ORIGINAL PAPER

Segregating gas from melt: an experimental study of the Ostwaldripening of vapor bubbles in magmas

Nicole C. Lautze • Thomas W. Sisson •

Margaret T. Mangan • Timothy L. Grove

Received: 31 July 2009 / Accepted: 4 May 2010 / Published online: 18 June 2010

� US Government 2010

Abstract Diffusive coarsening (Ostwald ripening) of H2O

and H2O-CO2 bubbles in rhyolite and basaltic andesite melts

was studied with elevated temperature–pressure experi-

ments to investigate the rates and time spans over which

vapor bubbles may enlarge and attain sufficient buoyancy to

segregate in magmatic systems. Bubble growth and segre-

gation are also considered in terms of classical steady-state

and transient (non-steady-state) ripening theory. Experi-

mental results are consistent with diffusive coarsening as the

dominant mechanism of bubble growth. Ripening is faster in

experiments saturated with pure H2O than in those with a

CO2-rich mixed vapor probably due to faster diffusion of

H2O than CO2 through the melt. None of the experimental

series followed the time1/3 increase in mean bubble radius

and time-1 decrease in bubble number density predicted by

classical steady-state ripening theory. Instead, products are

interpreted as resulting from transient regime ripening.

Application of transient regime theory suggests that bubbly

magmas may require from days to 100 years to reach steady-

state ripening conditions. Experimental results, as well as

theory for steady-state ripening of bubbles that are immobile

or undergoing buoyant ascent, indicate that diffusive coars-

ening efficiently eliminates micron-sized bubbles and would

produce mm-sized bubbles in 102–104 years in crustal

magma bodies. Once bubbles attain mm-sizes, their calcu-

lated ascent rates are sufficient that they could transit mul-

tiple kilometers over hundreds to thousands of years through

mafic and silicic melt, respectively. These results show that

diffusive coarsening can facilitate transfer of volatiles

through, and from, magmatic systems by creating bubbles

sufficiently large for rapid ascent.

Keywords Ostwald � Ripening � Coarsening � Bubble �Magma � Transient

Introduction

Magmatic degassing can control volcano-related seismicity,

edifice deformation, and the explosivity of eruptions.

Degassing also ultimately led to Earth’s atmosphere and

hydrosphere paired with its volatile-poor crust dominated by

intrusive igneous rocks, and its nearly degassed upper

mantle; degassing from magma reservoirs is thus a basic

process of planetary differentiation. Degassing involves the

nucleation, growth, and possible segregation or escape of

bubbles of gas or supercritical fluid (dominantly H2O and

CO2, but also S, Cl, and F). Nucleation commences at and

then follows the point of volatile saturation in a magma; it is

promoted by decompression, heating, and by crystallization

of volatile-poor or volatile-free minerals (second boiling).

Bubbles grow, and thus become more bouyant and poten-

tially mobile, by the processes of decompressive expansion,

coalescence, second boiling, and by diffusive coarsening—

the last known as Ostwald ripening. This study looks in detail

at the potential rates and durations of bubble growth by

Ostwald ripening both in terms of theory and with direct

high-temperature, high-pressure experiments.

Communicated by J. Blundy.

N. C. Lautze (&)

Istituto Nazionale di Geofisica e Volcanologia-Roma,

Rome, Italy

e-mail: nicole.lautze@ingv.it

N. C. Lautze � T. W. Sisson � M. T. Mangan

U.S. Geological Survey, Volcano Hazards Team,

Menlo Park, CA, USA

T. L. Grove

Massachusetts Institute of Technology, Cambridge, MA, USA

123

Contrib Mineral Petrol (2011) 161:331–347

DOI 10.1007/s00410-010-0535-x

Ripening has been shown to enlarge minerals crystal-

lized from silicate melts over experimentally accessible

time periods (Park and Hanson 1999; Kile et al. 2000;

Ayers et al. 2003; Cabane et al. 2005), has been interpreted

to create megacrysts in some igneous intrusions (Higgins

1998; Higgins and Chandrasekharam 2008), and has been

demonstrated as a viable mechanism for the formation of

zircon overgrowths during high-temperature metamor-

phism (Nemchin et al. 2001). Diffusion of volatiles, par-

ticularly H2O, is orders of magnitude faster in silicate

melts, relative to other melt components (Watson 1994),

which raises the possibility that ripening could be highly

effective in increasing the sizes of bubbles and facilitating

their escape by bouyant ascent, their draining by coales-

cence, and in establishing the initial size distributions of

magma reservoir bubbles preceding volcanic eruptions.

Studies considering bubbles in magmas have shown rip-

ening to be effective in decreasing the bubble number

density in medium viscosity melts (Yamada et al. 2008)

and to influence bubble textures in the products of basaltic

fire-fountain eruptions (Mangan and Cashman 1996). In

most if not all geologic studies that consider Ostwald rip-

ening, only theory for steady-state ripening is taken into

account. In this study, we consider both transient and

steady-state ripening rates, the former that explains

experimental results, and the latter that shows ripening to

promote significant bubble growth in both felsic and mafic

magma reservoirs on geologic timescales.

Our experimental approach was to hold rhyolite and

basaltic andesite melts saturated with mixed H2O-CO2

vapor or pure H2O (rhyolite only) for a range of durations

sufficient for bubbles to enlarge by diffusive coarsening,

measure the bubble sizes and abundances (number densi-

ties) by microscopy on 2-D sections, convert these to 3-D

distributions by stereology (Sahagian and Proussevitch

1998), and interpret results in terms of steady-state and

transient regime ripening theory drawn mainly from the

materials engineering literature (Ratke and Voorhees 2002,

and references therein). These theories are also applied to

generic mafic and felsic crustal magmas to consider the fate

of bubbles at time spans well beyond what can be simu-

lated in the laboratory.

Background—theory for steady-state ostwald ripening

Ripening leads to an overall coarsening of a dispersed

phase-crystals, bubbles, or liquid droplets. Coarsening

takes place as chemical species diffuse from smaller to

larger particles as the system minimizes surface energy for

the volume of the phase in question. Ripening can also be

understood in terms of the surface tension and pressure of

exsolved vapor bubbles in a liquid. Pressure p within a

spherical vapor bubble exceeds that within the host liquid

by p = 2c/r where c is the liquid–vapor surface tension and

r is the bubble radius. Thus, the pressure within a small

bubble exceeds that within a nearby large bubble, so vapor

components diffuse from the smaller to the larger bubble.

This causes the small bubble to shrink and the large bubble

to grow, resulting in an increase in particle (bubble) size

and decrease in particle number with time.

Theory predicts that with sufficient time, ripening

attains a steady state wherein the size distribution of the

dispersed phase, normalized by the mean particle size,

remains constant; at steady state, the mean particle radius

increases as time t1/3 when the ripening rate is controlled

by diffusion and as t1/2 when controlled by interface

kinetics; and that at steady state, particle number density

decreases as t-1 (Lifshitz and Slyozov 1961; Wagner 1961;

Marqusee and Ross 1984; DeHoff 1991; Ratke and

Voorhees 2002). For the simple case of a dispersed phase

undergoing steady-state ripening (therefore at long times)

limited by diffusion, the increase in mean radius r(t)

converges to:

r2 ¼ tK ð1Þ

where K is a rate constant and t is time. There are various

derivations for this rate constant (DeHoff 1991; Chen and

Voorhees 1993; Ratke and Voorhees 2002; Balluffi et al.

2005) but in all models coarsening scales with diffusivity.

Experimental and analytical approaches

Samples, sample syntheses, and run schedules

Bubble ripening was studied experimentally at elevated

pressures and temperatures using two natural lava samples

(Table 1): high-silica rhyolite obsidian MC78-58 from

Panum Obsidian Dome, California (Bailey et al. 1989;

Mangan and Sisson 2000), and a basaltic andesite

03S75M1 from Pavlov Volcano, Aleutian arc, Alaska, with

H2O and CO2 added in the laboratory to create bubbles.

