Chapter 1

Rheology of Partially Molten Rocks

David L. KOHLSTEDT, Quan BAIt, Zi-Chao WANG:t: and Shenghua MEY Department of Geology and Geophysics, University of Minnesota, MN 55455 U.S.A.

t Now at: Seagate Recording Media 3845 E. Coronado Street Anaheim, CA 92807 U.S.A.

1: Now at: Department de Geologie, University de Montreal C.P.6128 Succ. Central ville Montreal, (QC), H3C 3J7 CANADA

+ Now at: Department of Materials Science, University of Minnesota,Minneapolis, MN 55455 U.S.A. '"'

Key words: Rock rheology, partial melting, differential stresses, power law creep, grain size exponent, stress exponent, oxygen fugacity, microcracking

Abstract: Melt and water are two of the most important elements governing the viscosity of the rocks in regions of Earth's upper mantle such as beneath a mid-ocean

ridge and in the mantle wedge above a subducting plate. Over the past five

years, laboratory deformation experiments under controlled thermodynamic conditions have yielded quantitative relationships describing the dependence

of strain rate, t:, and, thus, viscosity, 11, on melt fraction, <1>, and hydrogen or

hydroxyl concentration, COH, as well as on differential stress, cr, grain size, d,

temperature, T, pressure, P, oxygen fugacity, f 02' and silica or pyroxene

activity, a opx ' These constitutive equations provide a critical part of the

framework necessary for modeling processes such as convective flow in the mantle and melt extraction from partially molten environments. To extend

flow laws to low differential stresses important in the mantle and to compare the high-temperature rheological behavior of partially molten rocks at total pressures 0.1 and 300 MPa, recent creep experiments were carried out on samples of olivine plus 3 vol% basalt with an average grain size of -30 11m

3

N. S. Bagdassarov et al. (eds.), Physics and Chemistry of Partially Molten Rocks© Kluwer Academic Publishers 2000

4 D. KOHLSTEDT et al: Chapter 1

under anhydrous conditions in compressive creep. In experiments performed

at 0.1 MPa, relatively small differential stresses of 0.5 to 3 MPa were used in order to minimize micro cracking that can occur in rock samples at low

confining pressures. These O.I-MPa experiments yield a stress exponent of n

= 1.0, an f 02 exponent of 117 and an activation energy of 530 kllmo!. To

eliminate the cavitation and microcracking that can occur during deformation

at 0.1 MPa, creep tests were performed at 300 MPa; in this case the differential stresses were in the range 14 to 224 MPa. At 1250°C, a transition from

diffusion creep (n " 1.0.) to dislocation creep (n " 3.5) occurs at a differential

stress of -70 MPa. The f 02 exponent detennined at 300 MPa agrees well

with that measured at 0.1 MPa. Creep rates obtained in experiments at 0.1 MPa are in good agreement with those determined at 300 MPa when

normalized to the same T - (J - f 02 conditions, indicating that contributions

due to cavitation and microcracking are, at most, minor in the lower pressure

experiments. The viscosities measured for partially molten olivine-basalt aggregates with 3 vol% melt deformed in both the diffusion and the

dislocation creep regime are 3 to 5 time smaller than values published for melt-free samples. These results imply that, if the melt fraction remains small

in the upwelling source rock beneath mid-ocean ridges, partial melting will not

dramatically modify the rheological behavior of this region of the mantle except as the melt depletes the hydroxyl content of the host minerals and

thereby eliminates water-weakening of the rock.

1. INTRODUCTION

Laboratory investigation of the rheological behavior of partially molten upper mantle rocks is important to the understanding of a variety of geodynamic processes, especially in regions beneath mid-ocean ridges and hot spots as well as in the mantle wedge above a subducting plate. A number of experimental studies have been carried out to measure the effects of temperature, differential stress, grain size, melt composition, melt fraction and water content on the viscosity of partially molten olivine-rich aggregates [Cooper and Kohlstedt, 1984, 1986; Borch and Green, 1989; Bussod and Christie, 1991; Beeman and Kohlstedt, 1993; lin et al., 1994; Kohlstedt and Chopra, 1994; Hirth and Kohlstedt, 1995a,b; Mei and Kohlstedt, 1999a,b]. The present paper reviews the results of these studies and extends them by combining results from experiments at moderate confining pressure with those from experiments at ambient pressure in order to investigate the dependence of creep rate on oxygen fugacity and the flow behavior at geologically relevant stresses over a wider range of temperature.

1. Rheology of Partially Molten Rocks 5

To address these aspects of the rheology for a partially molten olivine­rich rock, creep tests at a total pressure of 0.1 MPa were recently performed. Such experiments provide several advantages over experiments at high pressures:

(1) The oxygen fugacity dependence of creep rate can be easily measured in detail with a O.l-MPa creep rig, while with a high­pressure rig only two f 02 values (i.e., those set by NilNiO and

FeIFeO buffers) can be readily achieved within the f 02 stability

field of olivine.

(2) Due to the high resolution of stress and strain measurements in a 0.1 MPa rig, creep experiments can be performed at low stresses similar in magnitude to those producing flow in the upper mantle. These experiments are also much more time efficient than those at high pressures.

(3) Experiments can be performed over a wide temperature range at ambient pressure, while limitations in furnace design for gas­medium high-pressure deformation apparatuses presently restrict maximum temperatures to ~1300°C.

(4) Since a metal jacket is not needed as it is in high-pressure tests in order to isolate the sample from the confining gas, experiments in a 0.1 MPa creep rig can be performed under drained conditions, that is, with the melt phase is free to migrate out of the sample.

(5) Finally, a dry environment can be easily provided at 0.1 MPa, a situation difficult to attain in experiments carried out in a high­pressure vessel.

