Equilibrium geometry of a fluid phase in a polycrystalline aggregate

JOURNAL OF GEOPHYSICAL

RESEARCH, VOL. 105, NO. Bll, PAGES 25,937-25,953, NOVEMBER

10, 2000

Equilibrium geometry of a fluid phase in a polycrystalline aggregatewith anisotropic surface energies: Dry grain boundaries Didier LaporteandAriel Provost LaboratoireMagmaset Volcans,CNRS andUniversit6BlaisePascal Observatoirede Physiquedu Globe,Clermont-Ferrand, France

Abstract. The equilibriumdistributionof a fluid phasein a rockis controlledby the ratio of grainboundaryenergyto solid-fluidinterfacialenergy("surfaceenergy").A strong dependence of interfacialenergyon interfaceorientation(anisotropy)is the rule for solidfluid surfacesof geologicalinterest.We investigatedthe effectsof surfaceenergyanisotropy on the equilibriuminterfaceconfigurationat thejunctionof two crystalswith a fluid in a two-dimensional solid-fluidsystem.The two majoreffectsareto promotethe development of planarsolid-fluidinterfacesparallelto crystallographic planesof minimumenergyandto leadto largevariationsof the dihedralanglefrom onetriplejunctionto the other(as a functionof the orientationof crystallinelatticesrelativeto the grainboundary).Despitethese largevariations,the frequencydistributions of equilibriumdihedralanglesremainunimodal; alsothe relationshipbetweenthe meanfluid dihedralangleandthe ratio of grainboundaryto surfaceenergyis very closeto the isotropiclaw. Even in the mostanisotropicsystems,the fluid dihedralangleis thereforea parameterof primaryimportancefor predictingthe grainscalegeometryof a low-volumefractionof fluid andits mobility. 1. Introduction

solid-fluid triple junction is a fundamentalcharacteristicsof the systemand is given by [Bulau et al., 1979; von Bargen Geologicalsystemsare oftencharacterized by the presence and Waft, 1986; Watsonand Brenan, 1987] of a volumetricallyminor fluid phase:eithera C-O-H fluid or 2cos(0/2) = 7SS/7SF. (1) a partialmelt (silicate,carbonate,or metallosulfidemelt). The viscosityof this fluid phaseis 10 to 20 ordersof magnitude At low percentagesof fluid 0, the equilibrium fluid lower thanthat of the matrix;it is thereforepotentiallymobile distributionis critically dependentupon 0: (1) for 0 < 60ø the and may play a significantrole in a variety of geological fluid forms a continuously interconnected network of processes, includingplanetarydifferentiation, magmagenesis, channelsalong grain edges,even at an infinitely small fluid metasomatism, and heat transport.A low viscositydoesnot percentage(i.e., 0c = 0 vol %); and (2) for 0 > 60ø and 0 < 0C suffice, however, to ensure large-scalefluid transport:an the fluid occursas isolatedpocketsat grain cornersand along equallyimportantconditionis that the fluid be interconnected grain edges and grain boundaries. For 0>60 ø, fluid in the directionof potentialtransport.At the low percentages interconnectionis only establishedwhen 0 exceedsa critical of interestin this study(from almostno fluid to --10 vol %), value 0c increasingfrom 0.6 vol % at 0 = 60ø to =28 vol % at manygeologicalfluidsmay indeedoccuras isolatedpockets 0= 180ø [Wray, 1976; Bulau et al., 1979; von Bargen and at thegrainscaleandthereforebe unableto escapefrom their Waft, 1986; Laporteand Provost,2000]. host rocks. Accordingly, understandingthe grain-scale The possibility to discriminate interconnected from distributionof fluids in polycrystallineaggregatesis a noninterconnectedfluid geometriesby measuring dihedral prerequisiteto modelingfluid transferand to predictingthe angles, motivated numerous experimental studies of the effective transport properties of fluid-beating rocks. A wettingbehaviorof geologicalfluids: Waft and Bulau [1979] parameterof primary importancein this context is the for basaltic partial melts, Laporte [1994] for silicic melts, permeabilitythreshold0c, that is, the volumepercentage of Watson and Brenan [1987], and Holness [1993] for C-O-H fluid at which fluid interconnection is established. fluids, among others(see Laporte and Provost [2000] for a At high temperatureand under conditionsof hydrostatic review of dihedral angles for silicate, carbonate, and stressthe fluid phase is expectedto adopt an equilibrium metallosulfide melts). configurationthat minimizesthe interfacialenergyper unit In additionto a single-valueddihedral angle the idealized volumeof the system.In an idealizedsystemin which both model of isotropicsurfaceenergy predictssmoothlycurved thegrainboundaryenergyper unit area7ssandthe solid-fluid solid-fluid interfaceswith a constantmean curvature [e.g., interfacialenergy (or surfaceenergy) per unit area 7sv are Bulau et al., 1979]. Near-equilibriumtexturesin all silicateisotropicand single-valued, the dihedralangle 0 at a solid- fluid systemsinvestigatedso far are, however,characterized by the extensive development of crystal faces, that is, Copyright2000 by theAmericanGeophysicalUnion. crystallographically controlled,planarsilicate-fluidinterfaces: olivine-basalt[Cooper and Kohlstedt, 1982; Waft and Faul, Papernumber2000JB900256. 0148-0227/00/2000JB 900256509.00 1992: Faul, 1997; Cm[ral et al., 1998]; albite-water [Laporte 25,937

25,938

LAPORTE

AND PROVOST:

ANISOTROPIC

WETTING

and Watson,1991]; quartz-, biotite-, and hornblende-silicic The ECS of •-quartz in a H20-saturated silicicmelt at melt [Laporte, 1994; Laporte and Watson, 1995]; and 900øC-1GPais a rounded bipyramidin whichplanarfacesof plagioclase-silicate melt [LonghiandJurewicz,1995;Laporte the form {101 1} are joined by well-developed rounded et al., 1997]. The systematicobservationof crystalfaces regions[Laporteand Provost,1994].For manysilicatesthe indicatesthatthe assumption of isotropicsurfaceenergiesis roundedregionsof the ECS are expectedto be morelimited

not a goodone for silicate-fluidsystems.In this paper,we investigate the effectof anisotropy of surfaceenergieson the equilibriuminterfaceconfigurationat the junction of two

(or evenabsentleadingto a perfectpolyhedron with sharp edgesandcorners). Equilibriumtextures in manysilicate-fluid systems are indeed characterizedby the systematic crystalswith a fluid in a two-dimensional(2-D) solid-fluid developmentof planar faces and by the near absenceof system.The caseof equilibriumconfigurations comprising a smoothlycurvedsolid-fluidinterfaces.On the otherhand,no dry grainboundaryanda dihedralangle> 0ø is developed in texture showing only smoothly curved silicate-fluid thispaper.A futurepaperwill be devotedto partiallyor fully interfaces,and suggestive of a sphericalECS, hasyet beenwettedgrainboundaries that are observed for ratiosof grain reported. boundary energyto surfaceenergyslightlylargerthan2. A planarface of unit outwardnormalnF can only be presenton the ECS if the surfaceenergyfunctionYSF(n)is minimumandnot differentiableat n = nF [e.g., Wortis,1988, 2. Dependenceof SurfaceEnergy pp. 389-390]' in other words, only a cuspedminimumof on Surface Orientation surfaceenergycan give rise to a planar face on the ECS. First-orderinformationson the dependenceof surface Combiningthisrelationshipandthe texturaldatasummarized energyon orientationmay be obtainedfrom the equilibrium above,we mayanticipate thatthesurfaceenergyfunctions for crystal shape(ECS) of silicatesat high temperature.The silicatesat high temperatureare characterized by a small equilibriumshapeof a crystalof fixed volumeV, isolatedin a numberof symmetry-distinct cuspedminima: for instance, fluid phase,is the shapethat minimizesits total interfacial YSF(n)for •-quartzcomprises only one family of cusped energy,ffYsF(n)dS, whereYSF(n) is thesurface energyperunit minima.In the calculations below, the anisotropyof surface area of surface element dS of unit outward normal n. For energyof silicatesat hightemperatures is approximated by a isotropicYsF(n)the ECS is spherical.The main effect of 2-D analyticalexpression thatyieldsa singlefamilyof cusped anisotropyof YSF(n) on the ECS is to promote the minima and allows for a continuousrange of ECS from a development of planarfacesparallelto theplanesof minimum circleto a perfectpolygon.Surfaceorientationis definedby energy. theanglerpfromu, a directionof minimumsurfaceenergy,to ß

0,4 0,2

-0,2

= ß

1,2'i t,.xo1,1

.....

