Radial Melt Segregation During Extrusion of Partially Molten Rocks

Abstract

We performed a series of extrusion experiments on partially molten samples of forsterite plus 10 vol% of an anorthite-rich melt to investigate melt segregation in a pipe-extrusion geometry and test the predictions of two-phase flow theory with viscous anisotropy. The employed flow geometry has not been experimentally investigated for partially molten rocks; however, numerical solutions for a similar, pipe-Poiseuille geometry are available. Samples were extruded from a 6-mm diameter reservoir into a 2-mm diameter channel under a fixed normal stress at 1350°C and 0.1 MPa. The melt distribution in the channel was subsequently mapped with optical and backscattered electron microscopy and analyzed via quantitative image analysis. Melt segregated from the center toward the outer radius of the channel. The melt fraction at the wall increased with increasing extrusion duration and with increasing shear stress. The melt fraction profiles are parabolic with the melt fraction at the wall reaching 0.17–0.66, values 2 to 16 times higher than at the channel center. Segregation of melt toward the wall of the channel is consistent with base-state melt segregation as predicted by two-phase flow theory with viscous anisotropy. However, melt-rich sheets inclined at a low angle to the wall, which are anticipated from two-phase flow theory, were not observed, indicating that the compaction length is larger than the channel diameter. The results of our experiments are a test of two-phase flow theory that includes viscous anisotropy, an essential theoretical frame work needed for modeling large-scale melt migration and segregation in the upper mantle.

Key Points

• Partially molten rocks were extruded in a Poiseuille flow geometry
• Melt segregates toward wall of the pipe
• Radial melt segregation is consistent with prediction of two-phase flow theory with viscous anisotropy

1 Introduction

The effect of stress on melt segregation in partially molten rocks is of particular importance due to the fact that melting and deformation occur hand-in-hand in many tectonic settings and contribute to the physical and chemical differentiation of planets. Numerical models of melt segregation are based on two-phase flow theory as formulated by McKenzie (1984). Comparison of experimental observations to theoretical predictions offers a unique opportunity to refine such physical models.

The first theoretical analysis (Stevenson, 1989) predicted that mesoscale melt segregation will occur in a partially molten rock undergoing large-scale deformation if shear viscosity decreases with increasing melt fraction as previously demonstrated experimentally (Cooper & Kohlstedt, 1986). The first theoretical studies of this process focused on the formation of melt-rich bands (sheets in three dimensions; Stevenson, 1989; Spiegelman & Kelemen, 2003; Katz et al., 2006). Without imposing an unrealistically large value for stress exponent, two-phase flow theory predicted that sheets would form at 45° to the shear plane, antithetic to the shear direction, or, in other words, parallel to the largest principal stress, σ₁. General shear and torsion experiments on a series of partially molten rocks, however, consistently yielded a much lower angle of 15–25° (Holtzman et al., 2003; King et al., 2010; Kohlstedt et al., 2010). This discrepancy between theoretical predictions and experimental results motivated a refinement of the two-phase flow theory, leading to the next generation of analyses of deformation of partially molten rocks incorporating anisotropic viscosity (Takei & Holtzman, 2009).

The viscosity of a deforming partially molten rock becomes anisotropic due to stress-induced, grain scale alignment of melt pockets (Takei 2001, 2010). The viscous anisotropy hypothesis hence provides a bridge between grain-scale processes and a continuum mechanics description of two-phase flow. With viscous anisotropy, two-phase flow theory successfully predicts the emergence of melt-rich sheets at a low angle to the shear plane in both simple shear as well as torsional shear in accord with experimental results (Butler, 2012; Katz & Takei, 2013; Takei & Holtzman, 2009; Takei & Katz, 2013, 2015).