Experiments were designed to be either vapor-saturated

throughout their durations (rhyolite) or to saturate with

vapor bubbles upon a large decompression step followed

by constant pressure–temperature conditions (basaltic

andesite).

Four experimental series were performed overall, each

consisting of multiple runs of varying durations at constant

pressure–temperature conditions. Three of these series

were on the rhyolite (series 1, 3, 4) and one on the basaltic

andesite (series 2). The majority of experiments (series 1,

2, 4) were performed at 400 MPa in piston cylinder (PC)

presses at the US Geological Survey in Menlo Park, Cali-

fornia using 2.54 cm diameter vessels, graphite furnaces,

332 Contrib Mineral Petrol (2011) 161:331–347

123

and CaF-Pyrex assemblies, with the sample capsule ori-

ented horizontally. Fine pressure control was achieved by

connecting an argon reservoir to the hydraulic fluid line

that drives the piston. Replicate bracketing of the melting

curve of CsCl at 300–800 MPa indicates that pressure can

be reproduced with this adaptation to ±2.5 MPa. Series 3

(rhyolite) was the exception, being performed at 100 MPa

in water-pressurized cold seal (CS) vessels at the Massa-

chusetts Institute of Technology.

Series 1

For each rhyolite synthesis in series 1, 5 wt% liquid H2O

and 5 wt% high-purity oxalic acid (C2H2O4) along with 90

wt% powdered natural obsidian (40–50 mg) were loaded

into 3 mm diameter Au capsules and sealed by welding.

Each sample capsule was then centered in a 5 mm diameter

Pt outer capsule packed with *200 mg of Ni and NiO and

5 wt% oxalic acid that was then welded closed. The aim of

this recipe was to produce rhyolitic liquid with *5 wt%

dissolved H2O coexisting with *5 wt% vapor with

CO2:H2O of *2:1 M, as guided by the H2O solubility

model of Moore et al. (1998), with fO2 buffered at Ni-NiO.

Oxalic acid is oxygen deficient relative to stoichiometric

2:1 CO2:H2O but our experience was that trapped air, ferric

iron, and possibly oxygen diffusion from the surrounding

buffer were sufficient to prevent graphite precipitation. The

starting material rhyolite glass powder was prepared with a

range of grains sizes (to *150 lm) to create a range of

initial bubble sizes as well as areas of bubble-free glass that

could be examined by infrared spectroscopy. These rhyo-

lite H2O-CO2 experiments were brought directly to pres-

sure (400 MPa) and temperature (850�C) in the piston

cylinder and run for durations of 1, 7, 14, and 28 days

before quenching by shutting off power to the furnace

(B10 s to cool through the glass transition temperature).

Series 2

In series 2, powdered basaltic andesite was mixed with 50

wt% crushed plagioclase crystals. This began as an inves-

tigation into the influence of crystals on the escape of vapor

bubbles from basaltic andesite magma. In prior crystal-free

experiments on the basaltic andesite, H2O-CO2 bubble

foams nucleated and segregated upwards rapidly upon the

sudden decompression of the liquid from 1.2 GPa to

400 MPa (Mangan et al. 2006). In the present study, cru-

shed plagioclase crystals were added to serve as a barrier

for foam ascent. To find the initial conditions for the

crystal-bearing experiments, the basaltic andesite compo-

sition was first determined to grow trace amounts of calcic

plagioclase at 1.2 GPa and 1,125�C with 2 wt% H2O in the

melt. Excess plagioclase could therefore be added at those

conditions without modifying the melt characteristics

severely. Basaltic andesite starting materials were prepared

by synthesizing *300 mg batches of bubble-free glass

(plus quench amphibole) with 10 wt% dissolved H2O, by

weighing-in liquid H2O and then fusing and quenching at

elevated pressure and temperature. This hydrated basaltic

andesite was then crushed and mixed with powdered nat-

ural crystalline basaltic andesite, previously dried at 120�C,

to achieve a bulk H2O concentration of 2 wt%. The 2 wt%

H2O basaltic andesite mix was combined with 50 wt%

Crystal Bay bytownite (An80, Table 1) that had been cru-

shed, washed, and sieved to 80–140 lm, and the resulting

mix was loaded (*40 mg) and welded into 3 mm diameter

Ag70Pd30 capsules along with 3.9 wt% silver oxalate

(AgCO2). The resulting bulk composition contained 1 wt%

each of H2O and CO2, equivalent to 2 wt% each in the non-

plagioclase fraction. The sealed sample capsules were

centered in 5 mm diameter Pt capsules packed with Ni,

NiO, and a mixture of silver oxalate and oxalic acid in

proportions to produce a vapor of 3:1 CO2:H2O (molar),

guided by the mol fraction H2O in vapor expected for the

basaltic andesite plus volatiles at 400 MPa and 1,125�C

from the solubility equation of Moore et al. (1998). This

outer capsule was sealed by welding and centered with its

long axis horizontal in the hot spot of the furnace.

The basaltic andesite run schedule consisted of 20–24 h

at 1.2 GPa and 1,125�C, followed by a rapid decompres-

sion to 400 MPa to nucleate bubbles, and then holding for

1, 3, 10, 20, 40, 70, and 100 h. Decompressions were

accomplished by abruptly opening the piston cylinder

hydraulic line to a reservoir that had been charged with

argon at a pressure previously determined to bring the run

to 400 MPa. Tests with alkali-halide melting show that the

sample undergoes 80–90% of the decompression within

Table 1 Rhyolite composition from Mangan and Sisson (2000),

basaltic andesite from Mangan et al. (2009), plagioclase composition

is average of 25 electron microprobe analyses

Rhyolite MC78-58

(panum crater)

Basaltic andesite

03S75M1

(Pavlof)

Crystal bay

bytownite

(An80.0)

SiO2 75.6 52.0 48.1

TiO2 0.08 1.20

Al2O3 12.4 18.1 33.4

FeO 0.94 9.55

MnO 0.07 0.19

MgO 0.03 4.66

CaO 0.54 8.82 16.3

Na2O 4.17 3.36 2.25

K2O 4.72 0.61

P2O5 0.01 0.29

Total 98.6 98.8 100.0

Contrib Mineral Petrol (2011) 161:331–347 333

123

2 min with this procedure, but that the remainder of the

decompression may take up to 1 h. By 100 h at 400 MPa

and 1,125�C the bytownite crystals were observably cor-

roded (scanning electron images) due to the onset of

decompression melting, so runs of longer duration were not

attempted. Runs were quenched by shutting off power to

the furnace and cooled through the glass transition tem-

perature in B10 s.

Series 3

As discussed subsequently, bubbles coarsened slower than

the theoretical rates for steady-state ripening in the mixed

H2O-CO2 experiments (series 1 and 2), probably due to

slow diffusivity of CO2 through the melt. To investigate

this, two series of H2O-saturated experiments were also

conducted. Series 3 was performed in H2O-pressurized

cold seal (CS) vessels at the Massachusetts Institute of

Technology. Runs were performed in duplicate at 100 MPa

and 750�C for durations of 1 day, 1 and 2 weeks, and 1, 2,

and 4 months. Powdered rhyolite obsidian (275 mg) plus 6

wt% H2O were welded into 4 mm diameter Au capsules

and brought directly to pressure and temperature using

established methods (Elkins and Grove 1990). No explicit

oxygen buffer was employed, but the vessels are Ni-rich

alloys, so fO2s were near or slightly above Ni-NiO. Runs

were quenched by removing vessels from their furnaces

and cooling in a jet of compressed air.

Series 4

Series 4 was another H2O-saturated rhyolite series con-

ducted in the piston cylinder apparatus at 400 MPa and

850�C, similar to the series 1 except that 15 wt% H2O was

loaded with the rhyolite powder, and pure H2O was loaded

in the outer capsule along with Ni-NiO buffer. Experiments

in this series were held at pressure and temperature for

1 day, and 1, 2, and 3 weeks before quenching.