•

1 0,9 0o

30 ø

60 ø

90 ø

120 ø

Figure 1. The surfaceenergyfunction.(a) Surfaceorientation is definedby the angle(pmeasured fromthe referenceaxisu (parallelto a directionof minimumenergy)to v (the actualsurfacedirection).(b) Surface energyYSF(rP), asdefinedby equation(2), for an hexagonal lattice(N = 6) with k = 0.5 (thickline) andk = 1

(thinline);the firstderivativeY•F(rP)is shownfor k = 0.5. (c) Equilibrium crystalshapes (thickline) and gammaplots(thinline)for energyfunction(2) withN = 6, k = 0.5 and1; thegammaplotis obtained by plottingthesurface energyfunctionin polarcoordinates, withYSF(rP) astheradialdistance and(Pn = (p+90 øas the polarangle.

LAPORTE

AND PROVOST:

ANISOTROPIC

WETTING

25,939

v, the actualsurfacedirection(Figure la). For a crystalwith Table 1. Maximum, Maximum AngularDerivative, and lattice symmetryof orderN and a singlefamily of cusped WeightedAverageof YSF(rP) for N = 4 and6, minima,surfaceenergyYSF(rp) is periodicof period2•N, and and k = 0.5 and 1a it is characterized by cuspedminimaat rp= 2n•dN(n integer) k=0.5 k= 1 and rounded maxima at rp= (2n + 1)•N. Below, surface energyis takenas a piecewisesinusoidalfunction: max(y) max(y')

YSV((p) = g{l+ k[cosrp'+tan(•r/N)sinrp'-l] }, (2) where rp'= rp-2n•dN with n an integer such that 0 < rp'< 2•N, k is the anisotropyfactor(0 < k < 1), and g is theminimumsurfaceenergy. For k = 0, surfaceenergyis isotropic:YSF(rP) = g. Surface energyfunctionsand the corresponding ECS are shown in Figures lb and lc for N=6 (i.e., hexagonallattice) and k = 0.5 and 1. The ECS is a roundedpolygonfor k < 1 and a polygonwith sharpcornersfor k = 1. The angularderivativeof YSF(rp) is of specialrelevancein the calculationsbelow. For nonsingularorientations(that is, all orientationsthat do not correspondto a cusp of surface energy)the derivativeof equation(2) is

N = 4 N = 6

- g k tan(It/N)< y•F(cp) < g k tan½/N).

0.500 0.289

1.057 1.024

max(y) max(y') 1.414 1.155

1.000 0.577

Y-SF 1.000 1.000

aThe parametermax(y) is the maximumof the surfaceenergy function(at q0= (2n + 1)g/N, n integer);max(y') is the maximumof the angular derivative (at q0=2n•/•; the weighted average of ysF(q0), ?sF, is definedin the text. All numbersare normalizedto the minimumg of thefunction,thatis, max(y)= 1.207meansthatmax(y) equals1.207 timesg.

or, using AB = AM cosrp+ MB cos[(2•N)-rp] and sinrp/MB = sin[(2•N)-rp ]/AM,

YSF(rp)< g [coscp + tan½/N)sinrp].

y•F(q9)= g k [tan½/N)cosrp'-sinrp'].(3) Equation (3) fails at singular orientations(i.e., rp= 2n•N) where the slope of YSF(rp)changes abruptly. For these orientationsthe angular derivative is multivalued [Hoffman and Cahn,1972,pp. 376-377], havingany valuein therange

1.207 1.077

Y-SF

(6)

For k < 1 in (2) this relationis satisfiedfor all valuesof rp.On the contrary,it cannotbe satisfiedfor k > 1: a surfaceelement suchas AB is thereforeunstableand must rearrangeinto a hill-and-valleystructure(Figure 2b) whosemacroscopically averagedinterfacialenergyper unit areais

(4)

YSF ((p)= g[cosrp+tan½/N)sinrp]. (7)

We emphasizethat k must not be taken >1 in equation(2). With k > 1, any surfaceelementwith a nonsingularorientation would be unstableand would form a macroscopic array of planar facets, giving rise to a so-called hill-and-valley structure[Herring, 1951a; Wortis, 1988]. The conditionfor a surfaceelementAB of orientation0 < rp< 2•N to be stable relativeto decomposition in planarfacetswith orientations0 and2•N (Figure2a) is

Equation(7), which is equivalentto (2) with k = 1, gives the maximumsurfaceenergyper unit areathat a surfaceelement of orientationrpcan have. Numericalvaluesof the maximumof YSF(rP) andits angular derivativeare reportedin Table 1. For a hexagonallattice the maximum value of YSF(rp)is only 7.7% larger than the minimum g for k = 0.5, and 15.5% larger for k = 1. In the discussionof the computationalresults(section4), we used two differentparametersto quantifythe solid-fluidinterfacial energy:the first oneis the minimumenergyg; the secondone is a weightedaverageof YSF(qg), •SF, definedas the ratio of the total surfaceenergyof the ECS to the total surfaceareaof the ECS. Note that fSF is equal to g for k=0 and k=l (where the ECS only comprisesfaces of energyg) and just slightlylargerthang for intermediatevaluesof k (Table 1).

YSF½) AB< g (AM+MB)

(5)

a

b B

B

3. Equilibrium Interface Configuration at the Junction of Three Phases' Theoretical

Considerations

The equilibriumconfigurationat the junction line of three anisotropicphasesmay be found by equating to zero the

change 8E of interfacial energy that results from an infinitesimal displacementof the junction line [Herring, 195lb] or by an appropriatebalanceof surfacetensionforces. These two methods are discussed in sections 3.1 and 3.2.

3.1. Herring's Solution

Figure 2. (a) Hill-and-valley rearrangementof surface elementAB (the equilibriumcrystalshapeis shownby the thickshadedline). (b) If k > 1 in equation(2), AB is not stable and must decomposeinto a macroscopicarray of facets parallel to AM and MB.

The problem of determining the local equilibrium configurationat the junction of three anisotropicphaseswas first solvedby Herring [195lb]. Herring's methodis applied below to the specificcaseof a 2-D triple junctioninvolving two crystalsand a fluid phase (Figure 3a). We considera

25,940

LAPORTE

a

AND PROVOST:

Y

ANISOTROPIC

b

c

"1/ •i•: fluid

SS •2

WETTING

1

x

O 2

Figure 3. Determining the localequilibriumconfiguration at thejunctionof anisotropic phases: Herring's method.(a) Sketchof thepointjunctionO of two crystalswith a fluidphase(lightlyshaded); SS is thegrain boundary,SF1andSF2 arethe solid-fluidsurfaces;01 and02definetheorientationof the solid-fluidsurfaces, and•1, q2 the orientation of the referenceaxesul, u2 of crystalline latticesrelativeto the grainboundary. AlsoshownaretheaxesOx, Oy of thereference frameandtheconvention for measuring anglesin crystals1 and 2. (b and c) Interfaceconfigurations that are stableto infinitesimaldisplacements OA and OB are the equilibriumones.Deflections•, al, anda2 areexaggerated for clarity.

smallsurfaceelementcenteredat the triplejunctionO, sothat the grain boundarySS and the two solid-fluidsurfaces, and SF2, are planar (althoughthe modelis two-dimensional, we shall still refer to solid-fluid interfacesas surfaces).The orientationof the two crystallinelattices being fixed, the problemis to determinethe orientationof the threeinterfaces that correspondto a stableconfiguration.There is, however, an infinite numberof solutionsaboutany pointjunctionO [Hoffman and Cahn, 1972, p. 384]: accordingly,the orientationof oneinterfacemay be chosenarbitrarily,whence the other two are fixed. In the calculations below, the

orientationof the grainboundarySS is chosenarbitrarilyand definesthe axis Ox of the referenceframe (Figure 3a); the orientationof crystallinelattice i is definedby the angle cPi, measured from

Ox to the nearest direction

of minimum

YSS =YSF1 cos01 -Y•F1sin01+YSF2 cos02-Y•F2 sin02.

(9a)

If SF1 and/orSF2are singularsurfaces(thatis, if Y•F1 and/or Y•F2 are multivalued;see section2), the two following stabilitycriteriamustbe satisfied:

8E/OA>0forOA>0

(i.e.,al, a2 >0;Figure3b)

8E/ OA < 0 forOA < 0 (i.e.,Oq,Eg 2 < 0).

