The rates and extent of textural equilibration in high-temperature fluid-bearing systems

Chemical Geology 162 Ž2000. 137–153 www.elsevier.comrlocaterchemgeo

The rates and extent of textural equilibration in high-temperature fluid-bearing systems M.B. Holness a b

a,)

, S.T.C. Siklos

b

Department of Earth Sciences, UniÕersity of Cambridge, Downing Street, Cambridge CB2 3EQ, UK Department of Applied Mathematics and Theoretical Physics, SilÕer Street, Cambridge CB3 9EW, UK Received 19 March 1998; received in revised form 1 January 1999; accepted 17 March 1999

Abstract The geometry of fluid-filled pores in texturally equilibrated aggregates can, in theory, be uniquely determined given the porosity and the equilibrium dihedral angle for all possible orientations and combinations of the constituent solid phases. While it is useful to be able to do this, one should also ask whether, and to what extent, textural equilibrium is actually attained during fluid-present intervals in high-temperature rocks. In fact, textural equilibrium may be only rarely attained during melting of crustal rocks, and equilibration under mid-ocean ridges is possibly incomplete to depths as great as the garnet stability zone. There is very little published information on the rates of attainment of textural equilibrium in fluid-bearing geological environments. The available experimental data for the rates of growth of equilibrated domains are consistent with a power-law relationship between domain size and equilibration time with an exponent between 2 and 3. A simplified theoretical treatment of pore shape change governed by the Gibbs–Thompson relationship demonstrates that an exponent of 3 is to be expected for equilibration by diffusion within the fluid phase, with an effective diffusivity consistent with the available data. q 2000 Elsevier Science B.V. All rights reserved. Keywords: Textural equilibration; Fluid-bearing geological systems; High-temperature rocks

1. Introduction A fluid phase, either a silicic melt or a C–O–H volatile phase, is almost ubiquitous in the Earth, and the movement of such fluid has important consequences for mass and heat transport Že.g., Bickle and McKenzie, 1987.. The details of fluid flow and its effects can only be fully understood if we already have an understanding of pore geometry and porosity

)

Corresponding author

distribution in the relevant geological environments. During the last decade or so, many studies of fluid flow through high-grade rocks Ži.e., those undergoing partial melting or high-grade metamorphism. have been based on the assumption that the porosity at high temperatures is in textural equilibrium Že.g., Watson and Brenan, 1987; Holness and Graham, 1991; Laporte and Watson, 1995. and consequently much effort has been invested in determining both the equilibrium dihedral angle and the degree of surface energy anisotropy for systems of geological interest Žsee e.g., references in Table 1.. It is becom-

0009-2541r00r$ - see front matter q 2000 Elsevier Science B.V. All rights reserved. PII: S 0 0 0 9 - 2 5 4 1 Ž 9 9 . 0 0 1 2 4 - 2

M.B. Holness, S.T.C. Siklosr Chemical Geology 162 (2000) 137–153

138

Table 1 Time taken to equilibrate textures as a function of distance. Data from published studies. a s silicic melt in quartz; b s silicic melt in biotite; c s carbonatite melt in olivine; d s basaltic melt in olivine; e s Fe–Ni–S melt in olivine. f s estimated from photomicrographs by the present authors; g s given as final grain size; h s stated in paper. nd s not defined, as feature developed during the quench Time Žh.

Distance Žmm.

Temperature Ž8C.

Method

System

184 24 144–336 3 0.0014

11 1–5 50 1–2 0.5

850–1150 800 1250 920 nd

h h g f f

a b a a a

92 ) 120 200 40 52 381–549 32 8 2 1–10 3.4–4.1 0.8–1.1 4.9 1–10 10–20 4–100 72–120 161 5–50 0.02–0.04 3–5 91 235

5–20 63 45–62 10–20 40–60 27–66 2–5 2–5 4–10 1 6 3.25 8.7 5–14 5–10 35 10–20 10–20 50 10 11–28 30–50 20–30

1050–1290 1300 1250 1200

f h h f g g f f g f h h h g g g f f g f g f f

1300–1400 1250 1250 1050–1255 f 1100 1300–1325 1300–1325 1300–1325 1300–1500 1260–1350 1500 950–1150 900 1000 nd 600–800 300 600

ing increasingly clear, however, that many environments are not in textural equilibrium during hightemperature fluid-present episodes, even though the final textures observed by the petrologist may be in, or close to, equilibrium Že.g., Maaløe and Printzlau, 1979; Hunter, 1987; Holness and Graham, 1995; Wolf and Wyllie, 1995; Bryon et al., 1995, 1996; Lewis et al., 1998.. In this contribution we will discuss the extent of textural equilibrium in a variety of fluid-bearing high-temperature environments, arguing that textural equilibrium will only occur during the fluid-present event if the rate of porosity production or destruction is sufficiently slow for equilibration to keep pace. We also derive a relationship for the distance over which equilibration occurs as a function of time for a

References

Laporte Ž1994. Laporte and Watson Ž1995. Jurewicz and Watson Ž1984. Connolly et al. Ž1997. J.D. Clemens Ž1998., personal communication c Hunter and McKenzie Ž1989. c Minarik and Watson Ž1995. d Waff and Bulau Ž1982. d Jin et al. Ž1994. d Watson Ž1982. d Faul Ž1997. d Daines and Kohlstedt Ž1993. d Daines and Kohlstedt Ž1993. d Riley and Kohlstedt Ž1991. d this paper d Cooper and Kohlstedt Ž1984a. d Cooper and Kohlstedt Ž1984a. d Cooper and Kohlstedt Ž1984a. d Cooper and Kohlstedt Ž1984b. e Shannon and Agee Ž1996. e Minarik et al. Ž1996. H 2 O-quartz Watson and Brenan Ž1987. H 2 O-pyroxene Watson and Lupulescu Ž1993. H 2 O-olivine Mibe et al., 1998 brine–calcite Holness and Graham Ž1991. air–calcite Olgaard and Fitz Gerald Ž1993. brine–halite Holness and Lewis Ž1997. H 2 O–calcite Holness Ž1997. and unpublished data

simplified model system and compare it with the available information on textural equilibration rates.

2. Textural equilibrium in fluid-bearing systems Textural equilibrium is the state attained when the internal energy of the system, bound up in grain boundaries and interphase boundaries, is minimised. In such a state the criteria enumerated below are met. Ži. Three interfaces Ži.e., grain boundaries or interphase boundaries. intersect along a line, and four of these interfaces intersect at a point. Žii. The dihedral angle, Q , is constant. In a fluid-bearing monomineralic rock this is the angle subtended at two-phase junctions between two solid

M.B. Holness, S.T.C. Siklosr Chemical Geology 162 (2000) 137–153

grains and fluid, and is a function of the relative magnitudes of the grain boundary energy, ggb , and fluid–solid interfacial energy, gsf ŽSmith, 1948.:

Q ggb s 2 gsf cos

2

Ž 1.

