Vesiculation of Rhyolitic Melts Under Oscillatory Pressure

Magma ascending in the Earth’s crust can undergo oscillations in pressure, from ultra-low frequency changes associated with tectonics, to relatively higher frequency oscillations associated with seismicity. Seismic waves travelling through shallow magma bodies can lead to a range of unrest phenomena and potentially trigger volcanic eruptions. The mechanisms by which pressure oscillations can induce unrest or eruption remain debated. Here, we experimentally impose pressure oscillations on magma and study how they affect vesiculation processes. We use cylindrical samples (4.00 mm long, 4.85 mm diameter) of hydrous rhyolitic obsidian (0.11 ± 0.01 wt% H2O) placed in alumina (AL23) crucibles and vary pressure by the uniaxial loading of an alumina plunger in a thermo-mechanical analyzer. We monitor vesiculation at temperatures of 950–990°C and confining pressure of 177 kPa. We perform two types of experiment: 1) “static” experiments (at constant pressure) and 2) “oscillating” experiments in which we impose sinusoidal pressure oscillations of up to 71 kPa upon the static pressure (i.e., between 106 and 250 kPa). In both cases, we dilatometrically observe sample expansion driven by vesiculation. Post-experimental bubble textures reveal that bubbles formed preferentially at the sample margins. For the oscillating experiments, the sample expansion rate is lower than in the static experiments, and there are fewer vesicles at the sample margins. We examine the constituent processes of bubble formation (nucleation, growth, coalescence) and gas loss (diffusion, permeable flow) occurring during static experiments and with the added element of pressure oscillations. The most likely mechanism responsible for reduced sample expansion is that pressure oscillations drive the sample in and out of water saturation conditions and thus reduce the fraction of residence time over which bubble nucleation and/or growth are driven. Future work will be needed to confirm this hypothesis. These results are relevant to the study of earthquake-volcano interactions, where a magma body that sits close to volatile saturation is subject to pressure fluctuations.

Pressure fluctuations are highly likely to occur during magma storage and transport. The source of pressure oscillations may be external (e.g., earthquake or tidal; Manga and Brodsky, 2006;Girona et al., 2018) or internal (e.g., unsteady and pulsatory magma flow or gas-filled cavity resonance; Chouet, 1996;Wylie et al., 1999;Longo et al., 2012). In particular, constraining the effects of seismic waves on vesiculating magma is one key to understanding earthquake-volcano interactions, as evidence indicates that seismic waves can trigger a range of volcanic responses (e.g., Hill et al., 2002;Manga and Brodsky, 2006;Eggert and Walter, 2009;Sawi and Manga, 2018) depending on specific conditions dictated by the volcano type and the earthquake characteristics (Seropian et al., 2021). Masotta and Keppler (2017) employed a hydrothermal moissanite cell, enabling direct optical observation of bubbles growing in a melt under pressure oscillations. They investigated mean pressures of 50 and 60 MPa, with oscillation amplitudes between 5 and 40 MPa, but their experiments were limited to samples <100 µm thick. We seek to extend their analysis to larger samples. Here, we employ thicker samples (4 mm), in experiments based on confined or "contained" dilatometry (Bagdassarov et al., 1996;Stevenson et al., 1997;Gottsmann et al., 1999). This dilatometric approach enables the volumetric monitoring of vesiculation. We report the results of a series of high temperature (950-990°C) experiments where pressure oscillations are imposed on vesiculating rhyolitic melt. We focus on how the various bubble processes (nucleation, growth, coalescence and gas loss) are affected by the oscillations. We then explore the relevant geological conditions where our results could be applied and examine the potential applications of this novel type of experiment.

Starting Material
All samples employed here were prepared from a single~20 × 10 × 10 cm block of aphyric, bubble-free peraluminous rhyolitic obsidian, collected from the AO sample site of Tuffen and Castro (2009) at Hrafntinnuhryggur, Krafla, Iceland ( Figure 1A; Table 1). We drilled cylindrical cores (4.85 ± 0.05 mm diameter) from the block, and cut them into 4.00 ± 0.05 mmlong samples using a diamond wire saw. A thin disc was also obtained from between each successive sample, and prepared as a free-standing wafer, polished on both surfaces, for Fouriertransform infrared (FTIR) spectroscopy.
Water content was determined via FTIR spectroscopy using a Bruker Vertex 70 spectrometer attached to a Bruker Hyperion 2000 IR-microscope at the University of Canterbury. The spectrometer is equipped with a Globar source, an MCT detector cooled with liquid N 2 and a KBr beamsplitter. The spectrometer and microscope were continuously purged with N 2 gas. Samples were resting on an IR-invisible BaF 2 window. Transmittance spectra were recorded at 10 aligned spots, with a 100 µm square aperture limiting the beam, in the wavenumber range 6,800-400 cm −1 , with a resolution of 4 cm −1 , using 128 scans and corrected against a background measured before each set of sample measurements. We measured the height of the 3,570 cm −1 peak above a linear baseline and used a density of 2,460 kg m −3 , calculated with DensityX, for the composition given in Table 1 at 20°C and 1 bar (Iacovino and Till, 2018). Similarly to Tuffen and Castro (2009), we assumed that all of the water is present as OH − and used a molar absorptivity coefficient of 100 ± 2 L mol −1 cm −1 (Newman et al., 1986). Reflectance spectra were also collected at all measurement locations and used to calculate local thickness where possible, following the method of Wysoczanski and Tani (2006) with a refractive index of 1.5, appropriate for rhyolite glasses (Long and Friedman, 1968). Alternatively, in cases where reflectance spectra did not allow thickness to be determined, we measured thickness manually with a digital displacement micrometer (following von Aulock et al., 2014). All our samples are homogeneous within measurement uncertainty with an average water content of 0.11 ± 0.01 wt% ( Figure 1B, full details in Supplementary Material).