Furthermore, analysis of rotary shear of a partially molten material yielded melt segregation at a broader, sample scale (Takei & Holtzman, 2009), consistent with results from experiments (Gourlay et al., 2007). This broader-scale segregation has recently been reported in torsion experiments on partially molten aggregates of olivine plus basalt (Qi et al., 2015; Qi & Kohlstedt, 2018; Kohlstedt et al., 2014). Further theoretical analysis of melt segregation in a number of different deformation geometries including simple shear flow, parallel-plate Poiseuille flow, torsional flow (Katz & Takei, 2013; Takei & Katz, 2013), and pipe Poiseuille flow (Allwright & Katz, 2014) demonstrated that, based on two-phase flow theory with viscous anisotropy, broad scale melt segregation is expected for the latter three deformation geometries.

Here we report the results from a series of experiments designed to extrude a partially molten rock in a geometry similar to pipe Poiseuille flow. Our study presents, therefore, an opportunity to test two-phase flow theory with viscous anisotropy in a flow geometry that has not been experimentally tested for partially molten rocks but where numerical solutions are available (Allwright & Katz, 2014).

2 Materials and Methods

2.1 Sample Preparation

To create partially molten rock samples, powders of forsterite (melting temperature T_m = 1890°C) and an anorthitic powder (T_m ≈ 1280°C) were synthetized from high purity nanopowders of Mg (OH)₂, SiO₂, CaCO₃, and Al₂O₃. To mix the anorthitic powder with the forsterite, the two powders were disaggregated in an agate mortar, then tumbled in plastic bottles with plastic coated iron balls and ethanol for at least 24 hr (Koizumi et al., 2010). This slurry was decanted into glass beakers, dried on a hot plate at 60°C, and then calcined for 3 hr at 1000°C. The melt phase, “anorthite”, had a pseudo-anorthitic, magnesium-rich, composition: 48 wt% SiO₂, 12 wt% MgO, 15 wt% CaO, and 25 wt% Al₂O₃. The forsterite was composed of 43.4 wt% SiO₂, 56.3 wt% MgO, 0.2 wt% CaO, and 0.1 wt% Al₂O₃. The composition of the powders was chosen in order to minimize chemical reaction between melt and solid; that is, at experimental conditions, all olivine remains solid and all anorthitic powder becomes liquid allowing for good control of the starting melt fraction. Samples were created with 10% by volume of anorthite as the melt phase. These mixtures were again calcined at 1000°C for 3 hr to remove contamination possibly introduced from the alcohol and the plastics. After this calcination step, the powder mixture was disaggregated in an agate mortar and then placed into a cylindrical, zirconia die. The powders were loosely pressed into a disc and then vacuum sealed in plastic. This package was placed inside a hydraulic pressure vessel and compressed at 150 MPa for ~10 min. After depressurization, these green bodies were vacuum sintered in Pt crucibles at ~10⁻⁵ bar and 1400°C for 5 hr. This last sintering step controls the microstructure of the starting material. The discs were >99% dense after sintering, with the anorthite phase homogenously distributed throughout the sample. The dihedral angle between forsterite and the anorthitic glass is <60°; hence, the melt was fully interconnected (von Bargen & Waff, 1986).

2.2 Extrusion Experiments

The deformation apparatus, illustrated in Figure 1, consisted of a clamshell furnace that maintains a hot zone 2 cm in length to ±2°C and a pair of SiC pistons; the top piston was fixed to the apparatus frame, while the bottom one was mobile. Relative movement of the pistons was measured with two direct-current displacement transducers (DCDTs). An external DCDT measured the displacement of the bottom piston relative to the fixed frame, and an internal DCDT measured the displacement between the two SiC pistons, using SiC extension rods to translate this displacement in the hot zone up to the DCDT that was located in a colder area. Temperature was measured with an R-type thermocouple placed next to the sample. Both temperature and displacement were digitally recorded at 1-s intervals.

Figure 1

Schematic illustration of one-atmosphere deformation apparatus with extrusion assembly in place. Extrusion experiments were performed at controlled temperature in Argon. Temperature and displacement, measured internally and externally, were recorded at 1-s intervals. DCDT = direct-current displacement transducer.

