Journal of Petrology Advance Access originally published online on November 6, 2007
Journal of Petrology 2007 48(12):2379-2406; doi:10.1093/petrology/egm065
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Stress-driven Melt Segregation and Strain Partitioning in Partially Molten Rocks: Effects of Stress and Strain
Department of Geology & Geophysics, University of Minnesota, 310 Pillsbury Drive SE, Minneapolis, MN 55455, USA
RECEIVED NOVEMBER 25, 2005; ACCEPTED OCTOBER 2, 2007
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ABSTRACT |
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 TOP ABSTRACT INTRODUCTIONEXPERIMENTAL DESIGNRESULTSDISCUSSION: OPEN QUESTIONSCONCLUSIONSAPPENDIX A: MEASURING THE...APPENDIX B: CALCULATING STRESS...REFERENCES |
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The evolution of melt segregation in deforming partially molten olivine-rich rocks has been studied in a series of laboratory experiments. During deformation, melt segregates into networks of anastamosing channels (or ‘bands’) surrounding lenses of melt-depleted material. We quantify the nature of the melt distribution in the samples, including thickness, angle, spacing, volume fraction, and melt fraction of melt-rich bands, to understand the dynamics of melt-network organization. Two series of experiments were designed to isolate the effects of (1) increasing shear strain (at similar stress levels), and (2) varying stress levels (deformed to similar shear strains). Melt-rich bands develop by a shear strain of unity. In samples deformed at varying stress levels, higher stress produces smaller characteristic band spacings. We relate these variations to the compaction length,
c, which varies only as a result of the reduction of matrix viscosity with increasing stress. Simple approaches to scaling from experimental to mantle conditions suggest that stress-driven melt segregation can occur in the asthenosphere; if so, it will significantly affect rheological, transport and seismic properties, with enticing consequences for our understanding of plate–mantle interactions. KEY WORDS: melt segregation; rock rheology; magma transport; self-organization; mid-ocean ridges
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INTRODUCTION |
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TOPABSTRACTINTRODUCTIONEXPERIMENTAL DESIGNRESULTSDISCUSSION: OPEN QUESTIONSCONCLUSIONSAPPENDIX A: MEASURING THE...APPENDIX B: CALCULATING STRESS...REFERENCES |
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In the Earth, large-scale deformation is often intense in regions where melt is being produced and transported. This generalization may apply to plate boundary settings (subduction zones and mid-ocean ridges), intra-plate rifts, plume heads and boundary layers such as the D'' region and shallow asthenosphere. Deformation and melt migration are, in their own rights, complex ensembles of many processes that interact with each other in feedback relationships (e.g. Kohlstedt & Zimmerman, 1996
; Kelemen et al., 1997). Even in simple systems, stress causes complex behavior in the self-assembly of melts during deformation (e.g. Rosenberg & Handy, 2000; Holtzman et al., 2003a). Considerable evidence exists for such interactions in lower crustal, and to a lesser extent, upper mantle rocks (e.g. Nicolas, 1986; Kelemen & Dick, 1995; Neves et al., 1996; Vigneresse, 1999). However, the physical causes and consequences of these interactions are poorly understood and difficult to study in the field, by theory, or by experiment. We need to understand these interactions in practical and testable ways to explore their implications for the seismic, transport, and rheological properties of important volumes of the Earth.
In this study, we build on our previous studies (Holtzman et al., 2003a, 2003b; Hustoft & Kohlstedt, 2006; Parsons et al., 2007) in which we demonstrated that as a partially molten rock deforms at high temperatures and pressures, melt segregates spontaneously and organizes into melt-rich networks. Here, we focus on the melt distribution and the processes that control its variation in time and space. A series of experimental observations are presented to understand the dynamics of stress-driven melt segregation and its scaling relation to conditions in the Earth. Holtzman et al. (2003a) showed that stress drives melt segregation in experimental samples with compaction lengths, c (Table 1), of the order of or less than the sample thickness but longer than the grain scale. This result implies that stress-driven segregation will occur in nature in any partially molten system if the compaction length is smaller than the spacing between any ‘boundaries’. For example, in the upwelling asthenosphere beneath a ridge, c is much shorter (
100–1000 m) than the width of the partially molten region (10–100 km) (Holtzman et al., 2003a). In this study, we explore in detail the dynamics of melt segregation and deformation processes quantitatively by characterizing the properties of the melt distribution and the influence of stress and strain on these properties. Holtzman et al. (2005) described the effect of strain partitioning on the energetics of deformation, providing a theoretical framework with which to integrate observations from this study. Predicting the kinetics of stress-driven melt segregation will lead to a better understanding of the transport, rheological, and seismic properties of partially molten regions of the Earth.
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In the Earth, melt moves from the asthenosphere to the crust or surface and, as it does, passes through many mechanical regimes, from predominantly viscous porous flow at the bottom of the melting column to flow through cracks and dikes approaching the surface (e.g. Kelemen et al., 1997
). Several lines of evidence suggest that mantle-derived melts, or a large portion of them, migrate upwards through chemically isolated channels in the mantle. For example, studies of active mid-ocean ridges and ophiolite complexes reveal that the uppermost mantle rocks are not in chemical equilibrium with the basaltic melts that form the overlying crust (e.g. Kelemen et al., 1995, 1997). To maintain this disequilibrium, melt must move largely through chemically isolated channels or networks (e.g. Spiegelman & Kenyon, 1992). Numerous mechanisms have been proposed for such transport. Depending on the driving force for melt movement and the rheological behavior and transport properties of the peridotite host rock, melt transport in channels may be coupled to viscous flow of the solid, via processes that involve compaction (e.g. McKenzie, 1984; Scott & Stevenson, 1984; Fowler, 1990) and/or dissolution–compaction (e.g. Aharonov et al., 1997). Melt transport may also occur in fractures in a solid (e.g. Nicolas, 1986; Rubin, 1998; Phipps Morgan & Holtzman, 2005), which are likely to occur in regions of the mantle with increased viscoelastic relaxation times, approaching the top of the melting region in the mantle. However, the initial stages of melt segregation, prior to transport in channels, are probably the least well understood aspect of the whole pathway. The three general classes of processes for the initial segregation of melt into channels include: (1) runaway dissolution as defined by Chadam et al. (1986) and applied to mantle materials (e.g. Daines & Kohlstedt, 1994; Kelemen et al., 1995; Aharonov et al., 1997); (2) nucleation of localized buoyancy-driven porosity waves beneath permeability barriers (e.g. Scott & Stevenson, 1984; Connolly & Podladchikov, 1998); (3) stress-driven melt segregation (e.g. Stevenson, 1989). Here, we focus on developing an understanding of (3).
Stevenson (1989) proposed a model in which deviatoric stress could drive melt segregation. In his one-dimensional (1D) pure-shear model, small variations in melt fraction cause small variations in viscosity as a result of the strong dependence of viscosity on melt fraction. These spatial variations lead to local variations in mean stress that drive melt into the weak regions, causing them to further weaken. An instability develops that continues until the melt can no longer segregate, and melt-rich bands are separated by melt-depleted regions with some characteristic spacing. Stevenson suggested that this spacing should be less than the compaction length but larger than the grain size. In an extension of a similar analysis to two dimensions, Richardson (1998) found that segregation could occur at high pure-shear strains, and had a similar limit on the preferred length scale. Hall & Parmentier (2000) sought this preferred wavelength by invoking strengthening upon dehydration of olivine during melting as a means to set a lower limit on length scale of melt segregation. Recently, a new generation of two-phase flow theory has emerged involving the physics of damage (the loss of energy in the creation of new surfaces in a deforming material), in which deformation accelerates the segregation of a fluid phase (e.g. Bercovici & Ricard, 2003). Speigelman (2003) and Rabinowicz & Vigneresse (2004), who also solved two-phase flow equations in 2D simple shear, found that the the fastest growth is at 45°, but did not find stability of the bands at any angle. In a fully non-linear finite-difference model of the compaction equations, Katz et al. (2006) observed that the band angle is controlled by the stress exponent in the constitutive relation describing matrix deformation, which is stress and melt-fraction dependent. As n increases from one to six, the band angles drop from 45° to 20°. The implication of these results is that the degree of strain partitioning increases with increasing n, so band angles are reduced with increasing strain partitioning. This conclusion is conceptually similar to that reached in an analysis of the energetics of strain partitioning by Holtzman et al. (2005) that used only Newtonian viscosity. At present, two-phase flow theory can explain many aspects of the segregation processes observed in experiments, but many questions are open.