(9b)

The initial configurationshouldalsobe stablewith respectto an infinitesimaldisplacement OB of thejunctionline normal to the grain boundary (both OB and the grain boundary deflectiontr0 are > 0 in Figure 3c). The changeof interfacial energyassociated with displacement OB is, to first order,

tSE=-(Y•S1- YSS2 + YSF1 cos01 + YSF1 sinOl

surfaceenergy of the lattice, ui (rotatinganticlockwisefor (10) -Y•F2cos02-Ysr2 sin02)OB, i = 1 andclockwisefor i = 2; seeFigure 3a for the convention used to measureangles) Angles 01 and 02 specify the where Y•;Sland Y•;s2arethe angularderivativesof Yss(01,02) with respectto Ol and02,respectively. The stabilitycondition orientation of solid-fluid surfaces relative to Ox.

is 8E = 0, that is, An equilibriumconfigurationshouldbe stablewith respect to any infinitesimaldisplacementof the triple junction O -YSS1+ YSS2=YSF1cos01+YSF1sin01-YSF2cos02-YSF2sin02 [Herring, 1951b]. Considerfirst a displacementvector OA (11a) parallel to the grain boundarySS (Figure 3b; solid-fluid if the three interfaces in the initial configuration are interfacesare supposed to remainplanarandto bendat anchor nonsingular, or points,C1 and C2, arbitrarilydistantfrom O). Displacement OA resultsin infinitesimal deflectionsa'l, a'2 of interfaces •E/OB > 0 forOB> 0 (i.e.,a0, a2 > 0,Eg 1< 0;Figure3c)

SF1andSF2. The changeof interfacialenergy,fiE, resulting

(lib)

from displacement OA is, to first order[Herring,195lb],

tSE/OB 0•iso(01iso =arccos[Yss/2g•]; see text for further explanation); T is theresultant of thetangential components of interfacial tensions (including Oss);N is the resultant of normalcomponents of surface tensions. (c) Sameasin Figure5bfor 0• < 0•iso.

4. ComputationalResults 4.1. Program Overview

We developeda programto computethe equilibriumfluid dihedralangleat thejunctionof fluid with two grains(thatis, the anglebetweenthe tangentsto the two solid-fluidsurfaces at theirpointof intersection: 0• + 02 in Figure3a). The input dataare the surfaceenergyfunctionsfor the two crystals,the latticeorientations •pl, 02, and the grain boundaryenergy (supposed isotropicin the calculations below).The program calculatesthe couple(s)(01, 02) for whicha balanceof surface tensionsexistsat the triplejunction.Threetypesof solutions aresearched for in thefollowingorder:(1) singularsolutions, thatis, equilibriumconfigurations in whichthe two solid-fluid

surfaceshave a singularorientation,Oi= Ipi+2nilr./Ni,ni integer;(2) mixed solutions,in which only one of the two solid-fluid surfaceshas a singular orientation;and (3) nonsingularsolutions,in which the two solid-fluid surfaces

arenonsingular. In a texturallyequilibrated fluid-bearing rock a singularsolutionwill appearas a dihedralanglesubtended by two flat crystalfaces,a nonsingular solutionwill appearas a dihedralangle subtendedby two smoothlycurvedsolidfluid surfaces, anda mixedsolutionwill appearas a dihedral angle subtendedby one smoothlycurvedsolid-fluidsurface and one flat crystalface. Dihedralanglessubtended by two smoothlycurvedsolid-fluidsurfaces,by onesmoothlycurved surfaceandoneflat crystalface,andby two flat crystalfaces are thereafterreferred to as CC, CF, anf FF angles, respectively.

Nonsingular solutions arethecouples (01, 02)thatsatisfy (17a)-(17b).The searchfor singularandmixedsolutions was

madeasfollows. Letconsider a singular configuration 01= •Pl + 2ninNy, 02= 02+ 2n2•N2.For thisconfiguration to be at equilibrium thesumof surface tensions at thetriplejunction must be null:

gss+ TC•SF• + TOSF2 + x•(mgSF1) ø+ x2(Nt•sF2) ø= 0.

(19)

With 01 and 02 being fixed, the two unknownsare the coefficients x•, x2 whichspecifythemagnitude of thenormal

components of surfacetensions. If thereexistsa couple(xl, x2)with 1 and•r2l< 1 (equation (18)) for whichvector equation (19) is satisfied, thenthesingular configuration (01, 02) is an equilibrium geometry. For a mixedconfiguration

with Oi singular and0i nonsingular (i, j= 1,2; i •j) the condition for mechanical equilibrium at thetriplejunctionis

IJSS +TIJSFi +TIJSF j +Xi(NtJsFi) ø+NO'SF j = 0,

(20)

where the two unknownsare the magnitudeof the normal component of IJSF i (asembodiedby xi) andthe orientation of

surface SF/,0j.If thereexists a couple (xi, Oj),with•ci[_180ø.Dihedralanglesup to 230ø havebeenobtained for largevaluesof k anda squarelatticesymmetry. The differentconfigurations thatmay be stableat a given

triple junctioncorrespond to local minima of the total interfacialenergyfunction.Becausewe cannotcomputethe actual level of these interfacial energy minima without

considering a closedfluid geometry(for instance,a fluid pocketat thejunctionof threegrains),it is notpossible to tell

where•SF1is theweightedaverageof function•'SFI(½),and which solution is the most stable one. Y-SF1 = Y-SF2 for a monomineralic aggregate. The ratio G is speciallysuitedfor analyticaldevelopments; Gois, however, 4.3. Caseof SymmetricConfigurationsql = 02 morerepresentative andis usedin the final comparison of The effectsof anisotropy on dihedralanglesare illustrated wettingbehaviors in isotropic andanisotropic systems. Note by considering symmetricconfigurations where the two thatGoequalsG forki = 0 andki = 1 andthatit isjustslightly contiguous crystalsare of the samemineralspecies andthe smallerthanG for intermediatevaluesof ki: for instance,Go= two latticeshave symmetricorientations with respectto the G/1.024for Ni = 6 andki = 0.5 (Table1). grainboundarySS (01= 02). The main advantage of the symmetric caseis thatthe equilibrium configuration maybe specified by a singleangle01(= 02).The equilibrium values 4.2. Multiple Solutionsat Low Valuesof G of 01 and 02 are hereinafter referredto as 01eqand 02eq, For most valuesof G the programyields either a unique respectively, and0eq= 01eq + 02eqstands for theequilibrium solution(01, 02) or no solutionat all (at high valuesof G, dihedral angle. wherethe equilibriumconfiguration is a fluid-coatedgrain 4.3.1. Effect of lattice orientation.We computed01eqas boundary; seebelow).Complications ariseat low valuesof G a functionof 01for differentvaluesof N1, kl, andG; 01eqwas whereseveralconfigurations may be stableat a giventriple calculated for 01increasing from 0ø to 360ø/N1by stepsof junction.The exampleof a triplejunctionwith G = 0.01 is 0.5ø.Figure6 showstherelationship between01eqandlattice givenin Table2 (inputparameters arelistedin thefootnotes): orientation01 for G = 1.732,N 1= 6 (hexagonal lattice),and there are five possibleequilibriumconfigurations at this kl rangingfrom0.2 to 1. For kl = 0.5, for instance, thereis a junction,all five corresponding to dihedralangl.es of--180ø widerangeof 01 (from17.6øto 49.8ø;Figure6b) overwhich

butdiffering by theirasymmetry coefficient [01-021/2, 01eq--01andtheequilibrium configuration is a FF angle;for which ranges from 10ø to 70ø. Multiple solutionsare reminiscentof the degeneratebehavior at G =0 in the isotropic(monomineralic) case.In this case,there are no normal componentsof surface tensions:equation (15)

01< 17.6øor 01> 49.8ø,the equilibrium configuration is a CC anglewith 01eq • 01.Theextended stabilityrangeof FF anglesin Figure6b may be understood by considering a symmetric geometrywhereSF1 and SF2 have a singular

simplifies to TIJSF 1+ TIJSF 2 = 0 andis satisfied whenever the orientationso that 01: 01, 02= 02, and 01= 02; whetheror two surfacetensionshave the same directionsand opposite notthisgeometry is stablemaybededuced froma balance of signs; thiscondition leadsto aninfinitenumber of solutions: interfacialtensionsat the triple pointO (Figures5b and 5c). 01+ 02= 180 ø (01,02> 0ø). (23) The grain boundarySS being supposedisotropic,the interfacial tensionOsshasa magnitude equalto Yssandliesin the plane of the grain boundary. Because of the singular The calculations show that when surface energy is

anisotropic, thedegenerate behavior at lowvaluesof G gives

orientationof surfacesSF1 and SF2, the magnitudeof the

components of surfacetensions is equalto gl, and rise not to a continuous rangeof equilibriumconfigurations tangential

themagnitude of the normalcomponents is multivalued and may take any value in the range -glkltan(rdN1) to correspond to largeequilibrium dihedralangles,closeto or (equation (4)). Themagnitude of theresultant evenlargerthan180ø. For instance, the maximumdihedral gl kl tan(zr/N1) T(01)of tangential components of interfacial tensions is anglein Table2 is 190.7ø.Only nonzeronormalcomponents but to a discrete number of solutions. All these solutions

of surfacetensionscan accountfor equilibrium dihedral

Table2. Exampleof MultipleSolutions at a Low Valueof GrainBoundary to SurfaceEnergyG = 0.01a Type

01

02

FF CC CC FF FF

75ø 117.4ø 51.4ø 135ø 15ø

95ø 73.1ø 139.3ø 35ø 155ø

01+02 170ø 190.5ø 190.7ø 170ø 170ø

(0•-02)/2 -10 ø 22.2ø -44.0 ø 50ø -70ø

r(01)-'291cos01 - •'ss.