Sample mounting, imaging, and image processing

After quenching, capsules were unloaded from assemblies,

checked for volatile retention (weigh, puncture, observe

visible bubbling, heat, reweigh), sectioned longitudinally

with a wafering saw, and a portion mounted in epoxy and

polished. The piston cylinder assemblies produce a slight

flattened dimple on the top of the capsule allowing verti-

cally oriented sections to be prepared. Polished experi-

ments were examined both by scanning electron

(backscattered and secondary electron) and reflected light

microscopy.

Multiple digital images of each polished sample were

obtained such that nearly all of the cross-sectional area was

captured, and the range of bubble sizes was resolvable. For

the three rhyolite experimental series, size distributions

were measured from reflected light images collected with a

digital camera attached to a petrographic microscope. A

109 objective lens was sufficient to resolve the bubble size

range in the rhyolite PC experiments (series 1, 4), whereas

images obtained using both 59 and 109 lenses were

required to encompass bubbles in the rhyolite CS experi-

ments (series 3). For the basaltic andesite PC experiments

(series 2), we used 1009 and 3009 backscattered electron

(BSE) images obtained with a JEOL 8900 electron

microprobe at the USGS in Menlo Park. Digital images

were converted to binary (bubbles and condensed pha-

se(s)), and then processed to output measurements of ves-

icle areas and abundances. For the basaltic andesite series,

two sets of binary images were created: bubbles and melt,

and bubbles and crystals. The area and number of both

were obtained. Based on replicate processing, we deter-

mined 1 lm as our limit of confident particle radius reso-

lution and set this as the lower bound for 2-dimensional

vesicle size distributions. Only the 1-day and 1-week runs

of series 4 had bubble sizes approaching this minimum

(Table 2). In the anomalous case where gas pockets formed

at the wall of capsules, the areas of such ‘vesicles’ were

measured and noted, yet not included in size distribution

analysis. The stereology method of Sahagian and Prous-

sevitch (1998) was applied to convert 2-dimensional

bubble areas and abundances to 3-dimensional size distri-

butions. Processing of different images from the same run

and replicate processing of the same images suggest

uncertainties of up to 10% relative for bubble size and

fraction values, and 20% on bubble number density (NV).

Results

General features

Spherical bubbles were produced in every experimental

synthesis, and were generally evenly dispersed in the melt,

with no measurable occurrence of bubble migration or

accumulation at the capsule roof (Figs. 1, 2, 3, 4). Bubbles

in rhyolite runs are present from the onset as an excess

volatile phase, whereas in the basaltic andesite runs

nucleation of bubbles occurred discretely during decom-

pression. Larger bubbles or vapor pockets are situated at

the capsule sidewalls in series 4 (400 MPa rhyolite with

pure H2O) and these generally increase in size with

experiment duration (Table 2), and at the capsule ends in

series 3 (100 MPa rhyolite with pure H2O). Small crystals

grew only in the series 3 rhyolite experiments, generally

unassociated with bubbles (Fig. 5). No crystal formation or

growth was apparent in the other rhyolite, nor in the

334 Contrib Mineral Petrol (2011) 161:331–347

123

basaltic andesite series. In the latter case, initially sharp

crystal margins became slightly irregular or rounded by

100 h due to incipient decompression melting (Fig. 2).

A range of bubble sizes is apparent especially in shorter

duration runs of all series. In the rhyolite, we attribute this

to the range of grain sizes in the starting glass—since liquid

water that initially interconnected between glass grains

later became isolated as vapor bubbles trapped in grain

interstices as the glass melted. Two distinctive features are

(a) bubble-free melt ‘pools’ that were probably regions of

maximum initial grain size of the starting powder (Fig. 1),

and (b) bubble-free melt ‘haloes’ surrounding some large

bubbles. Figure 4b illustrates the extreme of such haloes in

the longer duration 100 MPa rhyolite cold seal experi-

ments. Note that in the 2- and 3-week runs, large vapor

pockets line the capsule walls, with small bubbles restricted

to the capsule center, and separating these is a sheath

of melt lacking bubbles. Most notably, within each

Fig. 1 One binary cross section image from each run in series 1, as

labeled. In this and subsequent figures (2–4) bubbles are black, meltwhite, and the scale is the same for all images. From top to bottom,

numbers in boxes at bottom left of each image are: bubble fraction

(with values obtained from the average grayscale value of several

images), mean radius, and bubble number density. Small ‘‘9’’

designates center of melt pool in 1-day and 2-week images. Note

lower vesicularity of 4-week run, indicating loss of volatiles

Fig. 2 Top row shows one binary cross section image from select runs in series 2, as labeled. Bottom row shows the same, unprocessed images

in which plagioclase crystals are dark gray and melt is light gray

Contrib Mineral Petrol (2011) 161:331–347 335

123

experimental series there is a visually apparent trend of

increasing bubble size and decreasing bubble number with

increasing run duration (Figs. 1, 2, 3, 4).

Bubble size distributions

Series 1

Runs in the rhyolite H2O-CO2 experimental series at

400 MPa consist of crystal-free glass, abundant dispersed

spherical bubbles, and minor capsule wall bubbles (Fig. 1).

Quantitative information on the bubble fraction, bubble

number density (NV), nearest neighbor distance (NND), and

mean radius for all runs in each series is given in Table 2. In

series 1, NV deceases while NND and the mean radius increase

with run duration, consistent with a coarsening process.

Series 1 also shows a decrease in total bubble fraction in

the 4-week run (Fig. 1; Table 2), which exceeds estimated

measurement uncertainty. This could result either by dif-

fusive loss of volatiles through the capsule wall or by growth

Fig. 4 a One binary cross section image from each run in series 4, as

labeled. Nearly all gas diffused to capsule walls in the 2- and 3-week

runs, as shown also in Fig. 4b. b Photos of the 2- and 3-week runs in

series 4, as labeled, with capsule wall bubbles outlined in white. Note

that large bubbles occur at the top and bottom of the capsule wall,

confirming they do not result from bouyant rise, and that small

bubbles are clustered in capsule center

Fig. 3 One binary cross section image from each run within series 3

336 Contrib Mineral Petrol (2011) 161:331–347

123

of unimaged bubbles on the capsule wall at the expense of

bubbles in the interior of the sample. Diffusive volatile loss

would be dominated by H2 and H2O transport and so would

result in a reduced concentration of H2O in the melt, which

was investigated by Fourier Transform Infrared Spectro-

scopic (FTIR) measurements of H2O and CO2 dissolved in

glass. Transmission FTIR of glass pools in the 1-day and 4-

week runs (following methods in Lowenstern et al. 1997)

give 4.3 ± 0.2 wt% H2O, 1850 ± 40 ppm CO2, and

3.2 ± 0.2 wt% H2O and 2290 ± 80 ppm CO2 (±1-sigma),

respectively. Using the 1-day run as a reference standard,

reflectance FTIR (following techniques in Moore et al.

(2008); P. King, pers. comm.) yielded 3.9, 3.7, 3.2 wt%

dissolved H2O in order of increasing run duration to 3 weeks.

Dissolved H2O concentrations in the 1 day through 3-week

runs are indistinguishable at the 2-sigma level, but the 4-

week run clearly shows evidence of diffusive volatile loss

both in its dissolved H2O concentration and its reduced

bubble number density.