Experimental Apparatus and Procedure
Our experimental setup is illustrated in Figures 1C, D. The geometry is similar to that of Bagdassarov et al. (1996), Stevenson et al. (1997) and Gottsmann et al. (1999). The prepared obsidian samples were placed inside crucibles composed of an alumina ceramic tube (DEGUSSIT AL23 produced by Friatec) with outer and inner diameters of 8 and 5 mm (±5%), respectively, and lengths of~12 mm. Tight-fitting cylindrical bases of the same material were prepared and inserted at the bottom, in order to avoid leaks. A cylinder of the same material was inserted above the obsidian sample, inside the crucible, to act as a plunger. Plungers were cut to lengths of~10 mm and lathed to a diameter of~4.9 mm so that they could fit snugly inside the crucible but still slide without resistance. The pre-experimental smooth sliding indicates the existence of a narrow space between the plunger and the crucible, such that the sample assembly can be assumed not to be gas-tight during the experiments. Every piece of ceramic used in the experiments was first heat-treated at 1200°C for 3 h, to remove impurities, and minimize potential geometrical changes to the sample assembly during the experiments.
These dilatometric experiments were performed in a NETZSCH TMA 402F1 Hyperion thermo-mechanical analyzer (TMA) at the University of Liverpool. The TMA consists of a sample holding platform and a vertical alumina push-rod, all enclosed in a silicon carbide furnace. The digital resolution of the push rod vertical position is 0.125 nm and resolution of the applied force is <0.01 mN. A fixed frame limits the rod's upward motion, and serves as support for a K-type thermocouple, which is located within 5 mm of the sample, and exposed to the furnace gas. The crucible containing the obsidian sample and ceramic plunger was placed onto the   Tuffen and Castro (2009). The mean and standard deviation were computed from 100 analyses.

Oxide
Mean ( platform, resting on the oscillating rod. For each experiment, initially a calibration was performed using a ceramic standard with dimensions that were within 3% of those of the obsidian sample, in order to quantify the geometrical response of the system (i.e., push-rod, crucible, base plate and plunger) to the precise temperature path. These calibration data were then subtracted from the data obtained during experiments on the actual sample to constrain precisely the sample response. Experiments were run in an Ar atmosphere to avoid melt hydration and inhibit melt oxidation. In order to reduce discrepancies between experiments, we first applied a constant force of 1 N for 10 min, at a temperature of 50°C. Next, a load of 1.5 N was applied and maintained as the temperature was increased to the target isothermal value (between 950 and 990°C) at a constant heating rate of 10°C min −1 . The force applied by the rod, the vertical position of the rod, the furnace temperature and the Ar flow rate were monitored at a sampling rate of 2 Hz throughout the experiment. At the end of the experiment, the samples were cooled at 10°C min −1 .

Post-Experimental Analysis
After each experiment, the samples were mounted in epoxy and cut parallel to the ends into halves: one half for textural analysis and one to measure water contents. The first half was polished and gold coated for observation with a JEOL IT-300 scanning electron microscope (SEM) at the University of Canterbury. Backscattered electron (BSE) images were acquired with a 15 kV accelerating voltage and a 40 nA beam current at a 10 mm working distance. We quantitatively estimated the bubble fraction using the BSE images. We first manually delineated bubbles in Inkscape and produced segmented ternary images with pixel colors representing either glass (black), vesicles (white) or crucible (gray). Bubbles were further manually divided between core and rim bubbles. Bubble fraction was then computed from the number of relevant bubble pixels (core or rim) divided by the sum of bubble and melt pixels.
The second half of the sample was polished on both sides to create a wafer for FTIR spectroscopy. Post-experimental water contents were determined via FTIR spectroscopy following the same procedure as was employed for measuring pre-experimental water contents.

Experimental Conditions
Preliminary experiments were undertaken in order to assess our experimental methods, to optimize our sample and crucible preparation, and to identify "ideal" experimental conditions (i.e., windows of temperature and timescale where vesiculation occurs at a manageable rate and is reproducible). We chose temperatures in the range 950-990°C, yielding viscosities of 10 7.4 -10 6.9 Pa s, for a water content of 0.11 wt%, according to the viscosity model of Giordano et al. (2008). To ensure our samples reach a relaxed state during experiments, we compare the shear relaxation timescale τ rlx to the time of observation τ obs , via analysis of the Deborah number De τ rlx /τ obs . The Deborah number allows us to assess whether a material has time to viscously relax a stress in the time of interest (e.g., Wadsworth et al., 2018). At De >> 1, viscous relaxation is inefficient on the timescale of oscillation and elastic forces dominate. Conversely, at De << 1, the material can relax and elastic forces may be negligible. The shear relaxation timescale of the melt can be computed as τ rlx η/G, where η is the shear viscosity and G ≈ 10 10 Pa is the elastic shear modulus (Dingwell and Webb, 1989). At our experimental conditions, we obtain relaxation timescales in the range 10 −3.1 -10 −2.6 s. The highest oscillation frequency, f, experimentally employed here is 0.1 Hz, i.e., an observation timescale of 10 s. The Deborah numbers of our experiments therefore lie in the range 10 -4.1 -10 -3.6 . Our highest Deborah number is 10 -3.6 << 1, implying that we can neglect elastic effects. All the symbols used in the text are summarized in Table 2.
During an experiment, lateral melt expansion is contained by the presence of the crucible. The experiments have also been designed such that the space between the plunger and the crucible is too narrow to permit melt extrusion along the narrow annulus between the containers' end pieces and cylinder during these experiments (e.g., Gotsmann et al., 1999;Yoshimura and Nakamura, 2008). Thus, the melt cylinder is effectively trapped between the plunger above and the crucible end piece underneath, and the crucible surrounding it radially. Under such circumstances the uniaxial loading of the plunger generates a bulk stress, transmitted through the melt body resulting in its transient pressurization. The viscous state of the melt on the timescale of the pressurization step ensures that the pressure is homogeneously distributed throughout the melt according to Pascal's law. It can be estimated as the force applied by the rod divided by the contact area with the obsidian sample. Here, the contact area is a disk with a diameter of 5 mm, i.e.,~20 mm 2 . The force (and thereby volume stress or pressure) can be set to either a constant value or to follow sinusoidal oscillations, corresponding to static and oscillating experiments, respectively. In static experiments, a constant force, F 0 , of 1.5 N (i.e., a pressure P 0 = 76 ± 8 kPa) is continuously applied to the sample (including during the heating phase). Moreover, the furnace chamber is kept at atmospheric pressure (101 kPa); therefore the effective pressure during a static experiment is 177 kPa.
In the oscillating experiments, F 0 was continuously applied during heating and for 10 min of thermal equilibration at the isotherm. The force with time is then set to: where F 0 is the constant baseline force, ΔF is the oscillation amplitude and t is time. For the majority of the oscillating experiments, we set ΔF = 1.4 N, and the pressure therefore oscillates between 106 ± 1 and 250 ± 10 kPa, during an oscillating experiment. The maximum instantaneous pressure change rate dP/dt is then 45 kPa s −1 . Our experimental setup is impermeable to melt but permeable to water vapour due to the narrow annulus between the plunger and crucible, and the very low permeability of the alumina ceramic. Experiments were run in an Ar atmosphere, thus the ambient partial pressure of water is lower than atmospheric. Therefore, a water pressure gradient exists between the pressurized melt and the furnace atmosphere, which leads to water loss by diffusion through the sample's free surface. The implications of this are discussed in further detail in the discussion below (Section 4.2). Our samples initially contain 0.11 ± 0.01 wt% water so depending upon the water solubility in the melt at experimental P-T conditions, water will either exsolve (supersaturated case) or remain in the melt (undersaturated case). Water solubility in the melt can be estimated for our experimental conditions using the empirical model of Liu et al. (2005) (Figure 2). During a static experiment at 965°C and 177 kPa, our samples are within the saturation uncertainty of the model, which does not allow a priori assessment of whether vesiculation will occur or not. Therefore, we explored whether vesiculation occurs by comparing expansion during static experiments of 1) a standard borosilicate dry glass (NIST SRM 717a) and 2) our hydrous obsidian, as starting materials. During oscillating experiments, samples alternate between supersaturated and undersaturated conditions, beyond the uncertainty of the model ( Figure 2).

