Rheology of bimodal crystals suspensions: Resultsfrom analogue experiments and implications formagma ascent

Piero Del GaudioIstituto Nazionale di Geofisica e Vulcanologia, Via di Vigna Murata 605, IT-00143 Rome, Italy(piero.delgaudio@ingv.it)

[1] The effect of crystal size of bimodal suspensions on rheology of magmas at strain rates between 0 and1 s21 is studied. Suspensions consist of silicon oil and two populations of natural crystals with size 63–125 and 250–500 mm mixed in different proportions; the total solid fraction / of the mixtures is between0.25 and 0.5. At /� 0.4, finer, coarser, and bimodal suspensions display comparable viscosities. At/� 0.4, the viscosity of the bimodal suspensions is larger than that of the unimodal ones. The bimodalsuspension, made mainly of finer crystals, shows a stronger increase of viscosity with /. The addition offiner crystals to a suspension of coarser ones produces a more pronounced increase of viscosity withrespect to suspensions of coarse or fine crystals alone, and of finer crystals with added coarser ones. Thebimodal suspensions of coarser crystals develops yield stress at /� 0.25, the others at /� 0.4. It ismodeled the ascent velocity in a 20 m wide dike of magmas with bimodal and unimodal populations ofcrystals of different size. For /� 0.4, the crystal size has not effects on the ascent velocity of magmas.For /� 0.4, the velocity of a magma with growing phenocrystals decreases as / increases less than thatof a magma with forming microlites, and more of a magma with microlites and growing phenocrystals. Amagma with phenocrystals and forming microlites has the lowest ascent velocity.

Components: 5,492 words, 5 figures, 1 table.

Keywords: rheology; magmatic suspensions; analogue model; dike

Index Terms: 8439 Physics and chemistry of magma bodies: Volcanology; 8445 Experimental volcanism: Volcanology;8160 Rheology: general: Tectonophysics; 1236 Rheology of the lithosphere and mantle: Geodesy and Gravity; 7218 Litho-sphere: Seismology; 8032 Rheology: general: Structural Geology.

Received 3 October 2013; Revised 17 December 2013; Accepted 17 December 2013; Published 00 Month 2014.

Del Gaudio, P. (2014), Rheology of bimodal crystals suspensions: Results from analogue experiments and implications formagma ascent, Geochem. Geophys. Geosyst., 15, doi :10.1002/2013GC005078.

1. Introduction

[2] The rheological behavior of the magmas iscontrolled by several, interconnected parametersincluding the temperature of the melt and its com-position, the amount, size and shape of crystals,the bubble content, and the strain rate. Manyexperimental and theoretical studies have been

conducted to investigate the role played by suchparameters on magmatic suspensions [Lejeune andRichet, 1995; Thies and Deubener, 2002; Souleand Cashman, 2005; Sato, 2005; Caricchi et al.,2007; Ishibashi and Sato, 2007; Giordano et al.,2008; Costa et al., 2009; Petford, 2009; Castruc-cio et al., 2010; Mueller et al., 2010; Vetere et al.,2010; Cimarelli et al., 2011; Vona et al., 2011;

© 2013. American Geophysical Union. All Rights Reserved. 1

Article

Volume 15, Number 00

00 Month 2014

doi: 10.1002/2013GC005078

ISSN: 1525-2027

Mader et al., 2013]. As the concerns the amountof solid particles, i.e., crystals, dilute suspensions(solid fraction / � 0.02), shows a Newtonianbehavior, and the relative viscosity gr, which is theratio between suspension viscosity gsusp and theviscosity of the liquid gl, does not considerablychange with increasing / [Rutgers, 1962; Thomas,1965]; in semidilute suspensions (/� 0.2–0.25),gr slightly increases with increasing / , whereas at/� 0.2–0.25, gr significantly increases withincreasing /, and a non-Newtonian behavioroccurs. Besides, the shape of crystals plays a rolebecause, at a constant /, suspensions of elongatedand of angular solids show viscosities higher thanthose of the suspensions of spheres [Raiskinm€akiet al., 2000; Mueller et al., 2010]. The size of thesolids should also be taken into account because,at constant / and shape of particles, finer suspen-sions display a higher viscosity than the coarserones [Del Gaudio et al., 2013]. In addition, finersuspensions develop shear thinning at lower /.The above cited results from previous studies evi-dence how the amount, size, and shape of the crys-tals may affect the viscosity and control therheological behavior of natural (magmas, debris/mudflows) and artificial suspensions. As known,magmas are frequently characterized by crystalsof different size (phenocrysts, microphenocrysts,and microlites) ; with larger (cm to mm sized)crystals formed in reservoirs, and smaller crystals(mm to lm sized) grown during the magma ascentin the conduit and/or on the Earth surface in lavaflows [Marsh, 1988; Armienti, 2008; Salisburyet al., 2008]. The rheology of such magmatic sus-pensions is poorly known because of the complex,multimodal character of the crystal size. Previousstudies by Farris [1968] and Zhou et al. [1999]report that (a) a bimodal suspension with coarseand fine spherical particles has, at a constant /, alower viscosity with respect to a unimodal suspen-sion of fine particles, and (b) broad size distributedsuspensions have a higher yield stress than narrowsize suspensions. However, spheres cannot be con-sidered as analogue of crystals because of the dif-ferent shape. Therefore, additional work must bedone to better know the role of the crystal size inthe rheological behavior of magmatic suspensions.

