Rheology of bubble-bearing magmas

ELSEVIER Earth and Planetary Science Letters 166 (1999) 71–84

Rheology of bubble-bearing magmas

A.M. Lejeune a,1, Y. Bottinga b, T.W. Trull c,2, P. Richet a,*

a Lab. Physique des Geomateriaux, Institut de Physique du Globe de Paris, ESA CNRS 7046, Paris, Franceb Lab. Geochimie et Cosmochimie, Institut de Physique du Globe de Paris, URA CNRS 1758, 4, place Jussieu,

75252 Paris cedex 05, Francec Lab. Geochimie des Isotopes Stables, Institut de Physique du Globe de Paris et Universite de Paris VII, URA CNRS 196,

2, place Jussieu, 75251 Paris cedex 05, France

Received 13 May 1998; accepted 17 December 1998

Abstract

The physical effects of air or argon bubbles on the rheology of a calcium aluminosilicate melt have been measuredat temperatures ranging from 830º to 960ºC, at 1 bar pressure. The melt composition is SiO2:64, Al2O3:23, and CaO:13(wt%), while bubble volume fractions are: 0, 0.06, 0.13, 0.32, 0.41 and 0.47. Measured Newtonian viscosities range from1010 to 1014 dPa s. Melts with bubble fractions of 0.06 and 0.13 show with increasing temperature .T / an increasingrelative viscosity for T < 850ºC. However at T > 850ºC, for all bubble fractions the viscosity decreases markedly withtemperature. The observed maximum decrease of the relative viscosity is 75% for a bubble fraction of 0.47 at 907ºC. At allbubble fractions the viscosity is independent of the applied stress, which ranged from 11 to 677 bars. No clear indicationswere observed of non-Newtonian rheological behavior. Under our experimental conditions the relative viscosity of the twophase liquid depends primarily on the bubble fraction. Physical and volcanological implications of these measurements arediscussed. 1999 Published by Elsevier Science B.V. All rights reserved.

Keywords: aluminosilicates; melts; magmas; rheology; volcanology; exsolution

1. Introduction

Bubbles formed by the exsolution of volatiles playa major role in magma ascent and volcanic eruptions.Bubbles affect two fundamental properties of mag-mas which govern the dynamics of these processes,namely the viscosity and the density. An especiallyimportant feature is that volatile exsolution changes

Ł Corresponding author. Tel.: C33 14427 2488; Fax: C33 1442724871 Present address: University of Bristol, Dept. of Earth Sciences,Wills Memorial Building, Queens Road, Bristol BS8 1RJ, UK.2 Present address: Antarctic CRC, University of Tasmania, GPOBox 252C, Hobart, Tasmania, 7001, Australia.

the chemical composition of the melt. For water,which is usually the main gaseous constituent ofbubbles at low pressures, experiments have shownlong time ago that its dissolution in silicate meltscauses a marked viscosity decrease [1–4]. The ex-solution of water as a result of pressure decreaseshould increase the viscosity and slow down magmaascent rate, while at the same time acting against thiseffect the growth of bubbles increases the buoyancyof the melt.

In addition to the chemical effect on the melt vis-cosity, there is the purely physical effect of bubbleson the rheology of magmas. Based on the extensiveliterature on foams [5,6], it is frequently assumed

0012-821X/99/$ – see front matter 1999 Published by Elsevier Science B.V. All rights reserved.PII: S 0 0 1 2 - 8 2 1 X ( 9 8 ) 0 0 2 7 8 - 7

72 A.M. Lejeune et al. / Earth and Planetary Science Letters 166 (1999) 71–84

that the presence of bubbles increases the effectiveviscosity of magmas. This effect has been reportedand discussed recently for molten GeO2 [7]. How-ever, Bagdassarov and Dingwell [8] reported thatpresence of bubbles in a rhyolitic magma caused aviscosity decrease. The scarcity of relevant experi-mental data leads to these apparent contradictions,which illustrate that the influence of bubbles onthe rheology of silicate melts is still not very wellknown.

As part of a general study of the rheology ofmagmas [9], we present in this paper a series ofexperiments aimed at determining the physical in-fluence of bubbles. The very strong compositiondependence of the viscosity of silicate melts has ledus to exclude natural compositions for both bubblesand melts. We have chosen instead to study a syn-thetic calcium aluminosilicate liquid in which air orargon bubbles were trapped. As a model system, acomposition dubbed Ca70.15, containing 70% SiO2,15% Al2O3 and 15% CaO (in mol) was found sat-isfactory. By varying the volume fraction of bubblesup to a maximum value of about 50%, we documentin this paper the observed changes of the viscositywith increasing vesicularity.

