Earth and Planetary Science Letters 288 (2009) 138–148

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Earth and Planetary Science Letters

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A generalized formulation of interfacial tension driven fluid migrationwith dissolution/precipitation

Yasuko Takei a,⁎, Saswata Hier-Majumder b

a Earthquake Research Institute, University of Tokyo, Tokyo, 113-0032, Japanb Department of Geology, University of Maryland, 237 Regents Drive, College Park, MD-20742, USA

⁎ Corresponding author.E-mail address: ytakei@eri.u-tokyo.ac.jp (Y. Takei).

0012-821X/$ – see front matter © 2009 Elsevier B.V. Adoi:10.1016/j.epsl.2009.09.016

a b s t r a c t

a r t i c l e i n f o

Article history:Received 11 January 2009Received in revised form 21 August 2009Accepted 8 September 2009Available online 13 October 2009

Editor: L. Stixrude

Keywords:interfacial tensiondissolution/precipitationfluid infiltrationtwo-phase flow

We present an extended formulation for the interfacial tension driven melt migration by taking into accountdissolution/precipitation and diffusive matter transport through the liquid phase. Our results indicate thatthe melt migration is caused by two mechanisms. In the first mechanism, a change in melt fraction isaccommodated by compaction/decompaction of solid matrix, and in the second mechanism, a change inmelt fraction is accommodated by dissolution/precipitation. The latter mechanism is newly introduced inthis study. As spatial scale decreases, the dominant mechanism changes from compaction/decompaction todissolution/precipitation, and when the second mechanism is dominant, evolution of melt fraction isgoverned by a nonlinear diffusion equation. Therefore, critical scale of this transition is called “diffusionlength δd” below which melt fraction evolves primarily by diffusion. Diffusion length δd is usually smallerthan the compaction length δc. Important roles of the new mechanism are discussed on the basis of existingexperimental data on melt infiltration, shear-induced melt segregation into melt-rich bands, andrehomogenization of melt in the bands due to static annealing. Melt distribution in the mantle is brieflydiscussed on the basis of the new model.

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© 2009 Elsevier B.V. All rights reserved.

1. Introduction

Interfacial tension plays an important role in determining the grainscale geometry of melt and aqueous fluid (e.g., von Bargen and Waff,1986; Watson and Brenan, 1987; Hier-Majumder, 2008). Also,interfacial tension driven fluid migration, or fluid infiltration, candrive melt and aqueous fluid over distances much greater than grainscale (e.g., Watson, 1982). Although interfacial tension is important atsmaller scales than buoyancy force, it affects the small scale porositystructure, and hence affects the permeability of the two-phase system.Therefore, interfacial tension can significantly affect the buoyancy-driven flow. Experimental studies on the interfacial tension drivenfluid migration constrain physical and chemical properties relevant tothis process (e.g., Watson, 1982; Riley et al., 1990; Riley and Kohlstedt,1991; Nakamura and Watson, 2001).

Interfacial tension driven fluid migration was first formulated byStevenson (1986) within the framework of two-phase flow. It waspredicted that fluid distribution tends to be homogenized or localizeddepending on whether the dihedral angle is smaller or larger, re-spectively, than 60°. The qualitative result of his model explains wellthe experimental result that liquid phase with small dihedral angles(b60°) infiltrates into dry rocks, while liquid phase with large di-

hedral angles (N60°) does not (e.g., Watson, 1982; Nakamura andWatson, 2001).

Another model for the interfacial tension driven fluid migrationwas proposed by Riley and Kohlstedt (1991). They focused on the roleof compaction/decompaction of the solid matrix, which was omittedin the model of Stevenson (1986). While the rate-controlling factorsin the model of Stevenson (1986) are given only by melt transportproperties (permeability and melt viscosity), the rate-controllingfactors in the model of Riley and Kohlstedt (1991) are given not onlyby the melt transport properties but also by the viscosity of the solidmatrix. Models similar to Riley and Kohlstedt (1991) were alsoderived by Bercovici and Ricard (2003) and Hier-Majumder et al.(2006). Riley and Kohlstedt (1991) conducted a melt infiltrationexperiment and reported that the best agreement between the meltmigration profile and the simulation generated on the basis of theirtheory is obtained when permeability has a melt fraction exponent of1. Because the melt fraction exponent expected for the melt existingas grain-edge tubules is about 2, there still remains a difficulty tounderstand the result of this analysis consistently with the micro-structural observation of equilibrium melt geometry.

The purpose of this study is to extend the existing models ofinterfacial tension driven fluid migration by taking into accountdissolution/precipitation and diffusive matter transport through theliquid phase. Previous studies on reactive, buoyancy-driven flow inporous media demonstrated that dissolution/precipitation causeschanneling instability (e.g., Aharonov et al., 1995; Spiegelman et al.,

Fig. 1. 2D microstructural model with (a) hexagonal grains and (b) square grains. rrepresents the mean curvature radius defined in the text. Liquid fraction ϕ is calculatedas ϕ=shaded area/area of OABC.

139Y. Takei, S. Hier-Majumder / Earth and Planetary Science Letters 288 (2009) 138–148

2001). However, interfacial tension was not taken into account inthese formulations. In the development of equilibrium pore geometry,in which mean curvature of the solid–liquid interface is constant,important role of dissolution/precipitation and matter diffusionthrough liquid caused by interfacial curvature dependent solubilityhas been recognized (e.g., Bulau et al., 1979; von Bargen and Waff,1986). We incorporate this effect into the interfacial tension drivenflow by using Thompson–Freundlich equation (e.g., Kingery et al.,1960).

The generalized formulation proposed in this study predicts twomechanisms of interfacial tension driven fluid migration; one of thetwo is similar to that derived by Riley and Kohlstedt (1991), Bercoviciand Ricard (2003), and Hier-Majumder et al. (2006), and the other isnewly proposed in this study. Importance of the new mechanism isdiscussed for the experimental data of melt infiltration experiment(Riley and Kohlstedt, 1991), shear-induced melt segregation intomelt-rich bands (Holtzman et al., 2003; Holtzman and Kohlstedt,2007), and rehomogenization of melt in the bands due to static an-nealing (Parsons et al., 2008).

