Uranium and Thorium Diffusion in Diopside

ELSEVIER Earth and Planetary Science Letters 160 (1998) 505±519

Uranium and thorium diffusion in diopside

James A. Van Orman a,Ł, Timothy L. Grove a, Nobumichi Shimizu b a Department of Earth, Atmospheric, and Planetary Sciences, Massachusetts Institute of Technology, Cambridge, MA 02139, USA b Department of Geology and Geophysics, Woods Hole Oceanographic Institution, Woods Hole, MA 02543, USA Received 25 August 1997; revised version received 26 March 1998; accepted 6 April 1998

Abstract

This paper presents new experimental data on the tracer diffusion rates of U and Th in diopside at 1 atm and 1150± 1300ëC. Diffusion couples were prepared by depositing a thin layer of U±Th oxide onto the polished surface of a natural diopside single crystal, and diffusion pro®les were measured by ion microprobe depth pro®ling. For diffusion parallel . : : / . = /= : to [001] the following Arrhenius relations were obtained: log10 DU D 5 75 š 0 98 418 š 28 kJ mol 2 303RT . : : / . = /= : log10 DTh D 7 77 š 0 92 356 š 26 kJ mol 2 303RT. The diffusion data are used to assess the extent to which equilibrium is obtained during near fractional melting of a high-Ca pyroxene bearing mantle peridotite. We ®nd that the diffusion rates for both elements are slow and that disequilibrium between solid and melt will occur under certain melting conditions. For near-fractional adiabatic decompression melting at ascent rates >3cm=yr, high-Ca pyroxene will exhibit disequilibrium effects. High-Ca pyroxene will become zoned in U and Th and the melts extracted will be depleted in these incompatible elements relative to melts produced by equilibrium fractional melting. U and Th diffusivities in high-Ca pyroxene are similar, and diffusive fractionation of these elements will be limited. Numerical solutions to a dynamic melting model with diffusion-controlled chemical equilibration indicate that the activity ratio [230Th=238U] in a partial melt of spinel peridotite will be slightly less than 1 for a broad range of melting parameters. This result reinforces the already widely accepted conclusion that melting of spinel peridotite cannot account for 230Th excesses in mid-ocean ridge and ocean island basalts, and that garnet must therefore be present over part of the melting column.  1998 Elsevier Science B.V. All rights reserved.

Keywords: diffusion; diopside; pyroxene group; uranium disequilibrium; U-238=Th-230; igneous processes

1. Introduction the mantle U and Th are fractionated from each other. Clinopyroxene (cpx) and garnet are the prin- Radioactive disequilibrium between 238Uand cipal hosts of U and Th in upper mantle rocks and 230Th is widely used to infer the rate and style control the ¯uxes of these elements during melting. of melting and melt transport in the mantle [1±8]. Equilibrium mineral=melt partition coef®cients for Making full use of the 238U±230Th system requires U and Th between clinopyroxene and basaltic melt knowing in which minerals and at what depths in are small (¾103 to 102) and differ by only about a factor of 2, with U being the more incompatible Ł Corresponding author. Fax: C1 (617) 253-7102; E-mail: element [9±13]. The similarity between U and Th [email protected] partition coef®cients implies that clinopyroxene has

0012-821X/98/$19.00  1998 Elsevier Science B.V. All rights reserved. PII S0012-821X(98)00107-1 506 J.A. Van Orman et al. / Earth and Planetary Science Letters 160 (1998) 505±519 little ability to fractionate these elements. Even if Table 1 signi®cant fractionation were possible it would be Kunlun Mts. diopside composition 230 in the wrong sense to explain the Th excesses Oxide Weight (%) observed in nearly all recently erupted mid-ocean ridge basalts (MORB) [14,15]. On the other hand, SiO2 55.5 TiO2 0.06 U and Th partition coef®cients between garnet and Al2O3 0.88 basaltic melt differ by nearly an order of magnitude FeOtot 0.55 [11,16,17], with Th being the more incompatible MgO 18.1 element. The equilibrium partitioning data, then, ap- CaO 24.7 pear to require melting in the presence of garnet Na2O0.53 Total 100.3 to explain the excess 230Th in MORB. This is an important conclusion because it requires that melting begins at depths greater than ¾75 km, where garnet becomes the stable aluminous phase in peridotite near pure diopside (Table 1). Each crystal was sec- [18,19] or, alternatively, that garnet pyroxenite is an tioned perpendicular to the c crystallographic axis important component of the MORB source [20]. into slabs ¾0.5 mm thick. One side of each slab The conclusion that clinopyroxene cannot deliver was polished to 0.06 µm alumina grit and then cut excess 230Th to the melt is valid if chemical equi- into pieces 1±2 mm on a side. These pieces were librium between cpx and melt is maintained during cleaned ultrasonically in deionized water and then melting. If, however, diffusion in clinopyroxene is pre-annealed for 2 days at 1200ëC, with oxygen fu- slow relative to the melting rate, then equilibrium gacity controlled near the quartz±fayalite±magnetite may not be achieved. In this case fractionation be- (QFM) buffer. tween U and Th depends strongly on the relative The diffusion source material was deposited as diffusion rates of these elements [21±23]. A reversal an aqueous solution onto the polished surface of in the effective compatibility order of two elements the diopside. A dilute (0.05 M) nitric acid solution is possible if the more compatible element diffuses that contained dissolved U, Th, and Al in 1 : 1 : 4 signi®cantly faster than the incompatible element molar proportions was prepared from 10,000 µg=ml [21]. To produce excess 230Th in the melt by a ICP standard solutions and diluted with puri®ed disequilibrium mechanism would require that (1) U water to 270 µgU=ml. Approximately 1 µlofthis diffuses slowly enough in clinopyroxene that equi- solution was deposited onto the polished surface of librium between cpx and melt is not achieved, and the diopside, along with a small amount of methanol (2) Th diffuses signi®cantly faster than U. To deter- to reduce surface tension and allow the solution mine whether disequilibrium melting can in¯uence to spread uniformly over the sample surface. The U and Th partitioning, we performed a series of ex- solution was evaporated in air at 120ëC, which left a periments to measure the diffusion rates of U and Th thin layer of nitrates with concentration ¾ 5 ð 1010 in diopside in the temperature range 1150±1300ëC. mol=mm2. At the conditions of the diffusion anneal, with fO2 at the QFM buffer, the nitrates decompose to Al2O3 and nearly stoichiometric UThO4 [24,25]. 2. Experimental and analytical techniques Because U and Th have the same formal charge in the oxide (C4) that they presumably have in 2.1. Sample preparation diopside, no further redox reactions are necessary to introduce these elements into the diopside lattice. Diffusion experiments were performed on gem There was no evidence in any of the experiments quality natural diopside crystals from the Kunlun that the tracer layer reacted with the diopside. We Mts., China (American Museum of Natural History calculate that less than 0.01% of the U and Th samples #100242 and #104501). The crystals were diffused into the diopside crystal during any of the light green, transparent, free of cracks and visible in- experiments, and thus the tracer layer provided an clusions, and had major element compositions very effectively in®nite reservoir of U and Th. J.A. Van Orman et al. / Earth and Planetary Science Letters 160 (1998) 505±519 507