2. Experimental methods

2.1. Choice of the melt composition

The viscosity of silicate melts depends severelyon chemical composition, therefore we took care tolimit possible compositional changes of the liquidphase during both the experiments and the prepa-ration of the samples. We decided to investigatethe influence of air and argon bubbles in view ofthe small solubility of these gases in silicate melts(for Ar see [10,11]). To avoid effects due to par-tial volatilization of alkali elements or iron redoxchanges during heat treatments at high temperatures,we considered synthetic melts in the system CaO–MgO–Al2O3–SiO2, especially in the SiO2-rich part,where the viscosity is high enough that bubbles canbe kept in the melt at elevated temperatures. TheSiO2 content must not be too high, however, be-cause of either too high liquidus temperatures, orliquid–liquid unmixing which develops below the

liquidus [12], right in the temperature range of ourexperiments, and affects markedly the rheology ofmelts [13]. Having a liquidus temperature of 1440ºC[14] and a glass transition temperature of 861ºC,the Ca70.15 composition with 70 mol% SiO2, 15%Al2O3 and 15% CaO was found adequate on all thesecounts.

2.2. Preparation of the samples

The Ca70.15 base product was synthesized fromdry CaCO3, Al2O3 and SiO2 reagent grade mate-rials. Electron microprobe analyses of the materialshowed good agreement with the nominal compo-sition, yielding averages of 63.90 (9) wt% SiO2,24.00 (15)% Al2O3, 12.85 (6)% CaO, with a totalof 100.92 wt% that includes 8 other elements whoseabundances were less than 0.06%. Vacuum fusionanalyses made over the years on a number of relatedglasses prepared under similar conditions indicatethat the H2O and CO2 contents are of the order of afew ppm at most.

The synthesis conditions of bubble-bearing sam-ples are summarized in Table 1. A first series of ex-periments was made with samples vesiculated withair. We prepared these samples by heating the veryfinely ground calcium aluminosilicate glass for shortperiods of time at 1550ºC. With this method, themaximum porosity achieved was only 10%. Sam-ples with low argon vesicularities were prepared inthe same way. To achieve higher porosities, we dis-solved argon at high temperature and pressure inCa70.15 melt, quenched the product, and then re-heated it at room pressure in order to let bubblesof exsolved argon grow. An internally heated vesselwith a graphite resistor was used for these experi-ments [15]. The finely ground glass was loaded in5 mm i.d., 20 mm long platinum tubes whose lowerend was plugged with a graphite disk. The tubeswere held on graphite plates in the vessel whosetemperature was slowly raised to 500ºC under a con-stant flow of argon in order to eliminate traces ofatmospheric water. The argon pressure was then in-creased up to the run pressure, and the temperatureraised to 1475ºC. After 25 min, the power was shutoff to quench the material which cooled to belowthe glass transition temperature in about 160 secondswhile argon was forced into the vessel to maintain

A.M. Lejeune et al. / Earth and Planetary Science Letters 166 (1999) 71–84 73

Table 1Synthesis conditions of bubble-bearing Ca70.15 samples a

Ps Ts ts ² ý �bubbles Gas(bars) (K) (min) (cm3=g) (vol%) (µm)

B1-1-A 1 1550 0 2.516 0.0 š 0.1 – airB1-1-B 1 1550 0 2.516 0.0 š 0.1 – airB1-2-A 1 1550 0 2.380 5.4 š 0.1 15–680 airB1-2-B 1 1550 0 2.363 6.1 š 0.1 20–570 airB1-2-C 1 1550 0 2.365 6.0 š 0.1 30–650 airB1-2-D 1 1550 0 2.376 5.6 š 0.1 15–500 airB1-3-A 1 1550 10 2.327 7.5 š 0.1 15–600 argonB1-3-B 1 1550 10 2.332 7.3 š 0.1 10–620 argonB1-4-A 1 1550 10 2.192 12.9 š 0.1 20–500 argonB1-4-B 1 1550 10 2.194 12.8 š 0.1 15–510 argonB200-A 200 1475 25 1.711 32.0 š 0.2 30–620 argonB302-A 302 1475 25 1.479 41.2 š 0.2 60–450 argonB302-B 302 1475 25 1.503 40.3 š 0.2 60–450 argonB452-A 452 1475 25 1.334 47.0 š 0.3 15–550 argon

a Ps, Ts and ts are the pressure, temperature and duration of the run, ² and ý the density and vesicularity of the samples, and � thebubble diameter.

the pressure of the experiment constant. In this way,partial vesiculation at the platinum–melt interfacecould be avoided during the quench.