Numerous experimental deformation studies have been performed on partially molten rocks or analog materials in the ductile, or near-ductile regime, in compression, shear, and torsional geometries, in drained (open) and closed conditions (e.g. Dell’Angelo & Tullis, 1988; Rutter & Neumann, 1995; Zimmerman et al., 1999; Rosenberg & Handy, 2000; Hirth & Kohlstedt, 2003; Mecklenburgh & Rutter, 2003; Renner et al., 2003; Schmocker et al., 2003; Zimmerman & Kohlstedt, 2004; Takei, 2005; Gourlay & Dahle, 2006; Hier-Majumder & Kohlstedt, 2006; Hustoft & Kohlstedt, 2006). Several of these studies have produced melt or fluid segregation similar to that observed in this study, compared in detail in the Discussion. These experiments provide theoretical models of stress-driven segregation with physical benchmarks; a nominally successful theory should be able to simulate the basic experimental observations.
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EXPERIMENTAL DESIGN |
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TOPABSTRACTINTRODUCTIONEXPERIMENTAL DESIGNRESULTSDISCUSSION: OPEN QUESTIONSCONCLUSIONSAPPENDIX A: MEASURING THE...APPENDIX B: CALCULATING STRESS...REFERENCES |
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The experiments were performed in a gas-medium deformation apparatus (Paterson, 1990
) using a method similar to that of previous studies on olivine + mid-ocean ridge basalt (MORB) samples (e.g. Hirth & Kohlstedt, 1995a, 1995b; Kohlstedt & Zimmerman, 1996; Mei et al., 2002). The materials used here are similar except for the addition of chromite to the samples, which reduces the permeability by partially blocking the pore throats between connected melt pockets, without significantly changing other physical properties of the rock (Holtzman et al., 2003a). We performed several series of shear experiments, all at 1523 K and 300 MPa, in which melt fraction, finite strain, loading conditions, and piston type were systematically varied (Fig. 1). The melt distribution in each shear experiment was determined by image analysis.
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Experimental procedure details
The starting material for all experiments reported here is composed of San Carlos olivine, chromite from the Semail Massif in the Oman ophiolite, and MORB from the Mid-Atlantic Ridge, with an olivine tholeite composition (Cooper & Kohlstedt, 1984
); compositions are reported in Table 2. Olivine and chromite powders were mixed in a volume ratio of 3:1 to which MORB powders were added. The powder grain sizes were about 6 µm for olivine and 2 µm for chromite. In all cases, powders were dried at 1423 K for 8 h at an fO2 near the Ni–NiO buffer to remove any water and then isostatically hot-pressed for 4 h at 300 MPa and 1523 K.
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All experiments were performed at a confining pressure of 300 MPa and a temperature of 1523 K. The samples were placed between thoriated tungsten pistons cut at 45°, with grooves machined into them to improve gripping between sample and piston (Fig. 1), referred to as ‘direct’ shear experiments. This deformation geometry approximates a simple shear, but has some degree of flattening as well. One experiment (PI-902) was performed with smooth pistons made from olivine single crystals to test if the grooves had a significant effect on the melt distribution and band formation. Prior to deformation, some samples were cut in half normal to the shear plane and a thin piece of Ni-foil was placed between the two halves as a marker of the strain distribution in the sample.
Experiments were performed with either a constant load or a constant displacement rate applied. These boundary conditions are not exactly equivalent to constant stress or constant strain rate conditions, respectively. When a constant load is applied, the stress increases smoothly with increasing displacement because the area of contact between the pistons decreases. Constant displacement rate is much closer to a constant strain rate condition, but because the thickness of the sample decreases slightly during deformation, strain rate increases slightly. (For simplicity, we refer to them as constant strain rate experiments.) Under constant strain rate (
) conditions, the load varies with time in response to changes in the effective viscosity of the sample.
After deformation, samples were cut normal to the shear plane in the shear direction for image analysis. The samples were polished in colloidal silica and then etched in dilute HF–HCl solution for 10 s to aid in viewing the melt pockets. Samples were then photographed using an optical microscope with reflected light and a scanning electron microscope (SEM) with backscattered (BSE) or secondary electrons (SE). High- and low-resolution photomicrographs were used to obtain measurements of the melt distribution.
Organization of shear experiments
We investigated the effects of melt fraction, strain, and stress level on the patterns of segregation of melt and the formation of melt-rich bands. To isolate (as much as possible) the effects of these parameters, experiments are grouped into two series, summarized in Table 3. The first series focuses on the evolution of the melt distribution with progressive strain (referred to simply as the ‘strain series’). The second series focused on the effect of variable stress level applied to the samples (referred to as the ‘stress series’). Compaction length was varied in the strain series by varying melt fraction and in the stress series by varying the stress-sensitive solid viscosity, addressed further in the Discussion.
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Strain series
The purpose of this set of experiments is to understand how the melt distribution evolves with strain or time. We systematically varied finite strain (
1, 2, 3) keeping values of applied load similar for most experiments (except that the samples deformed to high-strain were subjected to several load levels during the experiment to obtain rheological data). These six experiments were divided into two sets, defined by their total melt fractions, 
t = 0·02 and t = 0·06, performed to observe the evolution in two systems with different values of c, which depends strongly on melt fraction.
Stress series
The purpose of this series of experiments is to understand the effect of the level of stress applied to a sample on the distribution and self-organization of the melt. We deformed five samples with = 0·04 to the same finite strain ( 3·5). These samples had no Ni strain marker because, in the strain series, the impermeable marker significantly altered the melt distribution and thus the dynamics of melt migration. The samples in this set were also generally thicker than those in the strain series. Three samples were deformed under constant load conditions (CL subset). Two samples were deformed under conditions of constant displacement rate (CR subset). Experiments were performed under these conditions to explore the influence of the variation of load with time (the ‘loading path’) on melt distribution and the evolution of rheological properties. An additional experiment was performed with a very low melt fraction, t = 0·005 (PI-1070).
Methods: quantifying melt distribution
In this study we define and measure two different groups of parameters that describe properties of the melt ‘configuration’. These properties include the distributions of band angles (
), spacings (sp), thicknesses (th), area fraction of melt bands (ab), normalized melt content of the bands (b) and the degree of segregation, (St). , th and sp are the ‘spatial’ properties of the melt network. The other data, ab, b and St constitute the ‘average’ properties of the network, so called because they are effectively averaged across the sample and contain no length-scale information. We make these distinctions to provide links to the theories used to describe these kinds of systems. The spatial properties are used for comparison with continuum two-phase flow theory. The averaged properties will be used to calculate macroscopic effective properties such as elasticity and viscosity. Thus, we want to know the relation between these two kinds of data.
Several methods of image analysis were used to measure the parameters discussed above. Length-scale data (spacing, thickness and angle) are measured directly from a ‘linear intercept method’ (LIM) applied to photomicrographs; others are measured with calculations performed on 1D curves of melt distribution, and some are measured with both methods, as discussed below and in Appendix A, and shown in Fig. 2. For both series of experiments, we compiled numerous reflected-light images from each sample. We analyzed images from locations with minimal influences on the distribution of melt caused by end-effects. (As will become apparent, avoiding these influences is easier in the samples without a strain marker.) To visualize only the melt, we applied a threshold to the images such that all but melt (darkest grey values) was eliminated, resulting in a binary image of the melt distribution. To the binary melt images, we applied a smoothing algorithm in which a circular filter matrix of radius r is convolved with the binary image, and then divided by the number of points in the smoothing tensor, as illustrated in Fig. 2b and c. The averaging radius is chosen to be about twice the average grain diameter, large enough to smooth the distribution and small enough to provide representative detail. Three lines were then extracted from each of these images (Fig. 9c and d) and operated upon, as discussed below.
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Spatial quantities
(band angle): the angle of each band relative to the shear plane is measured using the LIM (Appendix A).
th (band thickness): the band thicknesses are measured with the LIM, at the same location as the angle, as far from a junction with another band as possible to avoid repetitive measurements.
sp (band spacing): using the LIM, the largest bands are generally clear and we record the maximum spacing between them, normal to their orientation, from center to center, as far as possible from junctions of large bands. Using the autocorrelation method (Appendix A), this band spacing is defined as the lag associated with first maxima.
Average quantities
We define bands as any region with a greater-than-average melt fraction and lenses (or ‘non-bands’) as any region with less-than-average melt fraction. Then, we calculate the following.