Al was added to the tracer solution as a possi- 2.3. Analyses ble charge-balancing species. However, as discussed below in the Analyses section, we were unable to Concentration pro®les in the annealed samples detect Al diffusion pro®les in any of our samples, were measured using the Cameca IMS 3f ion micro- and it is unclear whether Al was actually transported probe at the Woods Hole Oceanographic Institution. into the crystal during any of the diffusion anneals. A primary beam of O ions was focused onto the surface of the sample, producing secondary ions that 2.2. Diffusion anneals were continuously analyzed in a mass spectrome- ter. In this procedure successively deeper layers are Diffusion anneals were performed at constant sampled as the primary beam sputters through the temperature in open Pt crucibles placed in the sample, allowing concentrations to be measured as hotspot of a Deltech DT31VT vertical gas mixing functions of depth. The primary beam was acceler- furnace, with anneal times ranging from 2 to 24 ated under a potential of ¾8.2 kV and was focused to days. Run temperature was attained within approx- a diameter of ¾20±30 µm, with beam currents usu- imately 5 minutes after introducing the charge into ally in the range 10±20 nA. By analyzing `zero-time' the furnace and was continuously monitored using experiments, we found that optimum resolution was aPt±Pt90Rh10 thermocouple calibrated against the achieved by rastering the primary beam over a square melting points of NaCl, Au, and Pd. Fluctuation in area 150±200 µm on a side and inserting a circu- temperature over the course of each anneal was gen- lar mechanical aperture 68 µm in diameter into the erally within š2ëC, with the exception of a single secondary optics. The aperture was centered over the experiment (UTh1200b) in which the thermocouple sputtered area so that secondary ions were collected degraded during the anneal; temperature for this ex- only from the central, ¯at portion of the pit. This periment was assigned an uncertainty of š5ëC. Oxy- con®guration optimized depth resolution and min- gen fugacity was controlled near the QFM buffer imized contamination of the diffusion pro®le with by mixing CO2 and H2 gases and was continuously material from the tracer layer. In analyses of `zero- monitored with a solid ZrO2±CaO electrolyte oxy- time' experiments, U and Th counts dropped sharply gen sensor calibrated against the Fe±FeO, Ni±NiO, after penetration of the tracer layer, and fell to back- and Cu±Cu2O buffers. Variation in fO2 over the ground levels at a depth of approximately 40 nm. course of each experiment was within š0.1 log unit. Secondary ions were measured in a series of Experiments were quenched by removing the sample cycles in which electron multiplier counts were col- from the furnace and allowing it to cool in air. Heat- lected in sequence on masses 27Al (3 s), 44Ca (5 s), ing and cooling times were insigni®cant compared 232Th (15±30 s) and 238U (15±30 s). The 44Ca signal to the anneal durations, and therefore no attempt was monitored to determine the position of the inter- was made to account for diffusion during run-up or face between the tracer layer and the diopside. 44Ca quenching. counts increased sharply after the tracer layer was In addition to the diffusion anneals, we performed penetrated, which was usually within the ®rst two `zero-time' experiments in order to evaluate system- or three cycles, and then decreased very gradually atic errors associated with analysis of the diffusion due to charging of the sample surface. To correct pro®les. Diffusion couples were prepared as above, for the charging effect and for small ¯uctuations taken to 1200ëC and held there for approximately in primary beam intensity, 232Th and 238U counts ®ve minutes, then quenched in air. were normalized to 44Ca, and the ratios 232Th=44Ca After quenching, samples were rinsed ultrasoni- and 238U=44Ca were used in place of raw Th and U cally in deionized water, then ethanol. They were counts to model diffusion coef®cients. then mounted in epoxy with the diffusion source Aluminum concentration pro®les in annealed layer facing up and coated with a thin layer of gold, samples were indistinguishable from those from which provided a conductive surface for ion micro- `zero-time' experiments: Al counts dropped sharply probe depth pro®ling. over the upper 20±30 nm and were nearly constant over the remaining depth interval. 508 J.A. Van Orman et al. / Earth and Planetary Science Letters 160 (1998) 505±519