The lack of published deformation results for polycrystalline samples of olivine deformed without a confining pressure reflects two factors: First, without confining pressure, cavities develop in many materials due to local tensile stresses that arise due to grain boundary sliding [e.g. Lange et al. 1980; Tsai and Raj, 1982]. Second, without confining pressure, micro fractures can form in olivine aggregates due to anisotropy in the thermal expansion and elastic moduli of this orthorhombic mineral [e.g., Evans, 1978]. Therefore, creep experiments were also performed under a confining pressure of 300 MPa to assess the effects of cavitation and microfracturing on creep at a confining pressure of 0.1 MPa. The combined results from both types of experiments demonstrate that deformation experiments at 0.1 MPa pressure can be successfully performed to measure the rheology of dry, partially molten olivine-basalt aggregates at high

6 D. KOHLSTEDT et al: Chapter 1

temperatures and low differential stresses over a spectrum of chemical environments under drained conditions.

The present paper builds on the reviews presented by Kohlstedt [1992] and Kohlstedt and Zimmerman [1996] on the rheology of partially molten rocks at low melt fractions. In addition to introducing new data on rheology under anhydrous conditions at low differential stresses, it incorporates recent results quantifying the dependence of strain rate on melt fraction and on water concentration. The resulting constitutive equations provide a basis for modeling the viscosity-temperature profiles in a wide variety of mantle settings, including beneath mid-ocean ridges and in the mantle wedge above a subducting slab.

2. CONSTITUTIVE EQUATIONS

Results from deformation experiments on plastic flow of rocks and minerals are frequently analyzed in terms of a power-law creep equation of the form

. A an (Q+PV) E = - exp ---=----dP R T

(1)

in order to describe the dependence of strain rate, £, on differential stress, cr, grain size, d, pressure, P, and temperature, T. In Eq 1, A is a materials parameter, Q is the activation energy for creep, and V is the activation volume for creep. For diffusion creep, n = 1; if grain matrix diffusion dominates, p = 2, and if grain boundary diffusion dominates, p = 3. For dislocation creep, normally, n = 3-5 and p = O. For olivine-rich rocks with and without melt under anhydrous and hydrous conditions, p = 3 in the diffusion creep regime and n = 3.5 in the dislocation creep regime.

A number of years ago, Bai and co-workers [1991] argued that this creep equation must be modified to take into account the effects of oxygen fugacity and oxide activity. Based on an extension of analyses used to understand the point defect chemistry of transition-metal bearing oxides and silicates, they presented a creep equation of the form

r aopx Q+VP~

R T ) (2),

1. Rheology of Partially Molten Rocks 7

where f 02 is the oxygen fugacity and aopx is the orthopyroxene activity.

In most experiments, the aopx is fixed at unity by the presence of

orthopyroxene.

To take into account the influence of melt on creep rate, Eq 1 must again be modified. In their analysis of the role of melt on diffusion creep of olivine-basalt aggregates, Cooper and Kohlstedt [1986], Cooper et al. [1989] emphasized the importance of (i) local stress enhancement due to replacement of part of each grain with melt and (ii) enhanced transport kinetics resulting from rapid or 'short-circuit' diffusion through the melt. These authors concluded that deformation rate in the diffusion creep regime was limited by ionic diffusion through melt-free grain boundaries though enhanced by the two factors noted above. Their model, however, underestimates the effect of melt on strain rate by an amount that becomes significant at melt fractions, <p, greater than 0.03 to 0.05. In a paper with Keleman and co-workers [1997], Hirth noted that the data in both the dislocation creep regime and the diffusion creep regime could be well­described by modifying the power-law relation given in Eq 1 to

£ = A _(In exp( a </J) exp(- _Q_+_P_V_) dP R T

(3a)

with a = 45 for anhydrous experiments. Recent work in our laboratory suggests that the effect of melt on strain rate might more meaningfully be written as [Mei and Kohlstedt, 1999c]

£ = A _(In [_1_1 exp( a </J) exp(- -c:Q,---+_P V_) dP 1- </J ) R T

(3b)

where the term (J - </J yf3 arises from local stress enhancement effects and

the term exp( -a </J) enters due to enhanced grain boundary diffusion kinetics resulting from the changes in grain boundary composition that occur in melt-bearing samples. For values of P = (3n + 1)/2 [Ashby 1983; Chen, 1985] of 2 and 6 in the diffusion and dislocation creep regimes, respectively, a ~ 26 for water-saturated olivine-basalt samples [Mei and Kohlstedt, 1999c].

If water is now added, Eq 1 must be further modified. To quantify the effect of water on viscosity or strain rate, experiments were recently carried out under water-saturated conditions [Bai and Green, 1998; Mei and

8 D. KOHLSTEDT et al: Chapter 1

Kohlstedt, 1999a,b]. Water fugacity was varied by deforming samples at different confining pressures, following the approach used by Madewell and Kohlstedt [1990] and Bai and Kohlstedt [1992] in their investigations of diffusivity and solubility of water in olivine. The resulting flow law is of the form

£ = A (j"D f~ 0 exp (_ Q + P V ) dP 2 R T

(4a)

where the water fugacity exponent s is 0.7-0.8 in both the diffusion and the dislocation creep regime. If the dependence of solubility on water fugacity [Kohlstedt et al., 1996] is now incorporated into Eq 4a, then the dependence of creep rate can be expressed directly in terms of the water, that is, OR concentration, COH, as

£= A - COH exp -----• (j"D 1 (Q+PV) dP R T

(4b)

Finally, a general form for the flow law can be written to include the effects of stress, grain size, oxygen fugacity, melt fraction, water concentration, pressure and temperature as