- ab (band area). The total area (in a 2D image) of melt-rich bands is defined as the area of sample that has a melt fraction higher than the average, divided by the total area of the sample. Actually, these measurements are made on lines, so they are 1D areas. The fractional band area,
, is calculated as
in which at is the total area, Nt is the total number of pixels in the 1D curve and N
(1) >t is the number of pixels for which > t, calculated from the smoothed melt distribution, for which i is the 1D index. The prime indicates that the value is normalized. t, where the t indicates ‘total’, can be defined in two ways: first, locally
or, second, prescribed as being constant for all lines from a sample. The
(2) t value is fixed as that volume fraction of MORB put into the sample. (For each line of data, 
is calculated as well.) ab is calculated locally for each line of data; the area fraction of the lenses, an, is calculated in an analogous manner.
- b (‘normalized band melt fraction’). The average melt fraction in the bands, b, is the sum of the melt fraction values at every point in the band, divided by the number of points in the band:
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t used, as discussed above. In all data presented here, we use the global t fixed for each sample. 'n is calculated in the same manner.
Mass balance can be written in terms of these parameters; ab, an, b, and n must be related within one line as follows:
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t, we obtain
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The segregation factor, St
As a deforming system evolves with increasing strain, the melt will progressively segregate. However, we will never be able to measure strain directly in partially molten mantle. The best we can hope for is to measure some physical property that is related to the degree of melt segregation. Here we introduce a new parameter, St. St is a scalar measure of the degree of melt segregation, defined as
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and 'b to estimate seismic properties (Holtzman & Kendall, in preparation) and effective viscosity (Holtzman et al., 2005), but cannot measure them independently in the Earth, we would like to constrain the relation between and 'b. (In the sense that St reflects the degree of order in the system, it can also be seen as an inverse measure of configurational entropy, S; the higher the value of St, the lower the value of S).
In a closed system, St must be between zero and unity, when measured over the volume of the sample. However, if melt flows into or out of the system, then any given cross-section can contain more or less than the average melt fraction such that St > 1 or St < 1, respectively, even if all the melt appears to be segregated. This accounting problem can occur when St is measured in only one plane (or part) of a sample, as is done in this study. To account for flow into or out of the plane of measurement (or in nature, an open system), we can normalize by the local total melt fraction (the melt fraction in the line or area of measurement),
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for a 2D section. Therefore,
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| (9) |
and 
, as demonstrated below. However, it is not obvious how to split the weighting between and 'b. For now, until the physics is better understood, we use the following expression:
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and 'b in terms of St. From experimental observations, we know that and 'b increase with increasing strain. The following equations, although empirical, adhere to mass balance (Holtzman et al., 2005):
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| (11) |
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| (12) |
'max are determined by fitting these functions to the experimental data. They describe possible relationships between the measured parameters and St. Hereon, we drop the primes on 'b and . |
RESULTS |
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TOPABSTRACTINTRODUCTIONEXPERIMENTAL DESIGNRESULTSDISCUSSION: OPEN QUESTIONSCONCLUSIONSAPPENDIX A: MEASURING THE...APPENDIX B: CALCULATING STRESS...REFERENCES |
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Starting materials
As shown in Fig. 3, the starting material is a homogeneous and isotropic distribution of olivine, chromite and basaltic melt–glass. The olivine grains grew during the 4 h hot-presses from an initial mean grain size of 6 µm to a final mean grain size of 8 µm. (The grains grew to about 10 µm during the shear experiments.) The chromite has a mean grain size of 3 µm (slightly smaller in the stress series). Small chromite grains (< 1 µm) are incorporated within olivine crystals and larger ones (> 1 µm) are located in three- and four-grain junctions, surrounded by melt. The melt distribution is heterogeneous at the local length scale of several times the grain diameter, but roughly homogeneous at longer length scales (i.e. bands do not exist in the hot-pressed starting material). The volume fractions of melt in the starting materials are close to the volume fractions of MORB added,
t = 0·02 ± 0·01 and t = 0·06 ± 0·01 for the strain series and t = 0·04 ± 0·01 for the stress series.
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Strain series
Here, we present the detailed observations of the first set of shear experiments, focused on the evolution of melt segregation with increasing shear strain. The data come in several forms, visual and quantitative, including (1) images of the melt distribution across the entire sample and (2) distributions of band angles, thicknesses and spacings.
In Fig. 4, the evolution of melt segregation in two series is shown, grouped by average melt fraction (t = 0·02 and t = 0·06) and increasing finite strain. In the t = 0·02 series, bands are well developed by 1; in the t = 0·06 series, bands are barely developed by 1 but are well developed by 2. Melt segregates first into narrow, closely spaced bands (PI-884 and PI-885). The thickness and spacing of these bands increases with increasing strain, as demonstrated by more quantitative measures presented below. It is not possible to tell from these experiments if the melt configuration has reached a dynamic steady state by 3·5 because the sample thickness (but not volume) is decreasing and melt is constantly rearranging itself in response to these changes.
Two boundary effects influence the organization of the bands: the pressure variations at the ends of the sample and the permeability variations associated with the strain marker at the center of the sample. The main end effect is due to low-pressure focal points at the ends of each sample where the Ni jacket meets the sample and the piston. The number of bands and quantity of melt increases toward these focal points (Fig. 4). Further towards the ends of the sample from these points, the sample is supported only by the Ni jacket, and thus the mean stress is significantly lower. Melt does not appear to flow past the point into the ends of the sample supported by the Ni jacket.
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The second major influence on the network geometry in these experiments is the essentially impermeable Ni strain marker. In general, the melt fraction decreases around the strain marker, presumably because the melt is drawn away by the network, towards the low-pressure points. At the rare locations where melt bands cross the strain marker, an offset on the strain marker (but not necessarily a discontinuity) is well developed. At locations where there is offset but no band, the offset probably represents the previous location of a band that has since moved elsewhere.
High-resolution (SEM) images of melt-rich bands
In this section, detailed views of one experimental sample (PI-902) are shown to illustrate the range of microstructures in the bands. This sample, with t = 0·06, was deformed to a shear strain of = 3·5 using groove-free pistons made from olivine single crystals. Groove-free pistons were used to eliminate the possibility that the segregation is caused by pressure gradients associated with the grooves (and that the band spacing and network morphology are determined by the groove spacing). The single-crystal pistons were oriented with [010] normal or parallel to the compression direction, minimizing the resolved shear stress on the easy slip system. The pistons did not appear to deform. During deformation, melt segregated strongly into bands, as shown in Fig. 5a–e.
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The image of the whole sample in Fig. 5a reveals well-developed bands on both sides of the strain marker. At higher magnification, Fig. 5b illustrates that some of the bands span the thickness of the sample, whereas others terminate in the middle. The bands have an average angle of 16·4 ± 6·9° with respect to the shear plane. The average amount of melt in the bands in this sample is
b = 0·10 (not normalized); in the melt-depleted lenses, n = 0·02. In Fig. 5c, a band at lower angle appears to connect two bands at higher angles. The observation that narrow bands occur at lower angles than wider bands appears in all samples. A higher magnification view of one band in Fig. 5d reveals several interesting features. First, in some places, the edges of the band are very well defined, marked by a dramatic change in melt fraction; in other regions, the transition from band- to non-band regions are more gradational. Furthermore, inside the band, variations in melt fraction are large, with islands of low melt fraction surrounded by melt-rich channels. In other words, a larger band may be viewed as a cluster of smaller bands. In Fig. 5d and e, it is clear that the melt fraction can be locally very high in a band. As annotated in the images, the olivine crystals in the bands appear to be both dissolving and growing by precipitation, as suggested by the presence of very small grains (< 1 µm), incised or corrugated grain boundaries of the large grains and the highly euhedral overgrowths on olivine grains, respectively.
Melt configuration statistics
The frequency distributions of band angles, thicknesses, and spacings are presented here (Table 4). The band angles in each sample have normal distributions, as shown in Fig. 6. As a function of strain, the band angles appear remarkably constant, at 20 ± 6° in the t = 0·02 set and 17 ± 6° in the t = 0·06 set. The angles increase slightly with increasing strain in the t = 0·02 set but not in the t = 0·06 set, as shown in the top right box of Fig. 6. Bands were not clearly visible in most of the sample with t = 0·06 deformed to = 1·1 (PI-885), so no data exist for low strain at t = 0·06 in Figs. 6–8.