2.4. Analysis `reversals' 2.6. Determination of diffusion coef®cients

In order to be certain that the ion microprobe Diffusion in these experiments is well described depth pro®ling technique was measuring the true by the equation for one-dimensional diffusion in a concentration pro®le in the sample, we devised an semi-in®nite medium, with constant interface con- analysis `reversal'. We wanted to determine whether centration [26]: the diffusion pro®le was contaminated during the Â Ã C.x; t/ C x analysis by material from the tracer layer. Therefore, 0 D erf p (1) we reanalyzed one experiment (UTh1200a) from the Ci C0 2 Dt reverse direction. The sample was extracted from its where C is the concentration at depth x after anneal- epoxy mount and remounted in high-strength thin ing time t, C0 is the concentration at the interface, Ci section epoxy on a glass slide, with the tracer layer is the initial concentration in the diopside crystal (es- facing down. It was then carefully ground to a thick- sentially zero for both U and Th), and D is the diffu- ness of ¾7 µm. After grinding, the thin section was sion coef®cient. A typical diffusion pro®le is shown polished with diamond and alumina pastes (to 0.06 in Fig. 1a. The ®rst few points in the pro®le were µm alumina) and prepared as described above for sometimes high due to contamination from the tracer SIMS depth pro®ling. Sputtering began on the pol- layer, and these points were disregarded in the ®tting ished surface and proceeded through the diopside procedure. To extract a diffusion coef®cient from crystal to the tracer layer. A higher primary beam the data, each pro®le was linearized by plotting the current (30 nA) and smaller raster area (100 µm inverse error function of the left-hand side of Eq. 1 ð 100 µm) were used for this analysis so that the against depth (Fig. 1b). The diffusion coef®cient was relatively thick sample could be penetrated in a rea- then calculated from the relation D D .4m2t/1, sonable time (approximately 10 h). In the discussion where m is the slope of the least-squares line ®t below, results of this analysis are referred to as a to the inverted data. On the inverse error function reversal. plot, the data scatter and tail off slightly at the deep end of the pro®le as the concentration drops toward 2.5. Depth measurement the detectability limit. We ®t only the shallower part of the concentration pro®le and stopped including The depth of each sputtered pit was measured points when count rates approached background lev- with a Sloan Dektak 8000 surface-contact pro®lome- els. Small adjustments in the interface concentration, µ ter equipped with a 2.5 m diamond-tipped stylus. C0, were made in the ®tting procedure to force the Prior to making the depth measurements, the gold line through the origin. coat was removed from the sample by ultrasonic rinsing in a KI solution. At least two orthogonal scans were made for each pit, and in each we consid- 3. Results ered the mean depth over the central ¾70 µm. Pits produced under the same primary beam conditions 3.1. Error analysis yielded consistent sputtering rate estimates, con®rm- ing that the sputtering rate during each analysis was There are two types of uncertainty in our data constant. Although the pro®lometer is capable of set, one associated with intra-run variability and the measuring depth to a precision of ¾1 nm, the un- second associated with reproducibility of diffusion certainty in the depth measurement was considerably coef®cients for multiple experiments run at similar greater, being controlled primarily by the roughness conditions. The source of uncertainty in the ®rst is of the sample surface. In most cases the vertical re- associated with error in measurement of the depth lief at the bottom of the sputtered pit was between 10 pro®le. In the second case there is the additional and40nm. source of systematic error associated with our ability to exactly reproduce the conditions of the experiment. J.A. Van Orman et al. / Earth and Planetary Science Letters 160 (1998) 505±519 509

Fig. 1. A typical diffusion pro®le (a) and its errorp function inversion (b). The diffusion coef®cient is calculated from the slope of the line ®t to the inverted data, which is equal to .2 Dt/1. The error function curve shown in (a) is a solution to Eq. 1 using the diffusion coef®cient calculated from the linear inverse error function ®t.