Tracking Vesiculation With Sample Expansion
As noted above, we tested whether the TMA can track vesiculation by comparing length changes (i.e., expansion) during static experiments with 1) a standard borosilicate dry glass (NIST SRM 717a) and 2) our hydrous obsidian as starting material (Figure 3). For each experiment, strain was computed by normalizing sample length to the initial length, whereby positive strain is defined as corresponding to expansion. Figure 3 shows typical examples of the strain evolution during experiments, including the heating and dwell phases. The initial trends are identical in both dry and hydrous experiments. Initially volume increases due to thermal expansion of the glass samples, increasing in rate as the samples cross the glass transition (e.g., Gottsmann et al., 1999). At temperatures above the glass transition, the samples viscously relax the applied stress, and as a result viscously adjust, accommodate themselves and spread inside the crucible, corresponding to a sudden phase of compressive strain ( Figure 3). As the isotherm is reached and the melt sample fills the crucible, the strain curve flattens. Once the samples are viscously accommodated within the crucibles, the strain remains constant with time in the case of the dry NIST sample ( Figure 3A), whereas the sample length begins to increase in the case of the hydrous obsidian ( Figure 3B), FIGURE 2 | Temperature-pressure conditions of our experiments. The predicted P-T conditions of saturation for a rhyolite containing 0.11 ± 0.01 wt% of water are shown as a dashed line with a blue shaded region representing uncertainty, using the model from Liu et al. (2005). Above and left of this region, the sample is supersaturated in water. Below and right of this region, the sample is undersaturated. The pressure during static experiments is P 0 = 177 ± 8 kPa and displayed as a red vertical dotted line (with red shaded regions representing uncertainty). The minimum and maximum pressures reached during oscillations, P min = 106 ± 1 and P max = 250 ± 10 kPa respectively, are shown as black vertical dotted lines (with gray shaded regions representing uncertainty). The horizontal double-headed arrow is located at T = 965°C, where the majority of experiments were performed. indicating vesiculation. The obsidian samples can therefore be concluded to be supersaturated in water at the test conditions (T = 965°C and p = 177 kPa).
In comparisons going forward we suppress the reporting of the heating phase, setting both time and strain to zero 5 min after the isotherm is reached (as indicated on Figure 3B), thereby accounting for thermal equilibration of the furnace. At this point, melt has accommodated its form to the crucible and can no longer expand radially; therefore longitudinal strain can be converted to volume change, whereby positive changes correspond to expansion.

Experimental Results
We report the results of 14 experiments, seven at static pressure conditions and seven under oscillating pressure conditions. The majority of experiments were performed at 965°C. All experimental parameters are summarized in Table 3. Figure 4 shows experimental data from two static and two oscillation experiments at 965°C. We first describe the results from static experiments, and then compare them with the results from oscillation experiments.

Vesiculation During Static vs Oscillating Experiments
The two static curves (obtained over 1-and 2-h periods) follow similar trends, reflecting our experimental reproducibility. Initially, the volume change curve remains flat, close to zero, as the sample lengths do not vary. After 18-25 min, volume increases as the samples expand. The expansion rate remains approximately constant for 1 h, before slightly reducing (observed in experiments with sufficient duration, Figure 4). Once started, sample expansion proceeds continuously until the end of the experiment. Therefore, longer experiment durations yield larger volumes, as indicated by the higher final volume change values in the 2-h experiments compared to those of 1-h experiments ( Figure 4). These results are consistent with previous static dilatometry experiments (Bagdassarov et al., 1996;Stevenson et al., 1997). Sample expansion is attributed to water vapour bubbles nucleating and growing in the melt. The initial flat segment, termed "incubation period" by Bagdassarov et al. (1996), corresponds to the time required for bubbles to nucleate and grow to a detectable size after supersaturation is established. The incubation period in our static experiments (~20 min, Figure 4) is similar to observations from Ryan et al. (2015) and Cáceres et al. (2020) who performed static vesiculation experiments with the same starting material, at similar temperatures and at atmospheric pressures.
During pressure oscillation experiments, the volume change curves are initially flat, similar to the static case, as the samples undergo a period of incubation of 20-40 min before expansion starts. After the incubation period is complete, the trend differs in two ways: 1) the expansion rate is unsteady with successive acceleration and deceleration, and 2) the volume change values are consistently lower than in the static cases. These observations suggest that, in the oscillating cases, the samples 3 | Summary of experimental conditions. T is temperature, t total is the total time spent at isotherm temperature, ΔF is the oscillation amplitude, f is the oscillation frequency and t osc is the duration of oscillation. For the final volume change, + symbol indicates an increased in volume (i.e., sample expansion). expand at a slower rate (as evidenced by the overall shallower slope of the curves) and more erratically (as evidenced by the greater range of the curves).
We also explored temperatures of 950 and 990°C ( Supplementary Figures S1-S3). At 950°C, the measured strains remained close to zero in both static and oscillating experiments, presumably because vesiculation was effectively inhibited. At 990°C, sample expansion occurred faster than at 965°C, with an incubation period of <5 min in the static case and in the oscillating case, sample expansion occurred again at a relatively slower rate and more erratically.