2. Formulation

2.1. Governing equations

The interfacial tension driven fluid migration is formulatedwithin the framework of continuum mechanics. Let ρL and ρS beliquid and solid densities, respectively, vL and vS be liquid and solidvelocities, respectively, PL and σij

S be liquid pressure (compressionpositive) and solid stress (tension positive), respectively, and ϕbe liquid volume fraction. The solid phase is assumed to consist ofcomponent A, while the liquid phase is assumed to consist ofcomponents A and B. Let c be the mass fraction of component A inthe liquid phase. For simplicity, both solid and liquid phases areconsidered to be incompressible. Hence, ρL and ρS are constants. Thevariables introduced here represent the “macroscopic” quantities,which are defined by the phasic average in Representative Elemen-tary Volume (REV). REV is larger than the grain size but is smallerthan the scales of melt migration. In comparison, we use the term“microscopic” or “local” to represent smaller scale heterogeneitiesthan the grain size.

Mass conservations of liquid and solid phases are written as

∂ðρLϕÞ∂t + ∇⋅ðρLϕvLÞ = Γ ð1Þ

∂fρSð1−ϕÞg∂t + ∇⋅fρSð1−ϕÞvSg = −Γ; ð2Þ

where Γ (kg/m3/s) represents the mass exchange rate between solidand liquid due to dissolution/precipitation (dissolution positive).Mass conservation of component A in the liquid phase is written as

∂ðρLcϕÞ∂t + ∇⋅ðρLcϕvLÞ = −∇⋅ðϕJÞ + Γ; ð3Þ

where density of diffusive flux of component A in the liquid phase, J(kg/m2/s), is given by

J = −ρLDL∇c; ð4Þ

and the diffusivity of component A in the liquid is DL (m2/s). Massconservation equation of component B in the liquid phase can beobtained by subtracting Eq. (3) from Eq. (1). The definitions of vL and Jare identical to those used in the previous studies (e.g., Aharonovet al., 1995; Spiegelman et al., 2001).

In the presence of interfacial tension, γsl, microscopic pressurefield is discontinuous at the solid–liquid interface. The discontinuity p(Pa) is given by

p =2γsl

r; ð5Þ

where r represents the mean curvature radius [(r1−1+ r2−1)/2]−1

(positive when the center of the curvature lies on the solid side). Wealso assume textural equilibrium within the REV. By using simple 2Dmodels given in Fig. 1, the mean curvature radius is calculated as

r =2ϕ

12d

f ðθÞ ϕbϕc; grainsincontact

2βð1−ϕÞ12d ϕ≥ϕc; nointergranularcontact;

8>><>>: ð6Þ

where ϕc is the disaggregation melt fraction above which adjacentgrains loose contact. The quantities d and θ represent the grain size

140 Y. Takei, S. Hier-Majumder / Earth and Planetary Science Letters 288 (2009) 138–148

and dihedral angle, respectively. The disaggregation melt fraction ϕc

and the factors f(θ) and β in Eq. (6) are given by

f θ≤π3

� �= 8cos

θ2sin

π6− θ

2

� �−2

ffiffiffi3

p π3−θ

� �� �12

f θ≥π3

� �= −2⋅3

14

θ2−π

6

� �−

tan θ2−π

6

� �1 + tanθ

2tan θ

2−π

6

� �24

35

12

β =

ffiffiffiffiffiffiffiffiffi3

p

2π

r

ϕc =f ðθÞ

4sin π6−θ

2

� �" #2

8>>>>>>>>>>>>>>>>>><>>>>>>>>>>>>>>>>>>:

ð7Þ

for hexagonal grains (Fig. 1a) and

f θ≤π2

� �= 4cos2θ

2−π + 2θ−4cosθ

2sinθ

2

h i12

f θ≥π2

� �= −2 θ

2−π

4

� �−

tan θ2−π

4

� �1 + tan θ

2tan θ

2−π

4

� �" #1

2

β = 1ffiffiffiπ

p

ϕc =f ðθÞ

2ffiffiffi2

psin π

4−θ

2

� �" #2

8>>>>>>>>>>>>>>><>>>>>>>>>>>>>>>:

ð8Þ

for square grains (Fig. 1b). The square root dependence of r on ϕ inour 2D model at ϕbϕc is similar to that obtained using a 3D geometry(von Bargen and Waff, 1986; Takei, 2002). Substituting Eq. (6) intoEq. (5), p is written as

p =

γsl f ðθÞϕ

12d

ϕbϕc; grains in contact

γsl

βð1−ϕÞ12dϕ N ϕc; no intergranular contact:

8>>><>>>:

ð9Þ

Because d, θ, and γsl are assumed to be constant throughout thesystem, p is a function of ϕ. This work is mostly concerned withaggregates with intergranular contact (ϕbϕc) and with θb60°. Hence,each solid grain is considered to have both solid–liquid and solid–solid interfaces.

We incorporate the effects of interfacial tension into the governingequations of two-phase flow. A simple thought experiment showsthat the forces applied on the solid phase by the liquid phase underthe condition that liquid pressure is PL and solid–liquid interfacialtension is γsl can be identified with those under the condition thatliquid pressure is PL+p and solid–liquid interfacial tension is zero.Therefore, effects of γsl can be easily incorporated into the two-phaseflow theory by replacing all PL included in the governing equations ofthe solid phase by PL+p. (For the system of governing equationssummarized in Takei, 2009, this replacement is needed in equations Gand Ev.) Momentum conservations of liquid and solid phases are,therefore,

0 = −∇ðϕPLÞ + ϕρLg−ηLϕ

2

kϕðvL−vSÞ + PL∇ϕ ð10Þ

0 = ∇ ð1−ϕÞσSij

h i+ ð1−ϕÞρSg +

ηLϕ2

kϕðvL−vSÞ−ðPL + pÞ∇ϕ; ð11Þ

where kϕ is permeability, ηL is liquid viscosity, and g is the grav-itational acceleration, and viscous constitutive relation for the solidframework is given by

εij =13ξ

∑kσSkk

3+ PL + p

!δij +

12η

σSij−

∑kσSkk

3δij

!; ð12Þ

where εij =12

∂vSi∂xj

+∂vSj∂xi

� �represents the strain rate tensor and ξ

and η represent bulk and shear viscosities, respectively, of the solidframework.