The uncertainties associated with measurement the pro®les was very similar (Fig. 2). A measure of of the diffusion pro®le can be assessed by compar- the inter-run reproducibility can be inferred by com- ing diffusion coef®cients extracted from individual paring experiments that were performed under the diffusion pro®les from the same experiment. The un- same temperature conditions. The range of values certainty in the reported value of D for each pro®le among duplicate runs is similar to that determined was estimated from a combination of the error in the from separate pro®les in a single sample (Table 2). crater depth measurement and the formal uncertainty from the linear ®t to the erf-inverted diffusion pro®le. 3.2. Time series When multiple diffusion pro®les were measured on a single sample, these yielded diffusion coef®cients If volume diffusion is the mechanism responsible that in nearly all cases were identical within error for U and Th transport, then the calculated diffusion (Table 2). A single diffusion coef®cient for each coef®cients should be independent of the duration of experiment was calculated by taking the weighted the experiment. We ran three experiments at 1200ëC average of the results from each diffusion pro®le, that ranged in duration from 71 to 278 hours (nearly with the weights being equal to the inverse square a factor of 4). U and Th diffusion coef®cients in of the uncertainty in each measurement [27]. These these experiments agreed within less than a factor of are the uncertainties reported in bold type in Ta- two (Fig. 3). Two experiments at 1300ëC also were in ble 2 and shown in the Arrhenius plots in Fig. 4. An excellent agreement, although they covered a much independent assessment of the diffusion coef®cient smaller range in time (56±86 h). measured in the `forward' pro®les is given by our `reverse' pro®le measurement. Diffusion coef®cients 3.3. Temperature dependence of diffusion extracted from the reverse concentration pro®le were in excellent agreement with the forward measure- The diffusion coef®cients calculated for each ex- ments of the same experiment, and the quality of periment were used to assess the temperature depen- 510 J.A. Van Orman et al. / Earth and Planetary Science Letters 160 (1998) 505±519

Table 2 Run conditions and experimental results

ID T Anneal time DTh š 1σ DU š 1σ (ëC) (h) (m2=s) (m2=s) UTh1150a 1150 576.80 1.59 š 0.31 ð 1021 4.54 š 1.55 ð 1022 UTh1150b 1150 456.61 1.18 š 0.21 ð 1021 9.52 š 1.69 ð 1022 1.55 š 1.09 ð 1021 5.78 š 6.98 ð 1022 1.20 š 0.20 ð 1021 9.44 š 1.66 ð 1022 UTh1200a 1200 70.65 4.56 š 1.21 ð 1021 a 3.33 š 7.27 ð 1021 a 6.38 š 2.37 ð 1021 2.54 š 1.99 ð 1021 5.06 š 1.00 ð 1021 2.39 š 0.70 ð 1021 6.87 š 3.33 ð 1021 4.53 š 2.28ð 1021 5.26 š 0.72 ð 1021 2.90 š 0.69 ð 1021 UTh1200d 1200 199.50 4.49 š 1.05 ð 1021 5.46 š 2.10 ð 1021 2.87 š 0.78 ð 1021 2.44 š 0.94 ð 1021 3.80 š 0.67 ð 1021 3.95 š 1.08 ð 1021 UTh1200b 1200 277.67 2.82 š 0.94 ð 1021 2.29 š 0.82 ð 1021 3.45 š 0.87 ð 1021 2.40 š 0.85 ð 1021 3.22 š 0.65 ð 1021 2.35 š 0.59 ð 1021 UTh1 1300 56.22 1.32 š 0.62 ð 1020 1.06 š 0.50 ð 1020 1.39 š 0.75 ð 1020 1.84 š 0.86 ð 1020 2.70 š 0.83 ð 1020 2.88 š 0.74 ð 1020 2.12 š 0.49 ð 1020 2.35 š 0.48 ð 1020 UTh2 1300 86.10 1.97 š 0.81 ð 1020 1.36 š 0.69 ð 1020 3.98 š 2.06 ð 1020 3.30 š 1.95 ð 1020 2.75 š 0.89 ð 1020 2.18 š 0.84 ð 1020

Values in bold type are weighted averages for each experiment. a This pro®le was measured in the reverse direction.

dence of U and Th diffusion, which can be described by the Arrhenius equation:

Ha=RT D D D0e (2)

where D0 is a frequency factor, Ha is the activation enthalpy, R is the gas constant and T is absolute temperature. Both U and Th show good Arrhenian behavior, exhibiting linear trends on a plot of log D vs. inverse temperature (Fig. 4). Linear least-squares regressions of the data [28] yield the following Ar- rhenius relations:

log DU D .5:75 š 0:98/ .418 š 28 kJ=mol/=2:303RT (3)

log D D .7:77 š 0:92/ Fig. 2. Comparison of diffusion pro®les measured from opposite Th directions on the same sample. The circles represent a concen- .356 š 26 kJ=mol/=2:303RT (4) tration pro®le measured in the `forward' direction, through the 2 surface diffusant layer. The squares represent a depth pro®le where the diffusion coef®cients are in units of m =s made in the reverse direction. and the uncertainties quoted are š1σ. Uranium and J.A. Van Orman et al. / Earth and Planetary Science Letters 160 (1998) 505±519 511

Fig. 3. Time series at 1200ëC and 1300ëC for U and Th diffusion anneals. The horizontal dashes represent individual measurements of a single experiment, and circles indicate the weighted average. Squares indicate diffusion coef®cients extracted from Fig. 4. Arrhenius plots of U and Th diffusivity between 1150 and `reverse' pro®les. 1300ëC. The circles represent average diffusion coef®cients for each experiment, and the solid lines are weighted least-squares ®ts to the data. Activation enthalpies and pre-exponential factors thorium diffusivities converge with increasing tem- are given in the text. perature and are very similar near the melting point of diopside. orders of magnitude higher than our results for U diffusion. His experiments involved diopside crystals 4. Discussion annealed in a diopside±albite±anorthite melt, and it is possible that growth of the crystals during the ex- 4.1. Comparison with other diffusion data in periment accounts for the high apparent diffusivity. diopside Experimental measurements of tracer-, self-, and inter-diffusion coef®cients have been reported for Seitz [29] reported a U tracer diffusion coef®cient several other cations in natural diopside, and these for diopside of 1016 m2=s at 1240ëC, which is four are in good general agreement with our results 512 J.A. Van Orman et al. / Earth and Planetary Science Letters 160 (1998) 505±519