Varying Oscillation Parameters
We investigated the role of three oscillating parameters on sample expansion kinematics: 1) oscillation amplitude ΔF, 2) oscillation frequency f, and 3) oscillation duration t osc . We use the 2-h experiments presented in Figure 4 (L19 and L20) as a comparative baseline to understand the effects of each parameter ( Figure 5). We observe that the general trend remains the same in all cases. All experiments started with an FIGURE 5 | Volume change as a function of time for varying (A) oscillation amplitude ΔF, (B) oscillation frequency f, and (c) oscillation duration t osc . Time is set to zero 5 min after the isotherm is reached. In (C), the oscillation period for the short oscillation duration experiment (L23) starts 10 min after the isotherm is reached (i.e., t = 5 min on the graph) and is shown as a double headed black arrow labelled t osc,L23 ; for all other oscillating experiments, the oscillation starts at t = 5 min and continues until the end of the experiment at t = 115 min, i.e., for a duration of t osc = 110 min. For each plot, data from a baseline static (L19, black line) and oscillating (L20, blue line) experiment are shown. These are the same two 2-h experiments presented in Figure 4. All experiments are performed at T = 965°C and F 0 = 1.5 N. All experimental parameters are reported in Table 3. incubation period of 20-40 min (at 965°C), before the onset of sample expansion. When the force was halved (ΔF = 0.7 N instead of 1.4 N, Figure 5A), the volume increased to values similar to those observed in the static experiment, despite a delayed onset. Hence for forces low enough, sample expansion is not affected by pressure oscillations. Thus the oscillation amplitude appears to exert a control on sample expansion, with larger amplitudes leading to lower total volumes. In contrast, with either a reduced oscillation frequency (f = 0.01 Hz instead of 0.1 Hz, Figure 5B) or reduced oscillation duration (15 min as opposed to 110 min, Figure 5C), the volume change curve matches the original oscillating experiment (L20). Therefore, pressure oscillations with either a low frequency or a short duration appear to reduce sample expansion. Thus, under our experimental conditions, oscillation amplitude exerts the greater influence on sample expansion kinematics.

Textural Analysis
Following our experiments, the samples recovered from both static and oscillating experiments contain bubbles, consistent with the interpretation that the volume increase is driven by vesiculation. Segmented images of horizontal cross sections from the final experimental products of the four experiments shown in Figure 4 are presented in Figure 6A.
In the static cases, bubbles are unevenly distributed. Most bubbles are located on the outer margin of the samples, at the boundary with the ceramic crucible, forming a rim of bubbles. The bubbles on the rim are densely packed, exhibit a wide range of diameters (0.001-1 mm) with larger bubbles generally deformed (i.e., not round) and showing evidence of coalescence ( Figure 6A; c.f. Castro et al., 2012). We also observe a few bubbles in the core of the sample; these tend to be isolated, spherical and of uniform size (~0.1 mm diameter).
The bubble textures in samples from oscillating experiments are broadly similar to the static cases. Again, bubbles are unevenly distributed with 1) a polydisperse, deformed and coalesced population on the outer margin, and 2) a more monodisperse spherical population in the core. Core bubble populations are indistinguishable between static and oscillating experiments but we observe fewer bubbles on the rim of the samples that have undergone pressure oscillations.
We used the segmented images of the samples to assess the relative proportions of core and rim bubbles ( Figure 6B). This quantitative analysis supports our observations that the majority of the bubbles are found on the outer margin and that longer experiment durations lead to larger volumes. Comparison between the bubble fractions formed during static versus oscillating tests confirms that 1) core bubble fractions are similar, and 2) bubble fractions on the rim are reduced in the oscillating case compared to the static case.

Post-Experimental Water Content
Water contents were measured post-experimentally, where possible, using FTIR spectroscopy (Figure 7, full details in Supplementary Material). The average post-experimental water content is 0.11 ± 0.01 wt%. Post-experimental water contents are therefore indistinguishable from pre-experimental water contents. Additionally, we do not observe any correlation between post-experimental water contents and distance to the nearest sample edge (Figure 7).