(24)

Theresultant N(01)of normalcomponents of surface tensions isparallelto Ox(bysymmetry), anditsmagnitude is

N(01)= 2 NO'SFl sin 01=2x1glkltan½/N1)sinO1, (25) where-1 _49.8ø). Although (29) only holds for symmetricCC anglesand for surfaceenergyfunction(2), its importanceis to showthat equilibriumdihedralanglesin an anistropicsystem depend on three independentgroups of parameters: (1) the ratio of grain boundaryenergyto surface energy,G; (2) a seriesof parametersdescribingthe surface energyfunction(k1 andN1); and(3) parameters specifyingthe

LAPORTE AND PROVOST: ANISOTROPIC WETTING

orientationof crystallinelattices relative to the grain boundary, •l and02. Only the first parameter is involvedin thecaseof isotropicinterfacialenergies (equation(1)); also,a fourthgroupof parameters wouldbe involvedif Ysswas significantly anisotropic (seesection5.1). 4.3.2. Effect of G on the equilibrium dihedral angle. The relationship between01eqand G for a singletriple junctionwith •1 (=02) = 10ø is shownin Figure 7a for kl = 0.5 andN• = 6. Thecurve0•eq(G)for a differentlattice orientation,• (=02)=40% as well as the "isotropic" relationship (equation(1)) are shownfor comparison. The

curve01eq(G) for k• =0.5 and•l = 10ø displays a general trendof decreasing 01eqwithincreasing G thatparallels the isotropic trend(Figure7a). It is characterized, however,by plateau of singular solutions, 0•eq= • and01eq = lPl+ 2m'N•,

100ø I

•'"•

50 •

100 ø

0

G = 2.07 for k• = 0.5, whereasit is only stablefor G = 1.97 for kl = 0. To explainthis behavior,we againconsiderthe balance of

interfacial tensions for

0•=•p• +360ø/N •

0o

which mark a major deviationfrom the isotropiccase.The

solution01eq = 10ø, for instance, is stablefromG = 1.87to

25,945

......

02=½2 +360/N 2

,

,

0,5

1

,

1,5

,

2

G

2,5

a symmetric FF

configuration (01= 02= • = 02). For G < 2cos0•,T(O1)> 0 (equation(24)) and actsto increasethe dihedralangle.For G> 2cos0•, T(0•)1 q360ø/N• b

for 2.0969< G < 2.1127. Partiallywettedgrain boundaries (PWGB)makethe junctionbetweenclosedsolutions (for G 0ø) or 01eq0ø).In Figure 8b, solutionswith 02eq < 0ø areobtainedfor G rangingfrom 2.0969to 2.1127. The exampleof a triplejunctionwith 02eq =-9.4 ø and (Figure6d): 01eq=•Pl at large valuesof G and 01eq= 01eq=13.6ø (G=2.112) is sketchedin Figure 9e. With q)•+ 2•N• at lower valuesof G. For q)•(= 02)= 10ø,kl = 1, increasing G, both01eqand02eq decreases until01eq = -02eq and N• =6, for instance,the solution01eq=•Pl (that is, = 11.5ø (thus0eq- 0ø) at G = 2.1127.For configurations with 0eq= 20ø) is stablefrom G = 2.1701 to G = 1.7691,andthe negative01eqor 02eq the balanceof surfacetensions solution01eq = •Pl+ 2•N• (thatis, 0eq= 140ø) is stablefrom perpendicular to the grainboundary canonlybe satisfied if G = 1.7691 to G = 0 (for G > 2.1701 the stable solutionis a the normal components of surfacetensionsare nonzero; wettedgrainboundary;seesection4.5). therefore suchconfigurations areforbidden in the isotropic case.

4.4. NonsymmetricConfigurations

In a monomineralicaggregate,most grain boundaries 4.5. Grain BoundaryWetting separatetwo grainswith nonsymmetric orientations: q)•• 02. Thewettingbehavior at G > 2 is of primordial importance. The equilibriumfluid geometryat a nonsymmetric triple In anisotropic systemwithG > 2, all thegrainboundaries are junction will in generalbe characterizedby two distinct "wetted"by the fluid phase,and two solid-fluidinterfaces

valuesof 01eqand02eq.Figure8a showsthecurves01eq(G) cannotjoin eachotherto form a triplejunction.A wetted and 02eq(G)for a monomineralic triplejunctionwith N• = 6, grainboundary(WGB; Figure10a) is a grainboundary kl = 0.5, q)•= 15ø, and 02= 35ø. The generalform of the containing a layerof fluidwhosethickness is suchthat(1) the curvesOieq(G)is similarto that describedfor a symmetric physicalpropertiesof the fluid are characteristic of the bulk

triple junction (Figure 7), with portionswhere Oieq fluidphase and(2) theinterfacial energy of theWGBmaybe equatedto the sumof the interfacialenergiesof the two parallelsolid-fluidsurfaces. The caseof verythinfluidfilms

monotonouslydecreaseswith increasingG, and plateau corresponding to a singularorientationof SF• or SF2. The overall decreaseof the equilibriumdihedral angle with increasingG is emphasized in Figure9. Note thatCF angles

F

(1-10 nm thick) in which these two conditionsare not

satisfiedis not considered in thispaper[seeHess,1994].A

c

•oF

ss

a:G=0.5

b..G=1.2

[c:G=I.8

F

c

ii0•- 15

0•= 13.6ø

ø ........ ":':C•*•'02 =6.1 ø • • b2=-9.4 d.'G=2.08 I

e:G=2.112

Figure 9. Equilibrium configuration atthejunction oftwolikegrains withnonsymmetric lattice orientations (k•= k2= 0.5,N1= N2= 6, q)•= 15ø,02= 35ø;fluidis in shaded area)for(a)G= 0.5;(b)G= 1.2;(c)

LAPORTE

AND PROVOST:

ANISOTROPIC

WETTING

25,947

randomorientationsOl and 02 of the crystallinelattices.The system being monomineralic,the same energy function YSFl(tp)was usedfor the two grainsin contact.The dihedral

YSFI(

angle0eqas well as the natureof the solid-fluidinterfaces (CC, CF, or FF) were determinedfor each triple junction.

Figure1l a showsthe frequencydistributions of 0eqobtained for a system in which the crystallinelattices have an

)tSF

2 b

hexagonalsymmetry(N• = 6) andan anisotropyfactorof 50% and 100% (k• = 0.5 and 1); in the calculation,G was set equal to 1.732, a value which corresponds in the isotropiccaseto a

single-valued angle 0eq= 60ø. For k• = 0.5, the 2000 triple junctionsshowa fairly widedistribution of 0eq,from35.6ø to 98.7ø, with an averagevalueof 68.0ø and a standarddeviation of 13.5ø. CC angles make up only 21.9% of the whole population,with the percentages of CF and FF anglesbeing

equal to 48.4% and 29.7%, respectively.For k• = 1, 0eq

Figure 10. Wetted grain boundariesfor (a) ki < 1 and (b) ki = 1 (fluid in shadedarea). For ki = 1 the two solid-fluid interfacesshowa hill-and-valleystructure.

rangesfrom 24.6ø to 140.4ø with an averagevalue of 72.3ø and a standarddeviationof 23.8ø; all the dihedral anglesare subtendedby two planar faces (FF angles). Despite the

dispersion of 0eqin Figure1l a, the averagevaluesof 0eqfor

20

a:

necessarycondition for a WGB to be stable is that grain boundaryenergy be larger than the sum of the two surface energiesYSFl(- q)•)and YSF2(-02):

YSS > YSF• (- 0•)+Ysv2 (- 02).

(31)

For any value of 0i, YSFi(-tPi)being just moderatelylarger than the minimum value gi (Table 1), wetting of a grain boundaryoccurs for Yss just slightly larger than the sum g• + g2. Therefore,in the caseof two abuttinggrainsof the same mineral species,grain boundary wetting occurs for Yss/gljust slightly larger than the isotropicvalue G > 2. In Figure 7a, for instance,the transitionfrom a triple junction with an unwetted(fluid-freeor "dry") grainboundaryand an equilibriumdihedralangle > 0ø to a wetted grain boundary occursat G > 2.085 for the symmetrictriple junction with 01 = 10ø and at G > 2.137 for 01 = 40ø. We emphasizethat evenfor k• = 1 the equilibriumtextureat high valuesof G is a wetted grain boundary. For 0• (= 02) = 10ø, k• = 1, and N• = 6, for instance,the transitionfrom the singularsolution

F (%)

G=1.732;N•=6

Oiso= 60ø.