Žiii. All grain boundaries and fluid–solid interfaces have constant mean curvature, k, given by: ks

1

1

1 q

2 r1

r2

Ž 2.

where r 1 and r 2 are the principal radii of curvature. ŽThe sign of the curvature of pore walls is taken to be negative if the centre of curvature lies outside the pore under consideration.. This constraint arises from the difference in chemical potential of a curved interface compared to a flat one Žthe Gibbs–Thompson relationship, see later., and results in the dissolution of material from areas of interface with high curvature and its precipitation in areas of relatively low curvature. In a monomineralic aggregate this necessarily also results in a constant grain size since small grains bounded by few interfaces of positive curvature are unstable with respect to neighbouring large grains bounded by many interfaces with negative curvature. For aggregates with isotropic interfacial energies, the equilibrium pore geometry is uniquely defined by the porosity and the dihedral angle Žvon Bargen and Waff, 1986; Cheadle, 1989.. In such a simplified system, complete pore connectivity occurs if the dihedral angle is less than a critical value of 608 ŽSmith, 1964; Beere, 1975; Bulau et al., 1979., theoretically even at vanishingly low porosities, leading to the possibility of pervasive fluid flow in stable interconnected tubules on grain edges. The existence of anisotropy of interfacial energies in fluid-bearing systems results in a more complex pore geometry in which planar pore walls are stabilised resulting in large penny-shaped pores or to pinching-off of grain-edge tubules ŽBrenan, 1993; Laporte and Watson, 1995; Minarik and Watson, 1995; Faul, 1997.. The pore geometry in significantly anisotropic systems is poorly understood at present, although in theory it should be possible to predict it from the variation of interfacial energy with crystallographic direction.

139

It is important to note that even if we were in a position to predict pore topology and geometry from dihedral angles and the degree of anisotropy, the resulting predictions would only be useful insofar as we can be sure that textural equilibrium did actually pertain while fluid was present. Just how likely is it that texturally equilibrated pore geometries occur in fluid-bearing rocks? Equilibrium will only be achieved if the rate of movement away from equilibrium due to chemical reaction andror deformation is slower than the rate of textural adjustment towards equilibrium. The relative importance of these competing processes can be estimated from the magnitudes of their driving forces. Consider first chemical reaction as the process destroying textural equilibrium. The driving force for reaction arises from the change in chemical potential, D mr , where: D mr s yDT D S

Ž 3.

DT is the temperature overrunderstep and D S is the entropy change on reaction, which is of the order of 80 J moly1 8Cy1 for a dehydration reaction ŽWalther and Wood, 1984. and 30–200 J moly1 8Cy1 for melting of common rock-forming minerals ŽRichet and Bottinga, 1984a,b.. The competing driving force for textural equilibration arises from the difference in chemical potential, D mc , between regions of interface with different mean curvatures Že.g., Bulau et al., 1979.. For a bump on an otherwise planar surface this is given by: D mc s

2gsf Vm r

Ž 4.

where: gsf f 0.1 J my2 ; Vm is the molar volume, which for silicates is about 2 = 10y5 m3 moly1 ; and r is the radius of curvature of the bump. If the rate-controlling step for reaction is the same as that for textural adjustment Ži.e., either diffusion or interfacial processes., the bump will be smoothed and textural equilibrium maintained during reaction if the radius of curvature of the bump is less than the maximum given by: rmax s

2gsf Vm DTD S

Ž 5.

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M.B. Holness, S.T.C. Siklosr Chemical Geology 162 (2000) 137–153

Comparison of numerical models of metamorphic events with natural examples demonstrates that temperature oversteps may reach some tens of 8C in the vicinity of small intrusions ŽLasaga, 1986. and may exceed 108C even in regional-scale events ŽRidley, 1986.. For an overstep of 18C, bumps with a radius of curvature of 10y8 m or less will be smoothed out, whereas those with a greater radius of curvature will remain. An assessment of the appropriateness of this estimate of the extent of textural equilibrium in a reacting environment can be made by examining crystal shapes in a variety of high-grade rocks. An extreme endmember in the spectrum of textural states is given by dendritic olivine grains which grew during solidification of komatiite flows. Such an environment is dominated by crystal growth with virtually no sub-solidus textural adjustment. The shapes of the olivine grains are very close to the original shape controlled by reaction and are very far from textural equilibrium. Although dendritic olivine crystals are known to grow in highly olivine normative melts at cooling rates of - 58C hy1 ŽDonaldson, 1976., the amount of undercooling during crystallisation is not known. Preliminary optical examination of the points of the highly elongate, attenuated crystals in a typical komatiite demonstrates that their radius of curvature is considerably smaller than the 1 mm width of the dendrite arms, and is probably in the region 10y8 – 10y7 m. This is up to two orders of magnitude greater than the minimum which would be expected to survive during crystallisation with an undercooling of 108C, consistent with the equilibration of regions with an initial r - 10y7 m. The smoothingout of bumps with r ) 10y7 m could only occur during much slower crystallisation or during a phase of prolonged sub-solidus heating. Similar observations can be made on metamorphic rocks in which reaction textures are preserved; grains of the phases involved in the reaction almost invariably have shapes controlled by growth or dissolution rather than minimisation of internal energies, with many areas of the grain surface having very small radii of curvature. The discrepancy between the likely magnitudes of the driving forces for the two competing processes in high-temperature rocks thus suggests that fluid-producing and fluid-consuming reactions will generally result in textures and pore topologies which are,

initially at least, controlled entirely by dissolutionr growth kinetics. Only when reaction has progressed significantly or is complete will textures begin to adjust towards a lower energy configuration. Consider now deformation as the process destroying textural equilibrium. The problem of pore geometry in rocks undergoing deformation is complex owing to the inter-relationship between fluid and the deformation mechanism, either via chemical processes such as hydrolytic weakening, or by mechanical processes related to changes in the effective pressure. However, in general, textural equilibrium will only occur if the rate of textural change due to deformation is sufficiently slow. Observations of pore topology in experimentally deformed partially molten mantle demonstrate that deformation in the diffusion-creep regime is sufficiently slow to permit melt-filled pores to retain the static equilibrium topology, but dis-equilibrium pore geometries are found during more rapid deformation in the dislocation creep regime ŽJin et al., 1994; Daines and Kohlstedt, 1997..