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In contrast to the distribution of angles, the statistical distributions of the band thicknesses are clearly not normal distributions; in log–log space (upper right corner of Fig. 7), the distribution appears to follow a power law,
, where nth is the frequency of occurrence of a given thickness of a band of thickness, th, where the mean value of m is 1·7 (values for each sample are listed in Fig. 7). We cannot test the robustness of the apparent power-law nature of this distribution because the thickness data only span up to one order of magnitude. In the images in Fig. 4, the thicknesses of the bands appears to increase between = 1 and = 2 in both the t = 0·02 and t = 0·06 series, but there does not appear to be a systematic difference between the data from samples deformed to 2 and 3. These visual inferences appear in the histograms in the high-end tails of the distribution, but do not appear in the mean values, as shown in Fig. 9a.
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A correlation was found between the band angle and band thickness, as shown in Fig. 8. The correlation coefficients (R2), shown in Fig. 8, decrease slightly with increasing strain. In the samples deformed to higher strain (
= 2–3·5), the trends appear to reach a cap at an angle of 30°, but the thicknesses simply increase with increasing strain whereas the bands appear unable to rotate past this angle. This correlation is the statistical expression of an anastomosing network of bands, as observed in the photomicrographs.
The spacings between the largest bands, sp, were measured with both LIM and the autocorrelation function (ACF) for all samples. The results of both methods are plotted as a function of in Fig. 9. The data show essentially that the band spacings increase only slightly with increasing strain for both series of initial melt fractions, although uncertainties are larger than observed variations. For the high-strain samples, the measurements of sp data from LIM and ACF methods are consistent. In general, the average value of sp increases with increasing strain (except for the drop in sp in the high-strain sample in the t = 0·02 set, as shown in Fig. 9). To what extent the average spacing in band networks will reach a steady state with increasing time or strain cannot be determined from this set of experiments, but broadly speaking, the melt distribution does not evolve dramatically after = 1.
Summary
- The band angles have normal distributions with a mean of 20°. The mean values of both sets appear to be independent of strain (Fig. 6).
- Band thicknesses do not have normal distributions; they appear to have a power-law distribution, but the data are limited to less than one order of magnitude of length scale. The thickness of the thickest bands, but not the mean thickness, appears to increase with increasing strain (Fig. 7).
- A positive correlation exists between band thickness and band angle; the increase in band angle with increasing band thickness is roughly linear, with a slight upturn to higher thicknesses at an upper limit of angle of 30° (Fig. 8).
- The spacing between large bands increases with increasing strain up to 2. This trend appears clearly in the reflected-light images and Fig. 9.
The most robust observation is simply that by a shear strain of 1, the melt has begun to segregate into clear networks; by a shear strain of 2, the networks are well developed. However, there are noticeable differences in the two subsets of experiments with different average melt fractions (t = 0·02 and t = 0·06). Melt segregation is more advanced at a shear strain of 1 in the set with t = 0·02 than in the set with t = 0·06. However, as a function of time, the networks develop more quickly in the t = 0·06 set. Because the applied stresses were similar in the two sets (Table 2), the quicker segregation in the samples with higher melt fraction was probably due to lower resistance to melt migration (increased permeability and decreased composite viscosity).
The primary problem with the above datasets is that the network quickly finds its physical constraints, which are created by the low-pressure end-points of the piston and the impermeable strain marker in the middle. Because these two constraints impose such tight restrictions on the organization of the melt, we found that the statistical measures of the band area and average melt fraction (Table 6), and to some extent spacings and thickness, are difficult to interpret. We hesitate to probe too deeply into the details of the band networks and and their scaling properties. With so little room in which the melt can organize, the spacings and average properties of the melt distribution at large strains are probably controlled more by the boundary conditions (the proximity of the piston–sample interfaces) than the thermodynamic state. To remove as many of these complications as possible, we eliminated the strain marker in subsequent shear experiments.
Stress series
The major improvements in the method from the strain series experiments are the removal of the strain marker and an increase in the sample thickness, both of which give the melt-rich networks more space in which to organize. The networks are much better developed because of the reduction in spatial constraints on their organization. The three experiments performed at constant, but different, load conditions are referred to as the ‘constant load’ (CL) subset; the two experiments performed under constant displacement rate conditions, which approximates constant strain rate, are referred to as the constant strain or displacement rate (CR) subset.
Images
In the images of the three samples of the CL set (Fig. 10) and the two samples of the CR set (Fig. 11), the effect of stress on the melt distribution is clear. In the CL set, with increasing stress, the bands become narrower and more closely spaced. In the CR set, with increasing strain rate (also increasing stress), the bands also become narrower and more closely spaced, although the differences are more subtle. The morphology of the band networks also depends on the stress level. The bands become straighter and the junctions more angular with increasing stress, as seen by comparing Fig. 10a and c. The same anastomosing patterns of narrow low-angle bands and wider higher-angle bands exist in these samples as in the strain series. The anastomosing patterns are more penetrative and have a finer scale at high stress than at low stress.
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The anastomosing pattern is three dimensional. In Fig. 12, two orthogonal reflected-light images of sample PI-1025 are shown in perspective, to give a sense of the 3D nature of the melt-rich networks. As indicated in the schematic drawing of 3D networks from Holtzman et al. (2003b
), the average length of lenses is greater in the plane normal to the shear direction than it is in the flow plane (the usual view). Also, some sample material has been extruded laterally (in the shear plane, normal to the shear direction) beyond the edges of the piston. It also appears that the melt fraction is higher in and near this extruded material, indicating that there may be some flux of melt from the middle of the sample towards the edges.
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The visual and statistical signature of the anastomosing networks is a bimodal distribution of band angles, as illustrated in Fig. 13. In this image, and in all of the samples, there are indications of some scale-invariant properties of the distribution of melt. The first-order structure is the largest bands, which are oriented at
15–25° to the sample shear plane (that is, the grooved piston surface) in Fig. 13c. The population of smaller bands, the second-order structures, is oriented at 5–15° to the sample shear plane. However, if one views these structures in a local reference frame of the lenses, such that the shear plane in the lenses is back-rotated by 10° to the sample shear plane (the piston surface), then the secondary bands are oriented at 15–25° with respect to this secondary shear plane, illustrated in Fig. 13c. In other words, the secondary narrow bands that cut across lenses are controlled by the local stress field in the lenses. This local rotation of the stress field in the lenses is suggested by the back-rotation of olivine b-planes (Holtzman et al., 2003b) and by the analysis of strain partitioning (Holtzman et al., 2005). Extending this point of view one order downward in scale, the melt pockets most visible in the large bands are also oriented 20° to the wall of the band, the third-order shear plane defined by the surfaces of the melt bands. Thus, from the sample scale to the grain scale, there are three levels of scale-invariance to the orientation of melt alignment relative to the local and applied stress tensors.
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Is there a lower limit to the melt fraction required for segregation to occur? The sample with
= 0·005 was deformed at moderate stresses (
f = 122 MPa). Well-defined bands formed, as shown in detail in Fig. 14. The bands are narrow and the lenses between them have almost no visible melt. Furthermore, the chromite grains in the lenses appear to be stretched and aligned, forming an apparent foliation. However, where a band is present, the chromite grains are larger, fewer and less elongated, as shown in Fig. 14. This pattern suggests that chromite grain growth (an Ostwald ripening process) is much more efficient in the presence of melt than in its absence. These variations in chromite morphology are not present (or nearly as clearly) in samples with more melt, suggesting that the presence of melt significantly enhances chromite grain growth.
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Melt configuration statistics
Below, we characterize the details of these various melt distribution parameters (Table 5) and plot each of them as functions of stress (the shear stress at the end of the experiment,
f) and time, to develop the hypothesis that some of these parameters are controlled by thermodynamics (stress) and some by kinetics (time). Again, the question remains as to whether or not these band systems achieve a steady state.
The distributions of band angles in this series of experiments, shown in Fig. 15, have different patterns from those in the strain series of experiments. In almost all samples (except for PI-1010), the distributions are bimodal, and within each population, the distributions look asymmetric, not Gaussian (however, there are too few data and too few decades of length scale to suppose a power-law distribution). However, as shown in the upper right-hand plot in Fig. 15, the distributions of angles (from the higher angle subset of the bimodal distribution) have very consistent mean values of 18°, independent of stress (or strain or time), similar to the mean of the strain series experiments.
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The distributions of band thicknesses,
th, in Fig. 16, also appear to have asymmetric and possibly power-law distributions. The mean values decrease as a function of increasing stress. The distribution of th in PI-1020 is the one exception, with no clear asymmetry. Interestingly, there is a clear difference in mean values between the CL and CR sets: the constant strain rate experiments tend to have narrower bands, even for the same value of f.