Static Experiments
In all static experiments, samples expanded (Figure 4), indicating that vesiculation occurred, as later confirmed in physical examination ( Figure 6). Vesiculation consists of three main processes: bubble nucleation, growth and coalescence (e.g., Burgisser and Degruyter, 2015). Most of the observed expansion likely results from the formation and/or accumulation of bubbles on the outer margin ( Figure 6). A rim of bubbles (also termed "fringe" of bubbles by Mangan and Sisson, 2000) is a common occurrence in contained vesiculation experiments (e.g., Mangan and Sisson, 2000;Mourtada-Bonnefoi and Laporte, 2002;Fiege and Cichy, 2015). These bubbles are interpreted to result from a combination of heterogeneous nucleation at the contact with the crucible (e.g., Mangan and Sisson, 2000;Mourtada-Bonnefoi and Laporte, 2002) and diffusion into interstitial air trapped at the sample-crucible interface (e.g., Wiesmaier et al., 2015). It is very unlikely that bubbles migrated within the sample during the experiments, due to the high melt viscosity (ca. 10 7 Pa s). Bubble migration velocity for a bubble with radius R can be estimated using Stokes' law as: where ρ is the melt density (=2,370 kg m −3 , computed with DensityX, at T = 965°C and P = 177 kPa; Iacovino and Till, 2019), η is the melt viscosity (= 10 7 Pa s) and g is the gravitational acceleration (=9.81 m s −2 ). This yields maximum velocities of 5 × 10 −10 m s −1 for a large bubble with R = 1 mm, corresponding to a maximum travelled distance of 3 µm in 100 min (representing a 2-h experiment with a 20-min incubation period), which is negligible. Thus we exclude the possibility that bubble migration led to accumulation at sample margins, and conclude that these bubbles must have formed in-situ at the margins of the samples. Bubble formation (i.e., nucleation and/or growth) is followed by coalescence, as evidenced by the textures observed on the rim of the experimental products ( Figure 6). Given the high viscosity of the melt, bubble coalescence most likely results from inter-bubble melt films stretching and thereby thinning, driven by bubble growth (e.g., Castro et al., 2012;Masotta et al., 2014;Giachetti et al., 2019).
Ostwald ripening (the spontaneous growth of large bubbles at the expense of smaller ones) might also occur due to the variety of bubble sizes observed in the rim, but, at such high viscosities, ripening effects generally become significant on timescales of days to weeks, i.e., orders of magnitude longer than the duration of our experiments (e.g., Lautze et al., 2011;Masotta et al., 2014).
As our experimental apparatus is permeable to volatiles, some gas may progressively escape from the system, diminishing the driving force for bubble growth. There are two possible escape routes for water vapor: either via direct diffusion through the narrow space between the plunger and the crucible, or via porous flow through the ceramic crucible. In the case of the former, there is a thin ring of melt in direct contact with the furnace atmosphere around the base of the plunger. Here, water will diffuse from the pressurized melt into the furnace, due to the low partial pressure of water in the Ar atmosphere. This diffusion process will lead to the formation of a low water content, bubblefree, impermeable skin which effectively seals the system (Yoshimura and Nakamura, 2008;von Aulock et al., 2017). The thickness of this skin scales with the diffusion length L D Dt total √ , where D is water diffusivity in the melt and t total is the total time spent at the isotherm (von Aulock et al., 2017). We use the diffusivity model of Zhang and Ni (2010) and estimate D = 6.0 × 10 −13 m 2 s −1 at T = 965°C, p = 177 kPa and a water content of 0.11 wt%. The diffusion lengths are then c. 50 and 70 µm for experiment durations of 1 and 2 h, respectively. A dehydrated skin will thus form wherever direct diffusion into the furnace atmosphere is allowed, but its thickness will remain <70 μm, which is below the resolution of our FTIR analyses.
The second gas loss mechanism is permeable gas flow through the crucible walls. We can rearrange Darcy's law to estimate the time necessary to evacuate a half-bubble of radius R via porous flow through the ceramic wall as (see geometry of the problem in Figure 8A): where the thickness of the ceramic crucible wall b = 3 mm, the viscosity of the gas η g 10 −5 Pa s, k is the permeability of the ceramic, P 0 is the pressure experienced by the sample (=177 kPa), and P H2O atm is the partial pressure of water in the furnace atmosphere. Experiments are run in an Ar atmosphere, so, we consider an ideal value of P H 2 O atm 0. In practice, P H 2 O atm is likely small but not zero, hence our calculations will provide minimum estimates of t Darcy (i.e., the fastest scenario). We attempted to measure the permeability of the ceramic material k using a Vinci Technologies benchtop gas permeameter with a Bronkhorst flowmeter, but the measurement was below the 10 -17 m 2 detection limit of the instrument. We thus consider a range of permeability values between 10 −25 and 10 −18 m 2 . The necessary t Darcy to evacuate half-bubbles with radii between 0.3 µm and 3 mm are shown in Figure 8B. The analysis suggests that gas loss by permeable flow through the ceramic crucible can be discarded altogether if the crucibles' permeabilities are below 10 −23 m 2 , but it can be efficient (t Darcy < 10 s) for permeabilities >10 −20 m 2 and R < 50 µm, which is smaller than most of the observed bubbles.

Oscillating Experiments
The same processes of bubble nucleation, growth, coalescence and gas loss from the crucible occur during the oscillating experiments, but each of these may potentially be affected by pressure oscillations. The volume change curves (Figures 4,5) are generally similar in the static and oscillating cases, suggesting that the effects of oscillations tested here remain moderate. Nonetheless, the expansion curves for the oscillating experiments lie systematically below those for static experiments (Figures 4, 5). This is observed regardless of the duration and frequency of pressure oscillations across the range tested, but depends on the oscillation amplitude, with higher amplitudes yielding more reduced volumes compared with the static case ( Figure 5). These relationships are preliminary and need to be explored and confirmed by more experiments, ideally accompanied by a quantification of gas loss from the system, and an analysis of its potential influence on vesiculation. Textural evidence ( Figure 6A) shows that bubbles in the sample cores are of similar size, shape and present in similar amounts in static vs oscillating tests, but that the amount of bubbles located on the rim is lower in oscillation experiments ( Figure 6B). We propose that these observations can be explained by either: 1) reduced bubble formation (nucleation and/or growth), 2) increased gas loss, or 3) a combination of the two. Below, we explore how pressure oscillations may impact the different processes occurring during an experiment.