3. Evidence for textural dis-equilibrium in hightemperature environments Any assessment of the extent to which textural equilibrium is approached in high-temperature fluidbearing environments is complicated by the continuum of textural adjustment, such as compaction and solidification, that happens both during and after the fluid-present event. The effect of compaction is particularly evident for metamorphic environments in which the C–O–H fluid released by reaction is expelled rapidly and all metamorphic porosity eliminated long before any petrographic examination is possible ŽGraham et al., 1997.. The extent of textural equilibration during metamorphism can thus only be assessed from experimental studies. Compaction and sub-solidus textural adjustment can also be a significant factor in partially melted rocks ŽMcKenzie, 1984; Hunter, 1987., although rapid quenching and solidification of melts in some environments permits inference of the extent of equilibration while melt was present. Paradoxically, although the initial impetus for the determination of texturally equilibrated pore topolo-

M.B. Holness, S.T.C. Siklosr Chemical Geology 162 (2000) 137–153

gies in geological materials was driven by a need to understand the segregation of mantle melts, it has become clear recently that textural disequilibrium may dominate flow of melt under mid-ocean ridges. Although expulsion of mantle melts is probably both complete and rapid ŽMcKenzie, 1984. and subsequent deformation at high temperatures obliterates any petrographic clues to erstwhile melt distribution, consideration of isotopic disequilibrium in the U–Th decay series constrains the minimum rate of flow to values much higher than can be accommodated by flow along a network of thin grain-edge tubules ŽRichardson and McKenzie, 1994.. This points to a dominance of fracture flow, possibly to depths as great as those of garnet stability, despite an equilibrium dihedral angle for basaltic melts in mantle materials of less than 608 ŽRichardson and McKenzie, 1994; Kelemen et al., 1997 and references therein.. Only in environments where both compaction and deformation are minimal, and where cooling is sufficiently rapid to freeze in the distribution of melt with insignificant sub-solidus textural adjustment, does it become possible to determine pore geometry directly. The distribution of melt in mantle xenoliths entrained in rapidly rising basaltic magma is a case in point. Mantle xenoliths are frequently observed to contain inclusions of glass. Although many suites of xenoliths contain several chemically distinct glasses, the source of each not always well resolved, it is clear that at least some glass is derived from in situ melting of the xenolith itself during entrainment. The specific example to be discussed here is from the Meerfelder Maar, an alkalic basaltic maar-type volcano in the West Eifel, well known for its variety of peridotitic and pyroxenitic xenoliths. Most of the West Eifel xenoliths are spinel peridotites which underwent a metasomatic enrichment event - 200 Ma ŽStosch, 1987., resulting in the formation of Cr-bearing pale brown pargasitic amphibole. The amphibole grains have embayed margins and are surrounded by a reaction rim comprising a colourless glass-bearing zone containing small, euhedral, grains of Cr-spinel, and euhedral clinopyroxene grains which frequently form groups in optical continuity, oriented with a crystallographic relationship with the relict amphibole ŽFig. 1a.. The spinel grains generally form on the margins of the reaction rims, and

141

often are totally surrounded by the marginal olivine. Following Frey and Green Ž1974. this texture is interpreted to have formed by amphibole breakdown during ascent via the reaction: amphiboles melt q olivineq clinopyroxene q spinel

Ž 6.

This reaction resulted in a volume increase with the consequent development of f 10 mm wide melt films on grain boundaries and in cracks radiating from the reaction sites ŽFig. 1b.. O’Connor et al. Ž1996. have suggested that the clear glass in the Meerfelder Maar xenoliths is the residue of frozen metasomatic melts which percolated through the mantle prior to entrainment in the host lavas Žsee also Edgar et al., 1989.; Zinngrebe and Foley Ž1995. suggested something similar for the nearby Gees xenoliths., but the textural observations above and the composition of the melt are consistent with a provenance from in situ breakdown of amphibole during decompression ŽFrey and Green, 1974; Stosch and Seck, 1980.. Experimentally determined melt-olivine and melt-pyroxene dihedral angles in partially molten mantle are in the region of 20–508 ŽWaff and Bulau, 1979; Bulau, 1982; Toramaru and Fujii, 1986; Fujii et al., 1986; von Bargen and Waff, 1988., suggesting that the observed grain boundary fluid films are not stable in this system. If the Meerfelder Maar grain boundary glasses were indeed in textural equilibrium, this must mean that their unusual composition Ži.e., silica-rich, aluminous and alkali; see analyses in O’Connor et al., 1996. results in a significantly different equilibrium topology compared to MORB. However, close examination of the films in the Meerfelder Maar nodules reveals that they are not a stable feature, and display irregularities of the solidmelt interfaces due to the instability of the film relative to a series of unconnected pores ŽFig. 2.. A further line of evidence that the grain boundary melt films are an unstable transient feature is the chemical disequilibrium between the matrix Ži.e., pre-existing. olivine and the melt. Glass in large pools surrounding amphibole grains has a magnesium number ŽMga. of 60 Žassuming all Fe is FeO. instead of Mga s 74 expected for equilibrium with the matrix olivine of composition Fo 90 – 91 ŽRoeder

142

M.B. Holness, S.T.C. Siklosr Chemical Geology 162 (2000) 137–153

M.B. Holness, S.T.C. Siklosr Chemical Geology 162 (2000) 137–153

143

Fig. 2. Melt-filled grain boundaries within an olivine–orthopyroxene region of a mantle xenolith. Three of the grains visible are orthopyroxene, with the uppermost grain being olivine. Melt occurs as a thick Žf 10 mm. planar film along all grain boundaries. Note the instability developing in the film at the lower left, which has developed bulbous irregularities with a constant wavelength.

and Emslie, 1970.. A limited approach to chemical equilibrium is observed in the grain boundary glasses in which the Fe content and Mga vary with distance from the wall ŽFig. 3.. The Fe content of the marginal olivine is altered only within about 5 mm of the interface. If the melt had been introduced a long time prior to entrainment one would perhaps expect it to have resided predominantly at olivine triple junctions with a consequent approach to chemical equilibrium at these points. However, compositional profiles across melt-filled pores at olivine triple junctions also show very little chemical modification of the marginal olivine, supporting the suggestion that the melt was introduced very shortly before quenching, consistent with formation during entrainment.

The timescale for melting and limited textural and chemical equilibration, constrained from the observed modification of the marginal 5 mm of olivine, must be of the order of 1–10 h Žusing the diffusion coefficients for Fe–Mg exchange given by Jaoul et al., 1995... This estimate corresponds well with the 8 h ascent time deduced using similar reasoning for mantle nodules from British Columbia ŽFujii and Skarfe, 1982.. Although the timescale for melting and textural adjustment in mantle nodules is particularly short, textural disequilibrium can also be observed in anatectic crustal rocks in the contact aureoles around small intrusions where the timescale is measured in 10 3 years or more. Preservation of the melt distribu-

Fig. 1. Ža. Reaction rim surrounding pargasitic amphibole in olivine Žmantle xenolith from Meerfelder Maar, Eifel, West Germany.. The melt is clear and contains euhedral diopsidic clinopyroxene and Cr-spinel. The scale bar is 100 mm long. Žb. Reacting amphibole grain with radiating melt-filled intra-grain and inter-grain cracks. The amphibole grain is 0.5 mm in diameter.