After the quench, the bubble free glass containingdissolved argon was left in the platinum tubing andtransferred to an electric muffle furnace maintainedat 1550ºC at room pressure. At this temperature,nucleation and growth of bubbles is very rapid andthe melt tends to leave the platinum tube containerby overflowing. To prevent this from happening themelt was heated to 1550ºC for only short periods oftime. Typically, it was withdrawn from the furnaceafter 1 min, visually examined, reheated for anotherminute, examined again, and treated during progres-sively longer periods until the size of the sampledid not change any longer. Because the internallyheated vessel cannot be operated below 130 bars at1475ºC, we could not obtain vesicularities of lessthan 20% with this method. The upper vesicularitylimit, namely 47%, was set by the loss of homo-geneity of the distribution of bubbles throughout theglass, due to coalescence. At higher vesicularities,the bigger bubbles had diameters equal to the in-ternal diameter of the platinum tubing, prohibitingprecise viscosity measurements. At the end of thesepreliminary experiments, the samples had cylindricalshapes being 4 to 8 mm in diameter and 8 to 14 mmlong.

Available measurements for closely related com-positions [16] indicate that the viscosity of Ca70.15is of the order of 103 dPa s at 1550ºC. The vesicu-lation treatment described in the previous paragraphwas terminated when the length of a sample did notchange by more than a tenth of a millimeter perminute, which is equivalent to a strain rate of lessthan 10�4 s�1. If the bubbles in the melt were stillgrowing, they would have exerted stresses of lessthan 10�2 Pa on the silicate melt. At the temperatureof the viscosity measurements the surface energy ofthe melt bubble interface should have been about 0.4J=m2 [17,18], hence the bubble pressure was alwaysless than a bar and very close to that of the sample.

2.3. Sample vesicularities

Part of the samples was sliced for thin sectionexamination under a petrographic microscope. Thebubbles were spherical (Figs. 1 and 2), their sizedistribution depending on the synthesis conditions(Table 1). In all cases, however, they appeared regu-larly distributed as also indicated by the fact that nodensity difference was found between the bulk of thesamples and the slices cut at their ends. These den-sities measured with an archimedean method wereused to determine vesicularities; toluene was theimmersion fluid. Neglecting the intrinsic density of


Fig. 1. Transmitted light photo of a thin section of sample B1-2-C, with 6 vol% bubbles, before the viscosity measurements; the heightof the photo is 1.5 mm in reality.

Fig. 2. Transmitted light photo of a thin section of sample B200-A, with 32 vol% bubbles, before the viscosity measurements; the heightof the picture corresponds to 1.5 mm in reality.


bubbles, the volume fraction of bubbles .ý/ is sim-ply given by ý D 1�²=²0, where ² is the measureddensity of a vesiculated sample and ²0 D 2:516g=cm3 that of the bubble-free glass at room-temper-ature. Vesicularities accurate to within 0.3% wereobtained with this method. Given the size differencebetween the samples and the bubbles, the volumefraction of bubbles touching the surface was small.In addition, the weight of the immersed samples didnot change significantly with time, indicating that nodiffusion of toluene was taking place through inter-connected bubbles. We thus believe that the reportedvesicularities are not beset by the influx of toluenewithin the surface bubbles.

Using the ideal gas law, we can calculate theargon solubility in Ca70.15 melt at 600 bars and1475ºC from the volume fraction of bubbles deter-mined from the density measurements. The obtainedvalues are about 1:6 ð 10�4 cm3(STP)=g, that is tosay, they are comparable to Ar solubility measure-ments on molten rocks [10,11].

As discussed in the previous section, bubble sta-bilization took place at high temperatures and onebar. Because of the low solubility of air and argonin silicate melts at one bar, the residual amount ofdissolved gas after stabilization was very small. Nosystematic dependence of the measured viscositieson the nature of the gas and on the preparation pro-cedure of the sample was noticed. Hence this smallamount of residual dissolved gas did not affect in asignificant way the rheology of the samples.

2.4. Viscosity measurements

The measurements have been made under uniaxialstresses with the apparatus described by Neuvilleand Richet [19], whereby the deformation rate ofa sample is measured as a function of the appliedstress at constant temperature. With the geometry ofour samples, the viscosity is given by:

� D .¦=3/ d ln l=dt (1)

where l is the length of the sample, t the time and ¦the stress [20]. Two sets of viscosity measurementson National Bureau of Standards glasses (NBS 717,NBS 710 and NBS 710a) have been published, thedata for these standards obtained with our apparatusfall between these published values [19] and indicate

that the results obtained with this setup are accurateto better than 10% of the reported value.