Extrusion experiments were carried out using the above described apparatus at a pressure of 0.1 MPa and a temperature of T = 1350°C in argon. Oxygen fugacity did not need to be buffered since our samples did not contain any heterovalent cations such as Fe that could be reduced or oxidized. We conducted the experiments under Argon to prevent the oxydation of the Mo rod. Each sample was shaped into a 6.3-mm diameter, cylindrical pellet and extruded through a 2.0-mm diameter channel in a molybdenum rod, as shown schematically in Figure 2. Once the temperature reached 1350°C, the mobile piston was loaded with weights for 1 to 60 hr, thus forcing the molten rock pellet to extrude from the reservoir into the channel in the rod. The sample was then cooled under load at 5°C/min to preserve the melt distribution as the pseudo-anorthitic liquid turns into glass during cooling. The length of the extruded channel was calculated based on the displacement measured by the internal DCDT, since the external DCDT captures the variation in length of both the sample and the deformation apparatus. The thermomechanical conditions for all of our experiments are listed in Table 1. Load and extrusion time were varied. Additionally, in one series of experiments conducted at an extrusion stress of ~75 MPa, the sample was annealed either before or after the extrusion test. The main aim of annealing the sample before extrusion was to test if this procedure could eliminate gas-filled pores in the sample. Also, annealing before the extrusion should lead to grain growth and thus a higher permeability as well as higher shear viscosity during extrusion provided that the partially molten rock deforms by a grain size-sensitive deformation mechanism; a more detailed discussion is presented in section 4.1. Annealing after extrusion should again lead to grain growth and surface tension driven redistribution of the melt segregation profiles established during extrusion (Parsons et al. 2008; King et al., 2011). Since the chemical composition of the powders was chosen to be at equilibrium at the experimental conditions, no melting or crystallization is expected to occur during the annealing steps.

Figure 2

Sketch of sample assembly. (a) Before extrusion: A Fo-An vacuum-sintered pellet was placed in the reservoir chamber of a Mo rod. (b) After extrusion: The partially molten rock was extruded from a 6.3-mm diameter reservoir into a 2.0-mm diameter channel at 1350°C. The temperature was then decreased while maintaining the load in order to preserve the stress-driven melt distribution.

Table 1. Experimental Conditions of Extrusion Runs

Sample ID	Temperature (°C)	Load/stress (kg/MPa)	Extruded length (mm)	Extrusion time (hr)	Anneal Time (hr)
Mx1	1350	30/94	4.5	12	48 (before)
Mx46	1350	12/38	0.1	40	none
Mx47	1350	24/75	2.9	48	12 (before)
Mx49	1350	18/56	2.0	66	none
Mx50	1350	30/94	4.4	20	none
Mx52	1350	24/75	5.2	5	none
Mx54	1350	24/75	3.6	2	96 (after)
Mx55	1350	24/75	8.3	2	none

2.3 Characterization Techniques

After an experiment, the sample was embedded in a low-viscosity resin under vacuum, cut parallel to the loading axis, and polished down to a 0.5-μm particle size using diamond embedded lapping films. Some samples were subsequently cut perpendicular to the loading axis to permit examination of the melt distribution in the third dimension. Mosaics were made by stitching together a series of backscattered electron (BSE) images obtained with a JEOL JXA-8900 electron microprobe at an operating voltage of 15 keV, a beam current of 40 nA, and a magnification of 400X (1 pixel = 0.283 μm). In BSE images, anorthite is brighter than forsterite due to its high calcium content. The melt fraction was evaluated from the digital image using Matlab™. To remove high-frequency noise coming from subtle pixel-to-pixel grayscale variations and facilitate segmentation, a median filtering of the grayscale image was performed. The image was then converted into a binary image of melt and solid by gray level slicing. For each pixel, the local melt fraction was averaged over an area of 50 × 50 μm using a convolution algorithm. Grain size was measured from the images by manually tracing grains and re-calculating the grain area to an equivalent circle of a given diameter, d_equ. No sectioning correction factor was applied.

