Volume 49, Issue 19 e2022GL100040
Research Letter
Open Access

Crystal Shape Control on the Repacking and Jamming of Crystal-Rich Mushes

Susana Hoyos

Corresponding Author

Susana Hoyos

Department of the Earth, Atmospheric and Planetary Sciences, Massachusetts Institute of Technology, Cambridge, MA, USA

Correspondence to:

S. Hoyos,

[email protected]

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Darien Florez

Darien Florez

Department of Earth, Environmental and Planetary Sciences, Brown University, Providence, RI, USA

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Matej Pec

Matej Pec

Department of the Earth, Atmospheric and Planetary Sciences, Massachusetts Institute of Technology, Cambridge, MA, USA

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Christian Huber

Christian Huber

Department of Earth, Environmental and Planetary Sciences, Brown University, Providence, RI, USA

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First published: 22 September 2022
Citations: 1

Abstract

The rheology of crustal mushes is a crucial parameter controlling melt segregation and magma flow. However, the relations between mush dynamics and crystal size and shape distribution remain poorly understood because of the complexity of melt-crystal and crystal-crystal interactions. We performed analog experiments to characterize the mechanisms that control pore space reduction associated with repacking. Three suspensions of monodisperse particles with different geometries and aspect ratios (1:1, 2:1, 4:1) in a viscous fluid were tested. Our results show that particle aspect ratios strongly control the melt extraction processes. We identify two competing mechanisms that enable melt extraction at grain scale. The first mechanism leads to continuous deformation and melt extraction and is associated with “diffuse” frictional dissipation between neighboring particles. The second is stochastic, localized, and nearly instantaneous and is associated with the development and destruction of force chains percolating through the granular assembly.

Key Points

  • We identify two coexisting modes of mush deformation: a short-range (continuous) and a long-range (stochastic) granular repacking

  • The relative contribution of these two modes of deformation depends on crystal shape

  • Force chain disruptions can lead to a reduction of the overall resistance to melt extraction

Plain Language Summary

Mushes are magmas with a high crystal content. Melt segregation and extraction from mushes by compaction is a fundamental process that controls the physical and chemical evolution of magmatic systems. Hence, understanding how crystal mushes evolve dynamically bears on our ability to predict volcanic hazards or interpret the magmatic rock record. We carry out experiments with suspensions to investigate how particle-particle-liquid interactions impact fluid extraction. As the mush deforms as a granular medium, we find that there are two competing regimes of deformation, one is short-range (localized) particle-particle interactions, and the other consists of longer-range stress-carrying networks of particles (force chains). The relative proportion of the deformation accommodated by each mechanism depends on the shape of the particles involved. The findings from this experimental study have an application to our understanding of the mechanisms that facilitate melt extraction at grain scale.

1 Introduction

The chemical and physical evolution of crustal magma reservoirs are governed by the mechanical processes that control the separation between melt and crystals. This separation can be driven by gravity (density contrasts) and by external (e.g., regional tectonics) stresses (Bachmann & Bergantz, 2004; Bachmann & Huber, 2016; Davis et al., 2007; Koenders & Petford, 2000; Petford et al., 2020; Quintanilla-Terminel et al., 2019; Solano et al., 2012). Dufek and Bachmann (2010) argued that melt-crystal separation is most effective at intermediate crystallinity (between 0.4 and 0.7) when the crystalline framework resists deformation and forms what is often referred to as a crystal mush. Deformation of a mush generally leads to compaction, here loosely defined as any mechanical process that leads to the decrease of porosity by the expulsion of interstitial melt. Over a range of crystal content and under conditions relevant to crustal magmatic systems, compaction is likely to proceed differently than at high-temperature, high-pressure, and low melt fractions like in the upper mantle (McKenzie, 1984).

Textural and microstructural evidence (Holness, 2018; Holness et al., 2017), as well as rheological models (Bachmann & Huber, 2019; Petford et al., 2020; Vigneresse et al., 1996), suggest that compaction at intermediate crystallinity is a granular process whereby differences in pressure between the melt phase and the crystalline framework lead to deformation of the latter through reorganization/reorientation of the grains rather than their internal deformation. This granular compaction is referred to as mechanical compaction by Holness (2018) or repacking by Bachmann and Huber (2019). While the description of steady-state deformation in granular systems has improved significantly over the past decades (Boyer et al., 2011; Guazzelli & Pouliquen, 2018), a transient description that can be upscaled to the spatial and temporal scales relevant to melt segregation in magma reservoirs remains elusive. On the one hand, a granular description is necessary to understand the importance of local interactions between grains as well as non-local effects, such as the formation of force chains during deformation (Bergantz et al., 2017). On the other hand, these processes are short-lived and stochastic in nature; it remains unclear how and perhaps if they should be parameterized in continuous models looking at these processes over large spatial and temporal scales (Huber & Parmigiani, 2018; Jackson et al., 2018; Keller & Suckale, 2019; Solano et al., 2012).