The cylindrical samples (5 mm diameter, 20 mmheight) were annealed overnight at 1135 K to re-move internal stresses. Samples entirely prepared at1 bar were cored with a diamond drill into massivediscs of glass. The samples prepared at high pres-sure were made ready for viscosity measurements bypeeling off the walls of the platinum containers. Inboth cases, the opposite faces of the cylindrical sam-ples were ground and polished in order to achieveparallelism to within a few µm.

The application of Eq. 1, requires the volume ofthe sample to be constant during the deformationexperiment. To verify this condition we comparedthe density of each sample measured before and af-ter the viscosity measurements with its volume asgiven directly by its length and cross section. Ingeneral the before and after density determinationsagreed to within 5%. Viscosities were deduced fromthe measured rate of deformation of the cylindricalsample at constant stress and temperature. Duringthe experiments continuous plots of the logarith-mic deformation against time were obtained. For theviscosities reported in this paper, these plots were al-ways linear under conditions of constant temperatureand stress. The viscosity measurements were succes-sively performed on the same sample at temperatureswhich were varied irregularly. At each temperature,measurements were made at different stresses. Noanomalous scatter in the measurements has been ob-served as function of either stress or temperature,which indicates that the results did not depend onthe deformation history of the samples. As a pre-caution, measurements on a sample were terminatedwhen the cumulated sample deformation amountedto about 20%. The total deformation each sample hasundergone, that is to say, the ratio of the differenceof the final and initial lengths of a sample over itsinitial length expressed as percent is listed in Table 2.

The most important sources of errors in our mea-surements are related to the determination of thedeformation and its duration. All of our measure-ments were on highly viscous liquids, hence thedeformations were small, ranging from 0.005 to 0.1mm. However, the time span during which defor-mation took place could be long, it ranged from afew seconds to several tens of minutes. The overall


maximal lack of precision of the measured data isless than 10%.

3. Results

The viscosity measurements were made at tem-peratures between 1099 and 1235 K and ranged fromabout 1014 to 1010 dPa s The bubble sizes varied from15 to 680 µm as imposed by the adopted experimen-tal method. The stress ranges used for each sampleare listed in Table 2, the first figure referring to theminimum stress applied at the lowest viscosities, thesecond to the maximum stress at the highest viscosi-ties. Within the precision margins the observed strainrates plotted against the applied stress were in goodagreement with Eq. 1. Two examples of such plotsare shown in Fig. 3, one for a fairly high viscosityat low strain rates, the other example at a lowerviscosity permitting larger stresses and strain rates.Measurements made at different stresses but constantT and ý showed small deviations from Newtonianbehavior, which could be due to various physical ef-fects. These deviations did not vary consistently withý, �, ¦min, ¦max (Table 2) or strain rate. Extrapolationof plotted observations (Fig. 3) to a zero deforma-tion rate suggested small yield stresses, which rarelywere larger than 6 bars. Repeated measurements andobservations at the same ý but a slightly differentT , did not reproduce quantitatively these deviations,

Table 2Vesicularity, maximum deformation, minimum and maximumstresses applied to the Ca70.15 samples

ý Def ¦min ¦max

(vol%) (%) (bars) (bars)

B1-1-A 0.0 23 11 213B1-1-B 0.0 11 23 710B1-2-A 5.4 21 11 186B1-2-B 6.1 29 10 185B1-2-C 6.0 40 5 677B1-2-D 5.6 5 17 129B1-4-A 12.8 20 23 396B1-4-B 12.8 22 12 503B200-A 32.0 18 20 308B302-A 41.2 19 12 389B302-B 40.3 19 13 119B452-A 47.0 14 24 248

but gave viscosities in excellent agreement with theinitial measurements. Hence rheologically our highviscosity bubbly silicate melt did not behave like theBingham model. For practical purposes these smalldeviations from a strict accordance between the dataand Eq. 1 can be ignored safely.

For a given temperature the data, listed in Table 3,are Newtonian viscosities representing the average of15–25 measurements made at various stresses span-ning at least one order of magnitude (Fig. 3). Thestandard deviations from the mean for these isother-mal measurements were always smaller than 7%.The results plotted (Fig. 4) show that the viscositydecreases with increasing vesicularity, but the effectis relatively modest since a bubble fraction of 47%results in the same viscosity change as a temperatureincrease of about 10 degrees, i.e. a viscosity decreaseby a factor three.