15

;'•:

k•=0.5

10

[ "• .... kl= 1 ,

0o

,

,

,

,

•

•

,

60ø

!

!

I

,

,

i

120 ø Oe q

i

180 ø

20

F(%) b.G= 1;k•=0.5 0iso = 120ø

15

01eq = tP1to a wettedgrainboundary occursat G = 2.1701.In this case, however, the two solid-fluid interfaces that line the

grainboundaryshowa hill-and-valleystructure(Figure10b). The transitionto WGB with increasingG is not as simple for nonsymmetricconfigurations'In Figure 8b, for instance, solutions with 0eq->0øarestableup to G = 2.1127,butwetted grain boundariesare not stable below G= 2.1328 (from equation(31) with • = 15ø and 02 = 350)ßWe will show in our futurepaperthatthe gapbetweenthesetwo majortypesof solutionsis filled by a particularkind of equilibriumfluid geometryreferredto as partiallywettedgrainboundariesand madeof alternatingunwettedandfully wettedsegments[Kim et al., 1994].

4.6. StatisticalStudy

10

5 ß

Nl=4

i

0ø

60ø

120 ø 0eq180 ø

Figure 11. Frequencydistributionsof 2000 equilibrium

dihedralangles0eqin a 2-D monomineralic, fluid-beating

rock.(a) G = 1.732,N• = 6, andk• = 0.5 (thickline) or k• = 1 To illustratethe importanceof surfaceenergyanisotropy, (thin line) (from Laporte and Provost [2000], Figure 10, p. hereby kindpermission of KluwerAcademic we computed the frequency distribution of equilibrium 129,reproduced dihedralanglesfor a monomineralic, fluid-bearingrock. We Publishers).(b) G = 1, kl = 0.5, and N• = 6 (thick line) or first createda populationA of 2000 triple junctionswith N1 = 4 (thin line).

25,948

LAPORTE

AND PROVOST:

kl = 0.5 and 1 arejust slightly different from the isotropic value of 60 ø (for G = 1.732).

The frequencydistributions of 0eqfor G = 1, kl = 0.5, and' two differentsymmetryorders,N1 = 4 andN1 = 6, are shown in Figure 1lb. The averagevaluesof the two distributionsare nearly identical (119.6ø for N1 =4; 120.6ø for N1 =6) and basicallyequal to the isotropicvalue (120ø for G = 1). The

spreadof 0eqis, however,muchlargerfor the squarelattice symmetry than for the hexagonal lattice symmetry: the standarddeviationis 20.3ø for N1 = 4 versus13.1ø for N1 = 6. The statisticalcalculationswere repeatedfor ratiosof grain boundaryenergyto solid-fluidinterfacialenergyrangingfrom 0 to more than 2. The minimum, maximum, and average

ANISOTROPIC

WETTING

a

Grain 1 60ø-•p• .......

-:........... !½21 O--•} 3

•..•,,,,, Grain 3

Grain 260-•

SS•3

valuesof 0eq are plotted as a functionof Go for N1 = 4 andN1 = 6 in Figure 12 (Go is the ratio of grain boundary

SF•

2400 1

a'k•

= 0.5;N•-

6

180 ø

$F•

ANISOTROPIC

120 ø

100%

60 ø

•

WGB

•

SS•

Figure 13. (a) Sketch of a three-grain junction (grain boundaryanglesare supposedto be equal to 120ø). Lattice orientations are definedby the angles•l, •, and • relativeto

thegrainboundaries SSij.(b)Fluidpocket (lightshading) ata

o

three-grainjunction.The equilibriumsurfaceconfigurations at

0

0,5

1

1,5

2 Go 2,5

the threeSSFi/ pointjunctionscan be computed using Herring's method.

240 ø

180 ø

%.••

b:kl=0.5; N1 =4

average0eq in the anisotropicsystem.The agreementis

ROPIC

120 ø

energy to the weightedaverageof surfaceenergy;equation (22)). The salientfeaturein Figure 12 is the good agreement betweenthe isotropiccurve and the trend defined by the speciallygoodfor N1 = 6 (Figure 12a): for mostvaluesof Go the anisotropicaveragedepartsfrom the isotropicvalue by

much less than 10ø; also, the average0eq falls to 0ø at Go= 2.1 ascomparedto Go= 2 in the isotropiccase.

60 ø

At low-volume fractionsof fluid the fluid phaseis located o

0

0,5

1

1,5

2

Go

2,5

at three-grainjunctions.The equilibriumdihedralanglesat solid-solid-fluidpoint junctions SSF12, SSF23, and SSF31 (Figure13) maybe computedindependently from oneanother by applyingHerring'smethodor balancingsurfacetension

forces at eachSSFi/junction. A particular feature of a threeFigure 12. Average value of 2000 equilibriumdihedral angles(solidcircles)as a functionof Go in a monomineralic aggregatewith kl = 0.5. (a) N1 = 6 (Go = G/1.024; Table 1). (b) N1 =4 (Go= G/1.057). The minimum and maximum valuesof 0eqare shownby the thin lines.The isotropictrend is plottedfor comparison.At low valuesof Go (opencircles), severalequilibriumconfigurationswere obtainedfor someor all the triple junctions(see section4.2): in thesecases,only the two leastasymmetricsolutionswere takeninto accountin the calculationof the average, minimum, and maximum valuesof 0eq.At largevaluesof G0,partiallyandfully wetted grainboundariesare countedas 0ø anglesin the calculationof the averagedihedralangle.

grainjunctionis that the lattice orientations(relativeto the

grainboundary) at junctionSSFijarenotindependent from thoseat junction SSF/k: theorientations of thegrainsbeing definedby the angles•l, 02, and • (< 60ø; Figure 13a), latticeorientations at junctionsSSF12,SSF23,and SSF31are (•l, 02), (60- •, •), and (60- •, 60- •l), respectively.A populationB of 2001 solid-solid-fluid junctionswasobtained by creating667 three-grainjunctionswith randomlattice orientations •l, 02, and •. The dihedralanglestatistics for populationB werecomputed for N• = 6 anddifferentvalues of G andkl. In all cases,the statistics wereidenticalto those obtainedfor population A (2000 independent solid-solid-fluid junctions).For instance,for G = 1.732 and k• = 0.5 the mean

LAPORTE

AND PROVOST:

ANISOTROPIC

WETTING

25,949

dihedralangle, the standarddeviation,and the minimum and maximumdihedralangleswere 67.9ø, 13.2ø, 36.7ø, and 98.9ø, respectively,as comparedto 68.0ø, 13.5ø, 35.6ø, and98.7ø for

[1975] or equant-granuloblastic xenoliths of Harte [1977]) showsmoothlycurvedgrain boundariesand grain boundary anglescloseto 120ø.For thesemineralspeciesthe assumption populationA. The reasonfor this similarityis that although of constant7ssis probablya goodapproximation. lattice orientationsare interrelatedat a given three-grain junction,all orientations q•l, q2, q3, 60-q•l,60-q2, and 60-q3 5.2. Importance of YSFAnisotropyin Fluid-Bearing Rocks are equallyprobable(in the range0-60ø) and populationB is In a monomineralicsolid-fluid system with isotropic not statistically differentfrompopulationA. Accordingly,our interfacialenergiesthe equilibriumfluid dihedralanglesare resultsmay be used both at low-volume fractions of fluid predictedto have the samevalue at all triple junctionsand to (with fluid located at grain junctions) and at high-volume be subtendedby smoothly curved surfaceswith a constant fractions(with largerfluid poolsandisolatedgrains). mean curvature[Bulau et al., 1979]. Major deviationsfrom thesebasicfeatureshave been observedin all fluid-bearing rocksinvestigatedso far: for instance,CF andFF angleshave 5. Discussionsand Implications beendescribedin all partiallymoltenrocks(e.g., Cm[ral et al. 5.1. Assumptionof ConstantYss [1998] for the olivine-basaltsystem)and are even dominant In ourcalculations we assumed thatgrainboundaryenergy for mineralspecieswith a strongcrystallineanisotropysuch was constant.Low-angle grain boundarieshave, however, a

as amphibole [Laporte and Watson, 1995]. In addition, most

lower energy than high-angleones[Sutton and Balluffi, fluid-bearingrocks are characterizedby moderateto large 1995]:7ssrisesrapidlyfrom 0 if thetwo crystals(of the same variations of the fluid dihedral angle: for example, true mineral species)have the same orientationto a maximum dihedralanglesin the olivine-basaltsystemrange from 0 to value within the first 10-15 ø of tilt or twist. The fluid dihedral 40ø [Cm[ral et al., 1998] (the true dihedralangleis the angle

angleat thecontact of a low-angle grainboundary is larger measuredin the plane normal to the junction line of the fluid thanat a high-angleoneandapproaches 180ø for a very low degreeof lattice misorientation (75s--)0).The importanceof the misorientation effect dependson the frequencyof lowangle grain boundaries.In a 3-D monomineralicaggregate with randomlyorientatedgrains,90ø as comparedto medianvaluesof 10-

meanvalue [Jurewiczand Jurewicz, 1986].