Here, we performed a suite of analog experiments where compaction by repacking is imposed on granular suspensions immersed in corn syrup. The rate of compaction is imposed experimentally, and the resistance to deformation is measured over time during each experiment. These experiments provide a unique perspective on the nature of compaction in granular mixtures with intermediate solid fractions and highlight the respective role of local, distributed frictional interactions between neighboring grains versus longer range and heterogeneous force chains during repacking in crystal mushes.

2 Materials and Methods

Our experiments represent an analog system for a mush compaction column in the upper crust. The experiments were carried out in a loading stage with a holder for glass syringes (Figure 1a). The body of the syringe was fixed through a holder to the upper plate of the stage, and the syringe plunger was let to move upwards with the lower plate of the stage in a movement similar to an inverted French press. A 5 kN load cell measures applied load (Figure 1a), and a displacement transducer (LVDT) measures the position of the loading plate. All experiments were performed at a constant displacement rate of 0.3 mm/min.

Details are in the caption following the image

(a) Main components of the experimental setup. (b–d) Photographs of the used particles: (b) spheres, (c) 2:1 beads, and (d) 4:1 beads. (e) Table with particle characteristics.

A camera was used to record the motion of the plunger and particles inside the syringe. Video and mechanical data recorded for each experiment were time-synchronized to analyze changes in load and liquid extraction and to quantify the particle's movement. The fluid was extracted via a tube connected to a graduated glass cylinder that, with the camera recordings, was used to measure the extracted fluid. Mechanical data and video recordings of the experiments can be found in the online repository for this paper.

We conducted experiments with mixtures of rigid particles and corn syrup. The particles have two different geometries, spherical beads (Figure 1b, diameter, ø = 2 mm) and two elongated particles with length to width aspect ratios of 2:1 (Figure 1c, length, L:1.5 mm, width, W:3 mm) and 4:1 (Figure 1d, L:1.4 mm W:6 mm). The prolate particles have an inner hole along the long side of 0.4 mm for the 4:1 particles and 0.3 mm for the 2:1 particles. All the particles are made of quartz glass with a density of 2.2 g/cm3 (Figure 1e). This material is rigid at room temperature, and no permanent deformation is observed during compression probing; a similar regime of deformation to mechanical compaction or repacking, as suggested by Holness (2018). Corn syrup was used as the fluid in all the experiments as it is a Newtonian fluid, with a viscosity of 180 Pa·s and a density of 1.6 g/cm3 at 25°C. For each experiment, particles were weighed (Table S1 in Supporting Information S1), measured by volume, placed in a graduated silicate glass syringe with an internal diameter of 20.72 mm, and shaken 50 times to create random packing. Subsequently, corn syrup was added to the syringe from the bottom, bubbles were extracted, and an extra 1 ml of corn syrup was added to ensure a uniform starting volume across the experiments. The experiments were designed to let the particles settle to the bottom of the syringe before the matrix loading; therefore, the variation between initial and final solid fraction values can be attributed to repacking effects alone. A comprehensive description of experimental procedures and detailed data about all the experiments can be found in the supplemental material (Text S2 and Table S3 in Supporting Information S1).

Different bimodal suspensions were also tested by combining spheres and 2:1 ratio prolate particles and spheres and 4:1 ratio prolate particles (Table S1 in Supporting Information S1). Three different proportions of spheres to prolate particles were tested, where the proportion of spheres was increased progressively by 25%. The experiments performed on polydisperse suspensions followed the same experimental procedure as the monodisperse suspensions (Text S2 in Supporting Information S1).