One more equation, which is needed to close the system ofequations, specifies the reaction rate for dissolution/precipitation. LetμAS and μA

L be the chemical potentials of component A in the solid andliquid phases, respectively. At the solid–liquid interface, normalcomponent of traction of the solid phase (outward positive) is equalto −(PL+p). However, traction applied on the solid grains throughthe solid–solid interface, or through grain-to-grain contacts, is gen-erally different from −(PL+p)δij (e.g., Takei and Holtzman, 2009),and hence solid stress is not uniformwithin each grain. Therefore, it isimportant to clarify the condition of stress under which μAS isevaluated. Because dissolution/precipitation occurs locally at thesolid–liquid interface, μAS governing this process is evaluated for thelocal (ormicroscopic) stress at the interface,−(PL+p), rather than forthe macroscopic solid stress σ ij

S, which represents the average stressof the grain. Hence, μAS is written as a function of pressure andtemperature as μA

S(PL+p, T). In contrast, grain scale heterogeneity ofliquid pressure PL is relaxed in a short time scale (N~kHz, e.g.,Schmeling, 1985), and hence PL can be assumed uniform at the grainscale. Also, liquid composition c is assumed to be uniform at the grainscale. Therefore, μA

L is evaluated as μAL(PL, T, c). The dissolution/

precipitation rate Γ is assumed to be linearly dependent on the drivingforce given by the chemical potential difference,

Γ = −K μLAðPL

; T; cÞ−μSAðPL + p; TÞ

n o; ð13Þ

where K(≥0) is a constant (e.g., Lehner, 1995).In this study, we assume that the reaction rate described by K is

infinitely high (K→∞), and that Eq. (13) becomes

μLA PL

; T; c� �

= μSA PL + p; T� �

: ð14Þ

Let c0 be the equilibrium concentration with a planar solid–liquidinterface (r→∞ and p=0),

μLA PL

; T; c0� �

= μSA PL

; T� �

: ð15Þ

By assuming |c−c0|/c0≪1, μAL(PL, T, c) is written as

μLAðPL

; T ; cÞ = μL0A ðPL

; TÞ + RT In aðcÞ

= μLAðPL

; T; c0Þ +RT

aðc0Þdadc j

c0

� �ðc−c0Þ;

ð16Þ

where a(c) is an activity of component A in the liquid phase, which isa function of c. Also, μAS(PL+p, T) is written as

μSAðPL + p; TÞ = μS

AðPL; TÞ + pΩ; ð17Þ

where Ω (m3/mol) represents the molar volume of component A inthe solid phase. Then, by substituting Eqs. (15)–(17) into Eq. (14), weobtain

c = c0 1 +pðϕÞΩ

E

� �; ð18Þ

141Y. Takei, S. Hier-Majumder / Earth and Planetary Science Letters 288 (2009) 138–148

where E is a constant defined by

E = RTc0

aðc0Þdadc j c0

� �: ð19Þ

For an ideal solution, a(c) is equal to the mole fraction of A in theliquid. Then, Eqs. (18) and (19) are equal to Thompson–Freundlichequation (e.g., Kingery et al, 1960). Since p is a function of ϕ, c is afunction of ϕ, as well. T is constant in this study.

The governing equations used in this study are given by Eqs. (1)–(3),(10)–(12), and (18). Without interfacial tension (p=0), theseequations are similar to those used in the studies of reactive flow indeformable porousmedia [e.g., Aharonov et al., 1995; Spiegelman et al.,2001].Without dissolution/precipitation (E=∞, c=const, Γ=0), theseequations are similar to those used in the studies of interfacial tensiondriven fluid migration by Riley and Kohlstedt (1991), Bercovici andRicard (2003), and Hier-Majumder et al. (2006). Our formulation ismore general than these studies in that both effects of dissolution/precipitation and interfacial tension are taken into account.

2.2. Nondimensionalization

We use following nondimensionalization scheme,

x = Lx⁎

vS;L = v0vS;L⁎

t =Lv0

t⁎

η = η0η⁎

ξ = η0ξ⁎

kϕ = k0ϕn

p =γsl

dp⁎

g = gg⁎;

8>>>>>>>>>>>>>>>>>>><>>>>>>>>>>>>>>>>>>>:

ð20Þ

where superscript ⁎ represents the nondimensionalized quantities.Explicit forms of η0 and k0 are given in Section 3.3. Nondimensionalshear and bulk viscosities η⁎ and ξ⁎ are given by

η⁎ = e−Λϕ

ξ⁎ = αe−Λϕ;

(ð21Þ

where the values of Λ and α are given in Section 3.3. The quantities ξ⁎and η⁎ for a reference melt fraction ϕ=ϕ0 are written as ξ0⁎ and η0⁎.Reference length L is defined by

L =η0 ξ0⁎ + 4

3η0⁎

� �k0

ηL

0@

1A

12

; ð22Þ

and the characteristic velocity v0 is defined by

v0 =γsl

dLη0

: ð23Þ

In Section 3.3, we introduce the compaction length which is similar tobut not exactly the same with L.

Eliminating σ ijS and PL, governing Eqs. (1)–(3), (10)–(12), and

(18) can be rewritten in nondimensionalized form as

∂ϕ∂t⁎ + ∇⁎⋅ðϕvL⁎Þ� �

c0−1 + c0Wðp⁎ + ϕp⁎′Þ½ �

= c0W ϕ2p⁎′∇⁎⋅vL⁎ +1Pe

∇⁎⋅ðϕp⁎′∇⁎ϕÞ� �

ð24Þ

∂∂xj⁎

ð1−ϕÞ ξ⁎−23η⁎

� �∂vSk⁎∂xk⁎

δij + η⁎∂vSj ⁎∂xi⁎

+ ∂vSi ⁎∂xj⁎

!( )" #

+ ξ0⁎ +43η0⁎

� �ϕ1−nðvLi ⁎−vSi ⁎Þ = −Gð1−ϕÞρS−ρL

ρ0gi⁎ + ð1−ϕÞp⁎0 ∂ϕ∂xi⁎

ð25Þ

∇⁎⋅ ϕvL⁎ + ð1−ϕÞvS⁎h i

c0−1 + c0Wðp⁎ + ϕp⁎′Þ½ �

=ρS−ρLρS

c0W ϕ2p⁎′∇⁎⋅vL⁎ +1Pe

∇⁎⋅ðϕp⁎′∇⁎ϕÞ� �

;ð26Þ

where p⁎ and p⁎′ are explicitly given by

p⁎ = f ðθÞϕ−12

p⁎′ = −12f ðθÞϕ−3

2

ϕ bϕc; grains in contact

8><>: ð27Þ

and

p⁎ = β−1ð1−ϕÞ−12

p⁎′ =12β−1ð1−ϕÞ−

32

ϕ N ϕc; nointergranularcontact:

8><>: ð28Þ

The nondimensional parameters Pe, W, and G are defined as

Pe =Lv0DL

8>>>>>>

W =

γsl

dΩE

G =ρ0gLγsl = d

:

><>>>>>>>:

ð29Þ

As discussed in Section 3, interfacial tension driven flow is causedby two mechanisms, dissolution/precipitation and compaction/decompaction. The limit W→0 and/or Pe→∞ corresponds to oneend member case that dissolution/precipitation is negligible, whilePe→0 corresponds to the other end member case that dissolution/precipitation dominates over compaction/decompaction. In otherwords, when W→0 and/or Pe→∞, the generalized theory ap-proaches the previous theories without dissolution/precipitation(e.g., Riley and Kohlstedt, 1991; Hier-Majumder et al., 2006). WhenG→∞, buoyancy force dominates over interfacial tension, and whenG→0, interfacial tension dominates over buoyancy force. Effects ofbuoyancy force are beyond the scope of this paper. So, ρS=ρL isassumed, under which the term containing G in Eq. (25) becomeszero. Later in Section 5, we discuss the relative importance betweenbuoyancy force and interfacial tension by comparing the amplitudesof the two terms in the right hand side of Eq. (25).

2.3. Numerical analysis

In applying the present formulation to the one dimensional (1D)experimental systems in Sections 4.1.2 and 4.2, the governing equationswere numerically solved. We discretized the 1D version of Eqs. (24)–(26) employing a finite volume discretization scheme (Patankar, 1980).Once thevelocitieswere obtainedbysolvingEqs. (25) and (26), themeltfraction was updated from Eq. (24) using an Eulerian forward timeintegration. Under ρS=ρL, the barycentric velocity in Eq. (26) becomesconstant. If we also impose the boundary condition that velocities ofboth phases become zero at the ends of the 1D system, then we obtain

vLx = −1−ϕϕ

vSx⁎: ð30Þ

Table 1Material parameters for partially molten rocks.

Parameter Value

dE, m 10−3

kϕ, m2 10−9 ddE

� �2ϕn ðn = 2Þ

η, Pa s 2:5 × 1018 ddE

� �3e−Λϕ ðΛ = 26Þ

α(=ξ/η), nondim. 1.85ηL, Pa s 10γsl, J/m2 0.5E≃RT, J/mol 1.2×104

DL, m2/s 5×10−11

Ω, m3/mol 0.5×10−4

b in Eq. (36), nondim. 0.216

142 Y. Takei, S. Hier-Majumder / Earth and Planetary Science Letters 288 (2009) 138–148

The resulting system of algebraic equationswas solved using a Fortrancode.

3. Two inherent mechanisms

Thegeneralized formulation predicts a newmechanismof interfacialtension driven melt migration due to dissolution/precipitation, alongwith the previously reported mechanism due to matrix compaction/decompaction. To assess the relative importance of these two mechan-isms, we perform 1D linear analysis and derive growth rates of a smallperturbation in melt fraction.

3.1. Without dissolution/precipitation

We first consider one end member case in which contributions ofdissolution/precipitation and matter diffusion through liquid arenegligible (W→0 and/or Pe→∞). In the one dimensional case, for anequal density of themelt and thematrix, governing Eqs. (24) and (25)can be written as

∂ϕ∂t⁎ =

∂∂x⁎ fð1−ϕÞvSx⁎g

ð1−ϕÞp⁎′ ∂ϕ∂x⁎ =∂

∂x⁎ ð1−ϕÞ ξ⁎ +4η⁎3

� �∂vSx⁎∂x⁎

" #− ξ0⁎ +

4η0⁎

3

!1ϕn v

Sx⁎:

8>>>><>>>>:

ð31Þ

The melt fraction is slightly perturbed from a constant value ϕ0,which results in a perturbation velocity vx

S⁎ given by

ϕ = ϕ0 + �ϕeλ⁎t⁎e−ik⁎x⁎

vSx⁎ = �v⁎eλ⁎t⁎e−ik⁎x⁎ð�≪1Þ:

8<: ð32Þ

By substituting Eq. (32) into Eq. (31), the nondimensional growth rateλ* for aggregate with intergranular contact (ϕbϕc) is given by

λc=d⁎ = −

12ð1−ϕ0Þf ðθÞϕ

−32

0

ξ0⁎ + 43η0⁎

� �1 + 1

ð1−ϕ0Þϕn0k⁎

2

� � : ð33Þ

Subscript c/d implies that change in ϕ is caused by matrixcompaction/decompaction. The expression for growth rate inEq. (33) is identical to that obtained by Hier-Majumder et al. (2006).

3.2. Without matrix deformation

We consider another end member case in which contributions ofmatrix deformation vS⁎ and fluid flow vL⁎ are negligible, whichcorresponds to Pe→0. For an equal density of the melt and the matrix,we obtain, from Eq. (24),

∂ϕ∂t⁎ ½c0−1 + c0Wðp⁎ + ϕp⁎′Þ� = 1

Pec0W∇⁎⋅ðϕp⁎′∇⁎ϕÞ: ð34Þ

For 1D system, Eq. (34) is rewritten as

∂ϕ∂t⁎ =

1PeWb

1−Wbϕ−12

∂∂x⁎ ϕ

−12∂ϕ∂x⁎

!ð35Þ

with

b =12

c01−c0

f ðθÞ: ð36Þ

Eq. (35) is a nonlinear diffusion equation for ϕ. Once again, weperturb the constant reference melt fraction to obtain

ϕ = ϕ0 + � ϕeλ⁎t⁎e−ik⁎x⁎ð�≪1Þ: ð37Þ

By substituting Eq. (37) into Eq. (35), the nondimensional growth rateλ⁎ for aggregate with intergranular contact (ϕbϕc) is given by

λd=p⁎ = −k⁎2

1Pe

Wbϕ−12

0

1−Wbϕ−12

0

: ð38Þ

Subscript d/p implies that change in ϕ is caused by dissolution/precipitation. The growth rate λd/p

⁎ approaches ∞ as k⁎→∞, implyingthat the smallest wavelength perturbations are smoothed out bydiffusion the fastest.