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The correlation between
and th that was clear in the strain series experiments is not well defined in this series, as shown in Fig. 17. The correlation coefficients, R2, for each sample in both series, plotted in the upper right-hand corner of Fig. 17, show an increase with increasing stress, but also demonstrate low values relative to the strain series (whose values are also independent of stress). This decrease in R2 in the stress series relative to the strain series may occur because the angles have a more clearly bimodal distribution than the thicknesses do and because the networks are less constrained spatially and thus have more variability. The most ubiquitous and significant feature of these distributions is that does not exceed 25°.
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Because the samples are larger and the networks more continuous in the stress series than in the strain series, the statistics on spatial properties in the system are of better quality. In general, both
th and sp decrease with increasing stress, as shown in Fig. 18a and b respectively, in accordance with the visual observations. The sp vs data appear to collapse onto one trend as a function of stress, as shown in Fig. 18b. For sp, the agreement between the ACF and the LIM data is fairly good, except for one sample (PI-1096). When plotted against time, the mean values of sp for both the CL and CR series increase with increasing time, but the data from the two sets do not collapse onto one line. Thus, it appears that sp is controlled by stress, and it does not depend a great deal on the loading path or stress or strain history.
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Interestingly, the
th vs time data seem to collapse onto one line, much more so than the th vs data, as shown in comparison of Fig. 18a and c. The implication of this observation is that the thickness has not reached steady state.
In Fig. 19, the average properties of the melt distribution (ab, b, Table 7) are plotted as functions of shear stress and time. All values appear to decrease with increasing stress (Fig. 19a) and increase with increasing time (Fig. 19b). As functions of shear stress, all values are systematically lower in the CR series than in the CL series. However, when plotted as a function of time, the data appear to collapse onto single lines, suggesting that the average properties of the melt distribution have not reached steady state (similar to th but not sp).
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There are limitations to how far we can interpret stress control (steady state) vs kinetically limited (non-steady state) for these various parameters. One way of determining whether a sample is in steady state is to look at rheological data, shown in Fig. 20. In the CL set, the stress always rises with increasing strain. In the CR set, 1096 is clearly in a steady state, because both strain rate and stress are approximately constant. PI-1020 (CL set, lowest stress) was quenched at approximately the same stress level as PI-1096 (CR set, faster strain rate). Interestingly, they have very similar band spacings, consistent with the idea that band spacing is set by stress, but very different values for all other parameters, suggesting that PI-1020 is not in steady state. [Data from sample PI-1020 are often outliers, making interpretations ambiguous in such a small dataset. It is anomalous in its high values of ab and
b (Table 7). It has an inordinately high melt fraction in the measured plane (
= 0·09, much higher than 
= 0·04), which suggests that melt has flowed laterally into the plane of measurement. Therefore, the spacing may well be set by stress, but the other parameters, including th, ab, and b may be changing with time and affected by lateral flow of melt in the sample.]
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Summary
- The band angles have bimodal distributions. In the stress series, this bimodal pattern seems to be the key signature of a connected network (Fig. 15), whereas in the strain series, the signature was the correlation between and th (Fig. 8).
- Band thicknesses do not have normal distributions. They appear to have a power-law distribution, but the data are limited to less than one order of magnitude of length scale (Fig. 16). The mean of the thickness data in both CR and CL sets decreases with increasing stress.
- A correlation exists (although weaker) between band thickness and band angle. This correlation is much worse than that in the strain series, which may be an indication that in the thicker samples with no strain marker, the bimodality of the angles is configuration preferable to a strong correlation between angle and thickness (Fig. 17).
- The band spacings decrease with increasing stress, and data from CL and CR sets collapse onto one line as a function of stress, suggesting that band spacing is a direct function of stress. The band thicknesses seem to vary more consistently as a function of the duration of the experiment (Fig. 18).
- Both ab and b increase with increasing duration of the experiment. The data from the constant load samples and constant rate samples collapse onto one line as a function of time, not stress. This pattern is consistent with th changing with time, whereas sp may be set by the value of stress (Fig. 19), suggesting that the melt distribution in each sample is not in steady state.
- In the CL set, stress always rises; in the CR set, stress can vary while strain rate is approximately constant. Only one sample, PI-1096 of the CR set, appears to be in a macroscopic steady state as indicated by constant stress and strain rate.
Summary of experimental results from both series
The data from the strain series clearly showed that (1) the melt is significantly segregated and networks begin to form by a shear strain of 1, and (2) the networks are well formed by 2. In the stress series, with the strain marker removed and the sample size increased, differences in network morphology between samples become more visible and meaningful. In the data on geometric and spatial properties of the melt distribution, there are distinct differences between the tightly constrained samples in the strain series and those in the stress series. In the strain series, the band angles () have normal distributions, the band thicknesses (th) have log-normal or power-law distributions, and and th are tightly correlated. In contrast, in the stress series, the frequencies of bands at a given have bimodal distributions, the distributions of th appear to be more log-normal or power law, and and th are weakly correlated. Thus, in the strain series, the signature of the anastomosing network is the correlation of and th (Fig. 8), whereas in the stress series, the signature of the anastomosing network is the bimodal distribution of (Fig. 17).
In the stress series, all the variations within and between the constant load set and the constant rate set can be attributed to differences in either applied stress or total duration of the experiment, because strain was kept approximately constant. Whereas the distributions of are similar for all values of stress () or experiment duration (tf), there are significant differences in the other (average and geometric) properties of the melt distributions. We plotted the mean values of band spacing (sp) and band thickness (th), as well as the average properties of the melt configuration, ab and b, as functions of and tf for both sets. The data for sp from both sets collapse to one line when plotted against stress, unlike the data for all the other parameters, which collapse onto one line as a function of tf.
The results for St
The values of St and 
are given in Table 7. In Fig. 21, we plot ab and b (both normalized values) as functions of St and overlay the functions from equations (11) and (12). With this limited dataset, we are not able to look for similarities and differences in these paths as functions of thermodynamic conditions (such as stress and melt fraction) and loading path (such as constant-load and constant-strain rate paths). However, the fits to these empirical expressions are good enough to allow us to use them in applications to thermodynamics of deformation (Holtzman et al., 2005) and calculation of seismic properties (Holtzman & Kendall, in preparation).
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DISCUSSION: OPEN QUESTIONS |
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TOPABSTRACTINTRODUCTIONEXPERIMENTAL DESIGNRESULTSDISCUSSION: OPEN QUESTIONSCONCLUSIONSAPPENDIX A: MEASURING THE...APPENDIX B: CALCULATING STRESS...REFERENCES |
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In this study, we set out to address two broad questions: (1) How does the melt configuration evolve with increasing shear strain? (2) How does the applied stress level influence the melt configuration? We demonstrated that the bands form at shear strains of approximately
= 1 and increasing stress level causes a decrease in the band spacing. The questions raised by the experiments fall into the following groups.- Do the samples reach conditions of steady state deformation? All the following questions can be addressed most meaningfully if the data come from experiments that were quenched during steady state. The reason for this restraint is that thermodynamic driving forces are balanced.
- In the deforming system, what physical properties and/or driving forces control the observed melt distribution and degree of melt segregation? What combination controls the scaling of melt distribution parameters measured in experiments?
- Mechanically speaking, how does melt organize? What are the deformation and transport processes controlling the dynamics of the self-assembly and organization?
- How do we scale to the conditions of the Earth? How can we extrapolate inference of processes from these observations to natural conditions?
Steady state?
In the study of this dynamic system, the identification of a steady state is a goal in the experiments. In terms of irreversible (non-equilibrium) thermodynamics, steady state requires that the rate of entropy production be minimized and the related quantity of viscous energy dissipation is time-invariant (i.e. d
/dt = 0), (de Groot & Mazur, 1984, p. 43; Holtzman et al., 2005), which means that stress, strain rate, and effective viscosity are time-invariant. Because effective viscosity is dependent on the melt distribution in a system with strain partitioning between bands and lenses, the distribution should also be statistically time-invariant. Thus, the clearest macroscopic indication of steady state is that stress and strain rate are both constant. As shown in Fig. 20, only one sample (PI-1096) achieves this state unambiguously.
Understanding the conditions required for steady state would reveal a great deal about the subprocesses that must be in balance in the system. To achieve true steady state, the rate of reorganization of melt must perfectly balance the rate of solid deformation, such that, on average, the distribution of band and lens sizes does not change; locally there is always redistribution of melt relative to solid, but across the sample, the statistical distributions of all these properties are constant. Below, we present a view of the events and processes that must be occurring to achieve a constant band angle, one quantity we know is constant in all experiments.
