Viscous heating in rhyolite: An in situ experimental determination

Earth and Planetary Science Letters 275 (2008) 121–126

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Earth and Planetary Science Letters

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Viscous heating in rhyolite: An in situ experimental determination

Kai-Uwe Hess, Benoit Cordonnier, Yan Lavallée ⁎, Donald B. DingwellDepartment of Earth and Environmental Sciences, LMU - University of Munich, Theresienstrasse 41\III, 80333 Munich, Germany

⁎ Corresponding author.E-mail address: [email protected] (Y. L

0012-821X/$ – see front matter © 2008 Elsevier B.V. Aldoi:10.1016/j.epsl.2008.08.014

a b s t r a c t

Article history:

Editor: T.M. Harrison

Keywords:viscous heatingenergy dissipationnon-Newtonian rheologybrittle-ductile transitionstrain rate dependent viscositysilicate meltslavamagma

gmatic flow may play a major role in eruption dynamics. In order to documentviscous heating during deformation of magma, we have conducted a series of rheological experiments whereviscous heating is directly monitored via thermocouples in high-viscosity magmas. We observeexperimentally the strain rate dependence of viscous heating. Viscous heating becomes rheologicallysignificant in the highly viscous lavas investigated at strain rates above ca. 10−3 s−1. A simple analysis showsthat the temperature increase generated through viscous heating during deformation of melts withviscosities ranging from 108 to 1012 Pa·s can account for their apparent non-Newtonian rheology in theseexperiments. This thermal correction transforms apparent non-Newtonian, strain rate dependent rheology ofmagma to a Newtonian behavior over the range of conditions accessed in this work. In this manner, thisstudy provides an experimental basis for separating the relative roles of structural relaxation and viscousheating in the generation of apparent non-Newtonian rheology at high strain rates. Here, viscous heatingdominates and the observation of the structural onset of non-Newtonian behavior is precluded by a viscousheating-induced lowering of the Newtonian viscosity. The interplay of viscous heating and structuralrelaxation in melts in nature is discussed briefly.

© 2008 Elsevier B.V. All rights reserved.

1. Introduction

One of the most remarkable, unpredictable and (from the point ofview of monitoring) alarming aspects of explosive volcanic centers istheir ability to switch from low-risk effusive to high-risk explosiveeruptive behavior. A physical understanding the ductile-brittletransition is an essential aspect of distinguishing criteria for thechoice such a systemmakes between effusive and explosive eruptions.It is likely that competition between the strain rate and the structuralrelaxation timescale of a melt ultimately dictates whether theeruption will remain effusive or explode (Dingwell, 1996). Thestructural relaxation timescale (τ) is defined by Maxwell as:

τ ¼ ηsG∞

ð1Þ

where ηs is the shear viscosity and G∞ the infinite frequency elasticshear moduli (approximated at 1010+/−0.5 Pa for silicate melts(Dingwell and Webb, 1989)). High-temperature macroscopic rheolo-gical experiments have demonstrated that this transition, that of aNewtonian fluid to a non-Newtonian, strain rate dependent fluidrheology, can occur at a strain rate approximately one 1000th that ofthe structural relaxation timescale (Webb and Dingwell, 1990). For

avallée).

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high-viscosity silicate melts (e.g., 1010 Pa∙s), this would translate to anonset at ~10−3 s−1.

Silicate magmas flowing under such conditions are however, alsoexpected to generate significant temperature increase through theviscous heating which the deformation generates in them (Rosi et al.,2004; Tuffen and Dingwell, 2005). The strong temperature-depen-dence of Newtonian viscosity means that if significant viscous heatingcan occur then it will likely have rheological consequences for themagma (Shaw, 1969; White and Muller, 2000).

Viscous energy dissipation (also termed viscous heating, viscousdissipation, and shear heating) is the contribution to the totalenergy made by irreversible deformational work (Holtzman et al.,2005). In magmatic and volcanic systems, viscous heating iscommonly inferred in shear zones (Holtzman et al., 2005), dikes(Carrigan et al., 1992), along the conduit margins (Boyd, 1961; Fujiiand Uyeda, 1974; Hardee and Larson, 1977; Fujita et al., 2000;Mastin, 2002; Polacci et al., 2004; Rosi et al., 2004; Mastin, 2005;Vedeneeva et al., 2005; Hale et al., 2007), in lava domes (Smellieet al., 1998), and in low-viscosity lava flows (Pearson, 1977;Keszthelyi, 1995; Dragoni et al., 2002). Recent numerical modelingof viscous heating in a conduit has demonstrated the stronginfluence of viscous heating along the conduit margin and, as aresult, the importance of considering plug-like ascent profilesinstead of Poiseuille flows (Costa and Macedonio, 2003; Mastin,2005). Here we report results from a series of uniaxial compressionexperiments on natural and synthetic silicate melts that have been