3.3. Relative importance of the two mechanisms

In the presence of both mechanisms, (i.e. finite W and Pe), a smallperturbation in the melt fraction generates a perturbedmatrix velocity,as described in Eq. (32). The growth rate for this perturbation is givenby

λ⁎ =1−2bWϕ−1

2

0

1−bWϕ−12

0

λc=d⁎ + λd=p

⁎; ð39Þ

where the growth rate due to compaction/decompaction λc/d⁎ and

that due to dissolution/precipitation λd/p⁎ are given by Eqs. (33) and

(38), respectively. As discussed next, bWϕ−1

20 ≪1. Hence, λ⁎ is almost

equal to the sum of λc/d⁎ and λd/p

⁎ , indicating that the two mechanismsintroduced in Sections 3.1 and 3.2 work additively. Both λc/d

⁎ and λd/p⁎

are negative for θb60° and positive for θN60°.Material parameters determining the growth rates are summa-

rized in Table 1. The reference values of viscosity η0 and permeabilityk0 are given by

η0 = 2:5 × 1018 ddE

� �3ðPasÞ

k0 = 10−9 ddE

� �2ðm2Þ;

8>>><>>>:

ð40Þ

where grain size d varies from 10−6 to 10−2m, and dE=10−3mrepresents the grain size in the upper mantle. η0 represents ηwithoutmelt. For d=10 μm, η0 in Eq. (40) is 2.5×1012 Pa s, which is con-sistent with the shear viscosity of grain-boundary diffusion creepexperimentally measured at T=1523 K (η0=2.4×1012 Pa s atd=10 μm, Mei and Kohlstedt, 2000). Experimental data discussedin Section4weremostly obtained at 1523 K. The parameterΛ in Eq. (21)represents the melt fraction dependence of η and experimental data

143Y. Takei, S. Hier-Majumder / Earth and Planetary Science Letters 288 (2009) 138–148

show that Λ=26 (Mei et al., 2002). The parameter α in Eq. (21) isassumed to be 1.85 from theoretical result (Takei and Holtzman, 2009).The melt fraction exponent for permeability is fixed to be n=2, thevalue consistent with the microstructural observation of melt tubules.DL=5×10−11m2/s is estimated based on the data of diffusion ex-periment near 1500 K (10−10–10−12m2/s) (Hofmann, 1980) andfrom the analysis in Section 4.1. Factor b in Eq. (36) is estimated fromc0/(1− c0)≃1 and f(θ=30°)=0.433 (hexagonal model). The valueof E at T=1523 K is roughly estimated from RT by assuming melt asideal solution. The quantity Ω is estimated from density and molecularweight by assumingolivine as one-component system. In this paper, thevalues in Table 1 are referred to as reasonable values. Table 1 shows thatbWϕ−1

2

0 ≪1 holds even for d=5 μm and ϕ0=10−4.Dimensional growth rate is calculated as λ=λ⁎v0/L. In the

following discussions, θ is fixed to be 30° and hence λ is negativeindicating homogenization. Eqs. (33) and (38) show that the effects ofk⁎ on λc/d

⁎ and λd/p⁎ are different. Hence, the relative importance of the

two mechanisms is determined not only by the material parametersbut also by the perturbation wavelength. Let δd be the perturbationwavelength at which λc/d=λd/p for ϕ=ϕ0,

δd =2πk jλc=d =λd=p

= 2πdDL

γsl⋅ Wbϕ

−12

0

1−Wbϕ−1

20

⋅ξðϕ0Þ + 4

3ηðϕ0Þ

12ð1−ϕ0Þf ðθÞϕ

−32

0

0B@

1CA

12

:

ð41Þ

Wedesignate the length δd as the diffusion length, since the evolution ofperturbations at length smaller than δd is governed by diffusion. Fig. 2showsa contourmapof thehomogenization rate (=−λ) as a functionofgrain size d and perturbation wavelength 2π/k for ϕ0=0.01. Thewavelength equal to the diffusion length δd is shown by the dotted line.Interfacial tension driven fluid flow is dominated by compaction/decompaction for melt perturbations with wavelengths longer than δd,while dissolution/precipitation is the dominant mechanism for pertur-bations with wavelengths shorter than δd. Also shown in Fig. 2 is the

Fig. 2. Contours of homogenization rate,−λ (s−1), in log-scale as functions of grain size dand perturbation wavelength 2π/k, calculated for ϕ0=0.01. Dotted line shows thewavelength equal to the diffusion length δd, at which homogenization rate due tocompaction/decompaction (|λc/d|) is equal to that due to dissolution/precipitation (|λd/

p|), and dashed line shows the wavelength equal to the compaction length δc.

wavelength equal to the compaction length δc (dashed line), which isdefined by

δc = 2πð1−ϕ0Þ ξðϕ0Þ + 4

3ηðϕ0Þ

� �kϕðϕ0Þ

ηL

0@

1A

12

: ð42Þ

When perturbation wavelength 2π/k is equal to δc, the two terms 1and 1/((1−ϕ0)ϕ0

nk⁎2) in the latter parentheses in the denominator ofEq. (33) become equal. At this wavelength, the rate-controllingprocess for λc/d changes from matrix deformation to viscous drag ofliquid against solid. As shown in Fig. 2, δc is usually much larger thanδd. Hence, we assumed 1≫1/((1−ϕ0)ϕ0

nk⁎2) in deriving Eq. (41)from Eqs. (33) and (38).

Fig. 3 shows the cross-sections of Fig. 2 along the vertical lines ofd=5 μm and d=1 mm. Thick lines (labelled “this study”) show λgiven by Eq. (39) and thin lines (labelled “R&K (1991)”) show thecontribution of λc/d given by Eq. (33). Thin dotted lines (labelled“Stevenson (1986)”) show the prediction from the model ofStevenson (1986), which is discussed in Section 5. The wavelengthsequal to the diffusion length δd and the compaction length δc areshown for d=5 μm and d=1 mm. Fig. 3 clearly shows thecharacteristic behavior of the present generalized model, which issummarized as follows. At smaller spatial scales than δd, a change in ϕis dominantly accommodated by dissolution/precipitation. In thiscase, ϕ evolves by diffusion (Eq. 35), and the homogenization rate isproportional to the inverse square of the perturbation wavelength. Atlarger spatial scale than δd, a change in ϕ is dominantly accommo-dated by compaction/decompaction. In this case, if spatial scale doesnot exceeds δc, the homogenization rate is controlled by matrixdeformation and does not depend on the perturbation wavelength,and if spatial scale exceeds δc, the homogenization rate is controlledby viscous drag of liquid against solid and is proportional to theinverse square of the perturbation wavelength. With d=5 μm and atspatial scales smaller than 1 mm, the homogenization rate of thisstudy is much larger than that of R&K (1991), showing thatdissolution/precipitation dominates over compaction/decompaction,and hence that the theory presented in this work provides a morephysically relevant description of the experimental data.