(Fig. 5). The tracer diffusion data for cations that 4.2. Early partial melting? partition onto the M2 site are broadly consistent with trends observed in other minerals: diffusion rates Several studies have observed that Fe-bearing py- increase with decreasing ionic charge and with de- roxenes exsolve a silica-enriched melt phase at tem- creasing ionic radius. The in¯uence of ionic radius peratures far below their nominal melting tempera- on tetravalent and trivalent cation diffusion rates in tures [38,39]. This phenomenon, termed early partial diopside is of the same order of magnitude as that melting (EPM), appears to be related to cation va- observed in zircon [35,36]. In diopside, for example, cancies associated with Fe3C impurities [40] and is the diffusion coef®cient for Yb (0.985 AÊ radius in sensitive to the Fe content of the pyroxene and the VIII-fold coordination [37]) is a factor of ¾16 higher fO2 of the surrounding atmosphere. The diopside we than for Ce (1.14 AÊ ) at 1300ëC. In zircon, the diffu- used has less Fe (less than half by mole) than pyrox- sion coef®cient for Yb is a factor of ¾19 higher than enes in which EPM has so far been observed, and Sm (1.08 AÊ ) at the same temperature. In contrast, it is unclear whether EPM would be expected under ionic charge appears to be much less important in the conditions of our experiments. Optical inspection diopside than in zircon, particularly for the C3and of our samples revealed no evidence for EPM, and if C4 ions. In diopside the diffusion coef®cients for present, melt precipitates must be quite small in size. U4C (1.00 AÊ )andYb3C (0.985 AÊ ) differ by about We note that where EPM has been unambiguously an order of magnitude. In zircon the effect of ionic observed it has little effect on cation diffusion be- charge is much greater, with Yb diffusivity being cause the precipitates do not form an interconnected about 5 orders of magnitude faster than U [35,36]. network. Dimanov et al. [31] observed abundant This difference may be a consequence of the rela- glassy precipitates under the optical microscope in tively small energy of the divalent M2 lattice site in their experiments on Ca self-diffusion in Fe-bearing diopside compared to that of the tetravalent VIII-fold diopside, yet observed only a very small (one might site in zircon. argue unresolvable) in¯ection in the Arrhenius curve at the onset of EPM. We consider EPM effects absent in our data.

5. Equilibration during melting and melt transport

In this section we use the U and Th diffusion data presented above to estimate the time scale of chemical equilibration during melting and melt transport. We ®rst develop a scaling argument which shows that equilibrium between melt and the interiors of cpx grains is unlikely under most conditions appropriate to melting beneath mid-ocean ridges. We then use a numerical melting model to assess the degree of fractionation between U and Th during disequilibrium near-fractional melting.