The temperature dependence of the viscosity alsodepends little on the vesicularity (Fig. 5). For inter-polation purposes, viscosities are in general wellreproduced with the so-called Tammann–Vogel–Fulcher equation:

log � D A C B=.T � T1/ (2)

where A, B and T1 are adjustable parameters. Forour silica-rich material, however, the temperature in-terval of the experiments was too restricted to showsignificant deviations of log � from linear depen-dence on reciprocal temperature. Hence, Arrheniusequations were fitted instead to the data for eachbubble fraction:

log � D A C B=T (3)

The parameters A and B are listed in Table 4 for thevarious bubble fractions. As is frequently the case,A and B are correlated linearly, but neither A nor Bvaries monotonically with the bubble fraction ý.

Since the same results have been obtained forair and for argon at low vesicularities, the mostimportant factor is clearly the volume fraction ofbubbles, and not the difference between these twogases. To express the viscosity changes due to thepresence of bubbles, we used relative viscosities,�r.T / D �.ý;T /=�.0; T /, where �.0; T / is the vis-cosity of the bubble-free melt at temperature T . At agiven temperature, these relative viscosities (Fig. 5)have been calculated with Eq. 3 and the parameters


Fig. 3. Two examples of experimentally determined stresses and strain rates for bubbly Ca70.15 liquids, with ý D 41%; T D 1139:8 K,� D 1012:47 dPa s and ý D 6%, T D 1182:2 K, � D 1011:33 dPa s, showing linear relations.

Fig. 4. Illustration of the temperature dependence of the observed viscosity of Ca70.15 melts with vesicularities of 6, 13, 32, 41, and 47vol%.

listed in Table 4. The points (Fig. 5) are interpo-lations except for a few extrapolations. Because ofthe measured temperature constancy during the ex-

periments and the very good agreement between thedetermined data and Eq. 3, the error margins for �r

values are estimated to be less than 15%, i.e., 0.15


Table 3Viscosity (dPa s) of Ca70.15 samples with different vesicularities

0 vol% 6 vol% 13 vol% 32 vol% 41 vol% 47 vol%

T (K) log � T (K) log � T (K) log � T (K) log � T (K) log � T (K) log �

B1-1-A B1-2-A B1-4-A B200-A B302-A B452-A1148.1 12.51 1137.4 12.86 1 a 1117.9 13.64 4 1118.9 13.44 3 1099.7 13.90 7 1119.8 13.00 3

1156.9 12.22 1151.7 12.37 3 1128.0 13.29 5 1137.9 12.74 5 1121.1 13.10 3 1141.3 12.26 2

1166.6 11.90 1164.6 11.92 2 1136.4 12.98 3 1156.6 12.15 2 1139.8 12.47 4 1157.4 11.65 4

1177.0 11.58 1181.8 11.35 4 1148.8 12.51 7 1174.6 11.50 1 1158.8 11.83 2 1178.4 11.03 1

1185.3 11.32 1191.7 11.07 5 1172.8 11.72 1 1191.6 10.94 5 1167.7 11.39 6

1194.8 11.05 1200.4 10.78 6 1182.7 11.37 8 1177.0 11.20 1

1207.7 10.65 B1-2-B 1191.3 11.11 2 1194.9 10.54 5

1235.0 9.95 1129.3 13.19 2 1199.8 10.83 9

B1-1-B 1147.4 12.53 3 B302-B1129.4 13.14 1156.3 12.17 8 1099.1 14.02 6

1141.1 12.74 1165.1 11.91 1 B1-4-B 1109.9 13.59 5

1199.2 10.85 1175.0 11.57 7 1125.0 13.30 2 1129.8 12.84 4

1220.8 10.32 1183.4 11.32 6 1151.1 12.44 5 1148.6 12.16 3

1193.1 10.99 9 1174.7 11.65 1 1157.6 11.70 9

1201.8 10.76 4 1201.9 10.86 4 1167.1 11.42 7

1208.0 10.58 5 1225.4 10.09 3 1176.2 11.24 1

B1-2-C 1190.5 10.64 8

1182.8 11.33 1 1203.8 10.39 2

1209.6 10.56 2

1218.7 10.32 3

B1-2-D1139.7 12.76 1

1165.6 11.89 3

1167.3 11.86 2

a Superscripts indicate the sequences of measurements for the samples.

on the �r scale (Fig. 5). Therefore the maximum atý D 13 vol% and the temperature dependence of �r

(Fig. 5) are real.The measurements (Fig. 4) indicate clearly that,

for 0 < ý.vol%/ < 13, the suspension viscosity isonly affected in a minor way by the presence ofbubbles, and the sign of the variation of �r with ý istemperature dependent. Beyond this range, the rela-tive viscosity decreases with increasing vesicularity.The greatest viscosity decrease amounts to about75% relative to the bubble-free melt. Fig. 6 showsthe decrease of the relative viscosity at constant ýwith increasing temperature, i.e., �.ý;T / decreasesfaster with T than �.0; T /.