Bubble size distributions in most runs of all four experi-

mental series are fine skewed, probably imposed at the

inception of the experiments by the size distribution of the

glass powder. This skewness resembles a log-normal dis-

tribution, so reported (Table 2) and plotted (Fig. 6) mean

radii are calculated as geometric means. For completeness,

arithmetic mean radii are also given in Table 2. These are

slightly larger (by 20% relative, on average), have much

greater standard deviations, and give slightly poorer corre-

lations with run duration and only marginally different time-

related coarsening exponents (not plotted) than the geo-

metric mean radii. Figures 6 and 7 plot the time-dependant

trends of the geometric mean radius and NV, respectively, for

the 4 series. The size—time trends of the series 1 data are

well fit by the power relationships r = 2.1t0.16 (R2 = 0.97)

and NV = 3 9 106t-0.76 (R2 = 0.98) where r is mean radius

in microns, NV is bubble number density in mm-3 and t is

time in hours. Bubble nearest neighbor distance (NND;

calculated as NND = 0.554NV-1/3; Chandrasekhar, 1943 in

Table 2 Quantitative bubble data

Duration Bubble

fraction

NV (mm-3) NND (mm) Geometric

mean radius

(microns)

Geometric

standard

deviation (*/)

Arithmetic

mean radius

(microns)

Standard

deviation

Sample ID

Series 1 1 day 0.22 2.7 9 105 0.009 3.6 1.7 4.1 2.1 r2051

1 week 0.22 4.5 9 104 0.016 4.6 2.3 6.3 4.9 r2052

2 week 0.19 3.0 9 104 0.018 5.7 2.2 7.3 5.0 r2053

4 week 0.11 2.3 9 104 0.019 6.1 2.0 7.4 4.5 r2057

Series 2 1 h 0.17 9.1 9 105 0.006 2.3 1.7 2.4 2.0 r2018

3 h 0.11 3.2 9 105 0.008 3.2 1.7 3.4 2.6 r2049

10 h 0.13 2.3 9 105 0.009 3.1 1.9 3.5 3.0 r2020

20 h 0.10 7.8 9 104 0.013 3.9 1.9 4.5 4.2 r2032

40 h 0.15 1.6 9 105 0.010 4.5 1.8 4.9 3.6 r2050

70 h 0.16 1.2 9 105 0.011 4.0 1.8 4.4 3.7 r2056

100 h 0.16 2.6 9 104 0.019 4.7 2.6 8.2 7.8 r2045

Series 3 1 day (a) 0.14 2.3 9 104 0.019 5.2 1.8 6.4 5.6 MC78-58#4

1 day (b) 0.25 3.75 9 104 0.017 2.8 2.4 3.8 3.3 MC78-58#11

3 day 0.19 2.7 9 104 0.018 2.9 2.3 3.7 3.2 MC78-58#10

1 week 0.12 1.0 9 104 0.026 6.9 2.1 8.9 6.9 MC78-58#1

2 week 0.16 5.7 9 103 0.031 11.7 1.8 13.6 8.1 MC78-58#2

1 month (a) 0.21 2.4 9 103 0.041 17.4 1.7 20.2 12.5 MC78-58-X

1 month (b) 0.17 2.5 9 103 0.041 12.6 1.8 15.3 12.3 MC78-58#7

2 month (a) 0.10 4.9 9 102 0.070 27.0 1.5 29.4 14.1 MC78-58#9

2 month (b) 0.17 8.1 9 102 0.059 26.5 1.7 29.9 14.7 MC78-58#4

4 month (a) 0.13 3.4 9 102 0.079 29.9 2.0 35.5 29.9 MC78-58#8

4 month (b) 0.10 3.7 9 102 0.077 21.5 2.1 27.0 17.1 MC78-58#5

Series 4 1 day 0.08 4.7 9 105 0.007 1.4 1.9 2.2 2.0 r2109

1 week 0.17 2.7 9 105 0.009 1.8 2.7 3.7 3.5 r2118

2 week* 0.02 1.0 9 104 0.026 3.2 2.1 6.0 4.0 r2120

3 week* 0.01 4.7 9 103 0.033 4.5 3.2 7.1 5.1 r2119

* Interior bubbles only in 2- and 3-week runs

Contrib Mineral Petrol (2011) 161:331–347 337

123

Fig. 5 BSE microprobe 5009

images showing crystals in

series 3. Scale is the same for all

images. From top to bottom,

numbers at the top left of each

image are the following:

experiment duration, % crystals

by area, crystal mean diameter

(assuming a square). Brightwhite *equant crystals are

magnetite; light gray, elongate

crystals are plagioclase

Fig. 6 Plot of geometric mean

radius (r, in microns) versus run

time (t, in hours) for three

experimental series. Diamondsrepresent series 1; trianglesseries 2; and squares series 3

Fig. 7 Plot of bubble number

density (NV in mm-3) versus run

time (t, in hours) for three

experimental series. Diamondsrepresent series 1; trianglesseries 2; and squares series 3

338 Contrib Mineral Petrol (2011) 161:331–347

123

Russ, 1986) more than doubles from the 1-day (9 lm) to the

4-week (19 lm) run (Table 2).

Series 2

Series 2 was designed to investigate processes in crystal-

rich portions of magma reservoirs beneath active basaltic

andesite volcanoes such as Pavlof (Alaska), Arenal (Costa

Rica), Stromboli (Italy), and Cerro Negro (Nicaragua).

Spherical to sub-spherical bubbles formed in all runs, with

bubble fractions varying between 0.10 and 0.17 (Fig. 2,

Table 2). Despite the ease of bubble foam ascent in similar

decompression experiments on crystal-free basaltic andes-

ite (Mangan et al. 2006), we observed no tendency of

bubbles to accumulate near the tops of crystal-bounded

melt pockets, nor was there a tendency of bubbles to attach

to (‘‘wet’’) crystal margins. Inability of bubbles to ascend

probably results from the limited buoyancy of single bub-

bles, viscous drag due to the short distances between

bubbles and nearby crystals, and insufficient aggregate

buoyancy of bubbly melt pockets to displace the crystal

network. Instead of ascending, the bubbles ripened with a

general increase in mean radius and NND and decrease in

NV with increasing run duration. Given that crystals melted

slightly in the 100 h experiment, this was the longest run

performed. The geometric mean radius increases by a

factor of two from 1 to 100 h, and is fit by the relationship

r = 2.5t0.14 (R2 = 0.88, Fig. 6). NV decreases by[1 order

of magnitude and follows the relationship NV = 8 9

105t-0.59 (R2 = 0.79, Fig. 7). These rates are closely sim-

ilar to those derived for the rhyolite H2O-CO2 series, and

are significantly slower than steady-state ripening theory

predicts.

Series 3

All products of series 3 consist of glass that includes dis-

persed spherical bubbles, as well as relatively large vapor

voids located at the ends of the capsules, and a trace of

crystals. Figure 3 shows images from experiments with

durations of 1 day, 1 and 2 weeks, and 1, 2, and 4 months.

Bubbles get larger but less abundant with increasing run

duration. Replicate runs of 1 day, and 1, 2, and 4 months

duration were also measured to estimate overall repro-

ducibility. Quantitative results generally agree within

*30% (Table 2).

The total bubble volume fraction varies from 0.10 to 0.25

(Table 2), which probably stems from differences in the

fraction of vapor contained in voids at the ends of the cap-

sules that are of irregular shape and therefore are difficult to

quantify accurately. The geometric mean bubble radius

increases by a factor of six, NND increases by a factor of 4,

and NV decreases by almost two orders of magnitude

between 1 day and 4 months (Fig. 2; Table 2). Small crys-

tals of size 2–15 lm form between 0.2 and 1.7 area % of the

cold seal run products, with no discernable correlation

between abundance and experiment duration (Fig. 5).

Electron microprobe analyses and observations revealed that

*70% of such crystals are magnetite, *25% are plagio-

clase, and minor amounts of pyroxene, amphibole and apa-

tite are present. Magnetite crystals are*2–10 lm, generally

equant, and commonly are attached to bubble walls. Pla-

gioclase crystals are generally *5–15 lm, often show

overgrowth textures, and also are commonly attached to

bubbles. The majority of bubbles, however, have no asso-

ciated crystals.

The geometric mean radius and NV data (Table 2) are

well fit by the power relationship r = 0.86t0.44 (R2 = 0.88)

and NV = 8 9 105t-0.94 (R2 = 0.94) where r is mean

radius in microns, NV is bubble number density in mm-3

and t is time in hours (Figs. 6, 7).