5.1. Scaling argument Fig. 5. Summary of cation diffusion data in natural, near end- member diopside. The dashed line labeled CATS-Di refers to We follow the scaling approach adopted by interdiffusion of CaAl2SiO6 and diopside [30]. The dotted line shows Ca self-diffusion coef®cients [31]. Solid lines refer to Spiegelman and Kenyon [41] and Hart [42] to eval- tracer diffusion coef®cients for Sr [32], Pb [33], Yb and Ce [34], uate equilibration between clinopyroxene and melt and U and Th (present study). during melt transport. The question we address is J.A. Van Orman et al. / Earth and Planetary Science Letters 160 (1998) 505±519 513 whether partitioning of U and Th between clinopy- For a Peclet number > 1, equilibration will be roxene and melt will approach equilibrium under <83%. conditions relevant to the production and transport of Eq. 10 is equivalent to the expression derived mid-ocean ridge basalts. by Hart [42] (his Eq. 6), who considered the time For melt percolating upward through an upwelling scale for growth of a melt tubule lying along grain porous matrix, diffusive equilibration is governed by boundaries rather than the time scale for porous ¯ow the Peclet number, which is equal to the ratio of the through a network of melt tubules. These two time characteristic time required for diffusive transport in scales are the same because mass balance requires the solid grains of the matrix to the time required for that the melt production and extraction rates are melt to move through the system. For melt ¯owing equal. across a layer of thickness H, which for our purposes At 1400ëC, a temperature relevant to melting be- can be taken to be the height of the mantle melting neath mid-ocean ridges [45±47], the diffusion co- column, the characteristic advection time can be ef®cients for U and Th in clinopyroxene are both written: approximately 1:5 ð 1019 m2=s, extrapolating from H the Arrhenius relations given in Eqs. 3 and 4. Both ³ tadv w (6) U and Th have cpx=melt partition coef®cients in the where w is the melt velocity. range 0.001±0.01 [9±13]. Taking a partition coef®- As discussed by Hart [42], the characteristic dif- cient of 0.005, cpx grain diameter of 5 mm, mantle fusion time, or more speci®cally the time required upwelling rate of 3 cm=yr, melting degree of 15%, for the concentration of an element in the melt to and melting column height of 60 km, the Peclet reach 83% of its ultimate equilibrium value, is ap- number is ¾20, which is within the disequilibrium proximately: domain. Fig. 6 explores a more complete range of parameter values and shows that partitioning of U r 2 t ³ (7) and Th between clinopyroxene and melt will be a D . /1:5 D K disequilibrium process except at very slow spread- where r is the radius of a cylinder of melt, D is ing ridges. For example, a 5 mm grain at peridotite the diffusion coef®cient in the solid, and K is the solidus temperatures (¾1400±1500ëC) will approach solid=melt equilibrium partition coef®cient. In this expression, the characteristic diffusion time depends on the compatibility of the element being considered. An incompatible element (K < 1) requires longer diffusion times than a compatible element because a larger volume of solid must be tapped in order to provide its higher relative abundance in the melt. In a grain-scale porous network with wetting an- gle ¾50ë, the radius of a melt tubule is related to the grain size d and porosity by [42,43]: r ³ 0:13d./1=2 (8) Conservation of mass in a one-dimensional upwelling column requires that [44]: WF D w (9) where W is the solid velocity, w is the melt velocity, Fig. 6. Plot of cpx grain diameter vs. temperature showing and F is the melting degree. Combining Eqs. 6±9, the conditions under which U and Th partitioning equilibrium the Peclet number can be written: may be obtained during adiabatic melting. The curves represent solutions to Eq. 10, with tD=tadv D 1, for upwelling rates of t d2WF = Pe D D ³ : (10) 1 and 10 cm yr. Each curve separates regions of greater than 1:5 (above the curve) and less than 83% equilibration. tadv 59D.K / H 514 J.A. Van Orman et al. / Earth and Planetary Science Letters 160 (1998) 505±519 partitioning equilibrium only at upwelling rates of a rate different from that of the bulk rock, that rate ¾1cm=yr or slower. In contrast to the usual assump- depending on the stoichiometric coef®cient for cpx tion [3,7,42], we conclude that equilibrium may not in the melting reaction and on the abundance of cpx be achieved during melting or grain-scale porous in the rock. We consider U and Th ¯uxes between ¯ow for highly incompatible and slowly-diffusing clinopyroxene and melt only; in other words, we elements. assume that when melting begins all of the U and Th in the system reside in cpx and that the solid=liquid 5.2. Numerical model partition coef®cients for other solid phases are equal to zero. The scaling argument developed above allows The concentrations of 238Uand230Th in a cpx us to estimate the conditions under which disequi- grain are expressed by: ! librium is expected during near-fractional melting, @CU @2CU 2 @CU but it does not allow us to quantitatively predict cpx . / cpx cpx ½ U D DU t C UCcpx the degree to which elements are fractionated due @t @r 2 r @r to their different diffusion rates. Numerical models ! @CTh @2CTh 2 @CTh have shown that the extent of elemental fractiona- cpx D D .t/ cpx C cpx tion during disequilibrium melting is sensitive to the @t Th @r 2 r @r relative diffusion rates (e.g. [21±23]). We adopt the m ½ Th U ½ Th disequilibrium melting model of Qin [21], modi®ed U Ccpx ThCcpx (11) mU to allow for non-modal melting and for temperature- where CU and CTh are concentrations of 238Uand dependent (and therefore time-dependent) diffusion cpx cpx 230Th at time t and radial distance r. ½ and ½ coef®cients, in order to evaluate whether U and Th U Th are the decay constants and m and m the masses can be fractionated signi®cantly by clinopyroxene, U Th of 238Uand230Th, respectively. These are expres- and in particular whether there are any conditions sions of Fick's second law in spherical coordinates, under which their compatibility order may be re- modi®ed to account for decay of 238Uand230Th. versed. The diffusion coef®cients D and D depend on the The model we adopt is a near-fractional melting U Th temperature-time path followed during melting and model modi®ed to account for diffusion-controlled are thus functions of time. We assume that tempera- chemical equilibration between solid and its enclos- ture decreases linearly with time and specify initial ing melt. The solid is assumed to be composed and ®nal temperatures T and T . of spherical grains of equal size which retain their i f Initial and boundary conditions for Eq. 11 are: spherical symmetry during melting. These grains are U . ; / U assumed to be homogeneous in composition when Ccpx r 0 D C0 melting begins, with 238Uand230Th in radioactive ½ Th . ; / U U equilibrium. During melting, the interface between Ccpx r 0 D C0 (12) ½Th the solid and melt is always in chemical equilibrium, U . . /; / U. / and concentrations in the interiors of grains are con- Ccpx R t t D KUCm t trolled by diffusion. Melt remains with the residue Th . . /; / Th. / until a critical melt fraction is reached and is there- Ccpx R t t D KThCm t (13) U 238 after removed at the rate that keeps the porosity con- where C0 is the initial concentration of Uinthe stant. Extracted melt is pooled in a chemically iso- clinopyroxene and KU and KTh are the equilibrium lated reservoir, and both residual and extracted melts cpx=melt partition coef®cients for U and Th. The are assumed to homogenize continuously and instan- ®rst pair of equations describes the initial condi- taneously, an excellent approximation considering tion, in which 238Uand230Th are assumed to be that U and Th diffusion in silicate melts is ¾8orders in radioactive equilibrium and distributed homoge- of magnitude faster than in clinopyroxene [48]. neously in a grain of clinopyroxene. The second For simplicity, the melting rate is assumed to be states that the clinopyroxene rim is at all times in constant, but clinopyroxene is allowed to dissolve at equilibrium with the residual melt. J.A. Van Orman et al. / Earth and Planetary Science Letters 160 (1998) 505±519 515