The compressional deformation experiments onthe biphasic samples caused flattening of the bub-bles and flow of the liquid silicate phase, as shownin Fig. 7. This figure is the only one we have fora deformed sample, the results of this preliminary

deformation experiment are not reported here. Theplots, mentioned in the previous section, of the loga-rithmic deformation versus time show no signs of theoccurrence of a distinct deformation event associatedwith bubble deformation. Hence the viscosities wehave calculated from the deformation experimentsare due to the concomitant deformation of the bub-bles and the flow of liquid of the silicate liquid.

As far as we are aware, only the measurementsby Bagdassarov and Dingwell [8] and those by Steinand Spera [7] can be compared with our observa-tions. The Bagdassarov and Dingwell relative vis-cosities of about 0.1, for a rhyolite liquid at 850ºCwith ý ranging from 25 to 70 vol% are nearlyconstant, while for the remaining observations for0 < ý < 4 vol%, �r varies linearly with ý from1 to 0.43. The general trend agrees with our databut close inspection shows their measurements to bequite different from ours. At this moment, we have


Fig. 5. Relative viscosity �r.T / D �.ý; T /=�.0; T / as a function of ý at constant temperatures, see text. Plotted values were calculatedas described in the text. Encircled points indicate an extrapolation over less than 5 K of original observations, all the other points requiredonly interpolation of the measured values for the bubble containing melts (Table 3).

only speculative explanations for this difference. TheStein and Spera data are for liquid GeO2 with airbubbles, ý varies from 0.8 to 5.5 vol%, and thetemperature ranges from 1100 to 1175ºC. Their rel-ative viscosities increase with ý and vary from 1to 1.7. Qualitatively, our observations at T D 1100K for comparable bubble fractions agree with thoseby Stein and Spera [7]. However at T > 1120 K,our measurements show a decrease of �r with T atý D 6 vol%.

Table 4Parameters A and B of Eq. 3 and R values of the fit of Eq. 3 tothe data listed in Table 3 at constant bubble fractions

ý A B ð 10�4 R(vol%) (K)

0 �24.559 4.2548 1.0006 �26.088 4.4285 0.999

13 �26.685 4.5044 1.00032 �27.482 4.5809 1.00041 �28.425 4.6595 0.99947 �26.991 4.4780 0.999

Data in K and dPa s.

4. Discussion

4.1. Physical effects

The observed variation of viscosity with increas-ing vesicularity is not uncommon [21,22]. Largebubbles are easier to deform than small ones, be-cause the internal pressure of bubbles increases withdecreasing bubble size. Another effect is that a liquidphase close to a wall shows an increasing develop-ment of internal structure upon approaching the wall,this feature has been modeled by Urbakh et al. [23];it causes the viscosity of the melt to augment near thebubble–liquid interface. In terms of the Adam–Gibbs[24] cooperative relaxation theory, the decrease ofthe configurational entropy of a melt becoming morestructured results in increases of the structural re-laxation time and the viscosity [25]. Besides, withincreasing vesicularity bubbles start to interact mu-tually; shearing of the melt–bubble suspension pro-motes such interaction and may cause the formationof an ordered spatial distribution of the bubbles in themelt. This effect has been observed in suspensionscontaining small solid spherical particles [26]. At


Fig. 6. Relative viscosity versus temperature at constant ý. The overall trend is a diminution of �r with T , i.e., �.ý; T / decreases fasterwith T than �.0; T /. Hand drawn contours for constant bubble fractions are meant as trend indicators.

ý < 50 vol% and low shear rates, the particles werefound to be spatially distributed in a non-randomway; they were hexagonally packed in layers parallelto surfaces of constant shear. The extent of these lay-ers varied sympathetically with ý. Hoffman [27] be-lieves that at low shear rates in suspensions withý <

50 vol%, stress causes the sliding of the observedlayers, one over the other, in the flow direction. Theresult of this arrangement is a decrease of the relativeviscosity. Jogun and Zukoski [28] reported similarobservations for a suspension of plate-like particles.Colloidal suspensions too show an ordered distribu-tion of suspended particles when ý approaches 50vol% [29]. Clarke and Rennie [30] have describedin great detail the development of a particle orderedstructure in a flowing colloidal suspension with 12vol% of particles. They used a light scattering tech-nique, which permitted them to quantify their obser-vations. Their precise data are in essential agreementwith those by Hoffman [26], and indicate a slidingmovement of the layers of particles as a reaction tothe application of a shear stress. This ordering ofsuspended particles does not depend on the inter-particle potential because it has been observed forparticles with all kinds of mutual interactions. More-over ordering of suspended particles occurs also innon-colloidal systems, e.g. [31–33]. Experimentally,

this type of observation has not been reported forbubble-bearing silicate liquids. However, small bub-bles (5 � �.µm/ � 100) in glass inclusions oftenshow a regular two dimensional spatial distribution,see microphotograhs in Kurz et al. [34], Burnard etal. [35] or Schiano and Bottinga [36].