144

M.B. Holness, S.T.C. Siklosr Chemical Geology 162 (2000) 137–153

Fig. 3. Ža. Variation of total Fe content and Žb. Mga as a function of distance across a grain boundary melt film in an amphibole-bearing mantle xenolith from Eifel. Note that the Fe content of the melt varies smoothly across the film, whereas that of the olivine varies only within about 4 mm of the boundary. Concentrations of total Fe were determined as a function of distance across the melt films, and also across large melt-filled pores at three-grain junctions using a Cameca microprobe. A well-focused beam Ž3 nA current. with a nominal diameter of 1 mm was used, with step distances of 1–2 mm achieved using the automated stage. Counting times at the peak were 50 s. Errors on measurements are approximately twice the dimensions of the dots.

tion in small anatectic contact aureoles shows that melt is concentrated in grain boundary films and pools ŽPlatten, 1982; Kitchen, 1984; Emeleus, 1997, p. 89., consistent with experimental partial melting studies demonstrating that melting begins at polyphase boundaries ŽMehnert et al., 1973; Brearley and Rubie, 1990; Wolf and Wyllie, 1991.. On the timescale of the anatectic event, the distribution of the melt phase was clearly primarily controlled by the kinetics of the melting reaction and the distribution of reactants rather than minimisation of surface energies, consistent with predictions based on likely

temperature oversteps during metamorphism. That such disequilibrium textures may be common even for crustal melting on a much larger scale was suggested by Wolf and Wyllie Ž1995.. They argued that since chemical equilibration of major elements is achieved faster than textural equilibration, the commonly observed chemical disequilibrium of trace element and isotopic composition for large-scale migmatitic events Že.g., Sawyer, 1991; Harris et al., 1992; Watt and Harley, 1993. is consistent with textural equilibrium being very rarely attained during crustal melting.

M.B. Holness, S.T.C. Siklosr Chemical Geology 162 (2000) 137–153

4. Experimental constraints on textural equilibration rates There are clearly many environments in the Earth in which the rate of textural equilibration is too slow to accommodate the fluid in a pore network with a geometry controlled entirely by internal energy minimisation. The obvious question which now poses itself is: how fast is textural equilibrium attained? This question was first addressed by materials scientists interested either in the kinetics of attainment of the minimum energy shape for isolated crystals Žreviewed by Kern, 1987., or in the rates of development of the dihedral angle in a grain boundary groove on the surface of an initially planar bi-crystal ŽMullins, 1960., but has not yet been studied in any depth for the poly-crystalline systems relevant to geological problems. Although no detailed studies of textural equilibration rates in fluid-bearing geological systems have been published, attempts have been made to assess the kinetics of the problem using the results of experimental determinations of dihedral angles Že.g., Waff and Bulau, 1979, 1982; McKenzie, 1984; Cooper and Kohlstedt, 1984a.. Such experiments

145

involve the hydrostatic recrystallisation of a polycrystalline starting material in the presence of fluid. The polycrystalline solid phase is either crushed during sample preparation or grown from gel, and the starting materials in the experimental charge are far from textural equilibrium. During the experimental run the crystals adjust their shape to minimise internal energies, beginning with the establishment of the equilibrium dihedral angle at pore corners formed by the junctions of two contiguous grains. The subsequent surface of constant mean curvature propagates outwards away from the pore corner ŽMullins, 1957, 1960; Hunter, 1987; Laporte and Watson, 1995; Holness, 1997. by a coupled dissolution–transport–reprecipitation process. The distance over which the textural adjustment has propagated defines the ‘‘equilibrium domain’’ and increases with time. Complete textural equilibration occurs when the equilibrium domain attains the dimension of the grain size. Using the early experiments of Waff and Bulau Ž1982., Cooper and Kohlstedt Ž1984a. suggested that since the most probable rate-controlling mechanism for textural equilibration is diffusion, one would expect a relationship between d, the domain size,

Fig. 4. Compilation of best estimates for the time taken for textural equilibration to occur over a given distance in fluid-bearing systems, taken from published data ŽTable 1.. The two lines through the data cloud are the best fitting lines with slopes of 0.5 Žwith an undefined constant K 1 . or 0.33 Žwith a constant K 2 given by the term in square brackets in Eq. Ž17...

M.B. Holness, S.T.C. Siklosr Chemical Geology 162 (2000) 137–153

146

and t, time, in texturally equilibrated charges of the form: d s k Ž Dt .

1r2

Ž 7.

where k is a constant of proportionality and D is the diffusivity. Making the assumption that the domain size is equal to the final grain size in fully equilibrated experimental charges, they showed that the final grain size of texturally equilibrated experiments in the basalt–olivine system has a quadratic dependence on time and thus concluded that the above relationship holds for domain size. Subsequently, this result Žwhich was also independently suggested by McKenzie Ž1984.. has been extensively cited in the geological literature Že.g., Cheadle, 1989; Laporte and Watson, 1995.. How do other published experimental studies compare with this result? Compilation of published information on equilibration times ŽTable 1. is complicated by variation of the method used to assess the size of the equilibrium domain. Many studies which give an approximate time for equilibration follow Cooper and Kohlstedt Ž1984a. and use the final grain size as a measure of the equilibrium domain. For the purposes of the present study, we obtained other assessments either by measuring the distance over

which the surface of constant mean curvature has propagated from the pore corner using scanning electron photomicrographs Že.g., calcite–brine in Holness and Graham, 1991; halite–brine in Holness and Lewis, 1997., or by estimation of the domain size using published photomicrographs Žsee Table 1 for details.. The results of this compilation are shown in Fig. 4. All the available estimates form a cloud of points with a positive correlation between equilibrium domain size and time. No clear divisions of the data set are possible on the basis of composition, and the best fit line to the entire data set has a slope close to 0.5. When the data for the basalt–olivine system are considered alone ŽFig. 5. the best-fit line has a slope of 0.48. Although this result Žalbeit with a much smaller data set. has been construed as support for Eq. Ž7. ŽCooper and Kohlstedt, 1984a., we suggest that such a simplified treatment may be misleading. A close look at the basalt–olivine subset shows that the great majority of the data use the final grain size as a measure of the equilibrium domain ŽFig. 5.. Once the four points determined using other criteria Ždenoted by the shaded boxes in Fig. 5. are removed, the remaining data points lie close to a line with a slope of 0.5. The potential significance of this result

Fig. 5. Compilation of published equilibration rate data for the basalt–olivine system Žfrom Table 1.. No error bars are shown, but are given on Fig. 4. All data apart from the shaded points are derived assuming the equilibrium domain size is that of the final grain size.

M.B. Holness, S.T.C. Siklosr Chemical Geology 162 (2000) 137–153

to questions of textural equilibration is questionable, however, given that the relationship between final grain size and time in an ideal polycrystalline aggregate undergoing grain growth is also quadratic Žreviewed in Martin and Doherty Ž1976... Since no efforts were made to pin grain boundaries in the cited experimental studies, grain growth was certainly occurring in conjunction with textural equilibration. At present it is perhaps unwise to use these data as a basis for statements about rates of textural equilibration.

5. A model for textural equilibration in fluidbearing polycrystals As a simple two-dimensional model we consider a pore which is initially an equilateral triangle, with a lengthscale of L ŽFig. 6a.. Such a pore, bounded, for example, by planar growth faces of the surrounding crystals, could conceivably be formed immediately after the accumulation of crystals at the base of a magma chamber Že.g., Hunter, 1987.. The beginning of textural equilibration is marked by the establishment of the equilibrium dihedral angle, Q , at the pore corner ŽFig. 6b.. An equilibration front subsequently propagates along the initially planar pore wall, leaving a curved surface in its wake. Complete minimisation of internal energies in this simplified system is achieved when the six propagating equilibration fronts meet and merge, resulting in an equilibrium configuration consisting of circular arcs ŽFig. 6c.. The radius of the arcs, re , is a function of the pore area and the equilibrium dihedral angle.