Series 4

Bubbles ripened faster in series 3 (CS, rhyolite, pure H2O)

than in series 1 (PC, rhyolite mixed H2O-CO2). Potentially

this was due fast diffusion of H2O versus CO2 in silicate

melts (Watson 1994) controlling the ripening rate. To

investigate this, we conducted a rhyolite series using the

same conditions as series 1 but saturated with pure H2O

only. Experiment durations in this series were 1 day, and 1,

2, and 3 weeks. No longer durations were attempted

because H2O diffusion to capsule wall bubbles had nearly

eliminated interior bubbles by 3 weeks (Fig. 4). The 1-day

run has a high number density of micron-size, spherical

bubbles. The 1-week run has a notably bimodal bubble

population, a higher apparent vesicularity, and a lower

bubble number density than the 1-day run.

In the 2- and 3-week experiments, nearly all H2O had

diffused from interior bubbles to vapor pockets along the

capsule walls and ends, leaving few and small bubbles

restricted to the core of the glass slug. Capsule wall pockets

account for about 0.10 and 0.16 area fraction in the 2- and

3-week runs, respectively. Quantitative data (NV, NND,

mean radius) were obtained for the interior bubbles only

(Table 2), but a halo of glass completely lacking in bubbles

separates the capsule walls from the slightly bubbly core.

This is likely due to ‘‘completion’’ of the ripening process

near the capsule wall bubbles. Observations in the 3-week

run that, (a) the vapor fraction at the capsule wall is higher,

(b) the mean radius of interior bubbles is smaller, and (c)

the number density of bubbles is lower, than in the 2-week

run suggest that more H2O had diffused to vapor pockets at

the capsule wall in the 3-week run. Mean sizes and number

densities for these runs are not plotted on Figs. 6 and 7

because coarsening of capsule wall pockets overwhelmed

Contrib Mineral Petrol (2011) 161:331–347 339

123

interior ripening, with the result that measured size and

number results on interior bubbles do not produce inter-

pretable trends.

Theoretical analysis

Steady-state ripening

A quantitative theoretical description of Ostwald ripening

was developed by Lifshitz and Slyozov (1961) and Wagner

(1961), who considered dispersed particles of various sizes

as interacting individually with a mean concentration field,

rather than with one another. Strictly, this mean-field

approach considers ripening at steady-state for an infinite

dilution of dispersed particles. For steady-state ripening

controlled by diffusive mass transport, theory predicts the

mean particle radius increases as:

rðtÞ3 ¼ tKLSW ð2Þ

Here KLSW is identified specifically as the Lifshitz-

Slyozov-Wagner (LSW) rate constant. Predictions of Eqs.

(1) or (2) are that: (a) the logarithm of mean particle radius

as a function of the logarithm of time will approach a slope

of 1/3 at sufficiently long times, (b) at steady state, the

dispersed phase has a specific size distribution that is

unchanging with time if normalized to the mean radius

(coarsening is self-similar), and (c) the particle number

density NV decreases with t-1. It follows from these that

mean particle volume increases linearly with time, and that

for the mean particle radius hri to increase by a factor of n

requires an increase in time by a factor of n3. Many

modifications have been made of the basic LSW analysis to

account for non-zero volume fractions of the dispersed

phase. Generally, such analyses predict an increase in the

value of the rate constant up to an order of magnitude

relative to the LSW value, as well as a broadening of the

size distribution, but the hr(t)i proportional to t1/3 and NV

proportional to t-1 relations remain (Ratke and Voorhees

2002).

Coarsening can also be limited by interface kinetics as

components transfer from one phase to another. At steady

state (long times), such interface-limited coarsening fol-

lows the form (Ratke and Voorhees 2002):

hrðtÞi ¼ Kt1=2 ð3Þ

where K is a different rate constant. For steady-state

interface-limited coarsening a log–log plot of mean radius

(ordinate) against elapsed time (abscissa) would have a

slope of �.

Plots of the experimental results on bubble ripening in

silicate melts from this study show that none of the

experimental series follows the hr(t)i = (Kt)1/3 relation

expected for steady-state diffusion controlled ripening

(Fig. 6). The rhyolite and basaltic andesite mixed H2O-

CO2 series (series 1 and 2) coarsened almost identically

with t0.14-0.16, appreciably slower than the diffusion-lim-

ited steady state. In contrast, the rhyolite H2O series at

100 MPa (series 3) coarsened faster, with t0.44, close to the

theoretical exponent of 0.50 for interface-limited ripening.

Coarsening to capsule wall bubbles was so rapid in the

rhyolite H2O series at 400 MPa (series 4) that it precluded

our ability to measure and interpret by classical Ostwald

ripening theory.

Transient regime ripening

Failure to match theoretical steady-state time—size—

abundance relations is common in experimental studies of

ripening (DeHoff 1991; Snyder et al. 2001; Ratke and

Voorhees 2002) likely because such experiments have been

of insufficient duration to approach steady-state, implying

that growth of the dispersed phase was in a ‘‘transient

ripening regime’’. Chen and Voorhees (1993) analyzed

transient diffusion-limited ripening in an ideal two-com-

ponent system through numerical simulations of various

initial particle size distributions and volume fractions.

They normalize time by a factor s defined by:

s ¼ ðr3c ð0Þ=lacDÞðCb

eq � CaeqÞ ð4Þ

D is the diffusivity of the migrating component, rcð0Þ is the

critical particle radius at the onset of ripening (particles

smaller than the critical radius shrink, those larger grow),

Cbeq and Ca

eq are the equilibrium concentrations (mol

fractions) of the diffusing component in the dispersed band matrix a phases for particles of infinite radius (for

vapor bubbles, no excess pressure), and lac is the Gibbs–

Thomson capillary length in the matrix phase, defined as:

lac ¼ 2Vbmc

� ��DCeqG00a� �

ð5Þ

In (5) Vbm is the molar volume of the dispersed b phase, c

is the surface tension, DCeq is the concentration difference

as in (4), and G00a is the second derivative of the Gibbs free

energy of the matrix phase with respect to the diffusing

component. For ideal solutions with B as the diffusing

component this is:

G00a ¼ RT�ðxa

Bð1� xaBÞÞ ð6Þ

where R is the ideal gas constant, T is temperature and xaB is

the molar concentration of component B in phase a. Chen

and Voorhees (1993) use the mean particle radius in the

shortest duration experiment of various studies as a proxy

for rcð0Þ to calculate s and find that diverse initial scenarios

approach the classical LSW relation in Eq. (2) most com-

monly between 100 and 1,000 s and rarely in 10 s. Akaiwa

340 Contrib Mineral Petrol (2011) 161:331–347

123

and Voorhees (1994) find through additional numerical

simulations that transient regime coarsening characteristics

change substantially in the first *25 s, but the rate of

approach to steady-state then slows appreciably, and that

greater abundances of the dispersed phase further slow the

approach to steady state.

Duration of the transient regime

Considering rhyolitic melt undergoing coarsening of bub-

bles of a pure H2O vapor at mid-upper crustal conditions

using appropriate values of D * 10-6.5 cm2/s (Ni and

Zhang 2008), c *10-5 J/cm2 (Mangan and Sisson 2005),

VH2Om * 30–50 cm3/mol, T *750–850�C, Eqs. (4) and (5)

give lac of 0.6–0.06 nm (high limit uses high VH2Om , low T,

xmeltH2O 0.5; low limit uses low VH2O

m , high T, xmeltH2O 0.1), which

is equal in magnitude to lac estimates for other coarsening

systems (Ratke and Voorhees 2002). For the experimental

results in this study, using rcð0Þ of 3–5 lm (rhyolites,

Table 2) gives s of *12 min to 15 h. Then using 100–

1,000 s as the common duration of the transient regime

(Chen and Voorhees 1993) H2O-rich bubbles coarsening in

a typical rhyolitic melt at mid-upper crustal conditions

could remain from about 1 day to nearly 2 years

(1.8 years) to approach a constant (steady state) growth

rate.