3 Results

3.1 Mechanical Data

During an extrusion experiment, the length of the extrusion assembly was recorded via the internal DCDT that follows the advancement of the Mo piston into the Mo rod. At 1350°C, the partially molten sample flowed into the extrusion channel under the applied load. Material in the sample reservoir, far from the extrusion channel, compacts under the applied load. The shear strain rate, , of the material passing through the channel can be calculated as a function of the displacement rate, , and the radius of the channel r:

(1)

The stress was calculated by assuming that the load was distributed uniformly across the channel area. To determine the rheological behavior of the partially molten rocks being extruded, the sensitivity of strain rate to stress was calculated assuming that . The stress exponent of the forsterite + anorthite solid-liquid composite was n = 1.3 ± 0.2 as documented in Figure 3, indicating that the rocks predominantly deformed by diffusion creep. This value for stress exponent is slightly lower than that of n = 1.6 obtained by King et al. (2010) on samples of olivine + chromite + 4 vol% mid-ocean ridge basalt (MORB) deformed at 1200°C.

Figure 3

Shear strain rate vs shear stress data from five extrusion experiments. Only five of the eight experiments yielded rheological data due to various technical failings of the DCDTs recording displacement. For each load, the shear strain rate corresponds to the shear rate in the pipe assuming a no-slip boundary condition between the sample and the channel wall. The stress was calculated by assuming that the load was distributed uniformly across the channel area. The best fit to the experimental data yielded a stress sensitivity of n = 1.3 ± 0.2 with a coefficient of determination of R² = 0.86.

3.2 Mesoscale Melt Distribution

3.2.1 General Observations

In the following discussion, we will distinguish two main regions within a sample, (a) the sample reservoir and (b) the extrusion channel as defined in Figure 2. In Figure 4b, a map of the melt distribution confirms a homogeneous distribution for the starting material, as well as an average melt fraction of ~0.1 that is best appreciated in the melt profile in Figure 4c. The BSE images and melt distribution maps in Figures 5 and 6 characterize the reservoir and the extrusion channel after an extrusion experiment.

Figure 4

(a) BSE image of the starting material. The image is segmented based on the difference in grayscale between the melt (lighter gray) and the solid (darker gray). (b) Melt distribution map. The melt distribution, calculated in areas of 50 by 50 μm, is homogeneous on the scale of the extrusion channel with a mean melt fraction of ~0.1. (c) Melt profile averaged along the y axis calculated from the melt fraction map in (b). BSE = backscattered electron.

Figure 5

(a) BSE map and (b) melt distribution map of the portion of the pellet remaining in the reservoir for experiment Mx47. On average, the pellet is depleted in melt with an average melt fraction of 0.03 compared to a starting melt fraction of 0.1. Also, melt segregated toward the opening into the channel. A hemispherical, melt-rich region developed beneath the channel opening with a melt fraction >0.1. BSE = backscattered electron.

Figure 6

Melt distribution observed in the extrusion channels of two samples. (a) Sample is viewed parallel to the loading axis for sample Mx1, and (b) sample is viewed perpendicular to the extrusion direction for sample Mx46. (left column) In the BSE images, the anorthitic melt is lighter than the forsterite grains. Porosity (black) is distributed across the channel. (right column) The melt fraction maps were calculated by averaging the melt fraction determined from the BSE images in 50 by 50-μm squares. Note that the melt fraction is higher near the wall than at the center of the channel. BSE = backscattered electron.

Four main characteristics are observed in all samples from our extrusion experiments, as illustrated in Figures 5 and 6. First, the pellet remaining in the reservoir is always depleted of melt on average; the mean melt fraction in the sample reservoir in Figure 5, a good representation of all experiments, is ~0.03, much smaller than the starting melt fraction of 0.1 observed in Figure 4. Second, a plug of material with an average melt fraction close to that in the starting material is observed at the tip of the material extruded into the channel, as shown in Figure 6a. Third, roundish pores are present in the quenched melt within the channel, as visible in Figure 6. These pores are less abundant in the sample reservoir. Finally, the partially molten material in the channel exhibits clear segregation of melt from the center toward the cylindrical wall of the channel.