£= A _(j"D f qo a~px C~H (_1_1 exp(a ¢) exp( __ Q_+_P_V_) dP 2 1- ¢ ) R T

(Sa)

While the extrapolation of this equation from melt-bearing to melt-free conditions posses no mathematical difficulty, the extrapolation from hydrous to anhydrous conditions is clear not possible as it would yield £ = o. In the case of melt, Eq Sa describes the rheology of samples with <1> > 0 and samples with <1> = 0 implying that the addition of melt to a rock does not change the mechanism of deformation, such as in the model of Cooper and Kohlstedt [1986], Cooper et at. [1989]. If this condition is not met, then two separate equations are needed, one of the form given in Eq Sa and the second of the form

° £=A!!.- f q r O aopx dP 2

(Sb)

1. Rheology of Partially Molten Rocks 9

where it is understood that parameters such as A, n, p, q, r, Q and V will in general be different for melt-bearing and melt-free rocks. In the case of water, two separate flow laws are required, one for anhydrous conditions and one for hydrous conditions. It is not possible to write one flow law to describe flow in both dry and wet environments because the charge neutrality condition (i.e., point defect chemistry) is not the same when water is present as the charge neutrality condition that operates when water is absent [Kohlstedt and Mac/ewell, 1998; Mei and Kohlstedt, 1999a,b]. For samples deformed under anhydrous conditions, the flow law given in Eq Sa must be rewritten without the hydroxyl concentration term as

E= A _ern f6 a~px (_1_1 exp(a1J) exp( __ Q_+_P_V_) dP 2 I-1J) R T

(Sc)

To date, experiments on melt-free and melt-bearing samples under hydrous and anhydrous conditions indicate that Eqs Sa and Sc adequately describe flow in both the diffusion and the dislocation creep regime. That is, while separate equations are required to describe deformation in anhydrous and hydrous environments (i.e., Eqs Sc and Sa, respectively), additional equations are not required to characterize deformation of melt-free and melt­bearing rocks. Again, the reader is reminded that the parameters A, n, p, q, r, Q and V will differ from one creep regime to the next (e.g., from dislocation creep to diffusion creep) and from anhydrous to hydrous deformation conditions.

2.1 New Experimental Constraints

2.1.1 Samples

For the deformation experiments described below, samples were fabricated from San Carlos olivine and a mid-ocean ridge basalt. Olivine powder with an average particle size of - 30 11m was thoroughly mixed with the basalt powder with a particle size of -10 11m. This mixture was cold pressed into an Fe can. The can was then heated to IS0°C for 48 h in a high­vacuum chamber before an Fe cover was electron-beam welded onto it. A nearly fully dense polycrystalline aggregate was synthesized by hot isostatic pressing in a gas-medium apparatus at 200 MPa and 1300°C. By removing the air and adsorbed moisture before sealing the can, trapped porosity was minimized and pore growth observed in previous studies [e.g. Nichols and

10 D. KOHLSTEDT et al: Chapter 1

Mackwell, 1991] was effectively eliminated during subsequent high­temperature experiments at 0.1 MPa.

Table 1. Summary of the measurements of the effect of (J on [:: at 0.1 MPa

pressure.

run # T (0C) (J (MPa) f 0, (atm)

Log(An/ nt

B-3s 1300 1.2-3.0 10-6.0 -6.73 ±D.04 1.0 ±DJ

B-5s 1300 1.0-2.9 10-5.5 -6.80 ±D.03 1.1 ±DJ

B-6s1 1260 1.5 - 3.0 10-55 -7.24 ±D.06 1.1 ±D.2

B-6s2 1350 0.5-1.5 10-5.5 -6.21 ±D.D1 0.8 ±D.l

t The data were fit to the equation [:: = An an, where An is in MPa-n sol

Samples in the form of rectangular parallelepiped of dimensions 7x7x 12 mm3 were prepared for deformation experiments at 0.1 MPa, while cylindrical samples typically 10 mm in diameter by 15-20 mm in length were cored for deformation runs at 300 MPa. A relatively large sample size was used in the 0.1 MPa experiments so that the load greatly exceeded the frictional force between the piston and the guide bearings, yet the stress on the samples remained relatively small (-1 MPa). To fix the orthopyroxene activity at unity during the deformation experiments, the surfaces of the samples were coated with a layer of orthopyroxene powder.

Table 2. Summary of the measurements of the effect of f 02 on [::, at 0.1 MPa

pressure.

Run # TeC) (J (MPa) f 0, (atm) log(Am)t l

A-if 1300 2.0 1 0-X6 - 1 0-4.5 -5.7 ± 0.1 0.12± 0.02

A-4f 1300 2.0 1O-7.() _10-4.6 -5.3 ± 0.1 0.16 ± 0.02

A-5f 1280 4.0 10-5.4 _10-72 -5.3 ± 0.1 0.16 ± 0.02

A-6f 1290 3.0 1 O-R.4 -I 0-4R -5.55 ± 0.08 0.12 ± 0.02

t The data were fit to the equation [:: = Aq f 0; , where Aq is in atm-'! sol

1. Rheology of Partially Molten Rocks 11

2.1.2 Apparatus

Most of the creep experiments at 0.1 MPa were performed in a dead-load deformation apparatus [Bai et al., 1991]. A few runs were carried out in servo-controlled apparatus with a load cell directly attached to the bottom piston to maximize stress resolution [Wang et at., 1993]. In both rigs, an alumina tube is mounted vertically within a high-temperature furnace so that the f 02 around the specimen can be controlled with CO-C02 gas mixtures.

The f 02 is monitored with a stabilized zirconia sensor located next to the

sample. Sample temperature is controlled to ± 1°C and monitored by two thermocouples. Compressive load is transferred to the samples via SiC pistons. Slices of olivine single crystals were used as spacers to eliminate chemical interaction between the samples and the pistons; the olivine crystals were oriented with [010] parallel to the loading direction so that they contributed virtually zero strain during sample deformation.