Our resultsshowthatthe development of CF andFF angles and a spread of the true dihedral angle are the basic consequences of surface energy anisotropy.The effects of anisotropyare best illustratedin Figure 6 for the case of symmetrictriple junctions:the first effect is a large variation 20ø) only make up 2-5% of the total number of measured of the equilibrium dihedral angle as a function of the angles. orientation of the crystalline lattices relative to the grain In addition to the effect of lattice misorientation, another boundary;the secondeffect is to promotethe developmentof causethatmayinvalidatethe assumption of constantYssis the planarcrystalfacesand the formationof FF angles.Both the tendency of a grain boundary to lie parallel to a spreadof dihedral angles and the frequencyof FF angles crystallographic planeof one of the two neighboringgrains. increase with increasinganisotropyfactor. An important This type of behavior(referredto as 7ssanisotropy)is well contributionof the statisticalstudyis to showthat despitethe documentedin monomineralicaggregatesof amphibole, large variation of the equilibrium dihedral angles, their biotite,or pyroxenein high-grademetamorphic rocks[Kretz, frequency distributionremains unimodal with increasing 1966; Vernon, 1968]: most biotite grain boundariesare degree of anisotropy(Figure 11). This result is in good parallelto the (001) planein oneof the adjacentgrains;many qualitative agreement with the experimental findings: boundariesin monomineralicaggregatesof amphiboleor apparentdihedralanglesmeasuredin systemswith a strongly pyroxene are parallel to the (110) plane of one of the anisotropic surface energy show unimodal frequency neighboringgrains. Therefore,for mineral specieswith a distributions with a moderatespreadaboutthe medianvalue; strongcrystallographic anisotropythe grain boundaryhas a except for a few percent of very large anglespresumably minimum energy when it lies parallel to a specific associatedwith low-angle grain boundaries,most apparent crystallographic plane of one of the adjacentgrainsand the anglesfall within +_30 ø of the medianvalue [e.g., Laporte et assumption of constant 7ssis irrelevant[Kretz,1966].The Yss al., 1997]. anisotropyresultsin grain boundaryanglesthat may deviate We emphasizethat the frequencydistributions of apparent significantlyfrom the standardvalue of 120ø (at the junction dihedralanglescomputedfor idealizedsystemswith a singleof threelike crystallinephases). valued0eqare systematically sharperthan the experimental Unlike biotite, pyroxene, or amphibole, monomineralic frequencydistributions.For instance,the median apparent aggregatesof quartz, feldspar,garnet,and calcite in high- dihedralangleat thejunctionof anorthitegrainswith a watergrade metamorphicrocks [Kretz, 1966; Vernon, 1968] and saturatedmelt at 1000øC-1GPa is 28ø [Laporteet al., 1997] olivine in texturally equilibratedmantle xenoliths(the so- but the distributionof apparentanglesis distinctlybroader called mosaicequigranularxenolithsof Mercier and Nicolas than the theoreticaldistributionexpectedfor a single-valued

25,950

LAPORTE AND PROVOST: ANISOTROPIC WETTING

dihedralangle0eq= 28ø (Figure14a).The peakbluntingin Figure 14a may be due in part to the effect of measuring errors [Stickels and

Hucke,

1964].

The

systematic

development of planarcrystalfacesandFF dihedralanglesin the anorthite-liquid systemsuggests, however,thanthe main causeof peak bluntingis surfaceenergyanisotropy.To test the effect of anisotropyon apparent dihedral angles, we computedthe apparentfrequencydistributions for 3-D solidfluid systemshaving a distributionof 0eq equal to those calculatedby our 2-D model (the orientations of the solidsolid-fluidtriple junctionsare supposedto be all equally probable)[Harkerand Parker, 1945;Jurewicz: andJurewicz:, 1986]. The apparentdistributions for an anisotropic system

30-

F (%) 20

10

0

0o

60ø

120ø

p

180ø

p

180ø

with a mean0eqof 68ø (G = 1.732,kt = 0.5, Nt = 6; Figure 1l a) and an isotropicsystemwith 0eq= 68ø are comparedin Figure 14b. The two distributionsare unimodaland have the samemode,but the anisotropicone is muchmoreblunt due to

the spreadof 0eq.In general,even a moderatespreadof the true dihedralanglecan be responsiblefor the broadfrequency distributions of apparent dihedral angles observed in experiments[Jurewicz:and Jurewicz:,1986]. For instance,the

mean = 68o

20

• iso

F (%) 15

10

apparentdistribution computedfor a population of 0eqwith a mean

value

of

28.5 ø and

a standard

deviation

of

12.2 ø

SO

providesa reasonablygood fit to the distributionof apparent dihedralanglesin the anorthite-liquidsystem(Figure 14c). The importanceof surface energy anisotropyin silicate systemsis now well accepted[Waft and Faul, 1992; Laporte and Watson,1995; Cmfral et al., 1998]. Yet the interpretation 15 of equilibriumtexturesin fluid-bearingrocksin termsof •SF anisotropyoften leadsto the misconceptions that we discuss below. The first misconceptionis to use equation (1) to F (%) evaluatethe effect of surfaceenergyanisotropyor to estimate 10 the degree of anisotropyfrom the spread of equilibrium dihedral angles. Consider, for instance, a system with Yss= 1.732 anda surfaceenergydefinedby function(2) with ki = 0.5, N i = 6, and gi = 1: in sucha systemthe equilibrium dihedralanglesvary from 35.6ø to 98.7ø (Figure 11a). Using equation(1), one would computea minimumvalueof surface

0o

60ø

120ø

An22c

c

energyof 0.910 (from 0eq= 35.6ø) and a maximumvalueof 1.329 (from 0eq= 98.7ø).By comparison, the true rangeof surfaceenergy is only from 1 to 1.077 (for co=0ø and 30ø, respectively,in equation(2); Table 1). Clearly, equation(1) cannotbe usedto discussthe effect of •SF anisotropy. In their study of the olivine-basalt system, Cmfral et al. [1998, p. 342] recently claimed that melt dihedral angles subtendedby two planar faces are not controlled by "the relative energiesof the two F-faces and the grain boundary" (F facesareplanarcrystalfaces).They alsosuggested thatthis type of dihedral angles must be excludedfrom systematic dihedral angle measurements.These statementshave no physical basis: Assuming that textural equilibrium is established,CC, CF, and FF anglesare all controlledby the minimizationof interfacialenergies.When surfaceenergyis anisotropic,however, interfacial energy minimizationdoes not simplyconsistin minimizinginterfacialareas:in addition to reducing its area (as embodied by the tangential components of surfacetensions),any surfaceelementtendsto rotate toward an orientationof minimum surfaceenergy (as embodiedby the normal componentsof surfacetensions). That the equilibriumgeometrymay be a CC angle at one triplejunctionanda FF angleat an otherjunctiondepending on lattice orientationis well illustrated in Figure 6 and is discussed in detail in section 4.3.1. The basic fact that CC,

0ø

60 ø

V,

Figure 14. (a) Frequencydistributionof 118 apparent dihedralangles• at anorthite-anorthite-liquid triplejunctions in sampleQAn22c(run conditions:1000øC-1GPa-149hours; medianapparentangle of 28ø; from Laporteet al. [1997], Figure 6a, p. 44, reproducedhere by kind permissionof Kluwer Academic Publishers). The theoretical distribution

•isocomputed for a single-valued dihedralangleequalto 28ø is shownfor comparison (thinline). (b) Theoretical frequency distributions of apparentdihedralanglesfor an anisotropic

systemwith a mean0eqof 68ø anda standard deviation of 13.5ø (•aniso, thick line) and an isotropicsystemwith

0eq= 68ø (thinline).The3-D anisotropic system is supposed to havea distribution of 0eqequalto thatcomputed usingour 2-D model for G = 1.732, kt = 0.5, and N1 = 6 (see Figure 1l a). (c) Comparisonof the apparentdistributioncomputed

for a population of 0eqwith a meanvalueof 28.5ø and a standarddeviationof 12.2ø (•aniso,thick line; the distribution

of 0eqis equalto that computed for G = 2.000, kt =0.6, Nt = 8) andthedistribution of apparentdihedralanglesin the anorthite-liquid system(QAn22c,thinline).