3.2.2 Sample Reservoir

As illustrated in Figure 2, our experimental design involves extrusion of a partially molten sample from a 6.3-mm diameter reservoir into a 2.0-mm diameter channel. Examination of the melt distribution in Figure 5, typical of all experiments, reveals that the melt fraction increases from as low as ~0.01 at the outer diameter of the sample reservoir to ~0.2 near the channel entrance; the latter value is higher than the initial melt fraction in the pellet. This hemispherical volume enriched in melt near the channel has a diameter similar to that of the channel and extends ~1 mm down form the channel entrance into the reservoir. These observations demonstrate that the reservoir compacts far from and decompacts close to the channel entrance.

3.2.3 Extrusion Channel

BSE images of sample material in sections of a channel parallel to, Figure 6a, as well as perpendicular to, Figure 6b, the loading axis together with melt distribution maps, Figures 7a and 8, clearly document the segregation of melt toward the channel wall. This increase in melt fraction is complementary to the decrease in grain fraction, as illustrated in Figure 8b. Grain size is approximately constant at ~10 μm across the whole channel. Porosity is also approximately uniformly distributed across the extrusion channel, as noted in Figure 8c. The size of the pores is on average 0.5 to 15 μm. The volume fraction occupied by the pores is relatively small, between 0.02 and 0.07.

Figure 7

Melt fraction, grain fraction, and porosity fraction profiles for the extrusion channel in sample Mx1. (a) Melt fraction increases gradually from the center toward the wall of the channel. (b) The forsterite grain fraction follows a complementary evolution. (c) The porosity appears to be randomly distributed across the channel.

Figure 8

Melt profiles for samples extruded at different loads and for different times. For longer extrusion times, the melt fraction increases sharply near the wall with a melt fraction up to 16 times higher than at the channel center, giving the profile a box-like shape. For shorter extrusion times, the melt fraction increases gradually from the center to the edge of the channel; the melt fraction at the wall is two to four times higher than at the channel center.

The melt fraction at the wall of the channel is at least twice that at the channel center. Also, the shape of the melt segregation profile appears to be correlated with load and/or time at T = 1350°C. On the one hand, samples from lower load, slower extrusion rate experiments are melt depleted in the channel center (~0.05 melt fraction) relative to the starting material (0.1 melt fraction) with a sharp and significant increase in melt fraction at the walls (up to 0.8), that is, 16 times higher than at the channel center, giving the melt distribution profiles a box-like shape, as in Figure 8. On the other hand, samples from higher load and, therefore, faster extrusion rate experiments exhibit a more gradual increase in melt fraction toward the channel wall. The melt fraction at the wall is 0.2 to 0.3, a factor of 2 to 4 times higher than in the channel center, where the melt fraction is 0.05 to 0.15 (Figure 8). Currently, it is difficult to conclusively discriminate between the effects of time and load from our experiments as we lack a statistically significant number of samples for which only one of the variables is systematically changed.

3.2.4 Influence of Annealing Time on the Melt Segregation

In Figures 9 and 10, BSE images, melt distribution maps, and melt fraction profiles illustrate the degree of melt segregation that occurred for three extrusion experiments carried out at 1350 °C under the same load (24 kg, an average stress of 75 MPa acting on the channel area), but with different annealing histories. Sample Mx52 was quenched immediately after extrusion, sample Mx54 was annealed for 96 hr after extrusion, and sample Mx47 was annealed for 12 hr before extrusion. In Figure 10, all three extrusion runs reveal clear segregation of melt from the center toward the wall of the channel. The annealing time before extrusion had a large influence on the overall melt content: the channel is overall richer in melt, and the melt fraction at the wall is higher than 0.6. Also, the integrated melt profile of sample Mx47 displays more internal variation of melt fraction than the samples that did not involve an annealing step before the extrusion (Mx54 and Mx52), as documented by the amplitude of the peaks in the melt fraction profile. Annealing after the experiment did not significantly affect the melt distribution that developed during extrusion experiment; in the two cases, the melt segregation is similar. The grain size distribution (Figure 9, right) is lognormal for both samples that were not annealed before extrusion. The post-extrusion annealed sample has a mean grain size of 18 ± 8 μm, whereas the immediately quenched sample has a mean grain size of 13 ± 5 μm. That is, the standard deviations of these two grain size distributions are proportional, or in other words, the distributions are self-similar. Annealing before extruding has a more significant effect on the grain size and grain size distribution, which is better described by a normal distribution with a mean grain size of 21 ± 9 μm. Furthermore, the pores are larger, d_equ = 25–35 μm, and occupy a larger fraction, 0.05–0.07, in samples that were at temperature for longer times of 60–100 hr, than in the shorter experiment lasting 5 hr where the grain size, d_equ ≈ 12 μm and the volume fraction of pores is ~0.02, as documented in Figure 9.