Fig. 6. A series of cartoons showing the progressive equilibration of a two-dimensional fluid-filled equilateral pore of constant area and dimension L Ža. for which the equilibrium dihedral angle is less than 608. Equilibration begins at the pore corners by establishment of the dihedral angle Žb., and the consequent ‘‘hump’’ in surface curvature propagates away from the corner. Complete equilibration occurs when the interfaces have constant curvature given by 1r re , where re is the radius of curvature of the pore wall and is determined by the dihedral angle and the dimension of the pore Žc.. A similar series of cartoons for a system with Q )608 would result in a pore bounded by convex interfaces.

147

The driving force for the propagation of the equilibration fronts, which has already been introduced in the form of Eq. Ž4., is the change in chemical potential of the solid phase, ms , due to variations in

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148

the curvature of the fluid–solid interface. In two dimensions, this is given by: Vm gsf

ms s mso y

Ž 8.

r

ŽBulau et al., 1979. where mso is the chemical potential of a flat interface. Assuming chemical ideality, this difference in chemical potential can be related to a difference in the equilibrium concentration of the solid phase in the adjacent liquid, C, compared to that in liquid adjacent to a flat interface, Co : ln

C s Co

Vm gsf

Ž 9.

RTr

Since Vm gsf RT

f 10y10 < r

Ž 10 .

we can approximate the concentration C by:

ž

C f Co 1 q

Vm gsf RTr

of the pore, C Ž x,t . at position x and time t satisfies the two-dimensional diffusion equation: EC D= 2 C s

where D is the diffusivity of the solid phase in the fluid. We can essentially ignore the time dependence in this equation since the timescale for the establishment of the concentration gradient away from the fluid–solid interface is much faster than that for equilibration; in effect the concentration gradient in the pore can be regarded as being instantaneously in equilibrium with the existing curvature of the interface. The concentration of solid at any point in the fluid near the pore wall is determined by the local radius of curvature of the interface from Eq. Ž10.. In addition, conservation of mass across the pore wall and Fick’s law require that: 1 Ez Vm Et

/

Ž 12 .

Et

s Dn P = C

Ž 13 .

Ž 11 .

This relationship is variously known as the Gibbs– Thompson, Gibbs–Kelvin or Ostwald–Freundlich equation. It shows that solid material will dissolve from areas of the interface with high curvature and will precipitate at areas of low curvature until all variations in surface curvature Žand hence variations in solubility. are erased. Four mechanisms for mass transport will operate in a fluid-bearing system: volume diffusion in the fluid phase; volume diffusion in the solid phase Žlattice diffusion.; surface diffusion in the fluid–solid interface; and advection in the fluid. Discounting advection, the dominant diffusion process will be that in the fluid phase, and since this will be many orders of magnitude faster than either of the other two pathways, it will dominate the kinetics of the equilibration process. The problem of textural equilibration of the straight-sided pore described at the start of this section is highly non-linear but an idea of the timescales involved can be obtained using a simplified treatment. The concentration of solid material in the fluid

where n is the inward normal to the pore wall and z Ž x,t . is the displacement of the pore wall from its initial position in the n-direction ŽFig. 6.. ŽWe consider only one side of the triangle as conditions on the other sides follow by symmetry.. Initially the radius of curvature of the pore wall close to the pore corner will be small. In the equilibrium configuration, the final radius of curvature, re , depends on the dihedral angle, Q , and the original pore dimension, L, and is determined by the geometry. For example, if Q s 1808, then re s Ž3rp 2 .1r4 a where a is the length of the side of the original equilateral triangle. We expect that, if Q is not close to 608, equilibration starts with the creation of a region of high curvature which is rapidly dissipated, thus not contributing significantly to the timescale. For most of the evolution, the radius of curvature will be of order L. Hence, the excess concentration, DC, at the pore wall will be of order: DC s C y Co f

Co Vm gsf RTL

Ž 14 .

M.B. Holness, S.T.C. Siklosr Chemical Geology 162 (2000) 137–153

If we further approximate nP= Cf

C y Co

Ž 15 . l where l is the lengthscale over which the concentration changes away from the pore wall, we obtain: Ez

Co Vm gsf

Ž 16 . l RTL The lengthscale l is determined by solving the diffusion equation and is typically of the same order as the radius of curvature of the pore wall which we took to be l. Consideration of the two endmember cases given by values of 1208 and 408 for the equilibrium dihedral angle Ža likely range for geological materials, Holness, 1996. shows that equilibration of the twodimensional pore involves transporting an area of solid equal to about 10% of the total pore area. Hence, on average < z < has to reach about 0.1 L in order to attain the equilibrium shape. The time required for this, t , follows from Eq. Ž16., setting L s L: Et

f DVm

ts

RT 10 DVm2 Co gsf

L3

Ž 17 .

According to this approximation, the timescale to achieve an equilibrium pore shape by diffusive mass transfer in the fluid phase should vary as L3. The ‘‘extra’’ length Žcompared to what one would have expected in a diffusion equation. in the relationship is an inevitable consequence of the concentration gradient set up by spatial variations in solid solubility. This expression is very similar to that obtained for grain boundary grooving by diffusion in the fluid phase ŽMullins, 1960.. Comparison with treatments of Ostwald ripening of precipitate phases, which also has the same driving force, suggests that an exponent of 2 will only be seen if the kinetics of solutionrreprecipitation, rather than mass transport, are ratecontrolling ŽMartin and Doherty, 1976.. Grain boundary diffusion, which is rate-controlling in cases where the whole of the fluid–solid system is changing shape Že.g., Cooper and Kohlstedt, 1984a. but unlikely to be rate-controlling for pore-shape change, would result in a quartic relationship ŽMartin and Doherty, 1976.. A quartic relationship is also expected for grain boundary grooving if diffusion in

149

the fluid–solid interface is rate-controlling ŽMullins, 1957., but this is only likely to be the case for systems in which the concentration in the fluid phase is extremely low. The above estimate gives the timescale for equilibration as a function of pore size. We could ask a different question Žof little relevance to the existing data. namely, at what rate is equilibrium achieved in a given pore? Putting it another way, what is the timescale for the progression of the hump ŽFig. 6b. determined by z along a pore wall of fixed length L? The case of an equilateral pore when the dihedral angle is close to 608 is analysed in Appendix A. It is shown that z satisfies Žapproximately. the diffusion equation with a diffusion constant, DX , given by: X

Ds

DCo Vm2gsf

Ž 18 .

RTL

and in this case the equilibration time agrees, except for a numerical factor, with the estimate Ž17.. How does this result compare with previous work and the data compilations shown in Figs. 4 and 5? The expression in square brackets in Eq. Ž17. is equivalent to K 2 in Fig. 4. If a cubic fit is forced through the data for all melt-bearing systems ŽFig. 4., the intercept on the y-axis lies between 10y6 m and 10y7 m, giving 10y1 8 m3 sy1 - K 2 - 10y2 1 m3 sy1 . A cubic fit through the C–O–H fluid-bearing systems gives K 2 s 10y1 8 m3 sy1 . Using values of the various parameters given in Table 2, Eqs. Ž17. and Ž18. predict values of K 2 of 10y2 0 –10y22 m3 sy1 for melt-bearing systems and 10y2 0 –10y1 8 m3 sy1 for systems containing C–O–H fluids. The apparent correlation between these two sets of numbers Table 2 Symbols defined in the text. Diffusivities of cations and oxygen in silicate melts from Cooper et al. Ž1996., Liang et al. Ž1996., Linnen et al. Ž1996., LaTourette et al. Ž1996., and LaTourette and Wasserburg Ž1997.. Diffusivities of cations in C–O–H fluids from Brady Ž1983. and Balashov Ž1995.