Experiments in series 4 had a high dissolved H2O con-

centration, calculated as 8.8 wt% at the run conditions (Liu

et al. 2005), and from this effective H2O diffusivity is

*10-5.6 cm2/s (Ni and Zhang 2008). This is about a factor

of 5–15 faster than for similar melts with moderate dis-

solved H2O of 4–6 wt% (DH2O * 10-6.9–10-6.3 cm2/s).

Vapor-melt surface tension decreases with increasing melt

H2O concentrations (Mangan and Sisson 2005) but to no

more than about half the value considered previously for

moderate H2O concentrations. Together, these differences

decrease the estimated value of s by a factor of between 2.5

and 15. Therefore, whereas a bubbly moderate H2O rhyo-

lite might take nearly 2 years to attain steady-state ripen-

ing, no more than between 1 and 9 months may be required

for melts with dissolved H2O concentrations approaching 9

wt%.

Bubbly magmas with appreciable quantities of slower

diffusing volatile components, such as CO2, would take

longer to reach steady state. Watson (1994) notes that the

log of CO2 diffusivity increases roughly linearly with melt

H2O concentration, allowing estimates of DCO2of

*10-7.5–10-8 cm2/s for rhyolitic melts with dissolved

H2O concentrations and temperature typical for shallow

magma reservoirs (4–6 wt% H2O, 800�C). This is about

1–1.5 orders of magnitude slower than H2O diffusion. To

our knowledge there are no explicit estimates of the effects

on CO2 on melt–vapor surface tension for natural rhyolitic

compositions; however, the similar underpressures (over-

saturations) necessary to homogeneously nucleate bubbles

of H2O and of mixed H2O-CO2 in rhyolitic melt (Mangan

and Sisson 2000, 2005) suggest that c does not differ

greatly due the presence of appreciable CO2, at least for

melts with dissolved H2O of several wt%. Then following

the same approach as for H2O, if diffusion of CO2 were the

rate-limiting process, the transient regime would persist for

between 3 months and 100 years (assumes xmeltCO2

0.01–0.1,

Vvaporm 40 cm3/mol).

These theoretical and numerical simulations of the

transient regime suggest that approaching steady-state

could require days to years for magmas with high dissolved

H2O and months to centuries for magmas with appreciable

dissolved CO2. These estimates concord with the obser-

vations that none of the experimental series follows the

time1/3 coarsening and time-1 reduction in number density

appropriate for steady-state.

Ripening in the transient regime

Numerical simulations show that for most initial particle

size distributions and volume fractions, transient regime

ripening is slower than the steady-state rate at a com-

parable elapsed time (Chen and Voorhees 1993). The

\1/3 exponents of the power law fits to the mixed H2O-

CO2 series (Fig. 6) are therefore consistent with transient

regime ripening, but not likely representative of the

ripening rate in true bubbly magmas. Our calculations

suggest that in real magmas with lifetimes of months to

millennia the ripening rate would converge on the t1/3

KLSW growth law.

The [1/3 exponent of the fit to the H2O-only series

requires further discussion. Series 3 has a fitted power law

exponent of 0.44 (Fig. 6), and series 4 shows ripening to

capsule wall pockets was so rapid that it nearly eliminated

bubbles from the interiors of the melt in 2 and 3 weeks. In

these cases, ripening occurs at a faster rate than the 0.33

exponent predicted for diffusion controlled ripening, and

the trend of 0.44 approaches the 0.50 exponent for inter-

face-kinetic control. Interface control would indicate the

existence of a kinetic barrier to the transfer of H2O between

melt and vapor, and melt at the melt—vapor interface

would have a H2O concentration differing from the equi-

librium value for the pressure in the adjacent bubble.

Yet there is the possibility this experimental series also

remained in a transient regime. Numerical simulations of

initially bimodal particle distributions evolve through a

stage in which the ripening rate temporarily exceeds the t1/3

steady-state value for diffusion control (Chen and Voorh-

ees 1993). The exponent subsequently reduces (the evolu-

tion line flattens on a log–log plot of hri vs. t) as coarsening

destroys the bimodal population, then increases to the 0.33

Contrib Mineral Petrol (2011) 161:331–347 341

123

value as the system converges on the KLSW growth law and

particle size distribution. Since bubble populations are

notably bimodal in the shorter duration H2O-only runs

(Figs. 3, 4; probably an artifact of the size range of in

grains of the initial powder), this bimodality may have led

to fast transient ripening—not steady-state interface

kinetics.

Estimates of steady-state bubble coarsening

Static steady-state coarsening in rhyolitic magmas

After an interval of transient ripening lasting from days to

perhaps 100 years, the bubbles in a magma body can

evolve to a diffusion controlled hri3 = Kt rate law, with K

being the KLSW rate constant. Strictly, this rate relation

applies to bubbles that remain in place (static, steady state);

coarsening during bubble ascent is considered in a later

section. The classical rate constant is defined as (Chen and

Voorhees 1993; Ratke and Voorhees 2002; after Marqusee

and Ross 1983):

KLSW ¼ 4lacD�ð9DCeqÞ ð7Þ

with variables as defined previously (Table 3). Inserting

appropriate values: lac * 0.6–0.06 nm, D * 10-6.5 cm2/s,

and DCeq * 0.9–0.5 (where high value DCeq models low-

mass components in the melt (e.g. oxides) and low value

models high-mass components (i.e. mineral-like); e.g.

Burnham 1979) gives KLSW from 1.0 9 10-15 (minimum)

to 1.7 9 10-14 (maximum) cm3/s. For a high dissolved

H2O case (*9 wt%), values of *4910-15 (minimum) to

4 9 10-14 (maximum) cm3/s are estimated, calculated

using c = 5 9 10-6 J/cm2, Vvaporm = 30 cm3/mol,

xmeltH2O = 0.1–0.5, and 750–850�C. For reference, the high

and low rate constants would indicate *7 9 102–

3 9 104 years for bubbles to attain mm radii by steady-

state coarsening.

Evolution curves of mean bubble radius versus time for

these static, steady-state rate constants are plotted as color

coordinated solid lines in Fig. 8. These estimates are

applicable to rhyolitic magmas at *750–850�C with dis-

solved H2O *4–9 wt% (values likely for many mid-upper

crustal silicic magma bodies), and with no bubble motion.

A trend for a rate constant of ten times the high-H2O

rhyolite value is also plotted to encompass the potential

maximum effect of non-zero bubble fractions (Ratke and

Voorhees 2002).

Fig. 8 Plot of calculated mean bubble radius versus time, Solid linesare calculated for static, steady coarsening; dashed lines for steady-

state coarsening with bubble ascent. Black lines represent minimum

and maximum coarsening in rhyolite with 4–6 wt% H2O (moderate);

Red lines represent minimum and maximum coarsening in rhyolite

with 9 wt% H2O (H2O-rich); Orange line represents 910 the

maximum for H2O-rich rhyolite (to account for non-zero bubble

volume fraction); Purples lines represent the minimum and maximum

bubble coarsening in mid-upper crustal basalt

Table 3 Definition of variables

p Pressure

c Surface tension

r Radius

t Time

K Rate constant

NV Bubble number density

NND Bubble nearest neighbor distance

KLSW Lifshitz-Slyozov-Wagner rate constant

s Time normalization factor

D Diffusivity

rcð0Þ Critical particle radius

b Dispersed phase (bubbles)

a Matrix phase (melt)

Cbeq Equilibrium concentration of diffusing component

in dispersed phase

Caeq Equilibrium concentration of diffusing component

in matrix phase

lac Gibbs–Thomson capillary length in matrix phase

Vbm Molar volume of dispersed phase

DCeq Concentration difference between dispersed and matrix

phases

G00a Second derivative of Gibbs free energy of the matrix

phase with respect to the diffusing component

R Ideal gas constant

T Temperature

xaB Molar concentration of component B in phase a

g Acceleration due to gravity

g Viscosity

Dq Density difference between matrix and dispersed phases

342 Contrib Mineral Petrol (2011) 161:331–347

123

It is important to note that time for these coarsening

evolutions is relative to a hypothetical nucleation event that

commences with the steady-state bubble size distribution

and extremely small bubbles. Processes or events that

create larger initial bubbles, but with a near-steady-state

size distribution and concentration gradients, would appear

and behave as though they had coarsened for a long time.