The results described in the previous paragraph,obtained experimentally or by means of numericalsimulation, are not directly applicable to our ob-servations, because they were obtained for particlessmaller than 50 µm, while the maximum diametersof the bubbles in our experiments were nearly 700µm. However, the presence of arrays with submilli-metric bubbles have been reported by Stasiuk et al.[37] in a naturally sheared rhyolitic glass. Evidently,the interparticle forces used in the numerical simu-lations are not the same as those causing the bubblespatial distribution observed by Stasiuk et al. [37].These latter observations could have been caused bybubble interactions due to local deformation of themelt flow field due to the presence of these bubbles.The rotation of flattened bubbles in liquids subjectedto planar shear is another fluid dynamical effect ofordering of suspended particles. An example of this,observed in our sheared samples, is shown in Fig. 7.

Our results for the interval 0 < ý.vol%/ < 47indicate that �r depends on vesicularity as well as


Fig. 7. A thin section of a Ca70.15 melt sample (ý D 3 vol%) showing flattened bubbles after the viscosity measurements were done;the compression axis is vertical, total deformation is 33%, the vertical, the height of the picture is 1.5 mm. The data obtained from thisexperiment are not listed in Tables 1–4.

temperature. The data for ý < 13 vol% do not in-dicate an evident need for postulating the existenceof a yield stress. At ý > 13 vol% the observed de-crease of �r with ý cannot be extrapolated simply tolarge values of the vesicularity. The quantity of meltper unit volume not directly affected by the presenceof bubbles (at a constant bubble number density)diminishes with growing ý, whereas that of meltin boundary layers and bubbles interfaces becomeslarger. These changing volumetric relations will leadeventually to a change of �r with ý. One may sup-pose that the surface tension of the bubbles in ourexperiments affected the viscosity of the silicate meltsuspension. The surface tension of silicate melts inthe presence of an inert gas like N2 or Ar at 1250ºCis about 0.4 J=m2 [17,18]. An argon bubble in a sili-cate melt with a radius of 5 µm, the minimum radiuslisted in Table 1, has an excess pressure with respectto the pressure on the melt of 0.16 bar. This pressureis about 1% of the minimum applied stresses during

the viscosity measurements (see Table 1). There-fore the surface tension of the bubbles affects themeasured viscosities to an extent which is smallerthan the precision of our observations. Possibly, theobserved suspension shear viscosity diminution for13 D ý.vol%/ < 48 is due the progressive orderingof bubbles as a result of fluid dynamically transmit-ted mutual interaction.

In summary, one observes that there are at leastthree different effects influencing the shear viscosityof suspensions containing particles with sizes thatare small as compared with the typical dimensionsof the flow system. There is the effect of the bulkconstitution of the particles (i.e., gas, liquid, or crys-talline) with their inherent viscosities, which may belarger or smaller than the viscosity of the liquid ma-trix. Secondly, particles may show ordering in space,making possible the sliding of structured planes anddecreasing the viscosity at ý < 50 vol% and lowstress. The third effect is related to a small scale fea-


ture, namely the development of internal structure ofthe liquid in interfacial zones and associated bound-ary layers. This causes an increase of the rheologicalboundary, marked by a higher viscosity than the puremelt, and resulting in an effectively larger volumefraction of suspended particles.

4.2. Volcanological aspects

We first note that the effects of vesicularity onthe rheology of magmas are most relevant for theascent of melts in conduits and explosive bubblingduring eruptions, see for example Sparks [38]. Inthis context, the uniaxial stresses of our experimentsshould represent better natural stress conditions thanthose of previous shear viscosity measurements per-formed with rotating viscometers. At least in thestress ranges of our measurements, our results shouldthus be directly applicable to natural processes.

Although knowledge of the rheology of magmaticsuspensions is indispensable for a quantitative under-standing of volcanic eruptive phenomena, virtuallyall data on two-phase flow have unfortunately beenobtained on insufficiently described silicate systemssuch as natural lava flows. As a matter of fact, it isa priori difficult to obtain extensive rheological in-formation without detailed knowledge of bubble andcrystal size distributions, particle shape and num-ber densities, chemical composition (specially of thedissolved volatiles), temperature, pressure and thegeometry of the flow system. Therefore it would bepresumptuous to propose a recipe for the evaluationof the viscosity of magmatic suspensions based onthe available observational data.