Table 3. Summary of the measurements of the effect of Ton E at 0.1 MPa

pressure.

Run # T (0C) (J (MPa) f 02 (atm) Log(Ao)t Q (kJ/mo1)t

A-2T 1250-1340 2.0 10-51 15 ± 2 650 ± 60

A-3T 1230-1300 2.0 10-5.0 1O± 4 500 ± 100

A-4T 1240-1300 2.0 10-5.0 11.9 ± 0.1 550± 10

A-5T 1250-1330 2.0 10-60 9.9 ± 0.9 500± 30

B-4T 1260-1325 2.0 10-6.0 1O± 3 500± 80

t The data were fit to the equation E = A Q exp( - Q IRT ) , where AQ is in sol.

Creep experiments at 300 MPa were carried out in a gas-medium high­pressure apparatus [Paterson, 1990]. In this apparatus, a compressive load is applied with a servo-controlled electro-mechanical actuator and measured with an internal load cell; stress resolution is -1 MPa. A resistance-heated furnace constructed with three independent windings provides a hot zone -50 mm in length with a temperature gradient of "",O.l°C/mm and allows temperature control to ± 1°C. Oxygen fugacity is controlled near NiINiO or Fe/FeO by encapsulating samples in either Ni or Fe. The piston plus sample assembly is jacketed in Fe.

12 D. KOHLSTEDT et al: Chapter 1

2.1.3 Data analysis

The high-temperature creep results were analyzed in terms of two power­law relations of the type written in Eq 2 with aopx = 1. Because diffusion

creep and dislocation creep operate concurrently, the constitutive equation describing the sample creep rate, Ecreep' is the sum of two power law

equations, one each for diffusion creep, Ediff , and dislocation creep, Edisloc'

Ecreep = Ediff + Edisloc (6)

Without confining pressure, cavitation can occur during high-temperature creep as a result of local tensile stresses that develop, probably due to grain boundary sliding and diffusion associated with the resulting stress gradients. The creep strain (i.e., strain due to plastic deformation mechanisms that involves no change in volume), Cereep, can be calculated from the measured shortening strain, Emea , and the cavitation strain, Ccay, using the relation [Raj, 1982]

Eeay Eereep = Emea + 3 (7)

where Eeay = In (I + I'1p / p) ~ I'1p / p, the relative change in sample

density due to cavitation, is always negative. In this study, the mechanical data were analyzed based on the above three equations.

2.2 Microstructural observations

The microstructures of samples used in the experiments described below were analyzed using a scanning electron microscope (SEM). Micrographs obtained using backscattered electrons from an undeformed sample, a sample deformed at 0.1 MPa and one deformed at 300 MPa are shown in Figs. la, 1b and lc, respectively. The average grain size is -30 ,urn, as determined by multiplying the mean linear intercept length measured on electron micrographs by a factor of 1.5 [Gifkins, 1970]. Melt is present in all of the triple junctions and along some interfaces separating neighboring olivine grains. These wetted grain boundaries are generally straight, suggesting that they are low-indexed crystallographic planes [Cooper and Kohlstedt, 1982; Waif and Faul, 1992]. The starting samples and the samples deformed at a confining pressure of 300 MPa contain virtually no pores. In

1. Rheology of Partially Molten Rocks 13

contrast, those deformed to 12% strain at 0.] MPa have a porosity of ~7% due to cavitation during deformation. The cavities are isolated, and no fractures were observed. Based on Eq 7, the creep rate was ~ 1.2 times smaller than the measured axial strain rate.

Figure 1. Backscatter SEM micrographs showing microstructures of (a) a starting sample, (b) a sample deformed at P = 0.1 MPa and (c) a sample deformed at P = 300 MPa. The white regions at three-grain jnnctions and some two-grain boundaries are the basaltic glass phase. The black regions in (b) are pores produced mainly by cavitation during the creep experiment.

2.2.1 Mechanical results for samples deformed at 0.1 MPa

The effects of stress, oxygen fugacity and temperature on creep rate for the samples deformed as part of this paper are summarized in Tables 1, 2 and 3, respectively. The results from four experiments performed to measure the influence of differential stress on creep rate at an oxygen fugacity of 10-6 atm, temperatures of 1260°, 1300° and 1350°C, and differential stresses of 0.5 to 3.0 MPa are presented in Fig. 2. The experiments yield a stress exponent of n = 0.8-1.1, demonstrating that diffusion creep is the dominant flow mechanism.

The effect of oxygen fugacity on creep rate was measured in four experiments at stresses of 2.0-4.0 MPa and temperatures of 1280°, 1290°

14 D. KOHLSTEDT et al: Chapter 1

and 1300oe; the results are summarized in Table 2 and Fig. 3. A fit of these data to Eq 2 yields an oxygen fugacity exponent of q = 1/8-1/6.

10-6 P = 0.1 MPa

f02 = 10-6 atm

n = 0.S-1.1

~ .-, '" '-" 13500C

·w

10-7

100

cr (MPa)

Figure 2. Log-log plot of strain rate vs differential stress for the experiments at a pressure of 0.1 MPa, an oxygen fugacity of 10-6 atm and temperatures of l260 o-l3S0°C. Least squares fits of the data at to a power law relation yield values for the stress exponent of n = O.S-I.I, as indicated by the solid lines.