Their subsequent coarsening history would follow from

that apparent time, as given by their position on the

appropriate steady-state mean radius versus time trajectory.

For example, some scenario might quickly create a near–

steady-state bubble size distribution with a mean radius of

hri equal to the results after 1,000 years of steady-state

coarsening. Given the hri3 = Kt rate law [Eq. (1)], to attain

a mean radius of 10 hri by steady-state coarsening would

then require nearly a million years (i.e. 103 9 1,000–

1,000 years) irrespective of the true time since bubble

creation. The solid line trajectories in Fig. 8 show that

static, steady-state Ostwald ripening would be highly effi-

cient in eliminating micron-sized bubbles in mid-upper

crustal magma bodies, and would facilitate attainment of

mm-sized bubbles in reasonable geologic timescales

(*1,000 years), but that reaching cm-scale mean radii

would require prohibitively long times ([100,000 years)

by static coarsening alone.

Steady-state coarsening with bubble ascent in rhyolitic

magmas

Movement of bubbles through the melt disrupts and

steepens concentration gradients in the adjacent liquid,

thereby hastening the bubble coarsening rate. For bubbles

ascending at velocities derived from Stokes Law though an

infinitely similar medium, steady-state coarsening follows

(Ratke and Voorhees 2002):

hri3=2 ¼ Kt ð8Þ

This relation strictly applies where the Peclet number

Pe � 1, which corresponds to bubbles moving quickly

relative to the rate at which diffusion can relax

concentration gradients in the surrounding medium, and

does not consider expansion of the precipitate (bubbles)

with changing depth. Bubble radii of millimeters or larger

would be required for Pe � 1 in viscous rhyolitic melts.

The coarsening with ascent relation (8) shows that

increasing mean radius hri by a factor of n requires an

increase in time by a factor of n3/2, which is considerably

faster than the n3 relation for static steady-state ripening.

Steady-state coarsening at lower Peclet numbers is

intermediate between Eqs. (2) and (8) (Ratke and

Voorhees 2002). The rate constant in (8) is given by:

K ¼ 0:837½ðDg Dqj jðga þ gbÞ=ð3pgað2ga þ 3gbÞÞ�1=2lac=DCeq ð9Þ

D is diffusivity of the exchanging component, g is

acceleration of gravity, ga and gb are the viscosity of the

matrix and particle phases (in this case melt and bubbles,

respectively), Dq is the density difference between the

phases, and the other terms are defined as before. For vapor

bubbles in silicate melt at crustal pressures, ga � gb and

(9) simplifies to:

K � 0:837½Dg Dqj j=6pga�1=2lac=DCeq ð10Þ

Growth evolution curves of mean bubble radius with

elapsed time for ascending bubbles are plotted as dashed

lines in Fig. 8 for the rhyolitic melt scenarios considered

previously, under the simplifying assumption that pressure

changes due to bubble ascent are small relative to total

pressure so that Dq is approximately constant and r changes

only due to coarsening. As with steady-state coarsening,

elapsed time t is relative to a hypothetical nucleation event

with minutely small bubbles that would follow (8) and (9).

Figure 8 shows that to attain a mean radius of *1 mm by

Eqs. (8) and (9), necessary for Pe � 1 in rhyolite, would

require at least t * 3 9 102 (maximum for H2O-rich

rhyolite; red dashed line) to *29103 (minimum for

moderate H2O rhyolite; black dashed line) years. These are

minima because sub-millimeter bubbles would not have

Pe � 1 and therefore would coarsen slower. Times to reach

millimeter sizes are therefore bracketed between the

ascending and immobile results at 3–7 9 102 (high

dissolved H2O) to 2 9 103–3 9 104 (low dissolved H2O)

years. A subsequent increase in hri of an order of magnitude

(coarsening to centimeter-sized bubbles) would require

*69103–1 9 106 years for ascent assisted coarsening,

irrespective of the true time for hri to have initially

reached 1 mm. Large silicic magmatic systems have

periods and spans of activity on the order of 104–107 years,

so centimeter-sized bubbles are theoretically achievable by

coarsening during bubble rise through mid-upper crustal

intrusions. In a subsequent section we show, however, that

*millimeter size bubbles would rise too quickly through

liquid-rich magmatic systems to have time to coarsen to

centimeter sizes.

Bubble coarsening in basaltic magmas

Equations (7) through (10) also hold for coarsening of static

or rising bubbles in basaltic melts. Khitarov et al. (1979)

present melt–vapor surface tension results for basaltic liquid

from 0.1 to 500 MPa. At 0.1 MPa surface tension is

*49105 J/cm2 and is insensitive to temperature and

Contrib Mineral Petrol (2011) 161:331–347 343

123

coexisting vapor composition. Application of 100 MPa of

mixed H2O-CO2 vapor reduced surface tension to *0.17 J/

m2, and by 300 MPa surface tension deceased to*0.1 J/m2.

Increasing H2O-CO2 vapor pressure to 500 MPa failed to

reduce surface tension further. For their 1,200�C, 300 MPa

experiment, Khitarov et al. report *4.5 wt% H2O and

*3,000 ppm CO2 dissolved in the liquid, which are values

appropriate for many mafic liquids in subduction settings

(Wallace 2005). Using c = 0.1 J/m2, Vm = 41 cm3/mol

(ideal gas law at 1,200�C, 300 MPa), m *59103 Pa s, D

*6.3 9 10-6 cm2/s (Einstein–Roskow relation compared

with rhyolite), qmelt = 2,350 kg/m3, and qvapor = 550 kg/m3

produces the static and ascending bubble size evolution

curves plotted as solid and dashed purple lines on Fig. 8

(maximum and minimum limits use high or low mol frac-

tions of dissolved volatiles in melt). These estimates indicate

that *150–1,500 years of static coarsening would be suffi-

cient to produce *millimeter mean radius bubbles in mod-

erately hydrous basaltic liquid at magma reservoir depths.

Coarsening during bubble ascent would be faster, but as with

rhyolitic liquids and shown subsequently, bubbles could

transit crustal level basaltic magmatic systems faster than the

time necessary to coarsen to centimeter sizes.

Estimates of bubble rise times and distances

with and without coarsening

The effectiveness of coarsening in promoting bubble ascent

is evaluated by comparison with bubbles of initially equal

radius ro, but that do not coarsen. Using the expression for

Stokes movement of fluid droplets in a liquid of differing

density (bubbles b of radius r in melt m):

Fig. 9 Plot of calculated bubble

a scent distance versus time

(starting at time when bubble

reaches 1 mm) for scenarios in

which bubble is not coarsening

(dashed line); bubble is

coarsening slowly (thin solidline); bubble is coarsening

rapidly (thick solid line)

344 Contrib Mineral Petrol (2011) 161:331–347

123

v ¼ r2gð2ðqb � qmÞðgm þ gbÞÞ=ð3gmð2gm þ 3gbÞÞ ð11Þ

Because the viscosity of the vapor is� that of the melt,

(11) simplifies to:

v � r2gðDq=3gmÞ ð12Þ

Ascent distance can then be solved for by:

x ¼Z t

t�

vdt ð13Þ

For bubbles that do not coarsen (r : ro), ascent distance is

simply vDt. For bubbles that do coarsen, ascent distance is

determined for the mean radius by inserting (8), (10), and

(12), into (13), where t* is an apparent time that would be

required to attain the starting radius ro by (8) and (10), and t–

t* is the true subsequent elapsed time of ascent. For

consistency with the derivation of (9), these ascent

calculations assume that Dq is constant. In reality, the

bubbles will expand with decreasing depth, accelerating their

ascent, such that distances predicted by (13) are minima.