Co , mol my3 Vm , my3 mol gsf , J my2 T, K D, m2 sy1

Melt-bearing system

C–O–H fluid-bearing system

2500 2=10y5 0.1 1000 10y10 –10y12

200 2=10y5 0.1 1000 10y7 –10y9

150

M.B. Holness, S.T.C. Siklosr Chemical Geology 162 (2000) 137–153

is encouraging but its ultimate significance is unclear, particularly as the available data are of such uneven quality and comprise information from a variety of chemical systems. One subset of the data compilation in which only the minority of the distance estimates are derived from final grain sizes — the set for volatile fluids — shows a plausible cubic time dependence ŽFig. 4. and a close approximation to the expected value of K 2 . Clearly a set of careful experiments to determine the rate of propagation of a surface of constant mean curvature will be extremely helpful in testing the above predictions.

6. Discussion and conclusions The above approximate treatment of the rate at which pores will change shape in response to minimisation of internal energies shows some correlation with the limited, but perhaps inappropriate, available experimental data. In the absence of any direct examination of the rate at which surface curvature discontinuities propagate in fluid-bearing systems, we could perhaps feel confident about using the trend in Figs. 4 and 5 to give an indication of the times necessary to create equilibrium domains of any given size. However, it should be pointed out that fluid is a transient feature in geological environments, and the amount of porosity will not be constant on a geological timescale. A texturally equilibrated pore structure requires that the rate of equilibration be commensurate with that of the rate of change of porosity. If fluid production or loss is too rapid, the pore topology will be out of textural equilibrium. Similarly, deformation will result in a non-texturally equilibrated pore topology, and hence flow path, for strain rates greater than can be accommodated by diffusional processes. Direct observations of the pore geometry in some Žmelt-bearing. high-temperature environments in the Earth show that textural equilibrium is not attained during the melt-present event. This must be primarily due to the short or non-existent time interval during which the amount of porosity stayed constant, and the incommensurate rates of porosity creationrreduction and textural equilibration. The special cases mentioned in this paper certainly only include those in which the meltingrsolidification event had a rela-

tively short duration, but we consider it unwise to assume that fluid-bearing environments which we cannot observe directly are in a closer approach to textural equilibrium. Until we have more information on the rates of textural equilibration, we are not in any position to constrain adequately the extent of textural equilibrium in fluid-bearing environments in the Earth.

Acknowledgements We are grateful to Mike Cheadle, Dan McKenzie and Victor Balashov for discussions of the ideas in the paper, although any remaining misconceptions are our own. We thank John Clemens for providing unpublished photomicrographs of his melting experiments. Herbert Huppert, Dougal Jerram, and Matthew Jackson gave constructive and helpful criticism of an earlier version of the manuscript. This research was supported by NERC grant GR9r1589.

Appendix A An order of magnitude estimate of the time required to attain textural equilibrium of the equilateral pore shown in Fig. 6 can be obtained through linearisation, assuming that z is so small that terms involving z 2 can be ignored. This will be appropriate for systems in which the equilibrium dihedral angle is very close to 608 since then only small changes to the pore shape are required for equilibration. Linearising Eq. Ž13., and again assuming that C varies on lengthscale L, gives: Ez f Vm D

Et

ž

C y CO

/

L

Ž 19 .

and using the Gibbs–Thompson Eq. Ž11. leads to: Ez Et

f Vm D

gsf Vm Co E 2z RTL

Ž 20 .

Ex2

where we have approximated the curvature of the fluid–solid interface as follows: E 2z

1 s r

Ex2

Ez

1q

ž / Ex

2 y3 r2

E 2z f

Ex2

.

Ž 21 .

M.B. Holness, S.T.C. Siklosr Chemical Geology 162 (2000) 137–153

Thus, the disturbance propagates according to the diffusion equation, with an effective diffusivity given by: DX s

gsf Vm2 Co D RTL

Ž 22 .

This is dependent on the size of the pore, and leads to a cubic dependence of equilibration timescale with distance.

References Balashov, V.N., 1995. Diffusion of electrolytes in hydrothermal systems: free solution and porous media. In: Schmulovich, K.I., Yardley, B.W.D., Gonchar, G.G. ŽEds.., Fluids in the Crust. Chapman & Hall, pp. 215–251. Beere, W., 1975. A unifying theory of the stability of penetrating liquid phases and sintering pores. Acta Metallurgica 23, 131– 145. Bickle, M.J., McKenzie, D., 1987. The transport of heat and matter by fluids during metamorphism. Contributions to Mineralogy and Petrology 95, 78–93. Brady, J.B., 1983. Intergranular diffusion in metamorphic rocks. American Journal of Science 283 A, 181–200. Brearley, A.J., Rubie, D.C., 1990. Effects of H 2 O on the disequilibrium breakdown of muscoviteqquartz. Journal of Petrology 31, 925–956. Brenan, J.M., 1993. Diffusion of chlorine in fluid-bearing quartzite: effects of fluid composition and total porosity. Contributions to Mineralogy and Petrology 115, 215–224. Bryon, D.N., Atherton, M.P., Hunter, R.H., 1995. The interpretation of granitic textures from serial thin-sectioning, image analysis and three-dimensional reconstruction. Mineralogical Magazine 59, 203–211. Bryon, D.N., Atherton, M.P., Cheadle, M.J., Hunter, R.H., 1996. Melt movement and the occlusion of porosity in crystallising granitic systems. Mineralogical Magazine 60, 163–171. Bulau, J.R., 1982. Intergranular fluid distribution in olivine–liquid basalt systems. PhD thesis, Yale University, New Haven, CT, USA. Bulau, J.R., Waff, H.S., Tyburczy, J.A., 1979. Mechanical and thermodynamic constraints on fluid distribution in partial melts. Journal of Geophysical Research 84, 6102–6108. Cheadle, M.J., 1989. Properties of texturally equilibrated twophase aggregates. Unpublished PhD thesis, University of Cambridge. Connolly, J.A.D., Holness, M.B., Rubie, D.C., Rushmer, T., 1997. Reaction-induced microcracking: an experimental investigation of a mechanism for enhancing anatectic melt extraction. Geology 25, 591–594. Cooper, R.F., Kohlstedt, D.L., 1984a. Solution-precipitation enhanced diffusional creep of partially molten olivine–basalt aggregates during hot-pressing. Tectonophysics 107, 207–233.