Fig. 1. (a) Sketch of the uniaxial press: (1) load frame; (2) servo cylinder with LVDT; (3)load cell; (4) cooling jacket; (5) 3-zone split cylinder furnace; and (6) 6-inputthermocouple interface (for type K and S). The sample containing three thermocouplesis placed between the pistons. Two thermocouples monitor the ambient air and one thepiston temperature. (b) Photograph of a NIST 717a sample with three drill holes forthermocouples, alongside a deformed samples inserted by three thermocouples.Modified from Hess et al. (2007).

122 K.-U. Hess et al. / Earth and Planetary Science Letters 275 (2008) 121–126

performed in order to characterize in situ the heat generated byviscous dissipation during increasing strain rates. The viscosity-temperature dependence of the two selected melts (i.e., depoly-merized NIST 717a vs polymerized rhyolitic obsidian from Iceland)encompasses the variations given by natural melts (Mysen, 1987).Here, correction of apparent non-Newtonian rheology for thethermal influence of viscous heating reveals a rheology which isindependent of strain rate within experimental error.

2. Method

The present viscosity measurements were made with a uniquehigh-load, high-temperature uniaxial press (Hess et al., 2007) capableof ascerting viscosity within a precision of +/−0.06 log Pa s. In order toadequately quantify viscous heating effects in rheologically “simple”silicate melts, we have performed measurements on NIST referencestandard silicate glass SRM 717a (a standard silicate glass used tocheck test methods and calibrate equipment for the determination ofthe viscosity of glass in accordance with ASTM Procedure C 965-81)for which the temperature-dependence of viscosity is well-known(Hess et al., 2007) and on calc-alkaline rhyolitic obsidian for which wehave measured the temperature-dependent Newtonian viscosity(Table 1). The NIST viscosity standard was chosen as an essentialprerequisite for studies on natural rhyolite due to its extremely wellcharacterised rheology and homogeneity.

Cylindrical samples with a height of 40 mm and a diameter of20 mmwere prepared by cutting, drilling and grinding. Three holes of2-mm diameter, equally spaced along the samples axis, were radiallydrilled through to the centre in order to insert three thermocouplesfor precise monitoring of temperature variation during the experi-ments (Fig. 1). For this study we used shielded (Inconel) NiCr–Nithermocouples (T.M.H., Hanau, Germany, Type K, D=1.5 mm) with aprecision of +/−0.5 K and a thermal response of approximately 1 s toattain 63% of the exact temperature.

Samples were placed between the pistons and slowly heated up toa fixed temperature (NIST: 540–630 °C, and Obsidian: 685–850 °C).After slow thermal equilibration of the sample and pistons (8 h), ahigh loadwas applied andmaintained until a maximumof ~33% strainwas obtained. The resultant length changes (δh) were recorded by thedifferential transducer (LVDT) at a rate of 10 data per second (δt)which were then treated via Gent equation developed to calculate theshear viscosity (ηs in Pa s) of a cylindrical melt in parallel-platecompression measurements (Gent, 1960):

ηs ¼2πFh5

3Vδh=δt 2πh3 þ Vð Þ ð2Þ

where F is the force (N), h is the length (m), and V is the volume ofmelt (m3). Temperature changes produced by the deformation work

Table 1Chemical compositions of samples

NIST 717a Obsidian

SiO2 68 74.0Al2O3 3.5 12.9Na2O 1 5.2K2O 8 3.6CaO 0 1.0MnO 0 0.1MgO 0 0.2Fe203 0 0.7FeO 0 1.7TiO2 0 0.2Li2O 1 ndB2O3 18.5 ndWater 0 0.3Total 100 100.0

were measured in situ during the experiment; the ambient tempera-ture was maintained during deformation and the measured tempera-ture increase of the melt could be purely attributed to viscousdissipation. For the NIST 717a glass, we used the measured meansample temperature (Tmean see Table 2) to estimate the viscosity andcompare it to the measured viscosity. Below we show that, with thisexperiment, we have achieved the first in situ experimentalconfirmation of viscous heating during high-temperature, high-stressdeformation of dome lava.