10-6 P = 0.1 MPa 0" = 2.0 MPa

q = IIS-1I6

~

""7

'" '-" ·w

• A-If

12900 C * A-4f ... A-Sf

10-7 1280°C + A-6f

10-10 10-8 10-6 10-4

f02 (atm)

Figure 3_ Log-log plot of strain rate vs oxygen fugacity for the experiments at a pressure of 0.1 MPa, a differential stress of 2.0 MPa and temperatures of 12S0o-1300°C. The data for runs A-Sf and A-6f were normalized to a differential stress of 2.0 MPa from stresses of 4.0 and 3.0 MPa, respectively, using a stress exponent of 1.0. The solid lines represent least squares fits of the data to a power law relation, yielding an oxygen fugacity exponent q of liS - 116.

1. Rheology of Partially Molten Rocks 15

To measure the effect of temperature on creep rate, five experiments were carried out at oxygen fugacities of 10-6 and 10-5 atm and a stress of 2.0 MPa. The data plotted in Fig. 4 yield an average value for the activation energy of Q = 530 kllmol. This value agrees with the offsets in Fig. 2 between stress versus strain rate data sets obtained at different temperatures. An average of the results reported in Tables 1, 2 and 3 yields the flow law

. 3 6 11 10 1117 (530 kJ / mOl) E= . xlO (j' 0 exp ------2 RT

(8)

with E. ins-I, <JinMPa, 102 in atmand Tin Kelvin.

T (oC)

..... 14"'T'"0_0 __ 1_3.,....00 ___ l,20_0--, 10-5

,-.., .......

I CZl

'-" ·w 10-7

10-8

-5 fo = 10 atm ~

P=O.l MPa <J = 2.0 MPa

• + <II

~

* 6.0

~ * '\ A-2T '\

A-3T * A-4T A-5T B-4T

6.2 6.4 6.6 6.8 104fT (K-1)

Figure 4. Semi-log plot of strain rate vs inverse temperature data for the experiments at a pressure of 0.1 MPa and a differential stress of 2.0 MPa. The data at low oxygen fugacity (10-6 atm) and those at high oxygen fugacity (l0·5 atm) were plotted against the right and the left axes, respectively. Least squares fits of the data to a power law relation yield an activation energy for creep of - 530 kJ/mo\.

16 D. KOHLSTEDT et al: Chapter 1

10-4 P= 300MPa T = 12500 C

10-5

,-... -, <Zl

10-6 '-' ·w

10-7 n = 0.9-1.2 PI-69 •

FeIFeO • PI-8S x PI-219

Figure 5. Log-log plot of strain rate vs differential stress data for the experiments at a pressure of 300 MPa, a temperature of 1250°C and oxygen fugacities controlled at NilNiO and Fe/FeO buffers. The solid curves represent the results of the deconvolution regression analysis of the data based on Eq 7, which yielded a stress exponent of n = 0.9-1.2 for the low stress regime (diffusion creep) and n = 3.7-3.9 for the high stress regime (dislocation creep). The dashed lines represent the power law relations for diffusion and dislocation creep for experiment PI-85.

Table 4. Summary of the measurements of the effect of (] on £ at 300 MPa

pressure.

Run # T(°C) a (MPa) f 02 mechanism Log(An)t nt

1. diffusion -7.8UO.l 1.2 ± 0.1

PI-69 1250 14+211 NilNiO 2. dislocation -13.0 ± 0.8 3.7 ± 0.3

1. diffusion -8.2 ± 0.2 It

PI-85 1250 31+160 Fe/FeO 2. dislocation -13 ± 2 3.9 ±O.8

1. diffusion -7.54±O.03 0.9 ± 0.3

PI-219 1250 23+224 NilNiO 2. dislocation -13.0 + 0.4 3.7 + 0.2

t The data were fit to the equation £=A n, ern] + An2 ern2 , where the Ani are in

MPa·n S·I.

+ A value nl = I was used in the deconvolution calculation for this data set.

1. Rheology of Partially Molten Rocks 17

2.3 Mechanical results for deformation at 300 MPa

The results from three creep experiments performed at a pressure of 300 MPa, a temperature of 1250°C and differential stresses of 14 to 220 MPa are summarized in Table 4. The sample deformed in run PI-69 at f 02 ~

NiINiO (i.e., at an f 02 set by a NiINiO solid-state buffer) was deformed a

second time in run PI-85 at f 02 ~ FelFeO, in order to estimate the

influence of oxygen fugacity on creep rate. All three data sets lie on concave-upward curves in the log-log plot of £ versus (j' in Fig. 5, indicating that two creep processes operated concurrently. The data sets at f 02 ~

NiINiO were fit to Eq 6 using two power-law creep equations of the form given by Eq 2, through a nonlinear least squares regression in which both stress exponents and both preexponential factors were varied. The deconvolution calculation yields stress exponents of n = 0.9-1.2 and n = 3.7 for the creep mechanisms dominating at low and high stresses, respectively. These values indicate a transition from diffusion creep to dislocation creep with increasing stress. The creep rate was lower at f 02 ~ Fe/FeO than at

f 02 ~ Ni/NiO, especially in the low-stress regime. If a stress exponent of

n = 1.0 is used for the low-stress (diffusion creep) regime for the data set at f 02 ~ FelFeO, a deconvolution regression analysis based on Eq 6 yields a

stress exponent of n = 3.9 for the high-stress regime. The offset in creep rates between the data sets at the two oxygen fugacities corresponds to an oxygen fugacity exponent of -117 for the low-stress regime. In the high­stress regime, the data sets merge resulting in an f 02 exponent near zero.

These results at 300 MPa and 1250°C yield the constitutive equation

(9)

with £ in S-I, (j' in MPa and f 0, in atm. The first term on the right side of

this equation represents the contribution to the measured strain rate from diffusion creep while the second term represents that from dislocation creep.