151

Cooper, R.F., Kohlstedt, D.L., 1984b. Sintering of olivine and olivine–basalt aggregates. Physics and Chemistry of Minerals 11, 5–16. Cooper, R.F., Fanselow, J.B., Poker, D.B., 1996. The mechanism of oxidation of a basaltic glass: chemical diffusion of network-modifying cations. Geochimica et Cosmochimica Acta 60, 3253–3265. Daines, M.J., Kohlstedt, D.L., 1993. A laboratory study of melt migration. Philosophical Transactions of the Royal Society of London, Series A 342, 43–52. Daines, M.J., Kohlstedt, D.L., 1997. Influence of deformation on melt topology in peridotites. Journal of Geophysical Research 102, 10257–10271. Donaldson, C.H., 1976. An experimental investigation of olivine morphology. Contributions to Mineralogy and Petrology 57, 187–213. Edgar, A.D., Lloyd, F.E., Forsyth, D.M., Barnett, R.L., 1989. Origin of glass in upper mantle xenoliths from the quaternary volcanics of Gees, West Eifel, Germany. Contributions to Mineralogy and Petrology 103, 277–286. Emeleus, C.H., 1997. Geology of Rum and the adjacent islands. Memoir of the British Geological Survey, Sheet 60 ŽScotland.. Faul, U.H., 1997. Permeability of partially molten upper mantle rocks from experiments and percolation theory. Journal of Geophysical Research 102, 10299–10311. Frey, F.A., Green, D.H., 1974. The mineralogy, geochemistry and origin of lherzolite inclusions in Victorian basanites. Geochimica et Cosmochimica Acta 38, 1023–1059. Fujii, T., Skarfe, C.M., 1982. Petrology of ultramatic nodules from the West Kettle River, near Kelowna, Southern British Columbia. Contributions to Mineralogy and Petrology 81, 297–306. Fujii, N., Osamura, K., Takahashi, E., 1986. Effect of water saturation on the distribution of partial melt in the olivine–pyroxene–plagioclase system. Journal of Geophysical Research 91, 9253–9259. Graham, C.M., Skelton, A.D.L., Bickle, M.J., Cole, C. 1997. Lithological, structural and deformation controls on fluid flow during regional metamorphism. In: Holness, M.B. ŽEd.., Deformation-enhanced Fluid Transport in the Earth’s Crust and Mantle. Mineralogical Society Series, 8, 196–226. Harris, N.B.W., Gravestock, P., Inger, S., 1992. Ion-microprobe determinations of trace-element concentrations in garnets from anatectic assemblages. Chemical Geology 100, 41–49. Holness, M.B., 1996. Surface-chemical controls on pore-fluid connectivity. In: Jamtveit, B., Yardley, B.W.D. ŽEds.., Fluid Flow and Transport in Rocks: Mechanisms and Effects. Chapman & Hall. Holness, M.B., 1997. The permeability of non-deforming rock. In: Holness, M.B. ŽEd.., Deformation-enhanced Fluid Transport in the Earth’s Crust and Mantle. Mineralogical Society Series, 8, 9–39. Holness, M.B., Graham, C.M., 1991. Equilibrium dihedral angles in the system H 2 O–CO 2 –NaCl–calcite, and implications for fluid flow during metamorphism. Contributions to Mineralogy and Petrology 108, 368–383. Holness, M.B., Graham, C.M., 1995. P – T – X effects on equilib-

152

M.B. Holness, S.T.C. Siklosr Chemical Geology 162 (2000) 137–153

rium carbonate–H 2 O–CO 2 –NaCl dihedral angles: constraints on carbonate permeability and the role of deformation during fluid infiltration. Contributions to Mineralogy and Petrology 119, 301–313. Holness, M.B., Lewis, S., 1997. The structure of the halite–brine interface inferred from pressure and temperature variations of equilibrium dihedral angles in the halite–H 2 O–CO 2 system. Geochimica et Cosmochimica Acta 61, 795–804. Hunter, R.H., 1987. Textural equilibrium in layered igneous rocks. In: Parsons. I. ŽEd.., Origins of Igneous Layering. D. Riedel Publishing, pp. 473–503. Hunter, R.H., McKenzie, D., 1989. The equilibrium geometry of carbonate melts in rocks of mantle composition. Earth and Planetary Science Letters 92, 347–356. Jaoul, O., Bertran-Alvarez, Y., Liebermann, R.C., Price, G.D., 1995. Fe–Mg interdiffusion in olivine up to 9 GPa at T s 600–9008C; experimental data and comparison with defect calculations. Physics of the Earth and Planetary Interiors 89, 199–218. Jin, Z.-M., Green, H.W. II, Zhou, Y., 1994. Melt topology in partially molten mantle peridotite during ductile deformation. Nature 372, 164–167. Jurewicz, S.R., Watson, E.B., 1984. Distribution of partial melt in a felsic system: the importance of surface energy. Contributions to Mineralogy and Petrology 85, 25–29. Kelemen, P.B., Hirth, G., Shimizu, N., Spiegelman, M., Dick, H.J.B., 1997. A review of melt migration processes in the adiabatically upwelling mantle. Philosophical Transactions of the Royal Society of London, Series A 355, 283–318. Kern, R., 1987. The equilibrium form of crystals. In: Sunagawa, I. ŽEd.., Morphology of crystals. Terra Scientific Publishing and D. Reidel, pp. 77–206. Kitchen, D., 1984. Pyrometamorphism and the contamination of basaltic magma at Tieveragh, Co. Antrim. Journal of the Geological Society of London 141, 733–745. Laporte, D., 1994. Wetting behaviour of partial melts during crustal anatexis: the distribution of hydrous silicic melts in polycrystalline aggregates of quartz. Contributions to Mineralogy and Petrology 116, 486–499. Laporte, D., Watson, E.B., 1995. Experimental and theoretical constraints on melt distribution in crustal sources: the effect of crystalline anisotropy on melt interconnectivity. Chemical Geology 124, 161–184. Lasaga, A.C., 1986. Metamorphic reaction rate laws and development of isograds. Mineralogical Magazine 50, 359–373. LaTourette, T., Wasserburg, G.J., 1997. Self-diffusion of europium, neodymium, thorium and uranium in haplobasaltic melt: the effect of oxygen fugacity and the relationship to melt structure. Geochimica et Cosmochimica Acta 61, 755–764. LaTourette, T., Wasserburg, G.J., Fahey, A.J., 1996. Self-diffusion of Mg, Ca, Ba, Nd, Yb, Ti, Zr, and U in haplobasaltic melt. Geochimica et Cosmochimica Acta 60, 1329–1340. Lewis, S., Holness, M., Graham, C., 1998. An ion microprobe study of marble from Naxos, Greece: grain-scale fluid pathways and isotopic equilibration during metamorphism. Geology 26, 935–938. Liang, Y., Richter, F.M., Davis, A.M., Watson, E.B., 1996.