So then, if only one sample achieved a demonstrable steady state, how meaningful are the data? How far from steady state are the samples? Quantifying this question requires knowledge of the transient time scale for reorganization (relative to the inverse of the strain rate). We do not know yet, but we assume that the ratio is not too far from unity. The uncertainties in measurement and the crudeness of our current understanding of the scaling processes are within range of the variability in parameters caused by quenching the system at some distance from steady state. Thus, we proceed with a discussion of scaling by compaction length. We leave for future work the interpretation of experimental results in the framework of irreversible thermodynamics, as addressed by Holtzman et al. (2005).
Scaling by compaction length
With increasing stress, the band spacing sp decreases. The length scales of many structural properties of materials (such as grain size, distance between dislocations, fracture spacings, etc.) have this relationship. There is not a self-consistent theoretical explanation for this scaling for all of these processes; each is described somewhat independently. For band spacing, compaction theory works fairly well. In an earlier study (Holtzman et al., 2003a), we demonstrated that the spacing of the melt bands, sp, scales with the compaction length, c. One of the objectives of the stress series experiments was to provide a more controlled test of this scaling relationship.
c is defined by the material transport properties internal to the system, simplified as two different viscosities and a permeability. As stated above, we want to understand how the stress controls the melt band spacing. If the processes are governed by two-phase flow, a compaction–decompaction process, then the compaction length should influence the observed distribution. The compaction length is defined as
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is the shear viscosity of the composite, µ is the shear viscosity of the melt, is the melt fraction and is the shear stress. In the stress series of experiments, we varied compaction length by changing the stress-dependent solid viscosity, using increasing values of load to push the system further into the dislocation creep regime, thus lowering the effective viscosity. In our earlier study (Holtzman et al., 2003a), we calculated the initial compaction length, 
. In the present paper, all of the samples have the same initial melt fraction and grain size, and thus the same initial compaction length in the absence of stress. As soon as deformation begins and melt begins to segregate, the compaction length becomes stress-dependent and spatially variable. Thus, to compare the results from experiments with similar , we calculate the minimum compaction length at the end of the experiment, which occurs in the lenses where melt fraction is lowest. In our earlier study (Holtzman et al., 2003a), we demonstrated that at high , the increase in k with increasing approximately offsets the decrease in with increasing . In contrast, at low , the decrease in k with decreasing dominates, such that c drops quickly. Because the minimum compaction length in the system should set the spacing, we estimate c in the melt-depleted lenses. As discussed in Appendix B, we use the value of n measured with the average properties; grain size d is measured in the strain series experiments (Table 4) to constrain the grain growth parameter, to account for grain growth during the experiment; k is calculated with the method discussed by Holtzman et al. (2003a); the composite viscosity is calculated using the flow law parameters reported by Hirth and Kohlstedt (2003) based on the values of stress at the end of the experiment (f).
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When compaction length is plotted in this manner, as shown in Fig. 23, we provide the following interpretation. Four data points plot tightly on a line with a slope of 2/5, similar to the scaling coefficient of 1/5 estimated by Holtzman et al. (2003a
), if one data point (PI-1020) is considered an outlier. The difference in observed slope between the two studies (2/5 vs 1/5) may come from the difference in how c is calculated; here, we calculate c in the lenses at the end of the experiment (using f and a flow law, and the measured melt fraction in the lenses, n); in our earlier study (Holtzman et al., 2003a), we calculated an initial c (using a measured initial viscosity and the average t). Thus, 
. However, the basic idea that sp < c remains.
The fact that compaction theory has successfully simulated basic aspects of this segregation and organization process (Katz et al., 2006) lends further support to its use as a scaling parameter. The question now is why compaction theory works and what are its limitations. To address this question, we discuss our observations and intuitions into the processes active in these experimental systems.
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Dynamics of melt segregation and organization
Here, we develop a conceptual model for melt migration and the evolution of the melt network organization. This model is a recipe for the maintenance of a steady state, and in that context, a framework for evaluating what processes must be in balance in a steady state. Simple shear (as opposed to pure shear), because of its highly rotational nature, poses complications that are not understood yet: What causes the stability of the distribution of band angles? Why does the segregation instability stabilize, such that all the melt is not drained from the lenses and small bands appear to be persistent? What causes the formation of an anastomosing network geometry and why? Earlier (Holtzman et al., 2005
), we introduced a model for a ‘pumping mechanism’ by which the melt network can maintain a constant distribution of band angles, and ultimately a steady state in all quantitative aspects. Here we expand on this model based on experimental observations. This mechanism has three elements, (1) nucleation; (2) rotation and growth and (3) closure, all of which are occurring simultaneously in a deforming sample, as illustrated in Fig. 22.
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Nucleation
Why do bands nucleate and at what angle? Our observations from the lowest shear strain experiments in the strain series suggest that bands begin to form at 5–25°, with a mean and standard deviation of 18 ± 6°. In sheared samples of olivine + MORB, Zimmerman et al. (1999
) found that melt pockets were elongated and oriented at approximately the same angle, suggesting that this angle is ubiquitous and is stress controlled (not controlled by the physical properties of the matrix). These observations suggest that bands form at the same distribution of angles at which they are observed when mature, by the collection and accumulation of individual elongated pockets. Hier-Majumder et al. (2004) adopted a micromechanical approach to this question, looking at how shear stress caused the growth of some pockets at the expense of others, and speculated that pockets at 0–45° to 
1 open preferentially by stress corrosion, causing an average angle of 20°. Nucleation of small bands within the well-developed lenses occurs at lower angle (5–10° relative to the shear plane), reflecting the local back-rotation of the stress field as a result of strain partitioning between lenses and bands. This local modification of stress fields was discussed extensively in an earlier study (Holtzman et al., 2005). Once elongated pockets are nucleated, the stress-driven segregation instability is initiated.
Rotation and growth
As discussed by Holtzman et al. (2003a, 2005), Spiegelman (2003) and Katz et al. (2006), bands should rotate with shear if melt is not migrating relative to the solid. The apparent rotation is caused by the simple shear, but it is actually the shear parallel advection of a material line (melt-rich band) at an angle to the shear plane, causing the appearance of rotation. Spiegelman (2003) modified Stevenson's (1989) instability analysis from a pure shear to a simple shear geometry and showed that if significant strain partitioning does not occur, bands should rotate as they grow up to 90° to the shear plane, with maximum growth rate at 45°. As bands rotate with shear upwards towards 45°, both the shear and normal stress diminish on them, causing mean pressure and thus melt pressure to drop, and thus causing more melt to flow in (Fig. 22b). As discussed by Holtzman et al. (2005), strain partitioning reduces this tendency to rotate and can stabilize bands at the observed angles. This partitioning does occur in the numerical models of Katz et al. (2006). Melt migration through the network reduces the need for band rotation; the more effective the melt transport in the network, the less rotation should occur.
Closure
The observation from both sets of experiments is that there are no bands at angles >30°. Therefore, the bands must close and the melt move elsewhere (into a lower angle band) before a band rotates up to 30°. We suggest that this closure is caused by increasing normal stress on the bands as they approach 45° as a result of strain partitioning, as discussed by Holtzman et al. (2005). The cause of this increase in normal stress is that as less deformation is accommodated in the bands, the deformation must increasingly be accommodated in the lenses, causing them to flatten, as illustrated in Figs 22b, c. As they flatten, the converging walls of the lenses increase the melt pressure in the bands, driving their compaction. In Fig. 14, the rounded chromite grains suggest that melt was previously present in that location and has since migrated away into a nearby band, as evidence for these closures. This migration from high- to low-angle bands also occurred in the models of Katz et al. (2006). This closure is the key to the pumping mechanism: the melt is driven into the low-angle bands. Therefore, they decompact and continue to rotate until they too are sufficiently compressed. The stress-driven ‘instability’ is thus stabilized.
Of course, this distinction between events is somewhat artificial, because all are happening together; in steady state, it is possible that the rotations are very small. However, the idea may help explain the observation that in the strain and stress series of experiments, the signature of the anastomosing network was different. In the strain series, it was the correlation of band angles and band thicknesses (Fig. 8). In the stress series, the signature was the bimodal distributions of band angles (Fig. 17). The relevant difference between the two series is probably in the relative freedom to organize. The more constrained boundary conditions in the strain series experiments may have forced more rotation and made migration more difficult. In the stress series, the greater degree of freedom of the bands allows them to organize more effectively into two populations, which allow the system to be closer to a steady state and a smaller need for rotation of melt bands (Fig. 13).
