from this linear trend to varying degrees. It is impossible to determine how much of this non-Arrhenian behavior is real, and how

Table 6. Diagonal D fits atl473K. 1573K and 1723K. (lo-t*m%ec)

1473K 1573K 1723K

TiO;? 2.4 7.4 14.9

*‘2O3 2.5 5.0 27.9

Fe0 9.4 16.3 71.6

MgO 7.3 14.5 48.8

CaO 11.3 36.6 74.4

FIG. 4. Plot of natural log of EBDC estimates vs. inverse temperature for five components considered in this study.


Table 7. Polythermal model for diagonal diffusion matrix in liquid CRB composition. Na20 and KzO values are estimated from the data of Shfnn (1974).

D”i,i (m*/sec) Ai,i (kJ/mol)

TiO, 6.4x10-7 151.4

*‘2O3 4.1x10-5 205.2

Fe0 1.2x10-5 173.8

MgO 3.7x10-6 161.6

CaO 4.7x10-6 156.9

Na20 7.3x 1 o-5 169.4

K20 3.9x10-4 199.6

much is a reflection of the limitations of the EBDC formulation.

Estimated Dcao from this study is 5% higher than those of Medford ( 1973 ) near 1473 K but approaches 150% higher in the vicinity of 1723 K. Medford ( 1973 ) examined a mugearite composition, and it may be this composition difference which is responsible for the discrepancy between the results of these two studies. Our estimated value for Dbo is 77% lower than the value cited by Medford ( 1973), while the activation energy for Ca diffusion from this study is 23% higher than the Medford ( 1973 ) value.

EBDC values from Zhang et al. ( 1989) from experiments within 15” of temperatures considered in this study are between 0 and 2 orders of magnitude lower than values from this study. The compositions considered by Zhang et al. (1989) are similar to those considered here, being 5 wt% lower, on a relative basis, in FeO, 2% lower in Ti02, 2% higher in Na20, 4% higher in A1203, and higher in Hz0 to an unknown extent. The lower diffusivities observed by Zhang et al. (1989) may reflect a significant negative pressure dependence, as the Zhang et al. ( 1989) experiments were performed at 0.5-2.1 GPa pressure. However, experiments at 1573 K between 0.5 and 10.5 GPa from Zhang et al. (1989) do not indicate a measurable negative pressure-dependence for the diffusivity of any component considered in that study. It is also possible that the discrepancies between the two EBDC sets reflect limitations in the EBDC treatment.

Alkali Diffusion

Smith ( 1974) performed a large number of experiments specifically considering diffusion of alkali components in a CRB composition similar to those considered in this study. In most of the Smith ( 1974) experiments counter-gradients are set up between two different alkalis leading to diffusion of the alkali components in opposing directions.

Smith ( 1974) noted that alkali diffusivities are dependent on the concentration of the alkali in question over the total alkali content. He proposed a correction of the form:

1%” CD,) = m ;it& + b, (12)

where i, in this case, refers to alkali components. Smith ( 1974) also found that alkali EBDC can decrease an order of magnitude when considering couples in which alkali gradients are in only one direction. This would imply a significant negative D,,, where i and j are two different alkali components. According to Smith ( 1974), in the absence of a counter-gradient in some other mobile alkali component, charge-balance dictates that alkali diffusion must be balanced by counter- diffusion of some other, less mobile, component. This effectively decreases the EBDC of the alkali-component. The interpretation of Smith (1974) implies that cation diffusion takes place on a timescale which is rapid relative to anion (0’ ) diffusion. Such an interpretation is not consistent with oxygen chemical diffusion measurements of Wendlandt ( 1991) in a CRB melt at 1573 K which indicate oxygen diffusivities two orders of magnitude higher than cation diffusivities from this study. One would expect, therefore, that lo- cal charge balance could be more quickly established through anion diffusion than through cation diffusion. In any case, the observed dependence of EBDC for alkali components on the direction of the couple in composition space can only be prop- erly treated with a full D matrix formulation. This highlights a fundamental limitation of the EBDC treatment. Unfortu- nately, Smith ( 1974) did not present sufticient data to allow these flux-coupling effects to be quantified.