Bubble Nucleation
Bubble nucleation events do not produce a significant volume change and thus cannot be monitored directly with our experimental setup. We can nevertheless interrogate the classical nucleation theory (CNT) to assess the effects of pressure oscillations on the phenomenon, and build a hypothesis to be tested in future work (see Gonnermann and Gardner (2013) and Shea (2017) for shortcomings of the CNT for magmatic liquids). According to the CNT, bubbles nucleate from clusters of water molecules. CNT defines a cluster critical (minimum) radius R c above which clusters will form bubbles and under which they will dissolve by diffusion as (Hurwitz and Navon, 1994): where σ is the surface tension between a bubble nucleus and the melt, P n is the pressure inside the bubble nucleus and P is the ambient melt pressure. During oscillations, P varies with time according to: For the experiments presented in Figure 4, P 0 177 kPa, ΔP 71 kPa and f 0.1 Hz. We can then compare the evolution of the critical radius with time in an oscillating experiment R c (t) to the critical radius of a static experiment R c,0 as: In general, P n is given by (Cluzel et al., 2008): where P sat is the saturation pressure, Γ(P sat , T) is the fugacity of water at saturation, V H 2 O is the volume of water molecules and k B is the Boltzmann constant (=1.38 × 10 −23 m 2 kg s 2 K −1 ). Γ(P sat , T) and V H 2 O can both be easily computed (see Cluzel et al., 2008, and references therein), so we are only left with P sat to evaluate. The solubility model from Liu et al. (2005) indicates that, at T = 965°C and for a 0.11 ± 0.01 wt% water content, P sat lies in the range 125-190 kPa (Figure 2). We experimentally observed that samples vesiculated at 177 ± 8 kPa; therefore we suggest that P sat > 177 ± 8, and chose P sat ≈ 190 kPa. Entering those values into Eq. 7, we find that Γ(P sat , T) exp[ VH 2 O k B T (P − P sat )] ≈ 1, and thus P n ≈ P sat ≈ 190 kPa.
We can then use Eq. 6 to evaluate how the critical radius changes during pressure oscillations (Figure 9). While the FIGURE 9 | (A) Pressure, (B) critical radius for bubble nucleation R c (t) (Eq. 6) and (C) time to evacuate a half-bubble via permeable flow t Darcy (Eq. 10) as a function of time during an oscillating experiment (three oscillation cycles are shown). R c (t) and t Darcy are both normalized to the static case value (shown as horizontal dashed lines), and plotted on a logarithmic scale.
Frontiers in Earth Science | www.frontiersin.org March 2022 | Volume 10 | Article 812311 pressure oscillations P(t) are symmetrical about P 0 ( Figure 9A), the evolution of R c (t) is not symmetrical about R c,0 at all ( Figure 9B). When pressure decreases from P 0 towards P 0 − ΔP, supersaturation increases and R c (t) decreases by about 75%. A reduced critical radius implies that a greater number of clusters will be sufficiently large to grow into bubbles, and nucleation will be more efficient (c.f., Masotta and Keppler, 2017). When the pressure increases towards P 0 + ΔP however, the melt becomes undersaturated because P(t) > P sat and the critical radius becomes infinite: no cluster will be large enough to form bubbles, and nucleation will not occur. Hence there is an imbalance over each pressure oscillation: bubble nucleation is slightly enhanced during the lower half of the cycle compared to the static case, but it is completely shut off during the higher-pressure segment. This leads to the hypothesis that some of the water clusters resorb when the pressure is increased, eventually leading to overall reduced bubble nucleation. The process of cluster resorption is mainly controlled by the diffusivity of water. The diffusion length over a half-period of oscillation (5 s) is about 2 µm. We can compare this length to R c,0 , to assess whether diffusion would occur fast enough to dissolve water clusters. R c,0 can be computed using Eq. 4, which requires σ to be known. Macroscopic σ for water-poor rhyolite at low pressures is about 0.3 N m −1 (Bagdassarov et al., 2000), but this value should be reduced by up to an order of magnitude when considering the microscopic scale of bubble nuclei (e.g., Gonnermann and Gardner, 2013;Shea, 2017;Hajimirza et al., 2019). We thus consider σ to be in the range 0.03-0.3 N m −1 , which yields R c,0 values of 5-50 µm. This means that, according to the CNT, any existing bubble or cluster smaller than 5-50 µm will resorb. The lower end value (corresponding to more realistic σ; e.g., Shea, 2017) is comparable to the diffusion length scale over half a pressure oscillation. This preliminary analysis therefore indicates that pressure oscillations around saturation may effectively reduce bubble nucleation by momentarily annihilating some of the water clusters before they are able to form bubbles. We urge future studies to explore this mechanism further as it was not possible to assess bubble number density satisfactorily due to the coalesced bubble rim dominating the gas volume fraction. This problem could be avoided by using larger samples, hence reducing the relative importance of rim bubbles, or using a different pressurization mechanism that does involve an interface with a container (e.g., Masotta and Keppler, 2017).