3. Results

22 measurements were performed on the NIST glass to quantify asbroadly as possible the effects of viscous dissipation on the rheology ofsilicate melts (Table 2). A typical low-stress, or low-strain rate,experiment is characterized by a derived viscosity which initiallyincreases at a decreasing rate then stabilizes to a steady value. Thetemperature recorded by the embedded thermocouples remainedconstant throughout low-stress experiments. In contrast, medium- tohigh-stress experiments reveal an accentuating decrease in viscosityaccompanied by an internally recorded temperature increase of up to10 °C (Fig. 2).

Table 2Experimental results and calculated viscosities for the NIST referencematerial SRM 717a

Sample Stress(MPa)

Strain(%)

Strain rate(s−1)

Log measuredviscosity(Pa·s)

Tmean

(°C)Log calculatedviscosity(Pa·s)

VH1 Initial 46 0.0 1.6E−02 8.98 614.1 8.95End 46 19.4 1.3E−02 8.93 617.0 8.87

VH3 Intial 73 0.0 2.3E−02 8.99 611.1 9.03End 73 21.4 2.1E−02 8.84 617.2 8.86

VH4 Intial 110 0.0 4.0E−02 8.94 611.4 9.02End 110 17.3 6.4E−02 8.71 621.7 8.74

VH6 Intial 81 0.0 1.1E−04 11.31 540.4 11.39End 81 1.6 1.3E−04 11.32 540.0 11.40

VH6 Intial 120 1.6 1.9E−04 11.29 540.1 11.40End 120 4.7 1.5E−04 11.31 539.4 11.42

VH7 Intial 78 0.0 1.3E−03 10.27 570.1 10.29End 78 11.3 1.2E−03 10.25 570.2 10.29

VH9 Intial 140 0.0 3.1E−02 9.16 605.1 9.20End 140 13.5 4.1E−02 8.95 614.0 8.95

VH10 Intial 27 0.0 2.3E−02 8.58 627.1 8.60End 27 10.0 2.2E−02 8.55 628.3 8.57

VH11 Intial 28 0.0 2.3E−03 9.59 588.6 9.69End 28 14.1 1.9E−03 9.59 587.9 9.71

VH13 Intial 55 0.0 6.0E−03 9.49 593.6 9.54End 55 19.2 5.0E−03 9.41 596.4 9.45

VH14 Intial 50 0.0 6.0E−03 9.43 599.8 9.35End 50 22.9 4.8E−03 9.33 601.2 9.31

VH19 Intial 130 0.0 8.6E−03 9.69 591.3 9.61End 130 21.6 1.2E−02 9.42 599.7 9.36

Tmean was obtained by averaging the recorded temperatures of all three thermocouplesinside the samples.

123K.-U. Hess et al. / Earth and Planetary Science Letters 275 (2008) 121–126

The temperature increase generated during deformation of NISTreference melts correlates positively with the strain rate (Fig. 3). Thedeformation of natural rhyolitic melts produced a similar temperatureincrease (Table 3). In particular, the rate of heating shows apronounced increase as the strain rate approaches ~10−3 s−1. Duringthe experiments most temperature changes appear to be attributed toviscous heating and conductive dissipation of heat out of the sampleto the surrounding ambient air was not recorded by the thermo-couples around the samples.

The apparent Newtonian viscosity derived from the experimentaldata may differ significantly from the viscosity anticipated using thenominal or external experimental temperature. Thus the internalsample temperature (Tmean) was used to calculate the viscosity(applying the Fulcher equation from the NIST certificate), and thencompared to the experimental viscosity (Fig. 4). Correction for viscousheating using this approach completely accounts for the apparentdrop in viscosity. During all measurements, the precisions of themeasured viscosity changes remained within ~0.06 logarithmic unitof the calculated viscosity changes (in Pa s) independent of theconditions of applied stress, strain rate, and recorded heating (Fig. 5).

Fig. 2. Temporal profiles for three experiments showing the viscosity decrease (line) and a(a) 46 MPa, (b) 73 MPa, and (c) 110 MPa of applied stress.

The observed viscosity decreases in these experiments could betherefore attributed entirely to viscous heating.