Figure 9

BSE image (left), melt distribution map (center), and grain size distribution (right) for samples extruded under the same load but subjected to different static annealing conditions. (a) Mx52 was quenched immediately after extrusion, (b) Mx55 was annealed for 96 h after extrusion, and (c) Mx47 was annealed for 12hr before extrusion. Melt segregation toward the wall of the channel is evident in all three samples. BSE = backscattered electron.

Figure 10

Profiles of averaged melt fraction for samples Mx47, Mx52, and Mx54. Averaging was performed on the images in Figure 8 by integrating the melt fraction from top to bottom of each melt distribution map. All three samples were extruded at a stress of 75 MPa but with different annealing treatments. Mx52 was extruded for 2 hr but not annealed, Mx54 was extruded for 2 hr and then annealed for 96 hr, and Mx47 was annealed for 12 hr and then extruded for 48 hr. All anneals were carried out at T = 1350°C.

4 Discussion

4.1 Melt Segregation in the Viscous Anisotropy Theory

The numerical simulations based on two-phase flow theory with viscous anisotropy in Allwright and Katz (2014) yielded melt profiles with melt segregation toward the wall of the channel. The two-phase flow theory describes different regimes depending on the nondimensional ratio between compaction length and sample size, where the compaction length characterizes the intrinsic length scale over which pressure gradients in the fluid phase can be maintained (McKenzie, 1984). The compaction length is parametrized as

(2)

where k is permeability, ζ is bulk viscosity of the partially molten rock, η is shear viscosity of the partially molten rock, and μ is the melt viscosity. In equation 2, k is parametrized as

(3)

where d is grain size, ϕ is melt fraction, r is the melt fraction exponent, and C is a constant. For compaction lengths smaller than the sample size, Allwright and Katz (2014) predicted that melt-rich sheets should extend inward from the channel wall inclined at a low angle and antithetic to the imposed flow direction, similar to those predicted and observed for general shear and torsional deformation geometries (Takei & Katz, 2013; Holtzman et al., 2003; King et al., 2010; Kohlstedt et al., 2010; Qi et al., 2015; Qi & Kohlstedt, 2018). For compaction lengths larger than the sample size, only base-state segregation toward the wall of the channel was predicted (Allwright & Katz, 2014).

4.2 Melt Segregation in Our Samples

In all of our extrusion experiments, melt segregated toward the wall of the channel, but melt-rich sheets did not develop (Figures 6 and 9). Previous general shear experiments on ~0.5-mm thick samples of olivine + 4% MORB did not produce segregation of melt into melt-rich sheets either (Kohlstedt & Zimmerman, 1996; Zimmerman et al., 1999; Holtzman et al., 2003; Daines & Pec, 2015). However, the addition of small (~1 μm) particles of chromite to olivine + MORB mixtures lowered the permeability enough to shorten the compaction length below the sample thickness such that melt-rich sheets developed (Holtzman et al., 2003). To calculate the compaction length for our solid-liquid aggregates, we used a grain size of 20 μm and a starting melt fraction of 0.10. With the parameters C = 58 and r = 2.6 in equation 3 (Miller et al., 2014), k ≈ 2 × 10⁻¹⁴ m². The viscosity of the anorthitic melt at 1350°C is ~2 Pa s (Giordano et al., 2008). The bulk viscosity for partially molten rocks is taken as 5/3η (Cooper, 1990; Renner et al., 2003; Takei & Holtzman, 2009), and the shear viscosity, as determined from Figure 3, is ~1 × 10¹³ Pa s. Based on equation 3, these values yield a compaction length of ~10⁻¹ m, significantly larger than the sample size of ~10⁻³ m. As extrusion proceeds, both melt fraction and grain size (thus permeability and viscosity) will evolve. However, the relatively large compaction length insures that melt-rich sheets will not form.