Note that Eq 8 yields £ = 2.4x 10-7 (j 1.0 f ~~7 for diffusion creep at 0.1

MPa and 1250°C, which agrees well with the first term in Eq (9).

The results from creep experiments at 0.1 MPa are compared to those from creep experiments at 300 MPa in Fig. 6. Strain rates determined in the low-pressure experiments were normalized to a temperature of 1250°C and an oxygen fugacity corresponding to a NiINiO buffer using an activation

18 D. KOHLSTEDT et al: Chapter 1

energy of 530 kllmol and an oxygen fugacity exponent of 1/7, values determined in this study. These values of strain rate were reduced by a factor of 1.2 to account for the effect of cavitation and were corrected for the effect of pressure using an activation volume of 6x 10.6 m3/mol [e.g., Karato et al. 1993]. The normalized low-pressure data are in good agreement with the flow law determined at 300 MPa, extrapolated to lower stresses.

T = 12500C

10-4 P = 300 MPa d = 30 f.Lm f02-7 NiINiO

,-... 10-6 -,

rJj

'-' ·w

10-8

• H&K (1995a,b) o this study

10° 101 102 103

(j' (MPa)

Figure 6. Comparison of £: - (j data for aggregates of olivine plus -3% basalt obtained at P

= 0.1 and 300 MPa. The data obtained at P = 0.1 MPa (also plotted in Fig. 2) were

normalized to T = 1250°C, P = 300 MPa and f -NilNiO using an activation energy of 02

530 kllmol, an activation volume of 6x 10,6 m3/mol and an oxygen fugacity exponent q of 117.

These data, together with those from our experiments at P = 300 MPa, are plotted as open circles. The solid curve is an averaged flow law based on our experiments at P = 300 MPa

and f 02 -Ni/NiO (plotted as dots and crosses in Fig. 5). The creep data plotted as closed

circles are for runs PI-65, PI-38 and PI-145 from Hirth and Kohlstedt [1995a, b] reduced to a

grain size of 30 ,urn using a grain size exponent of p = 3.

3. DISCUSSION

3.1 New experimental results

At low confining pressures (i.e., confining pressures significantly smaller than the differential stress), cavitation in response to local tensile stresses

1. Rheology of Partially Molten Rocks 19

that arise due to grain boundary sliding and microfracturing in response to anisotropy in the thermal expansion and elastic properties of the olivine grains can result in an increase in sample volume and modification of the deformation rate. As addressed in Eq 7, the creep strain due to plastic flow can be calculated from the axial strain measured during the creep experiment and the cavitation strain determined from the density of the sample [Raj, 1982]; it is assumed that the cavitation strain varies linearly with the shortening strain.

To minimize microfracturing, samples were held at elevated temperatures before the load was applied in order to heal cracks that may have formed during heating or cooling. In addition, experiments at a confining pressure of 0.1 MPa were carried out at low differential stresses to minimize stresses between neighboring grains that arise due to the elastic anisotropy of olivine. Three lines of evidence suggest that microfracturing has little effect on the mechanical data obtained at 0.1 MPa:

(1) In each experiment after several creep tests had been performed, the sample was deformed a second time at the same cr, f 0, and Tused

in one of the earlier tests. In no case did the sample deform at a faster rate during the reloading test; that is, samples did not weaken with increasing strain.

(2) The measured value for the stress exponent, n = 1, is in good agreement with that predicted by theoretical models for diffusion creep [Nabarro, 1948; Coble, 1963], indicating that the strain is mainly produced via mass transport with associated grain boundary sliding. In experiments on Si3NJMgO alloys containing a few percent glass, the stress exponent increased from n = 1 to n = 2 when cavitation creep dominated over diffusion creep [Lange et at., 1980].

(3) The creep data obtained in experiments at 0.1 MPa are in good agreement with those measured in tests at 300 MPa in which cavitation and microfracturing were completely suppressed.

The creep data plotted in Fig. 6 indicate that two creep mechanisms operated concurrently in our creep experiments. At stresses between 0.5 and -70 MPa, the stress exponent is n = 1.0 characteristic of diffusion creep; at stresses between -70 and 224 MPa, the stress exponent is n = 3.7 characteristic of dislocation creep. As demonstrated in Fig. 6, our deformation results are in good agreement in both creep regimes with the data of Hirth and Kohlstedt [1995a,b] for samples of olivine plus -3 vol% basalt deformed at 1250°C and 300 MPa. The small difference between the two data sets in the n = 1 regime probably results from uncertainties in

20 D. KOHLSTEDT et al: Chapter 1

measurement of grain size, which has a strong effect on strain rate in the diffusion creep regime.

Our creep data are compared in Fig. 7 with the results of Hirth and Kohlstedt [1995a,b] for olivine aggregates without basaltic melt and the results of Bai et al. [1991] for olivine single crystals. In the diffusion creep regime, the data of Hirth and Kohlstedt were normalized to a grain size of 30 ,Urn using a grain size exponent of p = 3.

T = 12500C 10-4 f02-7 NiINiO

·w

(ll°le (Ollle

-- 3% melt, this study - - 0% melt, H & K (I995a,b) ....... single crystal, B, M & K (1991)

10° 101 102 103

(j (MPa)

Figure 7. A summary of I:- - (J results for the creep of olivine ± basalt. The solid curve was generated from Eq (9) for olivine plus 3% basalt. The dashed lines represent the creep data (low stresses: PI-17 and PI-82; high stresses: PI-72 and PI-82) from Hirth and Kohlstedt (1995a, b] for melt-free, coarse-grained polycrystalline olivine. The two dotted lines represent the creep data from Bai et al. (1991] for olivine single crystals oriented for deformation along [llOle to activate the easy slip system and along [Ollle to activate the hard slip system.