4. Interpretation

The heat generated during energy dissipation correlatespositively with the strain rate. We have observed that the heatingrate shows a pronounced increase when approaching ~10−3 s−1

(Fig. 3). Note that this strain rate records the measurement ofsignificant viscous heating in these experiments and not itstheoretical “onset”. Presumably, at lower strain rates, heat isgenerated at too low a rate to overcome heat loss by conduction,thus excluding the observation of viscous dissipation effects. Thisbalance of heat (in and out of the sample) is classically understoodin fluids dynamic as the viscous heating efficiency defined by thedimensionless numbers of Brickman or Nahme (Na). Na representsthe ratio between the heat gained by viscous heating and the heatloss by conduction, so that value greater than one would implythat temperature increases as a result of viscous heating. In a firstapproximation, Na can be expressed as:

Na ¼ σ dγVdAκdδT

ð3Þ

where σ is the applied stress, γ' is the strain rate, A is the area, κ is thethermal conductivity, and δT is the temperature gradient from thecentre of the sample to the end. We can rewrite this equation toexpress the strain rate at which viscous heating becomes experimen-tally important, to:

γV¼ NadκdδTσ dA

ð4Þ

For experiments onNIST717amelt (A=0.0314m2 andκ=1.3Wm−1 k−1

based on in situ thermal diffusivity, heat capacity and density measure-ments), we measure an onset of viscous heating (i.e., if a δT≥1 K, thenNa≥1) at applied stresses greater than ~20 MPa, which correspond to astrain rate higher than ~10−3 s−1. This value is in good agreementwith ourobservation in Fig. 3.

The viscous heating production rate is proportional to thetensorial product of stress and strain rate. Here, neglecting heatconduction term, we simplify the production rate of viscousdissipation (dT/dt) to:

dTdt

¼ σ dγVρdCp

ð5Þ

where σ is the applied stress, ρ the density, and Cp the heatcapacity of the melt. Since the stress applied on a viscous fluid

ssociated temperature increase (doted lines) of viscous dissipation. Experiments with

Fig. 3. Production rate of temperature versus strain rate and applied stress (colour coding). (a) Experiments for the SRM 717a glass. The coloured dotted curves show the trendcorrelating with the measured heating rate according to the experimental strain rate. (b) Comparison between the SRM 717a glass (blue) and an icelandic obsidian (red).

124 K.-U. Hess et al. / Earth and Planetary Science Letters 275 (2008) 121–126

equals the product of viscosity (η) and strain rate, Eq. (5) can bereduced to:

dTdt

¼ ηdγV2

ρdCpð6Þ

This formulation provides a first order explanation for the non-linear increase of heating rate observed in our experiment. It alsoyields an equivalence of behavior between the NIST glass and thenatural rhyolitic melts (Fig. 3). This relationship, theoreticallyanticipated and confirmed by the present results, may have profoundconsequences for magma ascent and the transition to high-dischargerates during eruption (e.g. Costa and Macedonio, 2003, 2005; Mastin,2005; Vedeneeva et al., 2005; Costa et al., 2007).

5. Viscous heating in the glass transition

When the strain rate of silicate melt deformation approaches thetimescale of structural relaxation, the melt can no longer behave as aNewtonian fluid. In experiments involving significant strain, suchmelts exhibit strain rate dependent viscosity (shear thinning) untilthey fail macroscopically (Simmons et al., 1982b; Webb and Dingwell,1990). Previous rheological works on the onset of non-Newtonianrheology in geological melts have not incorporated in situ monitoringof the sample temperature. The early work on fiber elongationprovided a simple parameterization of heat loss to argue for an

Table 3Experimental results for Iceland obsidians

Sample Stress(MPa)

Strain(%)

Strain rate(s−1)

Log measuredviscosity(Pa·s)

Tmean

(°C)

Ice3 Intial 100 0 3.3E−04 11.08 740.2End 100 10.8 3.3E−04 11.08 740.4

Ice4 Intial 74 0 2.0E−02 9.08 840.3End 74 20.9 1.9E−02 8.98 849.1

Ice5 Intial 77 0 2.2E−03 10.05 793.4End 77 24.9 2.1E−03 9.94 799.7

Ice6 Intial 129 0 3.9E−03 10.02 794.6End 129 22.3 5.6E−03 9.74 807.4

Ice7 Intial 120 0 4.5E−03 9.92 797.3End 120 12.3 6.8E−03 9.67 806.3

Tmean was obtained by averaging the recorded temperatures of all three thermocouplesinside the samples.