Fig. 3. Homogenization rate (−λ) versus perturbation wavelength 2π/k calculated ford=5 μm or 1 mm and for ϕ0=0.01. Wavelengths equal to the diffusion length δd andcompaction length δc are shown by arrows. Shaded area shows the conditions typical inexperimental studies.

Fig. 4. (a) Growth rate of perturbation in melt fraction under pure shear deformation, λ,is shown as a function of spatial scale 2π/k, for d=8μm and 1 mm and for shear stressτ=10 (solid lines) and 100 MPa (dotted lines). (b) Each of growth rate λdef andhomogenization rates −λc/d and −λd/p for d=8 μm and τ=10 MPa is showntogether with the total growth rate λ=λdef+λc/d+λd/p. (c) Measured (symbols) andpredicted (lines) band thickness versus shear stress.

144 Y. Takei, S. Hier-Majumder / Earth and Planetary Science Letters 288 (2009) 138–148

4. Application to experimental systems

The generalized theory is applied to the experimental data.We testthe applicability of the theory developed for a simple thermodynamicsystem to partially molten rocks. We also demonstrate the signifi-cance of dissolution/precipitation in the experimental systems.

4.1. Shear-induced melt segregation into melt-rich bands

Holtzman et al. (2003) and Holtzman and Kohlstedt (2007)experimentally observed the segregation of melt into melt-rich bandsin the partially molten rocks deformed under simple shear. Parsonset al. (2008) observed that the segregated melt starts diffusing uponremoval of the stress. We show that the interfacial tension drivenflow, especially the flow accommodated by dissolution/precipitation,plays an important role in these phenomena.

4.1.1. Formation of melt-rich bandsStevenson (1989) theoretically predicted the spontaneous segre-

gation of melt into melt-rich bands under pure shear deformation,demonstrating the important role of porosity-weakening viscosity inthis phenomenon. Spiegelman (2003) and Katz et al. (2006) madefurther refinements of the model for the case of simple shear andpower-law creep. Although the deformation type is different from theexperiment, for simplicity we apply the present generalized formu-lation to the model of Stevenson (1989) and demonstrate a role ofinterfacial tension driven flow in the formation of melt-rich bands.Following Stevenson (1989), we consider a stationary state of uniformpure shear (εxx=−εyy=ε0N0) under uniform melt fraction ϕ0 andperform a linear stability analysis for a small perturbation in ϕ intensile direction (=x direction), where strain rate in compressivedirection (=y direction) is kept uniform and constant. Using �≪1, ϕ,vS⁎, and vL⁎ are written as

ϕ = ϕ0 + � ϕeλ⁎t⁎e−ik⁎x⁎

vSx⁎ = ε0⁎x⁎ + � vS⁎eλ⁎t⁎e−ik⁎x⁎

vSy⁎ = −ε0⁎y⁎

vLx⁎ = ε0⁎x⁎ + � vL⁎eλ⁎t⁎e−ik⁎x⁎

vLy⁎ = −ε0⁎y⁎;

8>>>>>>>>>><>>>>>>>>>>:

ð43Þ

where ε0⁎(N0) represents the nondimensional strain rate for thestationary state; ε0⁎=ε0/(v0/L). Because of the background deforma-tion, k⁎ depends on time as k⁎(t⁎)=k0⁎(1−ε0⁎t⁎), where we assumeε0⁎t⁎≪1, for simplicity. By substituting Eqs. (43) into Eqs. (24)–(26),nondimensional growth rate λ⁎ is solved as

λ⁎ =1−2bWϕ

−12

0

1−bWϕ−1

20

ðλdef⁎ + λc=d

⁎ Þ + λd=p⁎

≃λdef⁎ + λc=d

⁎ + λd=p⁎ ;

ð44Þ

where

λdef⁎ =

2 ε0⁎η0⁎ð1−ϕ0Þ Λ + 11−ϕ0

� �

ξ0⁎ + 43η0⁎

� �1 + 1

ð1−ϕ0Þϕn0k⁎

2

� � ðN0Þ ð45Þ

is identical to the growth rate obtained by Stevenson (1989), and λc/d⁎

and λd/p⁎ are given by Eqs. (33) and (38), respectively.

Dimensional growth rate λ is calculated as λ⁎v0/L, where d=8 μmor 1 mm,ϕ0=0.04, and the parameter values given in Table 1 are used.Because kϕ of the experimental samples was reduced by mixing

chromite (Holtzman et al., 2003; Parsons et al., 2008), kϕ smaller thanTable 1 by a factor of 10 is used for d=8 μm (that is, kϕ=d2ϕ2/104).Growth rate λ versus perturbation wavelength 2π/k is shown inFig. 4a for shear stress τ=(σxx−σyy)/2=10 MPa (solid lines) and100 MPa (dotted lines). While λdef(N0) causes unstable growth of theperturbation, λc/d(b0) and λd/p(b0) reduce the perturbation. Each ofthe growth rate λdef and the homogenization rates −λc/d and −λd/p

145Y. Takei, S. Hier-Majumder / Earth and Planetary Science Letters 288 (2009) 138–148

is shown in Fig. 4b for d=8μm and τ=10 MPa. For all wavelengths,homogenization rate due to matrix compaction/decompaction,−λc/d,is smaller than the growth rate due to shear deformation λdef.However, at small wavelength, homogenization rate due to dissolu-tion/precipitation, −λd/p, exceeds λdef and the total growth rate λrapidly drops to b0. Under the existence of such upper cut-offfrequency, width or thickness of the developed perturbation has alower limit. We introduce here a simple assumption that the bandthickness (δth) is determined by the minimum wavelength belowwhich λ is negative (δth in Fig. 4b). By assuming that the effect of λc/d

on λ is negligible, δth can be solved as

δth =2πk jλ=0

= 2π DL⋅Wbϕ

−12

0

1−Wbϕ−1

20

⋅ξðϕ0Þ + 4

3ηðϕ0Þ

2τð1−ϕ0Þ Λ + 11−ϕ0

� �0BB@

1CCA

12

:

ð46Þ

Comparison between Eqs. (41) and (46) shows that δth is usuallymuch smaller than diffusion length δd. The assumption that λc/d isinsignificant compared to λ, which is used to obtain Eq. (46), isjustified by the fact that δth≪δd.

The theoretical prediction is compared to the experimentallymeasured δth of the bands developed under simple shear atT=1523 K (Holtzman and Kohlstedt, 2007). Eq. (46) shows that thepredicted δth is sensitive to DL. As shown in Fig. 4c, good agreementbetween the measured and predicted thickness can be obtained byusing a reasonable value of DL (=5×10−11m2/s). Also, the decreasein δth with increasing τ is explained well by the present model.