As melt is moving relative to the solid, all the matrix deformation processes required to accommodate local changes in melt fraction (i.e. deformation of the solid) must also be in balance. This balance is the principle of compaction theory (e.g. McKenzie, 1984). We have shown evidence that the mechanisms accommodating these deformations are different in the melt-depleted lenses and the melt-rich bands. In the lenses, the dominant deformation mechanisms are probably dislocation creep and dislocation-accommodated grain boundary sliding (GBS). Evidence for dislocation creep is the coexistence of a lattice preferred orientation (e.g. Holtzman et al., 2003b) and for GBS are small generally euhedral grains, as well as the measured values of n from 1 to 3·5 (increasing with increasing applied stress). Further discussion of these mechanisms has been given by de Bresser et al. (2001), Goldsby & Kohlstedt (2001), Hirth & Kohlstedt (2003) and Warren & Hirth (2006). In the melt-rich bands, diffusion creep is probably enhanced by the large melt fractions relative to dislocation creep. As shown in Fig. 5, melt fraction can locally be very high, and the deformation mechanism may be one closer to a granular flow, in which grains slide past each other with little internal deformation but are always in contact. This kind of mechanism would probably be only local and transient, and speculatively, moving in waves along a band. As shown in Fig. 5, growth and dissolution features suggest that dissolution-precipitation (pressure solution) might also be a locally important deformation mechanism, especially in the bands. There is no evidence for brittle fracture or the opening of air-filled pore space. Any local dilatency (i.e. decompaction) is balanced by local melt flow and mass-balanced compaction.
Relation to other experimental studies
Numerous deformation experiments on partially molten rocks have been performed in uniaxial compression, but very few in (approximate) simple shear or in torsion. Of those studies on simple shear deformation of partially molten rocks (e.g. Dell’Angelo & Tullis, 1988; Zimmerman et al. 1999), rock analogues (e.g. Rosenberg & Handy, 2000), or rocks with water (e.g. Schmocker et al., 2003; Hier-Majumder & Kohlstedt, 2006) conducted over a wide range of conditions within the ductile or semi-ductile regime, almost all display alignment of melt pockets controlled by the principal stress direction, but none to our knowledge have developed obvious melt-rich networks at length scales much longer than the grain size. In uniaxial compression experiments (e.g. Mecklenburgh & Rutter, 2003), melt pockets align but segregation at wavelengths longer than the grain scale does not occur either. Longer wavelength segregation may not occur because, in some cases, the compaction length is too small relative to the sample size (too close to the grain scale, cd) as a result of the high melt viscosity (e.g. granitic systems) and in others, because c is larger than the sample size (e.g. possible samples with fluid water). However, segregation similar to that occurring here has been observed in partially molten metals (Gourlay & Dahle, 2007).
Experiments on analogue materials (i.e. camphor + melt) have potential for insight in the comparison of the behavior of two ‘similar’ materials at similar scaled conditions. In several studies, segregation occurs during pure (Takei, 2005) and simple shear (e.g. Rosenberg & Handy, 2000). In the former case, thin melt-rich layers form parallel to 1 and normal to 3. In the latter case, pockets align at 20° to the shear plane, and melt segregates and strain localizes into one layer (not a network) parallel to the shear plane; melt then flows out of this layer to the edges of the sample. The experiments were unconfined and drained, and the rheology was semi-brittle, so the physics governing the melt distribution is different from ours; thus a well-constrained comparison cannot yet be made.
We iterate here that there are similarities between the physical processes causing melt segregation and organization in our experiments and those in the materials deforming in more brittle or granular regimes. For example, in torsion experiments on aggregates of quartz plus water at 350 MPa and 1300 K (Schmocker et al., 2003), conditions at which quartz deforms in a semi-brittle fashion, water-rich bands formed that look macroscopically similar to ours. Microscopically, the mechanisms are different, as they are in a semi-brittle regime, so compaction length may not be a relevant quantity. The shear bands in these materials, classified as Riedel shears, generally exhibit an angle and morphology similar to ours. However, a Riedel shear is a classification of structure based on its orientation, but is not necessarily associated with a particular process or mechanism of formation. Thus, whether all fluid-rich bands at a particular angle occur for the same reason is not clear, but if this is so, it is likely to be related to the relationship between shear and normal stress on the band. For example, in dilating materials, the separation of grains and creation of air-filled space may reflect the same stress state as the separation of grains caused by increasing melt fraction. In other words, local dilatation balanced by nearby compaction in our samples may affect the stress state in the shear band in the same way that a global dilatancy does.
Applications to the Earth
Here we expand somewhat on the potential applications of these experimental results to the Earth (Holtzman et al., 2003a, 2003b, 2005; Holtzman & Kendall, in preparation). First, can the stress-driven melt segregation process occur at the low melt fractions ( 
0·02–0·03) thought to occur in much of the partially molten region beneath a mid-ocean ridge (e.g. Kelemen et al., 1997)? We have demonstrated in experiments that the process occurs at very low melt fractions (t 0·005), suggesting that it is very effective at segregating very small melt fractions. However, the stresses at which it occurs are higher than those that exist in the asthenosphere. Therefore, an important question is whether the process requires a yield stress to be surpassed? Below, we readdress our compaction length scaling arguments, with an analysis of the importance of surface tension, raising the question of whether the stress-driven segregation process occurs only after a yield stress has been reached. Then, we review relevant field observations that were not pointed out in our earlier study (Holtzman et al., 2003). Finally we discuss a hypothetical application to ridge melting and melt transport in extensional settings, generalizable to other geodynamic settings.
Compaction length, surface tension, and yield stress
We have demonstrated here, with a series of better constrained experiments than used earlier (Holtzman et al., 2003a), that the compaction length is a relevant scaling factor, which works well for scaling internally. However, scaling to the mantle is still a large leap (but not any more bold than scaling a laboratory-derived flow law). We have two immediate questions, as follows.
First, what is the influence of the sample size on the scaling? It is clear that the melt-rich networks are ‘aware’ of the boundaries of the system. Because the sample thickness is approximately constant in the experiments and the network spacings are not, c is involved in determining sp. Thus, taking this scaling one step further will require larger samples (e.g. in torsion geometry) and well-calibrated numerical models and analysis from which more sophisticated scaling rules can be derived (e.g. Katz et al., 2006). For the moment, the compaction length is still the most reasonable scaling tool; the simple conclusion is that sp < c.
Second, for the segregation process to occur, does a yield stress have to be surpassed? If so, it is smaller than the lowest stress we achieved in these experiments (< 25 MPa). If a yield stress does exist, what determines it? One possibility is that surface tension plays a role. If so, the yield stress should be much lower in the mantle than in the experiments, based on the following analysis.
Surface tension is significantly larger in the small grain sizes of these experiments than in the Earth. To estimate this effect, we revive the scaling of Cooper (1990). He estimated the pressure gradient driving melt migration as the result of competition between two driving forces, (1) shear stress gradient, the first term on the right-hand side of the following equation, and (2) surface tension (sl), the second term on the right-hand side:
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is the bulk viscosity. In the following, the viscosity factor in parentheses is approximated as the melt-dependent composite shear viscosity.
The ratio of the two driving forces, viscous stress over surface tension stress, is related to the Capillary Number (C) (e.g. Stone, 2004), so we call it C*, where
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| (16) |
The ratio C* in equation (16) is plotted vs strain rate in Fig. 24. We keep grain size constant (i.e. it is an additional parameter, but does not vary within a sample), but vary d spatially. The results indicate that in the laboratory, at low strain rates or stresses, stress dominates over surface tension; if the stress were removed, surface tension would be a significant driving force that would work against melt segregation. Thus, in the experiments, the stress-driven segregation requires non-negligible energy to work against surface tension (e.g. Parsons et al., 2007). However, in the Earth, because of the much larger grain size, the surface tension force is much less significant, even relative to the much lower stresses. Thus, if surface tension causes a yield stress barrier for the process to occur in the Earth, the barrier is probably easily overcome in much of the asthenosphere. A value of C* > 1 means that, as melt segregates significantly, stress gradients in the sample are still large enough to keep melt segregated. This relationship may affect the compaction length scaling coefficient, but probably not the validity of the idea.