absence of viscous heating effects from those thin fibers (Webb andDingwell, 1990); postulating at the same time that the high-torqueconcentric cylinder work of Stein and Spera (1992) might have beeninfluenced by viscous heating. Yue and Bruckner (1994) laterreanalyzed the parallel-plate and fiber-elongation observations andformulated a correction for the strain rate dependence of viscositythat includes viscous heating effects explicitly. The question as towhether viscous heating may have had a measurable effect onprevious determinations of the onset of non-Newtonian viscosityremains. We cannot say explicitly because the sample geometries,stresses and strain rates have been very different in the experimentsto date. We can however reiterate two important observations. Firstly,the onset of non-Newtonian viscosity appears to have arrived earlier(approximately three orders of magnitude) than theoretical predictionbased on linear stress-strain approximations (the Maxwell model)(Webb andDingwell,1990). Thatmight imply a contribution of viscousheating. Secondly however, the onset of non-Newtonian rheology hasalways been recorded at approximately three orders of magnitudebefore the predicted strain rate of the structural relaxation time for

Fig. 4. Comparison between the measured values and calculated values of viscosities(based on Tmean) for the SRM 717a glass composition.

Fig. 5. Differences between the variations of measured viscosities and the calculated(certified) viscosity variations against (a) stress (b) temperature, and (3) strain rate. Thedotted line denotes the precision of the measurement equal to 0.06 logarithmic unit inPa⁎s. The absence of a preferential dependence to stress, temperature, or strain rateindicates that viscous heating could be the sole detectable cause of non-Newtonianbehavior in these experiments.

125K.-U. Hess et al. / Earth and Planetary Science Letters 275 (2008) 121–126

experiments performed under compression, simple shear andelongation, and for devices operating on very different samplethermal masses, viscosities and strain rates (Simmons et al., 1982a,1988; Webb and Dingwell, 1990). That observation might imply that

viscous effects are not significant at the onset. In any event, viscousdissipation alone cannot explain the subsequent failure of sampleswhich have been forced into non-Newtonian deformation.

In volcanic conduits, slow ascent rates likely yield low shearrates (e.g., b10−3 s−1), negligible viscous dissipation, and simpleNewtonian response of the liquid. At higher strain rates (e.g., N10−3 s−1),or upon strain localization along the conduit margin, viscousheating may become a primary rheological control. It has beenproposed that “viscosity reduction by viscous heating moves themagma away from the glass transition despite increasing shearrates.” (Gonnermann and Manga, 2007). Yet textural analyses inpumices have been interpreted as evidence for the occurrence ofviscous heating before brittle deformation (Rosi et al., 2004; Polacci,2005). Viscous heating does serve to decrease the viscosity, butthere may be thermal–fluid dynamic scenarios whereby theviscously heated magma can leak its heat more rapidly such thatthe rapidly deforming magma may fail even more effectively than inthe absence of a viscous heating history (Dingwell, 1996). Duringlaminar flow (e.g., Mastin, 2005) we suggest that lava flow wouldnot be incompressible and steady as viscous heating would thin themagma along the conduit margin and drive magma ascent. Thischain of reactions may be common inside volcanoes, especially formedium to high-viscosity magmas (104 to 1012 Pa∙s) as viscousheating may either accompany or immediately succeed the onset ofstructural non-Newtonian rheology.

6. Conclusion

Parallel-plate experiments have been conducted to characterize theeffects of the viscous dissipation on high-viscosity magmas. Viscousheating is observed to be strain rate dependent and becomes rheologi-cally significant, at these viscosities, above approximately 10−3 s−1. Simpleanalysis shows that the temperature generated through viscousdeformation in the experiments could be the sole cause of apparentnon-Newtonian rheology in these experiments. Once the temperaturegenerated is accounted for in the construction of the viscosity–temperature relationship, the rheology is consistent with a Newtonianregime. The interplay of viscous heating and structural onset of non-Newtonian rheology is likely to control the ascent of highly viscousmagmas in manner which is not yet fully understood and whoseunderstandingwould undoubtedly be enhancedby systematic controlleddeformation experiments with the capability for viscous heatingmonitoring.

Acknowledgments

The authors wish to express their gratitude toMarkus Sieber for hismeticulous work during the extensive sample preparation. They alsoacknowledge Hugh Tuffen for providing pristine rhyolitic obsidiansamples from Torfajökull, Iceland. Thermal diffusivity measurementswere performed at Netzsch GmbH. This research and K.U. Hess werepartially funded by the BMBF/DFG SonderprogrammGeoTechnologienKontinentalränder grant 03G0584A, GEOTECH #312. Financial sup-ports were provided to B. Cordonnier by the DFG-ICDP grant HE4565-1-1 and to Y. Lavallée by the THESIS program of the Bavarian EliteNetwork as well as the 2005 Girardin–Vaillancourt scholarship of theDesjardins Foundation.

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