Bubble Growth and Coalescence
Bubble growth is a complex, non-linear phenomenon and it is not, a priori, evident how pressure oscillations should influence it. It is important to note that, in our experiments, we monitor the whole sample volume change, which depends upon, but differs from, the growth rates of the individual bubbles. In general, for a given set of initial conditions, bubble growth rate (and thereby the whole sample expansion rate) is primarily governed by the melt viscosity and/or the diffusivity of water (e.g., Sparks, 1978;Bagdassarov et al., 1996;Proussevitch and Sahagian, 1998;Blower et al., 2001;Lensky et al., 2004). Here, the initial conditions refer to the initial degree of supersaturation and bubble number density (BND), which have been shown to exert significant controls on the resulting growth rate (e.g., Liu and Zhang, 2000;L'Heureux, 2007;Coumans et al., 2020). The effects of parameters such as temperature or composition of the melt are factored into the viscosity, diffusivity and degree of supersaturation (e.g., higher temperatures correspond to lower viscosities, higher degrees of supersaturation and higher diffusivities). Water diffusivity scales inversely with pressure, but in our case the pressure oscillations (with amplitude ΔP = 71 kPa) result in water diffusivity changes of <0.1% (using the model of Zhang and Ni, 2010). Additionally, melt viscosity is not significantly, nor systematically, influenced by pressure (e.g., Ardia et al., 2008). We therefore anticipate that the effect of pressure oscillations on water diffusivity and melt viscosity in our tests can be neglected.
Pressure oscillations could however significantly impact the degree of supersaturation and, potentially, the bubble number density. Concerning the latter, as discussed in Section 4.2.1, pressure oscillations may reduce the number of bubble nuclei. One of the consequences would be an overall reduction in bubble number density, and thus a reduced sample expansion rate (though not necessarily reduced individual bubble growth rates; e.g., Proussevitch and Sahagian, 1998;Coumans et al., 2020). Additionally, we saw that the pressure oscillations constantly change the degree of water supersaturation in the melt (see Figure 2 and previous section). The supersaturation increases in the low-pressure part of the cycle, but, in the high pressure part the degree of supersaturation decreases and eventually vanishes as the sample is driven into undersaturated conditions. In general, higher degrees of supersaturation produce faster bubble growth rates (e.g., Proussevitch and Sahagian, 1998;L'Heureux, 2007), yet previous research efforts have generally focused on either constant or continuously increasing degrees of supersaturation (e.g., Nowak et al., 2011). Therefore, bubble growth potentially accelerates when pressure is reduced, and decelerates when pressure is increased and bubble growth stops when the pressure increases past the saturation pressure and the sample becomes undersaturated. Bubbles may then start resorbing (e.g., McIntosh et al., 2014) but on a timescale beyond that explored in this study. As for the case of bubble nucleation then, it is probable that even symmetrical pressure oscillations lead to asymmetrical effects on bubble growth. If bubble growth cyclically stops as the melt becomes undersaturated, and if this effect is not counterbalanced by the bubble growth acceleration when pressure is decreased, then this mechanism may contribute to the observed reduced sample expansion rates (Figure 4). The importance of this mechanism remains to be fully evaluated. We note that for our experiments the diffusion length over a half-period of oscillation is only 2 µm. Hence it is possible that our pressure oscillations occur too fast compared to the diffusion of water, and thus bubble growth would remain unaffected.
Bubble growth may further be affected by rectified diffusion, if the sizes of the bubbles oscillate with pressure (Brodsky et al., 1998;Ichihara and Brodsky, 2006). The necessary timescale for a bubble to expand and contract in response to pressure perturbations can be estimated, after Ichihara and Nishimura (2009), as: .
( 8 ) In our experiments, τ e 190 s, which is much greater than the period of the pressure oscillations (10 s). We thus conclude that bubble sizes do not vary significantly as pressure oscillates, and rectified diffusion does not occur. This calculation also suggests that Bjerknes forces (i.e., translational forces arising from bubble volume oscillations that could potentially drive coalescence, e.g., Igualada-Villodre et al., 2018) are unlikely to play a significant role in our experiments. We note however that Masotta and Keppler (2017) observed significant changes in bubble sizes when imposing larger pressure oscillations (ΔP~10 4 kPa) on a haplogranitic melt, in their moissanite cell experiments. The effects of Bjerknes forces from bubble volume oscillations may deserve further investigation in studies testing a wider range of pressure oscillation conditions.
Finally, most coalescence processes directly depend on the BND and the individual bubble growth rates, with higher BND and growth rates generally leading to faster coalescence (Castro et al., 2012). Section 4.2.1 showed that pressure oscillations may reduce bubble nucleation, hence the resulting BND will be lower, and pressure oscillations potentially reduce bubble growth rates. Therefore, if pressure oscillations reduce both BND and growth rates, then the rate of coalescence should also decrease.

Gas Loss
The last processes that may potentially be affected by pressure oscillations are related to gas loss to the surrounding furnace atmosphere; namely, via direct diffusion through cracks between the plunger and the crucible, and via gas filtration through the crucible wall. While the former is likely to be negligible (see Section 4.1), the latter could be significant, contingent upon the ceramic material having a high enough permeability (k > 10 −20 m 2 , see Section 4.1 and Figure 8). In order to evaluate t Darcy in oscillating experiments, we replace P 0 by P(t) in Eq. 3 and obtain: where P(t) is given in Eq. 5. This implies that the gas loss rate depends non-linearly on pressure oscillations. We can divide Eq. 9 by Eq. 3, to compare t Darcy in the oscillating and static cases, yielding: assuming that P H2O atm 0. The evolution of this ratio with time is shown in Figure 9C. When pressure increases, so does the pressure differential between the melt and the furnace atmosphere, hence gas loss accelerates and t Darcy is reduced bỹ 30%. Conversely, when melt pressure is reduced, permeable gas flow is slowed down and t Darcy increases by~70%. There is again an imbalance between the higher and lower pressure parts of the oscillation cycle, with an average increase of t Darcy . Therefore, if gas loss occurs via filtration through the ceramic material (i.e., if k > 10 −20 m 2 ), then pressure oscillations would likely hinder the process, i.e., lead to reduced gas loss, rather than enhance it. On the contrary, our experiments show that there are less bubbles on the rim in oscillating experiments, which would require an enhanced gas loss mechanism, if indeed the differences observed result from more efficient gas loss rather than reduced nucleation. Gas escape through the ceramic crucible therefore does not provide an adequate explanation for our observations.

Limitations and Future Work
The experiments presented in this contribution are exploratory and in this sense very much in the spirit of other attempts to understand how melt behaves under disequilibrium conditions (e.g., Arzilli and Carroll, 2013;Masotta and Keppler, 2017;Kolzenburg et al., 2018). As such, further work will be needed to constrain and confirm the results presented here. Below, we highlight some current limitations and prospects for future work.
Firstly, our observations rely on a small number of experiments (14 on hydrous samples). While we strived for a reasonable level of reproducibility, the validity of our results is pending replication. Future experiments should also aim to explore saturation states more carefully by investigating smaller steps in temperature. Similarly, the role of the different oscillation parameters (amplitude, frequency and duration) needs to be more closely examined. While we have highlighted the importance of bubbles, in particular at the margin of the samples, observation of these processes is currently limited both in time and space. We only observe a snapshot of the bubble textures, in 2D, after quenching the sample at the end of an experiment. A crucial future step would be to have better temporal resolution, for example by monitoring bubble textures continuously during in-situ experiments, using X-ray tomography techniques for instance. Furthermore, as bubbles grow in the sample, the compressibility of the sample increases, which has been neglected in our analysis. We tried to mitigate this issue by focusing on short experiments, with low (<10%) exsolved phase volumes, but this could constitute an avenue for further work. Finally, we examined the importance of gas loss through the crucible walls. An accurate characterization of the permeability of the ceramic material used will be a critical step to constrain this mechanism further. Alternatively, a different material could be used for the crucible. For instance, Gottsmann et al. (1999) use containers made of Ni metal.

