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: [email protected]
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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