4.3 Differences Between Experiments and Numerical Simulations

While the radial segregation of melt is in agreement with the prediction of two-phase flow theory including viscous anisotropy for Poiseuille flow (Allwright & Katz, 2014), three results are worth emphasizing. First, our extrusion experiments only roughly approximate a Poiseuille flow geometry in which a deformable material flows through an infinitely long channel in response to a pressure gradient. With our extrusion design, the material in the reservoir compacts such that regions of the reservoir far from the extrusion channel are highly depleted of melt. A hemispherical region in the reservoir, immediately beneath the channel, with a radius approximately equal to that of the channel remains enriched in melt. The average melt fraction in the channel exceeds the melt fraction in the starting material.

Second, in several of our samples, the melt fraction at the wall of the extrusion channel exceeds the rheologically critical melt fraction or disaggregation threshold of ~0.3 for the olivine-basalt system (Scott & Kohlstedt, 2006). At this point, the microstructure corresponds to a magmatic suspension, as observed in Figures 7b and 10c. Two-phase flow theory relies on some grain-to-grain contract to transfer stresses in the solid skeleton. Once grains are fully surrounded by melt, the assumption of deformation by diffusion creep breaks down, and the bulk of the deformation could be taken up by the high-melt fraction layer close to the walls hence leading to a box-like strain and melt distribution profile. Figure 11a illustrates a prime example of a high melt fraction region with the solid grains surrounded by melt. No evidence of dendritic crystallization exists in these melt-rich regions in our samples, as the presence of grains stabilizes the melt phase (Raj, 1981). Although these high melt fraction areas likely take up a significant amount of strain, microstructural evidence indicates that the entire channel continues to deform. Even with initial melt segregation to the corners connecting the channel to the reservoir, the center of the channel still experiences deformation. The presence of the melt-enriched hemisphere in the reservoir suggests that the channel should start out melt enriched in the center. However, once material enters the channel, melt segregates toward the wall. This segregation continues as the sample moves up the channel. In Figure 11, the melt distribution is delineated with BSE micrographs taken at the edge (near the wall) and near the center of the channel at three different positions. The bottom pair of images, taken in the reservoir, reveal a higher melt fraction in the center than at the edge; in contrast, the top pair of images show the opposite; that is, the melt fraction is higher at the channel wall and lower at the channel center. As one moves up the channel, near the wall the melt fraction increases with increasing distance from the reservoir, while near the center the melt fraction decreases. This segregation of the melt toward the wall indicates that a stress gradient exists between the wall and the center of the channel, even if the boundary condition is not strictly a no-slip condition at the wall. In addition, most of our experiments display a parabolic melt distribution profile due to base-state melt segregation, as predicted for by two-phase flow theory with anisotropic viscosity (Katz & Takei, 2013; Takei & Katz, 2013).

Figure 11

Backscattered electron micrographs of a typical extrusion sample. The drawing at the bottom indicates the position in the channel or reservoir of each of the images. (a) Adjacent to the channel wall, and below the melt-depleted plug. Note the Mo wall is visible on the left hand side. A good example of the very high melt fraction at the well, completely free of dendritic quench crystals. (b) Center of the channel, just below the melt-depleted plug. A much lower melt fraction than its partner in (a) at the edge of the channel. (c) Base of the channel, adjacent to the wall. Note the lower melt fraction than (a), but higher than (e) and (d). (d) Base of the channel, in the center. Note the melt fraction is higher than (b), but lower than (c) or (f). (e) Directly below (a) and (b), in the reservoir. Just to the edge of the melt-enriched hemisphere. The lowest melt fraction of its column and row. (f) Centered below the channel, in the melt-enriched hemisphere. Highest melt fraction of its row and column.