Diffusion in silicate melts: I. Self-diffusion in CaO–Al 2 O 3 – SiO 2 at 15008C and 1 GPa. Geochimica et Cosmochimica Acta 60, 4353–4367. Linnen, R.L., Pichavant, M., Holtz, F., 1996. The combined effects of fO 2 and melt composition on SnO 2 solubility and tin diffusivity in haplogranitic melts. Geochimica et Cosmochimica Acta 60, 4965–4976. Maaløe, S., Printzlau, I., 1979. Natural partial melting of spinel lherzolites. Journal of Petrology 20, 727–741. Martin, J.W., Doherty, R.D., 1976. Stability of Microstructures in Metallic Systems. Cambridge Univ. Press. McKenzie, D.P., 1984. The generation and compaction of partially molten rock. Journal of Petrology 25, 713–765. Mehnert, K.R., Busch, W., Schneider, G., 1973. Initial melting at ¨ grain boundaries of quartz and feldspar in gneisses and granulites. Neues Jahrbuch fur ¨ Mineralogie Monatshefte 4, 165– 183. Mibe, K., Fujii, T., Yasuda, A., 1998. Connectivity of aqueous fluid in the Earths’ upper mantle. Geophysical Research Letters 25, 1233–1236. Minarik, W.G., Watson, E.B., 1995. Interconnectivity of carbonate melt at low melt fraction. Earth and Planetary Science Letters 133, 423–437. Minarik, W.G., Ryerson, F.J., Watson, E.B., 1996. Textural entrapment of core-forming melts. Science 272, 530–533. Mullins, W.W., 1957. Theory of thermal grooving. Journal of Applied Physics 28, 333–339. Mullins, W.W., 1960. Grain boundary grooving by volume diffusion. Transactions of the Metallurgical Society of the AIME 218, 354–361. O’Connor, T.K., Edgar, A.D., Lloyd, F.E., 1996. Origin of glass in quaternary mantle xenoliths from Meerfelder Maar, West Eifel, Germany: implications for enrichment in the lithospheric mantle. The Canadian Mineralogist 34, 187–200. Olgaard, D.L., Fitz Gerald, J.D., 1993. Evolution of pore structures during healing of grain boundaries in synthetic calcite rocks. Contributions to Mineralogy and Petrology 115, 138– 154. Platten, I.M., 1982. Partial melting of feldspathic quartzite around late Caledonian minor intrusions in Appin, Scotland. Geological Magazine 119, 413–419. Richardson, C., McKenzie, D., 1994. Radioactive disequilibria from 2D models of melt generation by plumes and ridges. Earth and Planetary Science Letters 128, 425–437. Richet, P., Bottinga, J., 1984a. Glass transitions and thermodynamic properties of amorphous SiO 2 , NaAlSi nO 2nq2 and KAlSi 3 O 8 . Geochimica et Cosmochimica Acta 48, 453–470. Richet, P., Bottinga, J., 1984b. Anorthite, andesine, wollastonite, diopside, cordierite and pyrope: thermodynamics of melting, glass transitions, and properties of the amorphous phase. Earth and Planetary Science Letters 67, 415–432. Ridley, J., 1986. Modelling of the relations between reaction enthalpy and the buffering of reaction progress in metamorphism. Mineralogical Magazine 50, 375–384. Riley, G.N., Kohlstedt, D.L., 1991. Kinetics of melt migration in upper mantle-type rocks. Earth and Planetary Sciences Letters 105, 500–521.

M.B. Holness, S.T.C. Siklosr Chemical Geology 162 (2000) 137–153 Roeder, P.L., Emslie, R.F., 1970. Olivine–liquid equilibrium. Contributions to Mineralogy and Petrology 29, 275–289. Sawyer, E.W., 1991. Disequilibrium melting and rate of melt–residuum separation during migmatization of mafic rocks from the Grenville front, Quebec. Journal of Petrology 32, 701–738. Shannon, M.C., Agee, C.B., 1996. High pressure constraints on percolative core formation. Geophysical Research Letters 23, 2717–2720. Smith, C.S., 1948. Grains, phases and interfaces: an interpretation of microstructure. Transactions of the Metallurgical Society of the AIME 175, 15–51. Smith, C.S., 1964. Some elementary principles of polycrystalline microstructure. Metallurgical Reviews 9, 1–48. Stosch, H.-G., 1987. Constitution and evolution of subcontinental upper mantle and lower crust in areas of young volcanism: differences and similarities between the Eifel ŽF.R. Germany. and Tariat Depression Žcentral Mongolia. as evidenced by peridotite and granulite xenoliths. Fortschritte der Mineralogie 65, 49–86. Stosch, H.-G., Seck, H.A., 1980. Geochemistry and mineralogy of two spinel peridotite suites from Dreiser Weiher, West Germany. Geochimica et Cosmochimica Acta 44, 457–470. Toramaru, A., Fujii, N., 1986. Connectivity of the melt phase in a partially molten peridotite. Journal of Geophysical Research 91, 9239–9252. von Bargen, N., Waff, H.S., 1986. Permeabilities, interfacial areas and curvatures of partially molten systems: results of numerical computations of equilibrium microstructures. Journal of Geophysical Research 91, 9261–9276. von Bargen, N., Waff, H.S., 1988. Wetting of enstatite by basaltic melt at 13508C and 1.0–2.5 GPa pressure. Journal of Geophysical Research 93, 1153–1158. Waff, H.S., Bulau, J.R., 1979. Equilibrium fluid distribution in an ultramafic partial melt under hydrostatic stress conditions. Journal of Geophysical Research 84, 6109–6114.

153

Waff, H.S., Bulau, J.R., 1982. Experimental determination of near-equilibrium textures in partially molten silicates at high pressures. In: Akimoto, S., Manghani, M.H. ŽEds.., Advances in Earth and Planetary Sciences. High Pressure Research in Geophysics, 12, 229–236. Walther, J.V., Wood, B.J., 1984. Rate and mechanism in prograde metamorphism. Contributions to Mineralogy and Petrology 88, 246–259. Watson, E.B., 1982. Melt infiltration and magma evolution. Geology 10, 236–240. Watson, E.B., Brenan, J.M., 1987. Fluids in the lithosphere: I. Experimentally-determined wetting characteristics of CO 2 – H 2 O fluids and their implications for fluid transport, host-rock physical properties and fluid inclusion formation. Earth and Planetary Science Letters 85, 497–515. Watson, E.B., Lupulescu, A., 1993. Aqueous fluid connectivity and chemical transport in clinopyroxene-rich rocks. Earth and Planetary Science Letters 117, 279–294. Watt, G.R., Harley, S.L., 1993. Accessory phase controls on the geochemistry of crustal melts and restites produced during water-undersaturated partial melting. Contributions to Mineralogy and Petrology 114, 550–566. Wolf, M.B., Wyllie, P.J., 1991. Dehydration-melting of solid amphibolite at 10 kbar. Textural development, liquid interconnectivity and applications to the segregation of magmas. Mineralogy and Petrology 44, 151–179. Wolf, M.B., Wyllie, P.J., 1995. Liquid segregation parameters from amphibolite dehydration melting experiments. Journal of Geophysical Research 100, 15611–15621. Zinngrebe, E., Foley, S.F., 1995. Metasomatism in mantle xenoliths from Gees, West Eifel, Germany: evidence for the genesis of calc-alkaline glasses and metasomatic Ca-enrichment. Contributions to Mineralogy and Petrology 122, 79–96.