Exsolution of a volatile phase in a magma maycause either an increase or a decrease of the shearviscosity. This depends on the size, the compositionof the bubbles the vesicularity, and the shear rate.The exsolution of H2O into bubbles causes an in-crease of the viscosity because of the large effectof dissolved water on the viscosity of naturally oc-curring liquid silicates [3,4,39–43]. Other volatilessuch as fluorine dissolved in magma may also causean important decrease of the viscosity of the meltphase [44]. However, many dissolved gases (CO2,N2, Ar, etc.) do not affect in an important way theviscosity of the melt fraction [42]. Therefore the for-mation of bubbles containing these gases may cause

a decrease of the magma viscosity for vesicularitiessmaller than 50 vol%.

Magma vesicularity increases because of diffu-sional flux of volatiles into bubbles and decom-pression. In a steadily upward moving magma therelative pressure drop per unit time becomes pro-gressively larger and the vesicularity increases ac-cordingly. When the pressure decreases, the growthrate of the vesicularity depends increasingly on therate of variation of the pressure instead of the diffu-sional flux of volatiles into a bubble. Consequently,in the absence of phenocryst formation and turbu-lent motion, the magmatic viscosity should diminishas long as ý is smaller than about 50 vol% uponapproaching the Earth’s surface.

Changes of vesicularity due to bubble nucleationare always small and gradual because of the micro-scopic size of bubble nuclei. The diffusional growthof bubbles, even at very low pressures and largesupersaturation will always be slow because of theslowness of diffusion processes in condensed phases.Therefore rapidly developing volatile supersaturationcannot lead to a large speed up of the bubble growthrate. Hence it is unlikely that vesicularity changesand concomitant viscosity variations are generatedabruptly because of these effects.

Gardner et al. [45] published an interesting con-clusion about the fragmentation conditions of vis-cous magmas. These authors explained their obser-vations on pumices of plinian deposits as being dueto fragmentation of magmas as a result of shearingat a vesicularity of 64 vol%. This vesicularity rangemakes physically good sense, because it coincideswith the upper limit for an arrangement of randomlyclose packed uniform spheres [46]. Bubble growthdue to decompression in an ascending magma withrandomly close packed bubbles causes the stretchingand thinning of melt films between bubbles. Con-sequently, the melt in intervesicular films becomesmore viscous. Under such conditions the structuralrelaxation time of melt in bubble interfaces increasesand the application of shear stress, even at a mod-erate rate, causes these interfaces to fractures likea solid instead of deforming as a liquid [47]. Thenet result will be fragmentation of the magma or thedevelopment of two phase slug flow.

In many publications on violent volcanic erup-tions, e.g. Sparks [38], non-Newtonian rheological


behavior of ascending lava is invoked as a causeof important pressure changes in the volcanic con-duit. In the same context one appeals also to magmayield strength. We would like to point out that weare unaware of experimental rheological researchon homogeneous silicate liquids done under labora-tory controlled conditions, which indicated the exis-tence of a yield strength [55]. The work by Lejeuneand Richet [48] on the rheology of crystal bearing(ý < 40 vol%) viscous silicate liquids showed thesuspension viscosity to be Newtonian. It was onlyfor ý > 40 vol% that these authors inferred the ex-istence of a yield strength as a cause of the observedstrain rate dependence of the viscosity. The work ona bubble containing (ý < 47 vol%) viscous silicatemelt discussed in this paper shows no sign of non-Newtonian rheological behavior. Strain rate or stressindependent viscosity is a steady state property. Thedevelopment of steady state flow takes time and ispreceded by transients during which the viscosity isstress or strain rate dependent, in particular when theliquid is very viscous. An ascending lava confinedto a volcanic conduit, can develop local stress accu-mulation and behave in a non-Newtonian way, whenit experiences viscosity changes, as a result of theexsolution of water or rapid bubble growth due todecompression. The result will be a discontinuousflow behavior, giving the impression of the existenceof yield strength, and may have explosive charac-teristics as described by Sparks [38]. The viscositychanges observed and discussed in this paper areimportant enough to cause such an effect.

Acknowledgements

We thank Marc Javoy, for the use of his labo-ratory, Sylvie Vergniolle, Mike Carroll and DanielNeuville for fruitful discussions at different stages ofthis work, and appreciated very much the detailedremarks by the two anonymous reviewers. [AC]

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Rheology of bubble-bearing magmas