Concentrations in the residual melt are described Table 3 by: Model parameters and values þ Ð þ Parameter Value(s) U @CU d VmCm 2 cpx þ D4³ R.t/ DU.t/ þ , critical melt fraction 0.001±0.01 dt @r þ = rDR.t/ KU,Ucpx liq partition coef®cient 0.005 KTh,Thcpx=liq partition coef®cient 0.01 × U . . /; / × U ½ U C m Ccpx R t t eCm UVmCm , cpx coef®cient in melting reaction 0.80 þ Xcpx, volume fraction cpx 0.15 Ð þ Th @CTh R0, cpx initial radius, mm 1±5 d VmCm 2 cpx þ D4³ R.t/ DTh.t/ þ H, melting column height, km 33±63 dt @r þ F, melting degree 0.10±0.20 rDR.t/ W, mantle upwelling rate, cm=yr 1±10 × Th . . /; / × Th C m Ccpx R t t eCm Ti, initial temperature, K 1588±1748 Â Ã Tf, ®nal temperature, K 1506±1514 m ½ Th U ½ Th C Vm U Cm ThCm (14) mU where CU and CTh are concentrations in the residual volume fraction of clinopyroxene in the solid. The m m × melt, V is the volume of residual melt, R(t)is melt extraction rate, e, is initially zero and is equal m to: the radius of a clinopyroxene grain, ×m and ×e are the volumetric melting and melt extraction rates, × 1 × e D m (17) respectively, and is the stoichiometric coef®cient 1 crit of clinopyroxene in the melting reaction. The terms when the melt fraction exceeds the critical value, on the right-hand side of Eq. 14 represent, in order, . The radius of a clinopyroxene grain, R(t), (1) the total diffusive current across the cpx=melt crit changes with time according to: interface at time t, (2) the U or Th mass ¯ux out of Â Ã the clinopyroxene by melting, (3) the mass ¯ux out × . / m t : of the system by melt extraction, and (4) changes in R t D R0 1 (18) V0 Xcpx mass due to decay of 238Uand230Th. In the pooled extracted melt, concentrations are Eqs. 11±18 were solved numerically using an given by: implicit Crank±Nicolson ®nite difference scheme. Ð The moving boundary between clinopyroxene and U d VMC melt was accommodated by keeping a ®xed number M D × CU ½ V CU dt e m U M M of radial grid points and rescaling the grid to the new Ð Â Ã grain radius at each time step. d V CTh m M M × Th ½ Th U ½ Th We performed a series of model runs using the D eCm C VM U CM ThCM dt mU parameter values listed in Table 3. In each, the radial (15) distribution of 238Uand230Th in cpx spheres and their concentrations in the residual and aggregated U Th where CM and CM are concentrations in the ex- melts are calculated as functions of time. Fig. 7a tracted melt and V M is the volume of melt extracted. illustrates the evolution of Th and U distributions in × The melting rate, m, is taken to be constant and a cpx sphere under melting conditions typical for fast is equal to: spreading ridges. Note that the interior of the crystal maintains high relative concentrations of U and Th V0WF ×m D (16) and is far from equilibrium with the residual melt H even at high degrees of melting. In contrast, at slow where W is the solid upwelling velocity, F is the spreading ridges (Fig. 7b) clinopyroxene crystals total degree of melting, and H is the height of approach equilibrium with the residual liquid. The the melting column. V0 is the initial solid volume, upwelling rate at which disequilibrium becomes im- = ³ 3= de®ned as V0 D 4 3 R0 Xcpx,whereXcpx is the portant depends on several factors, most importantly 516 J.A. Van Orman et al. / Earth and Planetary Science Letters 160 (1998) 505±519

rium develops even at upwelling rates as slow as 1 cm=yr.