In the dislocation creep regime, the creep strength of melt-free, coarse­grained dunite [Chopra and Paterson, 1984; Karato et al., 1986; Hirth and Kohlstedt, 1995b] is close to but smaller than that of olivine single crystals deformed along [Ollle [Bai et aI., 1991], suggesting that the creep rate is largely determined by the strongest slip system, (010)[001]. As shown in Fig. 7, creep rates of our partially molten aggregates in the dislocation creep regime are a factor of -5 larger than those of the melt-free olivine samples (with an average grain size of 20 ,Urn) measured by Hirth and Kohlstedt [1995b], reflecting the weakening effect of the melt. Further, the creep rates measured in this study for fine-grained samples lie close to those of olivine single crystals deformed along the softest orientation [llOle [Bai et aI., 1991]. This observation suggests that the rate-controlling step for

1. Rheology of Partially Molten Rocks 21

deformation of our partially molten olivine-basalt aggregates at low melt fraction is associated with dislocation movement on the easy slip system, (010)[100], and that the presence of the melt along triple junctions and some two-grain boundaries lowers the creep strength of olivine by reducing the contact area between olivine grains and enhancing grain boundary sliding.

In the diffusion creep regime, a comparison of our results with those of Hirth and Kohlstedt [1995a] for melt-free dunite indicate that the addition of 3 vol% basaltic melt results in an increase in creep rate by a factor of only ~ 3, Fig. 7. This strain-rate enhancement due to the presence of melt is consistent in magnitude with that calculated based on the solution­precipitation model developed by Cooper et al. [1989]. Hirth and Kohlstedt [1995a] measured a grain size dependence of p = 3 for both partially molten and melt-free olivine samples, indicating that grain boundary diffusion combined with grain boundary sliding is the dominant creep mechanism. In addition, components of the melt phase may segregate into the grain boundaries and increase the rate of grain boundary diffusion of all ionic species. Both this study and that of Hirth and Kohlstedt [1995a] determined activation energies for diffusion creep of Q = 530 - 570 kllmol, a value significantly larger than the activation energies for bulk and grain boundary diffusion of the major ions in olivine [e.g., Hermeling and Schmalzried 1984; Gerard and Jaoul 1989; Ryerson et ai., 1989; Houlier et al., 1990; Watson, 1991]. This discrepancy apparently occurs because the melt fraction increases with increasing temperature in the partially molten samples due to the presence of enstatite, which melts incongruently above 1250°C [e.g., Falloon et al., 1988]. Hence, the activation energy for diffusion creep reflects not only an increase in fundamental rate processes with increasing temperature but also an increase in melt fraction with increasing temperature [Hirth and Kohlstedt, 1995a; Kohlstedt and Zimmerman, 1996]. It should also be noted that Hirth and Kohlstedt [l995a] reported a value for the oxygen fugacity exponent of 0.15 in the diffusion creep regime, a value in good agreement with that measured in the present study. The effect of oxygen fugacity on the rate of Coble creep in olivine indicates that, as for grain matrix diffusion, the rate of mass transport through grain boundaries depends

The high-temperature creep behavior of fine-grained aggregates of Ni2Si04 plus a small amount of amorphous Si02 has also recently been investigated at 0.1 MPa [Wolfenstine and Kohlstedt, 1994]. As in the present study, deformation was dominated by grain boundary sliding accommodated by grain boundary diffusion. Cavitation contributed about 10% to the measured strain, and micro cracking was unimportant. However, in contrast to our results, the rate of diffusion creep was independent of oxygen fugacity. An activation energy for creep of 410 kllmol was reported. A

22 D. KOHLSTEDT et al: Chapter 1

comparison given in Fig. 8 of the creep data for (Mgo.9Feol)2Si04 plus basalt with those for Ni2Si04 plus silica of the same grain size and with the same fraction of melt suggests that the rate of mass transport through grain boundaries in the (Mgo9Feol)-olivine is one to two orders of magnitude larger than that in the Ni-olivine, possibly reflecting the higher silica content and thus higher viscosity of the melt/glass in the latter samples.

10-5

10-6

---,..... 10-7 I

rr.J '-" ·w

10-8

10-9

T (oC)

1500 1400 1300

cr = 2.0 MPa d = 30!lm

1200

- __ (MgO.9Feo.lhSi04 + basalt

Ni2Si04 + Si02 glass

5.5 6.0 6.5 7.0 104fT (K-1)

Figure 8. A comparison of diffusion creep data for (Mg,Fe)2Si04 olivine plus 3 vol% basalt (this study) with those for Ni2Si04 olivine plus 3 vol% amorphous SiOz [Woifenstine and Kohlstedt, 1994]. Both studies were performed on samples with a grain size of 30 ,urn at a total pressure of 0.1 MPa. The dominant deformation mechanism is Coble creep for both of the olivine-glass materials.

3.2 Implication for upper-mantle rheology

To illustrate the combined effects of melt and water on the rheology of partially molten rocks, results from the recent study of Mei and Kohlstedt [1999c] on diffusion creep of olivine-basalt aggregates under water-saturated conditions are presented in Fig. 9. In this log-log plot of stress versus strain rate, data for samples deformed under hydrous conditions are compared with

1. Rheology of Partially Molten Rocks 23

results for samples deformed under anhydrous conditions. Data obtained from both wet and dry samples with 0, 2, 8 and 12 vol% are included for direct comparison. Data measured for wet samples are plotted with closed symbols, while data determined on dry samples are plotted with open symbols. As an example, data from samples containing a melt fraction of <)I = 0.08 are plotted as either open squares, D, if the sample was dry, or as closed squares, ., if the sample was wet. To compare the results for two samples with the same melt fraction, one deformed under hydrous conditions and the other under anhydrous conditions, a vertical arrow is drawn connecting the two data sets. One striking feature in Fig. 9 is that the arrows connecting each of the four pairs of data sets are all very nearly the same length, where a pair is made up of two samples with the same melt fraction, one wet and the other dry. This observation implies that the effect of melt and the effect of water on the rheology of these partially molten rocks are independent.