3 Results

Examples of compaction curves are presented in Figure 2. The loading curve of the monodisperse suspensions exhibits three identifiable regimes (Figures 2a and 2b) determined by the change in their slope. (a) The first regime is the loading of the matrix, here defined as the collection of particles, where porosity decreases because of local particle rearrangements. This is recognizable by continuous (smooth) curves that follow a power-law behavior with displacement. (b) The loading regime seems to change in Figures 2a and 2b after 1.5–2 mm displacement as we observe an inflection and a more rapid increase in loading (second regime, present in all experiments). (c) Finally, experiments with prolate particles or bimodal mixtures with more than 25% prolate particles display at least one sudden load drop (third regime) after the sharp increase in load of the second regime (see Figure 2b). After a load drop, compaction curves return to the first regime, building up load with displacement until a new load drop occurs or the experiment concludes. Experiments with spheres do not display these sudden load drop events (regime 3) over the displacement range studied. However, the first two regimes can be identified in all the experiments (Figure S1 in Supporting Information S1).

Details are in the caption following the image

(a–b) Evolution of the applied load with displacement (upward motion of the plunger). Each line represents experiments with different sets of particles. Regimes are highlighted. A load drop in panel b is annotated to show δd and ΔL. (c–e) Colormaps of particle displacement in the x-z plane. Color is proportional to the number of individual displacement vectors in a cell. The average load change in these color maps is 10 N. Further details are provided in Text S2 in Supporting Information S1.

In order to quantify the impact of these sudden load drop events on liquid extraction, we compute the amount of displacement, δd, before and after the load drop event at a fixed (post load drop) load value (Figure 2b). Load drops are defined by a sharp negative slope in the force-displacement curves. δd is understood as the change in displacement accommodated by the load drop event. In experiments that display multiple load drop events, a cumulative displacement, Δd, caused by load drops is computed and compared to the net displacement, overall change in column height, imposed on the two-phase medium once the matrix resists the load. In all experiments, we discard the displacement data that takes place before the matrix is resisting compaction (Tables S2 and S3 in Supporting Information S1). Figure 3 shows how the contribution from load drops on the total displacement compares between experiments and reveals that load drops allow the matrix to accommodate a greater portion of the overall strain in experiments with 4:1 and 2:1 particles (Figure 3a). The trend is less clear with polydisperse suspensions; mixtures of 4:1 particles and spheres show a significant scatter, for example, (Figure 3b).

Details are in the caption following the image

Displacement accommodated during load drops (normalized to net displacement) as a function of particle shape. For (a) monodisperse and (b) bimodal suspensions. Error bars represent uncertainties in measurements associated with the experimental setup (e.g., load cell, mechanical stage).

Similarly, we calculate the work performed to extract melt from the experiments by integrating the loading curves with respect to displacement (approximated with a right Riemann sum (Figure 4 and Text S2 in Supporting Information S1)). Ideally, the calculations would be over identical ranges of fluid fractions. However, given that the fluid fraction at which the suspensions start loading varies from experiment to experiment, overlapping or similar ranges of fluid fractions are limited (see experimental limitations on the supplements Text S3 in Supporting Information S1). We normalize the work performed over the duration of a given experiment by the difference between starting (load onset) and ending porosity (end of experiment) for a given experiment. The exact displacement, porosity, and fluid/particle volume fraction ranges this window corresponds to for each experiment and details about the calculations are listed in the supplements.

Details are in the caption following the image

(a) Mechanical work done from onset of load until the end of the experiment. Values are normalized by the difference in starting and ending porosities (%) over this range. (b) Normalized work done by bimodal suspensions as a function of sphere fraction.

Some of the major load drops (ΔL > 10N), like the one in Figure 2b, can be correlated to particle movement using TrackMate (Tinevez et al., 2017), an image tracking software. To quantify the motion of the particles, we calculated the displacement vector of all the observable particles in the image plane before and after a load drop event. Detailed descriptions of post-processing techniques used to define the motions of the particles can be found in the supplemental material (Text S2 in Supporting Information S1). Figure 2 shows the magnitude of the displacement for the particles during a load drop event in the diameter-height plane. Particle movement during the load drop events is localized in the upper part of the granular assembly where fluid is expelled (Figures 2c and 2d). This observation is consistent for all the load drops, irrespective of the magnitude of the drop. In contrast, during loading regime 1 (smooth increase in loading), in the absence of load drops, displacement is distributed more homogeneously throughout the domain (Figure 2e). Note that out-of-plane movement cannot be measured.

4 Discussion

4.1 Two Coexisting Modes of Granular Mush Compaction

Particle tracking during background loading (regime 1) and load drop events (regime 3) reveal different modes of mush deformation. In the former, displacement and, therefore, pore space reduction are distributed homogeneously throughout the domain, as one would expect if the effective compaction length (McKenzie, 1984) is significantly greater than the height of the domain. Our results suggest that pore space reduction is accommodated by granular repacking through rotation and translation limited by friction between neighboring particles (short-range interactions). In contrast, load drop events are associated with a more heterogeneous movement of the particles near the top of the assembly, which, together with the short displacement (δd) covered during the event and dramatic change in loading observed, suggest that they reflect the build-up and disruption of force chains. These force chains are long-range interactions between particles that oppose deformation before yielding, which results in particle rotation and translation mostly along the former chain.