With these qualifications in mind, we use data from Smith ( 1974) to extend our polythermal diagonal matrix model to include Na20 and K,O. These estimates are likely to be on the high end of the range of EBDC values one would find in random orientations of the original compositional couple. Only the values of Smith ( 1974 ) for DNe10 and DKLo at c, / total alkalis = ‘/z are considered. These data were used to regress Dp and A, values for NazO and K20 in Table 7.

Composition Dependence

Diffusion coefficients have been shown to depend strongly on composition (Alibert and Carron, 1980; Baker, 1990; Ko- yaguchi, 1989). This is true, regardless of whether one is considering tracer, self, or chemical diffusion. 1723 K diffusion coefficients from this study are lower than those observed in simple analog liquids at 1723 K by Kress and Ghiorso ( 1993), but are within an order of magnitude. Calculated viscosities (Shaw, 1972) for liquids considered in Kress and Ghiorso (1993) are between a factor of two to an order of magnitude lower than estimated values for liquids in this study at similar temperatures. Thus, the difference in the magnitude of diffusion coefficients estimated in this study and in Kress and Ghiorso ( 1993) is qualitatively consistent with the

kJ Stokes-Einstein relation: D = -

37rqu ( Kirkaldy and Young,

1987), where kh is the Boltzmann’s constant, 7 is the melt viscosity, and 0 is the “particle diameter.” This qualitative inverse correlation between chemical diffusivities and viscosity in silicate liquids has been discussed extensively (Baker, 1990; Watson and Baker, 1991), and has been cited fre- quently in the context of tracer diffusion of oxygen in silicate


melts (Yinnon and Cooper, 1980; Dunn, 1982; Shimizu and Kushiro, 1984).

EBDC values for AlzOl, FeO, MgO, and CaO in dacites and rhyolites at one atmosphere ( SiOZ = 65-75 wt%; Baker, 1990) range from two to three orders of magnitude lower than values estimated in this study in basaltic compositions (SiO? = 54-60 wt%) at similar temperatures. This suggests that Si02 content exerts a significantly larger influence on diffusivities than suggested by Baker ( 1990). Predicted melt viscosities for melts considered by Baker ( 1990) are one to three orders of magnitude higher than those estimated for the melts in this study at 1473 K (Shaw, 1972). Thus, the observed difference in diffusivities between the two studies is in broad accord with the Stokes-Einstein relation.

CONCLUSIONS

Diffusion in natural basaltic melts is entirely consistent with a simple Fickian model for diffusion. This contrasts with the simple three- and four-component silicate melts considered by Kress and Ghiorso ( 1993), where diffusion features were observed which demand a more complicated formulation.

The model of Richter ( 1993 ) does not appear to success- fully predict D in CRB at 1573 K. This may reflect deficiencies in the input data rather than a failure of the model.

Compositional profiles predicted with full-matrix and diagonal-matrix fits differ in detail. For many applications these details will not be important, and the simpler diagonal-matrix model should be sufficient. Experimental evidence suggests that off-diagonal effects may be particularly important when considering alkali components (Smith, 1974; Watson, 1982; Koyaguchi, 1989). Thus, the EBDC model should be used with caution. Calibration of coupled multicomponent diffusion of alkali components in basaltic compositions will require further experimental study.

There appears to be no simple model by which diffusion coefficients can be accurately predicted on the basis of cur- rently available experimental data. It is likely that progress in this regard will require an Onsager-type formulation (Eqn. 2) combining an expanded multicomponent diffusion database with an accurate activity model for the silicate melt. Until such a formulation is available, diffusion coefficients should be applied only cautiously when considering compositions distant from those in which the values were experimentally calibrated.

Acknowledgmenfs-This text has benefited from thorough, construc- tive, and insightful reviews by Drs. J. S. Kirkaldy, F. J. Spera, and A. F. Trial. Their contributions are greatly appreciated. Research was generously supported by National Science Foundation grant EAR 9 l- 047 14 to M. Ghiorso.

Editorial handling: E. Merino

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Multicomponent diffusion in basaltic melts