Finally, pores are present in the melt phase. The amount of porosity is greater after (from 0.02 to 0.07) than before (0.01) extrusion. The pore distribution across the extrusion channel is nearly homogenous with no clear radial segregation within the channel (e.g., Figures 7a and 8); however, there is a gradient in bubble fraction along the extrusion direction with larger porosities at the end of the extrusion channel, and smaller porosities close to the melt reservoir. The pores are likely filled with air at the experimental conditions, which surrounds the samples.

4.4 Implications for Melt Segregation in Nature

Our experiments demonstrate that melt segregation in a Poiseuille flow setting leads to a clear segregation of melt toward the wall of the channel. Interestingly, such melt segregation was inferred based on strain distribution in paleo-melt channels as recently described in the ultramafic Twin Sisters Massif (Kruckenberg et al., 2013). While our experiments present a highly simplified system primarily aimed at testing two-phase flow theory with viscous anisotropy, some parallels with natural melt extraction can be drawn. Tabular dunite bodies are typically interpreted as pathways for melt migration in the upper mantle and are thought to form by a “reaction infiltration instability” in a deforming mantle (Aharonov et al., 1995, King et al., 2011a, 2011b, Pec et al., 2015, 2017). If flow through a mature channel, however, is described by parallel-plate Poiseuille flow, then the observed segregation of melt toward the walls could be a natural consequence of base-state segregation in the viscously anisotropic partially molten mantle rocks. Hence, detailed field studies might elucidate the melt fraction profile as a function of position, which then could be used to constrain the magnitude and orientation of viscous anisotropy in naturally deforming rocks. Furthermore, given that base-state segregation is not expected to occur in simple shear because no shear stress gradient is present (Katz & Takei, 2013; Takei & Katz, 2013), the distribution of melt could be used to assess the relative importance of simple shear and parallel-plate Poiseuille flow in a dunite channel.

5 Conclusions

Geodynamical models often take into account length scales much larger than the grain scale, and hence, such processes as mesoscale stress-driven melt segregation that arise from the grain-scale properties are not appropriately captured. Given the influence of melt topology and melt distribution on a number of transport properties of partially molten rocks, such as permeability, conductivity and viscosity, an appropriate description of the physics of melt migration is critical for advancing our understanding of fluid flow in a number of tectonic settings. The fact that two-phase flow theory incorporating anisotropic viscosity holds for a number of flow geometries tested in simple experimental settings with well-defined boundary conditions lends credence to the possibility of implementing the theory into larger scale geodynamical models. Observations of anastomosing dunite channels (e.g., Braun and Kelemen, 2002) with a topology similar to that formed by stress-driven melt segregation in experiments (Holtzman et al., 2003; King et al., 2010) together with inferred melt segregation toward walls of dunite channels in nature (Kruckenberg et al., 2013) provide additional clues that viscous anisotropy is important during migration of melt in partially molten rocks on the Earth and, by extension, on other silicate planetary bodies as well. The discrepancies between experimental results and theoretical predictions based on the viscous anisotropy hypothesis, for instance, the presence of large-scale structures not predicted by the model, are also of interest since they point toward future areas of research that will help better understand natural observations.

Acknowledgments

We would like to thank Anette von der Handt for the help with the electron microprobe maps, Mark Zimmerman for his assistance with the 1-atmosphere rig and the helpful reviews of Laurent Arbaret and Yaoling Niu. Funding by NSF proposals EAR-1520647 to D. L. K. and A. Q. T. and EAR-1753482 to M. P. is gratefully acknowledged. All data are available in the tables and figures.