Results are shown in Fig. 9 for H2O-rich bubbles ini-

tially of 1 mm radius in moderate-H2O rhyolitic melt (4–6

wt% dissolved H2O), and for mixed H2O-CO2 bubbles in

basaltic melt. These calculations utilize the maximum and

minimum coarsening with bubble ascent curves for mod-

erate-H2O rhyolite and for basalt in Fig. 8 (dashed black

and purple lines). Figure 9 shows that slow coarsening

estimates lead to ascent only marginally faster than for

non-coarsening bubbles in both rhyolite and basalt. The

reason for this is that apparent time t* is large in both cases

so the bubbles require long subsequent (true) times to grow

significantly. The fast coarsening estimates predict ascent

over *10 km distances, relative to the host liquid,

in roughly half the time required for millimeter radius

non-coarsening bubbles. While this is a significant increase

in ascent speed, a more important observation is that,

coarsening or not, *millimeter radius bubbles could

ascend multi-kilometer distances through magmatic liquid

on geologically relevant time scales. For reference, initially

1 mm radius bubbles have the potential to ascend *10 km

(a reference length scale of a large crustal magmatic sys-

tem) in less than 150–275 years through basaltic liquid and

in less than 2,500–6,000 years through moderately hydrous

rhyolitic liquid. True ascent times would be even briefer

due to enhanced buoyancy as the bubbles expand at shal-

lower depths. The more significant contribution of coars-

ening to degassing therefore is that in relatively short times

(hundreds to thousands of years) it can produce bubbles of

adequate size (millimeters) to ascend through magmatic

systems quickly. Once those bubbles attain sufficient sizes,

subsequent coarsening is only a minor factor in their ability

to rise through the system.

Bubble coarsening at low pressure

By (5) and (7), the large molar volume of magmatic vapor

at low pressure and the high melt-vapor surface tension of

H2O-poor melts (Khitarov et al. 1979; Mangan and Sisson

2005) are suggestive of large capillary lengths and espe-

cially rapid bubble coarsening for near-surface magmas,

particularly for basaltic liquids with high-volatile diffu-

sivities. Tephra bubble textures are consistent with Ostwald

ripening on the\1 min time scales of basaltic fire-fountain

eruptions (Mangan and Cashman 1996). A partly counter-

vailing factor is that the very low dissolved mole fractions

of the volatile component xaB imposed by low pressure

solubilities lead to large values for G00aðG00a !1as xa

B ! 0Þ limiting the steady-state KLSW rate constant.

For basaltic melt at 1,200�C, and 25 MPa, lc is *2 nm,

and KLSW is *4–5 9 10-13 cm3/s, or only slightly greater

than the maximum value estimated for especially H2O- and

bubble-rich rhyolite (basalt uses c = 0.4 J/m2, D =

6–7 9 10-6 cm2/s, xaB = 0.05). For these values, *50–

60 years would be required to produce millimeter radius

bubbles in basaltic melt by static, steady-state coarsening.

Coarsening with Stokes ascent would require similar time

periods to produce millimeter radii bubbles, and those

bubbles would have rise velocities of *50 m/year. Cen-

timeter radius bubbles could theoretically be attained in

1,700–2,000 years of coarsening during Stokes ascent,

though such bubbles would long previously have risen

through the magmatic system.

Comments on coarsening to crystal-wetting bubbles

Almost all magmas contain some crystals, as well as melt.

In silicic magmas bubbles adhere preferentially to certain

mineral species (FeTi-oxides) consistent with surface ten-

sion of mineral-vapor \ melt-vapor. Due to the release of

latent heat of crystallization, crustal magma bodies spend

much of their histories in a mushy, crystal-rich state, and

coarsening to bubbles wetting crystal surfaces is probably

widespread but is not considered by the relations reviewed

so far. From a qualitative standpoint, coarsening to crystal-

wetting bubbles can be expected to be faster than coars-

ening to bubbles freely suspended in melt. A consequence

of low mineral-vapor surface tension is that the pressure in

a bubble adhering to a crystal is lower than for a similarly

sized bubble that is surrounded completely by melt. To its

environment, the mineral-wetting bubble would appear as

though it were a much larger bubble, and it would form a

preferential sink for the diffusing volatile components. In

effect, the bubble population may act as though it were

bimodal with seemingly large (low pressure) crystal-wet-

ting bubbles and seemingly small (high pressure) bubbles

enveloped in melt. Numerical simulations of transient

Contrib Mineral Petrol (2011) 161:331–347 345

123

ripening show that bimodal populations coarsen faster than

the static, steady-state rate law (Chen and Voorhees 1993),

so crystal-wetting bubbles can be expected to coarsen rel-

atively quickly and at the expense of bubbles in liquid. An

example is the capsule wall pockets that coarsened at the

expense of interior bubbles in series 4 (Fig. 4). The result

in a crystal mush containing silicic liquid could be few but

relatively large bubbles attached to crystal surfaces.

Because of growth (second boiling) and shear, such bub-

bles might become interconnected, thereby precipitating

drainage of vapor. Wetting of crystals by vapor bubbles

appears to be less advantageous for systems with mafic

liquids, and rapid coarsening to crystal-wetting bubbles

may be delayed until the interstitial melt fractionates to

felsic compositions.

Conclusions

Experiments on rhyolite and crystal-bearing basaltic

andesite melts show that H2O and H2O-CO2 vapor bubbles

coarsen and decrease in number density by diffusive

(Ostwald) ripening in time periods accessible in the labo-

ratory. In no cases, however, did the bubbles follow the

classical time1/3 increase in mean radius and time-1

decrease in number density predicted by steady-state dif-

fusive ripening theory. Instead, bubbles are interpreted to

have coarsened in the transient regime, consistent with

literature simulations of non-steady-state ripening. Appli-

cation of transient regime theory to natural magmas sug-

gests that the transient regime may persist for days to years

in magmas with high H2O contents and for months to

centuries in magmas with high CO2 contents due to the

slow diffusion of CO2 relative to H2O. Given that ripening

in the transient regime is in general slower than steady-

state ripening, the low ripening rate in the mixed H2O-CO2

experiments can be attributed to this non-equilibrium

regime. Unexpectedly, fast ripening observed in the two

H2O-only series is most likely due to a unique character-

istic of the transient regime in which the coarsening rate

can temporarily exceed *t1/3 to eliminate initially bimodal

bubble size distributions.

From the duration of the transient regime, we estimate

that reservoir magmas could attain steady-state bubble

ripening between a few days and *100 years after the

onset of nucleation. Results for calculations of static (no

bubble ascent) steady-state bubble ripening in mid-upper

crust magmas show that the process is highly effective in

eliminating micron-size bubbles, and that it can produce

millimeter-size bubbles in *1,000 years in rhyolitic

magmas (centimeter-size in [100,000 years) and in 150–

1,500 years in moderately hydrous basalt. Ripening can be

significantly faster if bubbles ascend at close to their Stokes

velocities due to steepening of diffusive concentration

gradients, and faster still if non-zero bubble fractions are

taken into account. Stokes law predictions for the ascent

rates of bubbles show that while coarsening bubbles would

ascend faster than non-coarsening bubbles, the more sub-

stantial revelation is that once bubbles attain millimeter

sizes, they are capable of transiting magmatic system-scale

distances in 10 s to 1,000 s of years. The most important

contribution of coarsening to degassing therefore may be

that in geologically short times, it produces bubbles large

enough to rise rapidly through magmatic systems. Because

this is due to the physical–chemical properties of common

silicate melts (volatile solubility and diffusivity, melt-

vapor surface tension), diffusive coarsening does not

depend on case-specific magmatic histories.

Acknowledgments The authors are highly grateful to Ben Hankins

for his skillful guidance in the USGS Magma Dynamics Laboratory,

and to Shaul Hurwitz, Larry Mastin, and two anonymous reviewers

for constructive suggestions. Lautze acknowledges the USGS Men-

denhall Postdoctoral Program and Volcano Hazards Team for the

generous support that enabled this study, and support from the NSF

International Research Fellowship Program and INGV-Roma during

the review process.

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