4.1.2. AnnealingThe bands start reducing upon removal of stress. Fig. 5 shows the

reduction of the perturbation peak height reported by Parsons et al.(2008) and also calculated from our model for d=8 μm and for theparameter values in Table 1. For kϕ, the same value as Section 4.1.1 isused. Symbols in Fig. 5a show the data of the samples quenchedimmediately after simple shear at 1523 K, after 10 h anneal at 1523 K,and after 100 h anneal at 1473 K, where several bands of each sampleare shown by open symbols and the solid symbols show the average.Dotted line in Fig. 5a shows the numerical result of Parsons et al.(2008) calculated from the model of Riley and Kohlstedt (1991)without dissolution/precipitation. Although it can explain the data of100 h anneal with a reasonable value of η0 (=1.3×1012 Pa s), itcannot explain those of the 10 h anneal. To explain the data of the 10 hanneal without dissolution/precipitation, η0 must be as small asη0=8×1010 Pa s, which is substantially smaller than the reasonablevalue (solid line labelled (b) in Fig. 5a).With dissolution/precipitation,the data of 10 hr anneal can be explained by the reasonable values of η0(1.3×1012 Pa s for d=8 μm) and DL (5×10−11m/s2) (solid linelabelled (c) in Fig. 5a). Although the data of 100 h anneal are difficult tointerpret because of a decrease in the totalmelt fraction (Parsons et al.,2008), the peak heights after the 100 hr anneal look higher than theprediction with η0=1.3×1012 Pa s and DL=5×10−11m/s2. Thedifference in run temperature between 10 h and 100 h anneals mayaffect DL (activation energy ~100–200 kJ/mol, Hofmann, 1980) and η0(activation energy ~300 kJ/mol, Mei et al., 2002) by a factor of about2–3. The slower reduction of the 100 h data may be explained by thedifference in DL and η0 (solid line labelled (d) in Fig. 5a), where theeffect of DL is much larger than that of η0 at this spatial scale (bδd).Therefore, by using the generalized theory, the data of Parsons et al.(2008), especially the rapid homogenization by 10 h anneal, can beexplained without invoking some other driving forces than interfacialtension.

In summary, the experimental data on the thickness δth of thebands, which increases with decreasing stress, indicate the significantrole of dissolution/precipitation. Also, the data on the rehomogeniza-tion of melt in the bands due to static annealing indicate theoccurrence of dissolution/precipitation.

4.2. Melt infiltration experiment

We reanalyse a melt fraction profile of a melt infiltration exper-iment from partially molten olivine aggregate (ϕ=0.143) to dryolivine aggregate (ϕ=0) with d=4.5 μm, performed for t=2 h atT=1528 K (Riley and Kohlstedt, 1991). Riley and Kohlstedt (1991)showed that the best fit curve is obtained when the exponential factorn for kϕ is 1.We reanalyse the profile to checkwhether the generalizedtheory can explain the profile with more reasonable value of n.

Under given nondimensional parameters Pe and W, melt fractionprofile is simulated as a function of nondimensional distance x⁎ andnondimensional time t⁎. W is fixed to the value calculated fromd=4.5 μm and from the values in Table 1. Pe ranges from 105 to 0.01.Effect of dissolution/precipitation decreases with increasing Pe, andprofiles for Pe=105 are equal to those simulated by themodel of Rileyand Kohlstedt (1991). Exponential factor n of kϕ is fixed to be 2. Thesimulated profiles are redimensionalized by multiplying time L/v0 tot⁎ and length L to x⁎. Similar to Riley and Kohlstedt (1991), the valuesof L/v0 and L are not given a priori but are determined such that theprofile gives the least RMS misfit to the experimental data at t=2 h.To perform this procedure consistently, parameters η0, k0, and DL areconsidered as independent variables, rather than assuming the valuesof Table 1. Fig. 6a–c show the best fit profiles for Pe=105, 1, and 0.01.The values of η0, k0, and DL corresponding to the best fit profile areshown in Fig. 7 as functions of Pe. Shaded areas in Fig. 7b–d show thereasonable values of η0, k0, and DL calculated from Table 1.

Because RMS is not sensitive to Pe at Pe≥1 (Fig. 7a), it is difficult toestimate Pe from the values of RMS alone. Pe estimated from thereasonable value of DL in Table 1 is about 1–10. However, the best fitprofile for Pe=1 is very similar to that for Pe=105 (Fig. 6), indicatingthat the occurrence of dissolution/precipitation cannot be affirmed fromthis experiment. At small ϕ at distance N+0.2 mm, the experimentallyobserved increase in ϕ is larger than the best fit profile. For Pe≥1, theprofile at largeϕ at distance b0 is dominantly controlled by compaction/decompaction. Although the measured profile at distance b0 isexplained well by the best fit profile, the value of η0 corresponding tothis curve is substantially smaller than the value in Table 1 (Fig. 7b).Therefore, even if the generalized theory is applied, melt migrations inboth sink (distance N0) and source (distance b0) are faster than thoseexpected from the model with reasonable material parameters.

5. Discussion

Stevenson (1986) derived a diffusion equation of ϕ similar toEq. (35). However, his model does not take into account dissolution/precipitation, and the diffusion coefficient is quite different from thatof Eq. (35). The homogenization rate predicted from the model ofStevenson (1986) is shown in Fig. 3 (dotted lines), which is almostequal to the prediction from the model of Riley and Kohlstedt (1991)at wavelengths larger than the compaction length δc but issignificantly different at the wavelength smaller than δc. The majorcause of this difference comes from the different governing equationsrequired for the macroscopic pressures of solid and liquid, PS and PL,

PS−PL = p Eq: ð1Þof Stevenson ð1986ÞPS−PL = p−ξεkk Eq: ð12Þof thisstudy

(ð47Þ

Here, we emphasize that Eq. (47) describes the relationships be-tween the two “macroscopic” quantities PS and PL. The former equation

Fig. 5. (a) Reduction of peak melt fraction of shear-induced melt-rich bands due to static annealing. Open symbols show several bands of each sample and solid symbols show theaverage. Data from Parsons et al. (2008). Dotted line shows the numerical result of Parsons et al. (2008) calculated for η0 =1.3×1012 Pa s without dissolution/precipitation (readfrom Fig. 6 of Parsons et al., 2008). Solid lines labelled (b)–(d) show the numerical results of this study calculated for (b) η0 =8×1010 Pa s and DL=0 m/s2, (c) η0=1.3×1012 Pa sand DL=5×10−11m/s2, and (d) η0=3.9×1012 Pa s and DL=2.5×10−11m/s2, where evolutions of Gaussian perturbation corresponding to these lines are shown in Figs. 5(b)–(d)with a time interval of 10 h.