5.3. Deformation of high-Ca pyroxene during mantle upwelling

It is reasonable to question whether the model presented above is realistic for describing a solid mantle that deforms as it ascends. The model considers spherical high-Ca pyroxene grains that remain undeformed during decompression melting. Deformation of these grains would decrease the net diffusion length, thereby increasing the degree of equilibration between solid and melt. It is therefore important to assess the degree of deformation that is expected beneath mid-ocean ridges. The degree to which high-Ca pyroxene is de- formed during upwelling depends on its viscosity relative to the viscosities of olivine and orthopyroxene. Plastic ¯ow laws of the form " " . /m¦ n . = /; P DP0 fO2 exp Q RT (19) where "P is strain rate and ¦ shear stress, have been Fig. 7. Model Th and U diffusion pro®les developed across determined for olivine [52], orthopyroxene [53], a 2.5 mm clinopyroxene grain during progressive melting. (a) and diopside [54] and are given in Table 4. Un- For a mantle upwelling rate of 10 cm=yr cpx is in strong disequilibrium with the melt, even after 15% melting. (b) In der typical upper mantle conditions of T D 1400ëC, 7 contrast, at 1 cm=yr cpx is near equilibrium with the melt after fO2 D 4 ð 10 atm, and ¦ D 30 MPa, the viscosi- only ¾3% melting. Both ®gures show elemental rather than ties of olivine and orthopyroxene are, respectively, 12 13 isotopic concentrations. D 1%, Ti D 1395ëC, Tf D 1240ëC, 2:6 ð 10 and 7:4 ð 10 Pa s. Diopside, in contrast, H D 48 km. is several orders of magnitude stiffer than either of these minerals, having a viscosity of 1:9 ð 1017 Pa the grain size. The examples shown in Fig. 7 both s under the same conditions. We therefore expect used an initial grain diameter of 5 mm. This is within that olivine and orthopyroxene will accommodate the range of cpx grain sizes (¾2±10 mm diameter) most of the strain associated with upwelling and in fertile peridotite xenoliths from Pali-Aike, Chile, that high-Ca pyroxene will remain relatively unde- which geochemically and isotopically resemble un- formed. The assumption in our model that high-Ca melted MORB-source mantle [49]. Clinopyroxene pyroxene grains remain undeformed therefore seems grains from abyssal peridotites are typically smaller, reasonable. with diameters up to 2±3 mm [50,51], but these rocks represent melting residues and are more appropriate 5.4. Implications for 238U=230Th radioactive for estimating the ®nal, rather than initial, cpx radii. disequilibrium in MORB The actual size of cpx grains in unmelted MORB mantle is not well known, but a range of 2±10 mm It is clear that under certain melting conditions provides a realistic lower limit. In model runs using solid-state diffusion may exert strong control on the an initial cpx diameter of 2 mm (not shown), cpx ¯uxes of U and Th between cpx and melt. Can ex- is in moderate disequilibrium with residual melt for cess 230Th in the melt be generated under these con- fast spreading ridges (upwelling rate of 10 cm=yr). ditions? Fig. 8a shows activity ratios [230Th=238U] For an initial diameter of 10 mm, strong disequilib- in the aggregated melt for a model run with cpx J.A. Van Orman et al. / Earth and Planetary Science Letters 160 (1998) 505±519 517

Table 4 Flow laws for olivine, orthopyroxene, and diopside

a Mineral Orientation of stress axis "0 (σ in MPa) nm Q(kJ=mol) " Ref. Comments

Olivine (Fo91) [110] 0.02 3.5 0.36 230 "1 [52] buffered by opx 22 1.3 ð 10 3.5 0.10 1000 "2 [52] 1.2 3.5 0.15 290 "3 [52] 15 Orthopyroxene (En96) random 2.1 ð 10 3.9 0 880 [53] buffered by olivine Diopside [110] 7.0 ð 1018 6.0 0 48 [54] for T > 1130ëC a " " "1 "1 1 For olivine, D 1 C [ 2 C 3 ] .

progresses. 230Th excesses never occur because during the early stages of melting, at high temperature, U has a higher diffusivity than Th and is thus re- leased more quickly into the melt. The dashed curves re¯ect the uncertainty in the relative diffusivity of U and Th, and were generated by running the model with U and Th Arrhenius lines rotated within their 1σ uncertainty limits. The upper curve pairs the steepest Th Arrhenius line with the shallowest U Arrhenius line, and the lower curve vice versa. The uncertainty envelope covers a moderate range of activity ratios at small melting degrees, including a region of small 230Th excesses, but at higher degrees of melting becomes quite narrow and does not overlap the region with [230Th=238U] > 1. Fig. 8b shows a similar plot for an upwelling rate of 1 cm=yr. In this case cpx is close to equilibrium with the melt, and varying the U and Th Arrhenius parameters within their uncertainty limits has little effect on the activity curves. The activity ratio has a nearly constant value of ¾0.96 for melting degrees above ¾2%. We conclude that it is highly unlikely that high-Ca 230 =238 pyroxene could be responsible for generating the Fig.8.[ Th U] activity ratios in the aggregated melt. The 230 dashed curves de®ne an uncertainty envelope that is based on Th excesses observed in MORB. Most MORB 230 238 the error in the Arrhenius parameters for U and Th. Uranium lavas have [ Th= U] activity ratios of 1.1±1.4 and thorium are not ef®ciently fractionated under either (a) (e.g. [14,15]), well outside the uncertainty limits de- disequilibrium or (b) near-equilibrium melting conditions, and in ®ned by the upper and lower curves in Fig. 8a or both cases the activity ratio is near 1 throughout most of the b. Even at a very high degree of chemical dise- melting interval. Parameter values are the same as those used in the calculations shown in Fig. 7. quilibrium, with a cpx diameter of 10 mm and an upwelling rate of 10 cm=yr, the uncertainty envelope never extends to an activity ratio above 1.15. diameter of 5 mm and upwelling rate of 10 cm=yr. Clinopyroxene can therefore be ruled out as the The solid curve was generated using the Arrhenius source of the 230Th excess signal observed in ocean relations for U and Th given in Eqs. 3 and 4. The ¯oor basalts, as it is unable to produce signi®cant activity ratio is always below 1 and gradually con- 230Th excesses under either equilibrium or disequi- verges on a steady-state value of ¾0.97 as melting librium conditions. 518 J.A. Van Orman et al. / Earth and Planetary Science Letters 160 (1998) 505±519

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