LAPORTE

AND PROVOST:

ANISOTROPIC

WETTING

25,951

CF, and FF angles are all controlledby interfacial energy minimization strongly argues against the exclusion of FF angles (or CF angles) from systematicmeasurementsof dihedral angles in experimentalsamples.To determinethe effect of excludingFF angles,we comparedthe statisticsfor the bulk population of dihedral angles and for the subpopulations of CC, CF, andFF angles(Table 3). For G = 1 the mean dihedral angles for the three subpopulationsare nearlyequal,but for G = 1.732 andespeciallyG = 2 the mean CC angle is distinctlylarger than the mean FF angle. In all

1. Our model is only devisedto computethe equilibrium surfaceconfigurationsat the junction of two crystalswith a fluid phase(Figure 3a): given a certainnumberof intrinsic parameters(the lattice orientations relative to the grain boundary,the surfaceenergy functionsfor the two crystals, the grain boundary energy), it allows calculation of the orientationsof the tangentsto the two solid-fluid surfacesat the point where they join the grain boundary.The model cannot be used, however, to compute the shape and the

cases,however, the standarddeviations and the minimum and

dependon extrinsicparameterssuchas the presenceof a third grainin the nearneighborhood or the volumefractionof fluid. For the caseof a fluid pocket at a three-grainjunction, for instance,the equilibrium surfaceconfigurationsat the three solid-solid-fluidjunctions can be computedusing Herring's method,but the shapeof surfacesSFi and the exact position

maximum dihedral angles for the three subpopulationsare almostindistinguishable (a physicalexplanationto this result is providedin section4.3.1 in whichwe demonstrated that the minimumand maximumvaluesof CC andFF anglesare equal in the caseof symmetricconfigurations; Figure6). At present, it is difficult to compare these theoretical results with experimentsowing to the lack of detaileddata.In the quartzgranitesytem[Laporte,1994] we did not noticedifferencesin maximum or minimum angles for the CC and CF subpopulations (no FF angleswere found in this system).In the olivine-basaltsystem,Cmi'ralet al. [1998] concludedthat FF anglesare systematicallylarger (10-40ø) than CC and CF angles(0-10ø). This conclusionis based,however,on only six FF anglesand needsto be confirmedby more measurements. 5.3. Applicability of the Model

Two importantlimitationsof the model must be stated explicitly:

curvature

of

solid-fluid

surfaces

because

these

features

of thepointjunctions SSFi/alongthegrainboundaries SSij (Figure 13b) cannot be determined without incorporating curvatureconstraints[Cahn and Hoffman, 1974]. 2. The

second

limitation

is

that

the

model

is

two-

dimensionaland the surfaceenergyfunction(2) is presumably much more simple than real anisotropyfunctions.We argue that by using a simple function in which surface energy anisotropyis describedby a singleparameter,we are able to

bring to light the fundamentaleffectsof anisotropy.A 3-D anisotropy function able to reproduce any degree of anisotropyand any kind of lattice symmetrywas recently developedin our laboratoryand will be used in future calculations.

6. Conclusions

Table 3. DihedralAngle Statisticsa Parameters Speciation

G = 1.732

k• = 0.25

G = 1.732

k• = 0.5

G = 1.732

k• = 0.75

G = 1.000

k• = 0.5

G = 2.000

k• = 0.5

Mean

S.D. Minimum Maximum

bulk

64.5 ø

7.0 ø

46.7 ø

78.3 ø

CC (53.3) CF (39.3) FF (7.4)

65.8 ø 63.3 ø 61.3 ø

6.9 ø 6.8 ø 7.1 ø

46.7 ø 46.7 ø 47.2 ø

78.3 ø 78.0 ø 76.2 ø

Our resultsmay be summarizedin four points: 1. The •YSFanisotropyhas two major effects on the

equilibriuminterfacegeometryat thejunctionof two grains with a fluid phase:first, it is responsible for the development of planar solid-fluidinterfacesparallel to crystallographic planesof minimumsurfaceenergy;second,it resultsin a rangeof fluid dihedralangles,the equilibriumdihedralangle at a given triple junction being stronglydependenton the

bulk

68.0 ø

13.5 ø

35.6 ø

98.7 ø

CC (21.9) CF (48.4) FF (29.7)

72.5 ø 68.9 ø 63.2 ø

12.8ø 13.0ø 13.5ø

39.4 ø 36.5 ø 35.6 ø

98.0 ø 98.7 ø 98.3 ø

bulk

70.6 ø

19.3 ø

29.0 ø

120.2 ø

CC (5.1) CF (33.2) FF (61.7)

79.7ø 73.9 ø 68.1 ø

19.6ø 18.9ø 18.9ø

30.7ø 29.0 ø 29.7 ø

118.9ø 120.2ø 118.8ø

anglesobtainedin the statisticalstudyare distinctlyunimodal even at high degreesof anisotropy(Figures11a and lib). Increasing kl resultsin a major increaseof the standard deviation,but by comparison, the effecton the meandihedral angleis limited.

bulk

120.6 ø

13.1 ø

91.3 ø

153.6 ø

CC (21.6) CF (49.3) FF (29.1)

121.8ø 121.0ø 118.9ø

13.3ø 13.0ø 12.9ø

92.6 ø 91.3 ø 92.4 ø

153.6ø 153.1ø 153.6ø

bulk

35.1 ø

14.2 ø

0.5 ø

64.1 ø

CC (22.5) CF (49.1) FF (28.4)

44.0 ø 36.6 ø 25.5ø

12.4ø 11.9ø 13.5ø

4.5 ø 1.6ø 0.5ø

64.1ø 63.5 ø 60.4ø

aDihedralanglestatisticsfor a populationof 2000 triplejunctions with five different valuesof the ratio of grain boundaryto surface energy G and of the anisotropyfactor kl (all the grainsare of the samemineralspeciesand have an hexagonalsymmetry,N1 = 6). For each set of input parameters,the mean dihedralangle, the standard deviation(S.D.), and the minimumandmaximumdihedralanglesare given for the bulk populationand for the subpopulations of CC, CF andFF angles(the percentages of CC, CF andFF anglesare givenin

parentheses). In the computation for G = 2, solutions with 01eq< 0ø or 02eq< 0øamountto 7.3% of thebulkpopulation.

relativeorientationof the two crystallinelattices. 2. The frequency distributionsof equilibrium dihedral

3. Even in the mostanisotropic systems, the ratio of grain boundaryenergyto surfaceenergyis the fundamental factor controlling thevalueof thefluid dihedralangle:in particular, the meanfluid dihedralangledecreases with increasingGo accordingto a law very close to the isotropiclaw (0eq= 2 arccos(Yss/2YsF); Figure 12). 4. As in the isotropiccase, a transitionto wetted grain boundariesis observedat ratiosof grain boundaryto surface energyjust slightlylarger than 2. By analogy,the transition from a state dominatedby isolated fluid pockets at grain cornersto a state dominatedby wetted grain edges and an interconnectedfluid geometry in an anisotropic system containinga low fraction of fluid is predictedto occur at values of G just slightly larger than the isotropic value (G-1.732). At present, it is difficult to translate with certaintythisconditionin termsof a meandihedralangle.On

25,952

LAPORTE

AND

PROVOST:

ANISOTROPIC

the basis of the statistical data in Figure 12, anisotropic systemswith meandihedralangleswell below 60ø (say40ø or 50ø) are expectedto have a high proportionof wetted grain edgesat a very low volumefractionof fluid. From points 2 to 4, the mean fluid dihedral angle in a polycrystalline aggregate with anisotropic YSF must be considered as a parameter of primary importance for predictingthe grain-scalegeometryof a low-volumefraction of fluid and its mobility. Thus the measurementof dihedral anglesin texturallyequilibratedsamplesis usefuleven if the texture is indicative of a strongly anisotropicsolid-fluid

WETTING

Cmfral M., J. D. Fitz Gerald, U. H. Faul, and D. H. Green, A close

look at dihedral angles and melt geometry in olivine-basalt aggregates:A TEM study, Contrib. Mineral. Petrol., 130, 336345, 1998.

Cooper, R. F., and D. L. Kohlstedt, Interfacial energiesin the olivine-basaltsystem,in High PressureResearchin Geophysics, Adv. Earth Planet. Sci., vol. 12, editedby S. Akimoto and M. H. Manghnani,pp. 217-228, Cent.for Acad.Publ.,Tokyo, 1982. Defay, R., andI. Prigogine,TensionSuperficielleet Adsorption,295 pp., Desser,Liege,Belgium,1951. Faul, U. H., Permeabilityof partiallymoltenuppermantlerocksfrom experimentsand percolationtheory, J. Geophys.Res., 102, 10,299-10,311, 1997.

interfacialenergy.Finally, we emphasizethat dihedralangle Faul, U. H., Constraintson the melt distributionin anisotropic polycrystallineaggregatesundergoinggrain growth, in Physics measurementsonly provide a partial description of the and Chemistry of Partially Molten Rocks, edited by N.

distributionof fluids in real rocks. Complementarystudies Bagdassarov, D. Laporte,andA B. Thompson,pp. 67-92, Kluwer Acad., Norwell, Mass., 2000. (diffusionexperiments[Watsonand Lupulescu,1993], image analysis[Faul, 1997], etc.) must be performedin parallel to Harker, D., and E. R. Parker, Grain shapeand grain growth, Trans. ASM, 34, 156-195, 1945. determinequantitativelythe fluid transportpropertiessuchas Harte,B., Rock nomenclature with particularrelationto deformation the interconnection thresholdor the permeability. and recrystallisationtextures in olivine-bearing xenoliths, J. Geol., 85, 279-288, 1977.