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Geological evidence for stress-driven melt segregation
The other approach to testing the applicability of the processes occurring in these experiments to the Earth is to look for evidence directly (e.g. Holtzman et al., 2003b
). In general, geological evidence for stress-driven melt segregation is abundant in exposures of formerly ductile, partially molten lower crust in oceanic and continental settings, although proving the existence of a dynamic process is difficult. Many structures in migmatites that show hyper-solidus deformation may well be evidence of stress-driven melt segregation (e.g. Tommasi et al., 1994; Neves et al., 1996; Brown & Solar, 1998). Such preservation is more likely in the crust than in the asthenospheric and lithospheric mantle because, in the crust, melt viscosities are higher and temperature gradients are steeper. Evidence is more difficult to find in the mantle because melt extraction is generally more complete [for exceptions see Cannat et al. (1990) and references given by Holtzman et al. (2003a)]. Planar concentrations of plagioclase and/or clinopyroxene in dunite and harzburgite are suggestive of segregated, channelized melt (e.g. Boudier & Nicolas, 1995; Kelemen et al. 1997; Holtzman et al., 2003a). A comprehensive review of length scales of melt segregation features in the lower crust and upper mantle, looking for correlations with stress estimates, is necessary for developing these ideas further.
Braun & Kelemen (2002) have shown that dunites deep in the mantle section in the Oman ophiolite form by melt migrating from below into shallow mantle rocks with which they are not in equilibrium. As the melt migrated to shallower depths, it dissolved clinopyroxene and precipitated olivine, causing a feedback between increasing dissolution and increasing melt flux (Kelemen et al., 1995). However, modelling of the reaction infiltration instability that results in the formation of dunite channels predicts tubular, not planar, features (e.g. Aharonov et al. 1997). In Fig. 25, a tracing of a satellite photograph of a part of the Oman ophiolite with a large-scale melt-transport network illustrates that the smaller dunites have different orientations from the larger ones, and the smaller ones connect to the larger ones, both characteristics of the melt patterns observed in our experiments. We suggest that deviatoric stress and strain partitioning influenced melt distribution as the dunite-forming reaction progressed.
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Geodynamics: an example application to mid-ocean ridges
The global mid-ocean ridge system is organized to transport heat from the Earth's interior to its exterior. Within the roughly triangular region of partially molten asthenosphere, from about 60–100 km below the oceanic crust up to the Moho, melt is formed and transported in increasing disequilibrium with its host rock as it rises and focuses towards the ridge axis (Fig. 26a). Most of the melt volume is not in equilibrium with the shallow upper mantle. Based on this observation and the rapid rates of melt ascent, melt must travel predominantly in high-flux channels, chemically separated from the surrounding rock. Here, we reiterate our proposition that the stress-driven segregation process may be very effective at segregating the deepest small melt fractions and becomes increasingly important in the parts of the ridge system closer to the flanks or boundaries of the system, where strain rates are higher because of corner flow and the rheological transition occurs from lithosphere to asthenosphere, as illustrated in Fig. 26b. A potentially testable prediction is that the transport will be most effective, and thus melt–rock disequilibrium greatest, at the flanks of the melt region, where deformation is most intense. This kind of approach can be adapted to any geodynamic environment, by making predictions about the locations of the greatest melt productivity and highest strain rates (Holtzman & Kendall, in preparation).
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CONCLUSIONS |
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TOPABSTRACTINTRODUCTIONEXPERIMENTAL DESIGNRESULTSDISCUSSION: OPEN QUESTIONSCONCLUSIONSAPPENDIX A: MEASURING THE...APPENDIX B: CALCULATING STRESS...REFERENCES |
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We have demonstrated that deviatoric stress drives melt to segregate into melt-rich networks. They develop after small strains and, as they form, progressively concentrate strain into them. Higher stress levels cause the characteristic spacings of the networks to decrease. We relate this decrease to a decrease in the minimum compaction length in the system. We propose a conceptual model for strain partitioning and melt pumping to maintain steady-state melt distributions. Scaling analyses applied to Earth conditions suggest that the process should be possible at the low stresses in the partially molten mantle and produce networks with spacings of the order of 10–1000 m. This process, if it occurs in the Earth, will strongly influence the rheological, seismic, and melt transport properties of the upper mantle and lower crust.
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APPENDIX A: MEASURING THE MELT DISTRIBUTION |
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TOPABSTRACTINTRODUCTIONEXPERIMENTAL DESIGNRESULTSDISCUSSION: OPEN QUESTIONSCONCLUSIONSAPPENDIX A: MEASURING THE...APPENDIX B: CALCULATING STRESS...REFERENCES |
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Linear intercept method
With low-resolution photomicrographs, such as Figs. 2, 4, 10 and 11, we measured band angle (
) and band thickness (th) with a ‘linear intercept method’ (LIM). As illustrated in Fig. 9a, lines are drawn across the sample at an orientation that is approximately normal to the bands, spaced far enough apart to minimize repeated measurements on one band. (Qualitatively, we consider the properties of a band to change on one side of a large junction with another band.) Wherever the line crosses a band, we draw a right angle composed of a line tangent to the band and one that traverses the thickness of the band and measure the angle and thickness. In samples with a Ni strain marker, the melt bands form in regions between the nearest corner of a piston and the edge of the impermeable Ni strain marker. We analyzed the statistics of the melt distributions in these regions, cutting off the length equivalent to one sample thickness from each of the ends, to minimize measuring statistics of organization associated with these end-effects. In samples without a strain marker, we measured in the center of the samples.
Autocorrelation function and band spacing
To quantify the band spacings, sp, in a way that is less subjective than the linear intercept method (as a check), we extract linear profiles normal to the band lengths from the smoothed melt distribution images. The autocorrelation function is a method for extracting amplitude content at different frequencies by comparing a signal with itself
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| (17) |
, p. 492), where f is the signal, x is the spatial variable and r is the lag. The function f is multiplied by itself and divided by the total number of points (the meaning of the angular brackets). Then the signal is shifted by r and the summation and averaging is repeated for all values of r, from zero to N, where N is the total number of points in f. The result is a vector of length N, plotted as a function of the lag, r, as shown in Fig. 2e. |
APPENDIX B: CALCULATING STRESS-DEPENDENT COMPACTION LENGTH |
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TOPABSTRACTINTRODUCTIONEXPERIMENTAL DESIGNRESULTSDISCUSSION: OPEN QUESTIONSCONCLUSIONSAPPENDIX A: MEASURING THE...APPENDIX B: CALCULATING STRESS...REFERENCES |
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There are many parameters that go into the compaction length calculation. The main difference between what we have done here and in our earlier study (Holtzman et al., 2003a
) is to use the final compaction length in the lenses, calculated from measured melt fraction in the lenses and the final applied stress. The melt fraction enters into the permeability and the composite shear viscosity; the stress enters into the viscosity calculation.
Permeability
The permeability is calculated as discussed in the Appendix of Holtzman et al. (2003a). We use an anisotropic permeability model based on the observations of strongly aligned melt pockets in our samples and in those of Daines & Kohlstedt (1997) and Zimmerman et al. (1999), as discussed by Holtzman et al. (2003a);
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| (18) |
adm/b where m = 2 and values for a and b depend on direction. For k11, a = 3 and b = 25, approximated as sheets. For k33, a = 2 and b = 10 000, approximated as tubes. Because k33, the minimum, should be oriented roughly normal to the bands, we use this component as a scalar value in permeability in the compaction length.
Viscosity
The strain rate of a melt-free rock at experimental conditions may be described by a flow law of the form (Zimmerman & Kohlstedt, 2004)
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| (19) |
is the shear stress (the final value in any dataset), d is the grain size, R is the gas constant and T is absolute temperature. Grain sizes were measured in two samples using a linear intercept method, before and after deformation (PI-830: do = 9 ± 3 µm, df = 12 ± 5 µm; PI-833: do = 8 ± 3 µm, df = 12 ± 6 µm). We use the final value for all samples. In fact, the composite dislocation and diffusion creep flow law is required to fit the data:
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| (20) |
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| (21) |
). Finally, using the parameters listed in the following table,
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| (22) |
).
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ACKNOWLEDGEMENTS |
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We wish to thank Mark Zimmerman and members of the Kohlstedt laboratory for immeasurable help, Ben Greenhagen and Dan Moir for assistance with the melt distribution measurements, and Takehiko Hiraga and Florian Heidelbach for help with electron microscopy. Yasuko Takei generously and thoughtfully scrutinized this study. Valuable and critical reviews were provided by Julian Mecklenburgh, Michel Rabinowicz and Dan McKenzie. Funding for this research comes from NSF grants OCE-0327143 and OCE-0648020, and for B.K.H., a Fulbright Fellowship to France, UMN doctoral dissertation fellowship, and LDEO Postdoctoral Research Fellowship.
*Corresponding author. Present address: Lamont–Doherty Earth Observatory, Columbia University, Palisades, NY 10964, USA. Fax: 612-624-3819. E-mail: benh{at}ldeo.columbia.edu, dlkohl{at}umn.edu
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