Geological Implications
Our experiments investigate the influence of pressure oscillations on bubble growth in magma. Magma may experience pressure fluctuations in a range of natural scenarios, in particular in the case of incoming seismic waves (Hill et al., 2002;Manga and Brodsky, 2006;Seropian et al., 2021). Although the conclusions from our experiments remain preliminary, we describe below the conditions for which our findings would be relevant, in order to guide future applications.
Seismic waves are a common source for pressure oscillations on magma bodies. Typical frequencies for seismic waves are in the range 0.1-10 Hz, though frequencies as low as 0.01 Hz could be attained for very large earthquakes (Manga and Brodsky, 2006). The amplitude of the stress perturbation depends on many parameters, including the earthquakes' magnitude, distance to the epicentre, focal mechanism and local crustal structure. Nonetheless typical values lie between 10 3 and 10 4 kPa in the near-field (<100-500 km, depending on magnitude), and reduce to <100 kPa for distances >1,000 km from the epicentre (e.g., Manga and Brodsky, 2006;Walter et al., 2007Walter et al., , 2009Avouris et al., 2017;Farías and Basualto, 2020). Hence the pressure oscillations in our experiments (f = 0.01-0.1 Hz and ΔP = 71 kPa) are representative of natural seismic waves following a large distant or moderate local earthquake. Importantly, in our experiments, pressure oscillations last for 15-110 min, whereas typical seismic waves will produce oscillations for <5 min. Figure 5C suggests that sample expansion is reduced, even in the case of short pressure oscillations, but further work should aim at reducing this discrepancy between experimental and natural oscillation durations, to facilitate application.
The low confinement pressure (177 kPa), low water content (0.11 wt%) and composition of our experiments correspond to a very shallow or surficial silicic magma. This is one end-member of the spectrum of silicic magmas as they are generally stored at higher confining pressures (100-250 MPa, e.g., Huber et al., 2019), contain more dissolved water (typically 2-6 wt% in subduction zones, e.g., Wallace et al., 1995), and are likely to already contain bubbles. Nonetheless, we surmise that the asymmetrical response, and effective reduction of bubble nucleation would remain similar, provided that pressure oscillations drive the system in and out of water saturation. It is the pressure relative to the saturation pressure, rather than the absolute pressure that dictates how bubble nucleation will respond to pressure oscillations.
At higher water contents, however, melt viscosity will be significantly reduced. For instance, our rhyolitic melt, in equilibrium at 150 MPa and 775°C would contain 5.14 wt% H 2 O (Liu et al., 2005), and its viscosity would be 10 5 Pa s . The bubble expansion timescale τ e (Eq. 8) would then be significantly reduced, meaning that the volume of already existing bubbles could vary significantly during the oscillations (as observed by Masotta and Keppler, 2017). This would have particularly important implications for bubble coalescence, and possibly for bubble migration (e.g., Iwata et al., 2008;De Corato et al., 2019). Additionally, the contained nature of our experiments is also relevant to natural scenarios, where magma is contained and interacts with variably permeable country rock (e.g., Jaupart and Allègre, 1991).
It is commonly assumed that seismic waves may induce bubble nucleation in magma, increasing magma overpressure, and potentially leading to unrest or an eruptive episode (e.g., Manga and Brodsky, 2006;Rothery et al., 2007;Cannata et al., 2010;Hamling and Kilgour, 2020;Seropian et al., 2021), yet this mechanism has never been validated experimentally. We did not observe any evidence suggesting that pressure oscillations triggered bubble nucleation in our experiments. Our analysis of bubble nucleation suggests that the opposite could occur, that is pressure oscillations could hamper bubble nucleation. We emphasize that the lack of evidence for increased nucleation at our experimental conditions does not mean that seismic waves cannot nucleate bubbles in magma in general; our experiments featured high-viscosity, crystal-poor, low water content rhyolite, all of which may limit bubble nucleation processes (Shea, 2017). Hamling and Kilgour (2020) showed that the necessary P-T and water content conditions for seismic waves to promote bubble nucleation form a very tight window, with the optimal case being partially crystallized basalt. Therefore seismicallyenhanced bubble nucleation remains to be experimentally tested as a viable mechanism to trigger eruptions and our experimental data indicate that degree of supersaturation of the melt is an essential parameter to constrain for oscillating vesiculating experiments.

CONCLUSION
In this contribution we report a series of high-temperature experiments investigating the effects of pressure oscillations on magma vesiculation. We designed a novel experimental setup where melt is contained within a cylindrical ceramic crucible and pressurized with a piston-style plunger actuated in a thermomechanical analyzer; then vesiculation is studied in two types of experiments: static (where pressure is maintained) and oscillating (where pressure fluctuates). We carefully chose P-T conditions so that the samples would be slightly oversaturated in water and volatiles would slowly exsolve during an experiment. The experiments show that vesicles are mostly present near the sample edge in all cases, with very few vesicles in the sample core, irrespective of testing conditions. We observe that both sample expansion and the amount of bubbles present at the outer margin are reduced when pressure oscillates, compared to the static case. The oscillation amplitude exerts the strongest control on this effect. The observed reduced volumes in oscillating experiments can be explained by reduced vesiculation and/or increased gas loss. We evaluated different processes which may occur in the samples during the experiments, and discuss how they would be affected by oscillatory pressures. This preliminary analysis suggests that bubble formation could be hindered by pressure fluctuations if the melt oscillates between over-and undersaturated conditions. This work highlights the number of interacting processes happening as magma vesiculates, each of which is susceptible to modulation by external pressure perturbations. These processes are particularly relevant to the study of earthquake-volcano interactions, where a magma body sits close to volatile saturation and responds to pressure fluctuations.

DATA AVAILABILITY STATEMENT
The original contributions presented in the study are included in the article/Supplementary Material, further inquiries can be directed to the corresponding author.