4.2 Effect of Crystal Shape

Load drop events are not observed in the experiments involving only spherical particles. Although kinks in the loading-displacement curves seem to suggest that force chains are developing in these experiments, over the short amount of accumulated displacement, additional displacement did not require their disruption. However, the amount of displacement for these experiments was limited by the maximum loading the experimental assembly could withstand. Experiments with prolate particles accommodate a greater portion of the overall displacement and fluid extraction by sequences of jamming (regime 2) and load drops (regime 3) (Figure S3 in Supporting Information S1). Donev et al. (2004) show that elongated particles have higher rotational degrees of freedom and higher conjectured maximum packing density than spheres. These two effects should contribute to reducing the probability of reaching a jamming state and possibly the frequency of load drop events. Our experimental results suggest that another factor is responsible for the occurrence and higher frequency of load drop events for non-spherical particles. Along their long axis, the elongated particles used in our experiments are larger than the spheres and require fewer particles to be jammed into a force chain that spans across the domain. For example, assuming a uniform distribution of orientation for elongated particles, 2:1 and 4:1 particles require fewer particles to create a percolating network across the height or width of the container by factors of 1.5 and 2.4, respectively, compared to the spheres (Text S1 in Supporting Information S1). The greater probability of developing long-range force chains, rather than the degrees of freedom for rotation, explains why our experiments observed load drop events with elongated particles but not with spheres. Another consideration is that the load drop magnitude of the monodisperse suspensions during these events scales on average with the length and shape anisotropy of the particles (Figure S2 in Supporting Information S1), given that rotation of particles alone can significantly impact the packing efficiency.

4.3 Effect of Force Chain Disruption on Resistance to Melt Extraction

While our data might suggest that the mechanical work involved in extracting comparable volumes of melt over a similar porosity range is weakly dependent on particle shape (Figure 4), the most significant result is the ∼5-fold variation in work exerted for the 4:1 experiments with available fluid fraction data, with the work performed for SH010 being ∼410 mJ compared to, SH016 being ∼80 mJ per percent melt extracted (Figure 4). Upon inspection of the loading curves (Figure S1 in Supporting Information S1), the difference in mechanical work stems from the contrast in the efficiency of force chain disruption in reducing the load. As revealed by the occurrence of regime 2, force chains develop in all experiments and increase the resistance to matrix deformation and liquid extraction. Despite the need for further, more controlled experiments to get a qualitative understanding of how force chain disruption affects mush rheology during compaction, our results suggest that the resistance to repacking is modulated by the efficiency of force chain disruption, specifically the frequency and magnitude of the load drop events. Given that elongated particles are more likely to develop long-range force chains since fewer particles are necessary to span the full domain, the weak dependence of mechanical work on particle shape is likely explained by the larger load drops (more efficient repacking) that result from their disruption (Figure 4).

4.4 Implications for Melt Extraction in Crustal Magma Reservoirs

The rate and extent of gravity-driven phase separation in crustal magma reservoirs at mush crystallinities depend on the rheology of the crystal matrix (McKenzie, 19841985; Wickham, 1987). Though the main driver of phase separation in our experiments is an applied force on the piston, the evolution of the load curves similarly depends on the rheology of the matrix (Bercovici et al., 2001). Our results show that deformation of the mush is accommodated by repacking (Bachmann and Huber (2019); and Holness (2018)'s mechanical compaction) over a fluid fraction range of ∼0.7–0.4 (Figure S3 in Supporting Information S1) and resisted by force chains to varying extents depending on the particle size and distribution. Previous compaction models (Jackson et al., 2018; Solano et al., 2012) at mush crystallinities have employed effective matrix viscosities informed by viscosities associated with the deformation of individual matrix grains (Holness (2018)'s viscous compaction), which are likely much greater than the effective viscosity of compaction through repacking, even though the presence of force chains may act to increase the resistance of the mush to deformation during this regime. At lower fluid fractions, there is likely a rheological transition where the effective matrix viscosity sharply increases as further compaction must be accommodated by the deformation of the individual grains (Bachmann & Huber, 2019; Holness, 2018). By exploring deformation at mush crystallinities, we offer better constraints to parameterized phase separation models over a complete range of fluid fractions (i.e., Wong & Keller, 2022).