Notation

0 fluid dihedralangle. 01, 02 two components of a fluid dihedralangle(Figure3). u an orientationof minimumenergyof thecrystal. •0 anglefrom u to the actualsurfaceorientation. •pl, 052 anglesdefiningtheorientationof thecrystalline latticesrelativeto the grainboundary. 6E changeof interfacialenergyassociated with an infinitesimaldisplacement of the pointjunctionof the threeinterfaces(Figure 3). ?'SF solid-fluidinterfacialenergy(or surfaceenergy). Y-SF weightedaverageof surfaceenergy. 7ss grainboundaryenergy. g minimumvalueof surfaceenergyfunction(2). k anisotropyfactorin surfaceenergyfunction(2). N symmetryorderof thecrystallinelattice. G ratio of grainboundaryenergyto minimumsurface energy,equalto 7ss/g. Go ratio of grainboundaryenergyto averagesurface energy,equalto 7ss/•SF ß OSS grainboundarytension. OSF solid-fluidinterfacialtension(surfacetension). x a parameterthatmeasures the magnitudeof the normalcomponent of oSFfor a crystalface. T, N resultantof the tangentialandnormalcomponents of tensionsactingat thejunctionof threeinterfaces. The main subscriptsused in the text are 1 and 2 for variablespertainingto grains1 and 2, respectively; eq for an

equilibrium value(01eq,etc.)'isofor an equilibrium valuein the isotropiccase(0•iso).

Herring, C., Sometheoremson the free energiesof crystalsurfaces, Phys.Rev., 82, 87-93, 1951a. Herring, C., Surfacetensionas a motivationfor sintering,in Physics of PowderMetallurgy, editedby W. E. Kingston,pp. 143-179, McGraw-Hill, New York, 1951b.

Hess,P. C., Thermodynamicsof thin fluid films, J. Geophys.Res., 99, 7219-7229, 1994.

Hoffman, D.W., and J. W. Cahn, A vector thermodynamicsfor anisotropicsurfaces,I, Fundamentalsand applicationto plane surfacejunctions,Sutyfi. Sci.,31,368-388, 1972. Holness,M. B., Temperatureand pressuredependenceof quartzaqueousfluid dihedralangles:The controlof adsorbedH20 on the permeabilityof quartzites,Earth Planet. Sci. Lett., 117, 363377, 1993.

Jurewicz,S. R., and A. J. G. Jurewicz,Distributionof apparent angles on random sectionswith emphasison dihedral angle measurements, J. Geophys.Res.,91, 9277-9282, 1986. Kim, D.-Y., S. M. Wiederhorn,B. J. Hockey,C. A. Handwerker,and J. E. Blendell, Stability and surfaceenergiesof wetted grain boundariesin aluminum oxide, J. Am. Ceram. Soc., 77, 444-453, 1994.

Kretz, R., Interpretationof the shape of mineral grains in metamorphic rocks,J. Petrol.,7, 68-94, 1966. Laporte, D., Wetting behaviourof partial melts during crustal anatexis: the distribution of hydrous silicic melts in polycrystalline aggregates of quartz,Contrib.Mineral. Petrol., 116, 486-499, 1994.

Laporte,D., and A. Provost,The equilibriumcrystal shapeof silicates:Implicationsfor the grain-scaledistributionof partial melts,EosTrans.AGU, 75(16),SpringMeet. Suppl.,364, 1994. Laporte,D., andA. Provost,The grain-scale distribution of silicate, carbonateandmetallosulfide partialmelts:a reviewof theoryand experiments, in Physicsand Chemistry of PartiallyMoltenRocks, editedby N. Bagdassarov, D. Laporte,and A B. Thompson,pp. 93-140, Kluwer Acad., Norwell, Mass., 2000.

Laporte, D., and E. B. Watson, Direct observationof nearequilibriumporegeometryin syntheticquartzites at 600ø-800øC and 2-10.5 kbar, J. Geol., 99, 873-878, 1991.

Laporte, D., and E. B. Watson, Experimentaland theoretical Acknowledgments.This manuscript benefittedfromdiscussions with BertrandDevouardand EstelleRignaultand from constructive

constraints on melt distribution in crustal sources: The effect of

crystallineanisotropyon melt interconnectivity, Chem.Geol.,

124, 161-184, 1995. reviewsby GregHirth andan anonymous reviewer.Our work was supported by theInstitutNationaldesSciencesde l'Univers,through Laporte,D., C. Rapaille,andA. Provost, Wettingangles, equilibrium grants 92-DBT-4.29 and 98-Dorsales-02.This is INSU-CNRS meltgeometry,andthepermeabilitythreshold of partiallymolten contribution 236. crustalprotoliths,in Granite: From Segregation of Melt to Emplacement Fabrics,editedby J.-L. Bouchez,D. H. Huttonand W. E. Stephens, pp.31-54,KluwerAcad.,Norwell,Mass.,1997. References Longhi,J., andS. R. Jurewicz,Plagioclase-melt wettinganglesand Bulau, J. R., H. S. Waft, and J. A. Tyburczy, Mechanical and textures:Implications for anorthosites, Lunar Planet.Sci.,XXVI, 859-860, 1995. thermodynamical constraints on fluid distribution in partialmelts,

J. Geophys.Res,,84, 6102-6108, 1979. Cahn, J. W., and D. W. Hoffman, A vector thermodynamicsfor

anisotropic surfaces, II, Curvedandfacetedsurfaces, ActaMetall., 22, 1205-1214, 1974.

Mercier, J.-C. C., and A. Nicolas, Textures and fabrics of upper-

mantleperidotiresas illustratedby xenolithsfrom basalts,J. Petrol., 16, 454-487, 1975.

Smith, C. S., Some elementary principles of polycrystalline

LAPORTE

AND PROVOST:

microstructure,Metall. Rev., 9, 1-48, 1964.

Stickels,C. A., and E. E. Hucke, Measurementof dihedralangles, Trans.Am. Inst. Min. Metall. Pet. Eng., 230, 795-801, 1964. Sutton,A. P., and R. W. Balluffi, Interfacesin CrystallineMaterials, Monogr. Phys. Chem. Mater., vol. 51, 819 pp., Clarendon, Oxford,England,1995. Vernon, R. H., Microstructuresof high-grademetamorphicrocks at Broken Hill, Australia, J. Petrol., 9, 1-22, 1968.

von Bargen,N., and H. S. Waff, Permeabilities,interfacialareasand curvaturesof partially molten systems:results of numerical computationsof equilibriummicrostructures, J. Geophys.Res., 91, 9261-9276, 1986.

Waff, H. S., and J. R. Bulau, Equilibriumfluid distributionin an ultramaficpartialmelt underhydrostaticconditions,J. Geophys. Res., 84, 6109-6114, 1979.

Waft, H. S., andU. H. Faul, Effectsof crystallineanisotropyon fluid distributionin ultramafic partial melts, J. Geophys.Res., 97, 9003-9014, 1992.

Watson,E.B., Melt infiltration and magmaevolution,Geology, 10,

ANISOTROPIC

WETTING

fluidsand their implicationsfor fluid transport,host-rockphysical propertiesand fluid inclusionformation,Earth Planet. Sci. Lett., 85, 497-515, 1987.

Watson, E.B., and A. Lupulescu,Aqueousfluid connectivityand chemicaltransportin clinopyroxene-rich rocks,Earth Planet. Sci. Lett., 117, 279-294, 1993.

Wortis, M., Equilibrium crystal shapes and interfacial phase transitions,in Chemistry and Physics of Solid Surfaces VIII, editedby R. Vanselowand R. F. Howe, pp. 367-405, SpringerVerlag, New York, 1988. Wray, P.J., The geometry of two-phaseaggregatesin which the shapeof t.he secondphase is determinedby its dihedral angle, Acta Metall., 24, 125-135, 1976.

D. Laporte and A. Provost, LaboratoireMagmas et Volcans, CNRS et Universit6 Blaise Pascal, Observatoirede Physiquedu Globe, 5, rue Kessler, F-63038 Clermont-Ferrand cedex, France.

([email protected]; bpclermont.fr)

236-240, 1982.

Watson, E.B., and J. M. Brenan, Fluids in the lithosphere, 1, Experimentallydeterminedwetting characteristics of CO2-H20

25,953

(receivedJuly 27, 1999;revisedMay 25, 2000; acceptedJuly 6, 2000.)

[email protected]