146 Y. Takei, S. Hier-Majumder / Earth and Planetary Science Letters 288 (2009) 138–148

of Eq. (47) has a similar form to the Laplace condition. The Laplacecondition is satisfied “locally” at the solid–liquid interfaces, but is notalways satisfied by the “macroscopic” pressures PS and PL. As discussedin Section 2.1, macroscopic liquid pressure PL can be identified with thelocal ormicroscopic liquid pressure, because the relaxation time scale forthe liquid pressure heterogeneity at microscopic scale (≤ grain scale) isshort. However, this is not the case for solid. The quantity PS representsthe average pressure of each solid grain and is determined by thetractions applied to the grain through both solid–liquid and solid–solidinterfaces. When external forces exist, the tractions at the solid–liquidand solid–solid interfaces are generally different, and hence the averagesolid pressure PS can be substantially different from the local pressure ofsolid at the solid–liquid interface. Such stress heterogeneity within eachgrain asked for special attention in deriving Eq. (14). The macroscopicquantities PS and PL are not related by the Laplace condition but by the

macroscopic constitutive relation givenby the latter equationof Eq. (47).Contribution from the compaction term, −ξεkk, is negligible at longerwavelength than compaction length δc, where the liquid migration israte-limited byviscous dragof liquid against solid, but is nonnegligible atsmaller wavelength than δc. This is the reason for the large differencebetween themodels of Stevenson(1986)andRiley andKohlstedt (1991)at smaller wavelength than δc (Fig. 3). The homogenization ratepredicted from the model of Stevenson (1986) is several orders ofmagnitude faster at d=1mm and wavelength b100 m than ourprediction, and the model of Stevenson (1986) significantly over-estimates the effects of interfacial tension driven melt migration in themantle.

The present extension and correction of the previous modelscontribute to correctly estimate the effects of interfacial tension drivenfluid migration in the Earth and in the laboratory systems. Although

Fig. 6. Melt fraction profiles with least RMS misfit to the data of melt infiltration experiment by Riley and Kohlstedt (1991) at 1528 K are shown for (a) Pe=105, (b) Pe=1, and(c) Pe=0.01.

147Y. Takei, S. Hier-Majumder / Earth and Planetary Science Letters 288 (2009) 138–148

effects of gravity or buoyancy force were not considered in this study,not only interfacial tension but also buoyancy force give the drivingforces for fluid migration. Here, we roughly assess the relativeimportance of the two driving forces by comparing the relativeamplitude of the two terms in the right hand side of Eq. (25). Theamplitudes of these two terms can be roughly evaluated by G(ρS−ρL)/ρS and f ðθÞLkϕ−1

2 = 2. Therefore, the driving force by interfacial tensiondominates over the buoyancy force, if spatial scale 2π/k is smaller than

Fig. 7. Material parameters corresponding to the best fit profile to the data of meltinfiltration experiment shown in Fig. 6 versus Peclet number. (a) RMS misfit of the bestfit profile, (b) shear viscosity without melt η0, (c) permeability factor k0 calculated forηL=1, 10, and 150 Pa s, and (d) liquid diffusivity DL. Shaded regions in Fig. 7(b)–(d)show the values from Table 1.

~1 m in the mantle (d=1mm) and ~100 m in the fine-grain samplesused in the experiments (d≃10 μm).

To simulate the interfacial tension driven fluid migration with asmaller spatial scale than the diffusion length δd, it is important to usethe generalized formulation developed in this study. Importantapplication of the present theory to the actual mantle process is thedevelopment of porosity structure under stress, which significantlyaffects the effective permeability and matrix deformability of partiallymolten region. FromEq. (46) and Fig. 4a, thickness of the shear-inducedmelt-rich bands in the mantle is estimated as δth≃6 mm for d≃1 mm,τ≃10 MPa, and ϕ=0.04. Recent seismological data suggest theexistence of melt-rich bands in the mantle (Kawakatsu et al., 2009).This present result provides an important constraint to the thickness ofthe bands, which cannot be constrained by the seismological data.

The primary assumptions used in the present theory are as follows:

a) The thermodynamic treatment is simplistic, consisting of puresolid and binary liquid, which is further assumed to be an idealsolution in estimating the value of E.

b) Grain size distribution is neglected.c) Textural equilibrium is assumed in REV.d) Reaction rate for dissolution/precipitation is assumed to be infi-

nitely fast.

In spite of these simplifications, experimental data discussed inSection 4.1 can be explained well based on the reasonable values ofthe material parameters, showing the validity of the theory. However,the present theory cannot fully explain the data of melt infiltrationexperiment discussed in Section 4.2. Further refinement of the modeland further experimental studies are needed to clarify the cause ofthis discrepancy. Because melt infiltration experiment is simple, it isexpected to provide a valuable method to constrain physical andchemical properties of partially molten rocks relevant to this process.

6. Conclusion

Generalized formulation for the interfacial tension driven meltmigration is developed by taking into account the effects ofdissolution/precipitation and diffusive matter transport throughliquid. Partially molten rock is assumed as a simple binary systemand the Thompson–Freundlich equation is applied. The generalizedformulation describes the interfacial tension driven flow due to bothmatrix compaction/decompaction and dissolution/precipitation. Thelatter contribution is dominant at smaller spatial scale than “diffusionlength δd”, where melt fraction evolves being governed by a diffusion

148 Y. Takei, S. Hier-Majumder / Earth and Planetary Science Letters 288 (2009) 138–148

equation. Significant effects of dissolution/precipitation in theexperimental systems are demonstrated in the thickness of shear-induced melt-rich bands and the rate of their rehomogenization bystatic annealing. In spite of several simplifications of the theory, thesedata can be explained well based on the reasonable values of thematerial parameters, showing the validity of the theory.

Acknowledgments

Comments by D. L. Kohlstedt and an anonymous reviewer werehelpful to improve the paper. R. Parsons and T. Koyaguchi providedhelpful discussions. The research conducted by SH-M was supportedby ACS PRF grant #47853-G8. Acknowledgment is made to the Donorsof the American Chemical Society Petroleum Research Fund for partialsupport of this research.

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