For many silicic magma reservoirs in the upper crust, the phases comprising the matrix are assemblages of crystals with different shapes, with tabular crystals (e.g., feldspars, plagioclase), equant crystals (quartz), as well as other accessory minerals such as ilmenite or zircon (Claiborne et al., 2010; Petford et al., 2020). The exact phase assemblage and the resulting particle size distribution depend on the reservoir's thermodynamic conditions and cooling history. Our results suggest that the textures that form under these conditions contribute to the rheology of the matrix in that (a) increasing the effective radius of the particles comprising the matrix tends to increase the likelihood of forming longer-range force chains and that (b) the presence of equant phases in mixtures of elongate particles pins force chains, making them harder to break. However, our experiments likely represent an end member regime where force chains dominate the rheology of the matrix, as the largest dimension of the domain is only larger than the effective particle radius by a factor of ∼4. For crustal silicic magma reservoirs, the effective particle size is many orders of magnitude smaller than the largest dimensions of the domain. Furthermore, the compaction length associated with our experiments is much greater than the domain size. However, the compaction length for crustal silicic magma reservoirs is likely much smaller than the domain size (McKenzie, 1985), and the average strain rate of ∼2e−4 s−1 for the experiments is much greater than those found in nature. The impact of these disparities on the effective matrix rheology and the extent to which force chains play a part in resisting compaction are unclear but should be explored in future experiments with a larger domain size (or smaller particles), as well as experiments with shorter compaction length.

Furthermore, how the effective matrix viscosity and the prevalence of force chains, and therefore the efficiency of melt extraction, are augmented by shear is also unclear. For instance, Boyer et al. (2011) studied the rheology of granular media with shear stress-driven melt extraction experiments. The incorporation of shear stress in melt extraction experiments may likely aid in disrupting force chains. In crustal silicic magma, chambers, margins, and near contacts with new injection pulses can impose shear stresses which, through localized dilation, may enhance melt extraction (Daines & Pec, 2015; Kohlstedt & Holtzman, 2009). The effect of these more complex stress conditions needs to be further studied.

Applications of compaction models to experimental data such as those presented here will provide the opportunity to explore the systematic variation of effective matrix viscosity with particle shape and size distributions. It will also provide a framework to understand how load drop events affect the state of the particle assembly and therefore the compaction of the matrix. These studies are necessary to assess how particle-particle friction contributes to the matrix rheology, which is essential to the development of a suitable model to describe melt extraction at the intermediate crystallinities in magma reservoirs.

5 Conclusions

The experiment's loading curves (Figure 2 and Supplements) reveal a continuous component of granular deformation that we identify with nearest neighbor particle-particle interactions acting homogeneously throughout the domain as well as a stochastic component of deformation that we identify with long-range force chain disruptions. While the former can be parameterized within upscaled granular rheology (e.g., Boyer et al., 2011) within continuum models, the intermittent behavior of the latter would require a different strategy. As this study aims at investigating the different regimes of granular repacking and how they link to the nature of the particles forming a mush, it also brings up important challenges to continuum modeling efforts of mush compaction:
  • Granular repacking involves particle-particle friction, the rheology of which may differ significantly from the matrix rheology laws generally used when dealing with compaction at low melt fractions (Bergantz et al., 2017; Cooper & Kohlstedt, 1986; Hirth & Kohlstedt, 2003).

  • Force chain build-up and disruption, a typical process of granular deformation, introduce a stochastic component that can significantly impact the mechanical work employed to extract melt during repacking.

  • The frequency and magnitude of load drop events (cycles of force chain build-up and disruption) seem to depend on the particle's shape and size distribution. We observe, in general, a greater frequency and magnitude of load drop events for bigger and more elongated particles.

  • Further experiments are needed to better constrain the factors that control load drop events and study how they modify the rheology of the granular medium before and after disruption. Recent advances in developing stochastic rheology models in granular media (e.g., Bakarji & Tartakovsky, 2020) may allow for a continuum scale parameterization of these discrete and episodic events.

Acknowledgments

This material is based upon work supported by the National Science Foundation under Grant No. EAR-2021328. Laboratory technician support by NSFEAR- 2054414 is gratefully acknowledged.

    Data Availability Statement

    Datasets including videos of experiments, codes for the mechanical data evaluation and additional figures are available at: https://osf.io/qgte2/.