~ ...... I

Cf.l '--'

·w

10-4

10-5

T = 12500 C P = 300 MPa d = 15 Ilm

o dry (<I> = 0) • wet (<I> = 0) * dry (<I> = 0.02) * wet (<I> = 0.02) [J dry (<I> = 0.08) • wet (<I> = 0.08) t;,. dry (<I> = 0.12) A wet(<I>=0.12)

cr (MPa)

Figure 9. Log-log plot of strain rate versus differential stress for four dry samples and four wet samples with <I> = 0, 0.2, 0.8 and 0.12. The vertical arrows connect data from samples with the same melt fraction, one deformed under anhydrous conditions and the other deformed under hydrous conditions.

24

-7

-8

-9

·w -11 ~ ........

-12

-13

D. KOHLSTEDT et al: Chapter 1

T = 12500 C / P=300MPa /. d = 15 ~m

• cr= 60MPi ./a '£28.3

/ .....

-14 '--'-__ ---L ____ L...-_----I

0.00 0.05 <1> 0.10 0.15

Figure 10. Natural logarithm of strain rate versus melt fraction for one group of samples deformed under hydrous conditions and another group deformed under anhydrous conditions. Note that both the slopes of the two line are the same.

To illustrate this point, the results from several creep tests performed in the diffusion creep regime on partially molten rocks deformed either under wet or under dry conditions are presented on a plot of natural logarithm of strain rate as a function of melt fraction in Fig. 10. The slope of the line defined by the data obtained from wet samples, a. = 28, is the same as the slope of the line fit to the data from dry samples. Consequently, for samples deformed under anhydrous conditions, the flow law describing the creep process can be written as

e = Adry (J'~ exp(28 ¢) exp(- Qdry 1 d- R T )

(lOa)

For samples deformed under hydrous conditions, the flow law becomes

e= Awet (J'~ COH exp(2S ¢) exp(- Qwet ) d R T

(lOb)

Similar flow laws have been determined in the dislocation creep regime with n = 3.5 and p = 0 [Mei and Kohlstedt, 1999c]. With increasing water concentration, the transition from the flow law for anhydrous conditions (Eq lOa) to that for hydrous conditions (Eq lOb) takes place at a water

1. Rheology of Partially Molten Rocks 25

concentration of about 50 Hl106Si. The experimental results presented in Figs 9 and 10 could also be analyzed using an equation with a dependence of strain rate on melt fraction of the form given in Eq 5; however, for the purpose of extrapolating from laboratory to mantle conditions, Eq 10 suffices.

The constitutive equations for anhydrous and for hydrous environments, formed by combining the appropriate flow laws for diffusion creep and dislocation creep (Eq 6), provide the framework necessary to model geodynamical processes in water-rich and water-depleted regions of the upper mantle both with and without melt present. The creep data discussed above demonstrated that a small amount of melt does not dramatically weaken polycrystalline olivine in either the dislocation or the diffusion creep regime; addition of 3 vol% melt increases the creep rate (reduces the viscosity) by only a factor of about 3. Since the melt fraction in most partially molten regions in the upper mantle beneath a mid-ocean ridge is expected to be small «0.03) [e.g., Saiters and Hart, 1989; Johnson et ai., 1990], the effect of partial melting on the rheology of upper mantle appears in general to be modest. In addition, since the reduction in strain rate due to the presence of a small amount of melt is similar for both diffusion and dislocation creep, partial melting in the upper mantle should not promote a transition in deformation mechanism between dislocation creep and diffusion creep.

For melt directly to have a significant effect on mantle viscosity, the melt must be trapped, possibly by cold lithosphere above the region of partial melting, so that the melt can accumulate. If the melt fraction reaches 0.07-0.08, the viscosity will decrease relative to the melt-free value by a factor of -10. However, melt can indirectly impact the viscosity of partially molten rocks through its influence on water content, as originally suggested by Karato [1986] and subsequently analyzed in detail by Hirth and Kohlstedt [1996]. In their analysis of the effect of partial melting on the rheology of lithospheric mantle, these researchers argued that upwelling mantle beneath a mid-ocean ridge will be effectively dried out at a depth of -65 km due to partitioning of water from the minerals into the melt during pressure-release melting [Hirth and Kohistedt, 1996]. As a result, the viscosity of the mantle in and below the source region for mid-ocean ridge basalt (a depth of =lOO km) will be a factor of > 1 00 smaller than the viscosity of the water-depleted rocks above. This analysis suggests that the base of the oceanic plate corresponds to a compositional boundary layer defined by the extraction of water from mantle minerals during partial melting. In summary, our understanding of the rheological behavior of partially molten mantle rocks has progressed rapidly over the past five years. The dependencies of viscosity on melt fraction, water content, stress and temperature have been

26 D. KOHLSTEDT et al: Chapter 1

quantified. However, additional experimental research is needed to determine, for example, the effect of pressure on viscosity particularly in the case when both water and melt are present. Although the composition of the melt phase appears to influence the viscosity of partially molten rocks, possibly by its effect on the structure and composition of grain boundaries, little is known about the specific aspects of the melt chemistry that are important for influencing kinetic properties of partially molten rocks.

ACKNOWLEDGMENTS

Greg Hirth helped in performing two of the creep experiments, and Mark Zimmerman assisted with reducing the raw deformation data. The National Science Foundation supported this research through grants OCE-9529744 and EAR-9815039.

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