[3] Here, the results of rheological measurementsof bimodal suspensions of silicon oil and differentvolume fractions of two crystal population of dif-ferent size (maximum axis between 63 and 125mm, and between 250 and 500 mm) are presented.The collected data are combined with those of Del

Gaudio et al. [2013], who studied separately theunimodal suspensions with the same crystals andsize interval. This study treats the way in which abimodal distribution of crystals may affect the rhe-ology of a suspension when the size distributionsare combined in different proportions. The experi-ments are conducted at controlled shear (strain)rates in the same range of those expected in manyunfragmented natural flows like lava flows orunfragmented magmas within reservoirs orconduits.

2. Starting Material and ExperimentalProcedure

[4] Starting materials and experimental methodare the same than detailed in Del Gaudio et al.[2013], the only main material properties andexperimental steps are reported here. Rheologymeasurements were performed on suspensions ofnatural crystals immersed in a Newtonian siliconeoil. The used silicone oil is a Cannon N15000 cali-bration oil with viscosity g 5 41.32 Pa s (logg 5 1.62) and density 1870 kg/m3 at 25�C. The sil-icone oil behaves as a Newtonian fluid at tempera-ture and strain rate applied during experiments[Del Gaudio et al., 2013] and the viscosity at 25�Cis within the range of low-viscosity magmas atsubliquidus temperature [McBirney and Murase,1984].

[5] Magmatic crystals (sanidine and pyroxene)collected by a pyroclastic deposit in the StenzanoEruptive Unit (Latera caldera, Vulsini VolcanicDistrict, Italy) [Taddeucci and Palladino, 2002],were separated from other components by sievingand hand picking under binocular microscope, andthen rinsed in distilled water and dried at 110�C.After crystals separation by sieving, two crystalspopulations were prepared as starting material tobe mixed in different proportion: a populationwith crystals of size 63–125 mm (hereafter popula-tion A) and a population of size 250–500 mm(hereafter population B). The size and shape of thecrystals in the two populations were determinedby Del Gaudio et al. [2013] through image analy-sis (Figure 1). The average values of the axial ratio(Figure 1f), which is 1.52–1.53, and form factor(FF 5 4p surface/perimeter2) [Cox and Budhu,2008], are very similar, so excluding a possiblerole played by the shape in the rheological behav-ior [see Del Gaudio et al., 2013].

[6] Suspensions with total / of 0.25, 0.30, 0.35,0.40, 0.45, and 0.50 were obtained by mixing

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weighted amounts of crystals A and B added to adefined volume of oil in this way:

1. Starting from a solid volume fraction of 0.2 A, afraction /B of 0.05, 0.10, 0.15, 0.20, 0.25,and 0.30 B was added (bimodal suspensions0.2A/B).

2. From a solid volume fraction of 0.2 B, a fraction/A of 0.05, 0.10, 0.15, 0.20, 0.25, and 0.30 Awas added (bimodal suspensions 0.2B/A).

[7] Rheology of suspension was studied using arheometer AntonPaar Physica MCR301 equippedwith parallel plate geometry (diameter of 25 mm)with rough surface and a Peltier temperature con-trol system. Flow curves s( _c) and gsusp( _c) weredetermined by rotational rheometry in controlledshear-rate mode [Mezger, 2006] at 25�C, where sis the applied shear rate, gsusp the viscosity of thesuspension, and _c the strain rate. In all the experi-ments, _c increases from 0 to 1 s21 and decreases

Figure 1. (a–d) Images from field emission scanning electron microscopy of the used crystals; the imagesdisplay the following mixtures: (Figure 1a) 0.2A/B50.05, (Figure 1b) 0.2A/B50.15, (Figure 1c) 0.2A/B50.30,and (Figure 1d) 0.3A/B50.2. (e) Crystal size distribution of the two crystal populations used in the experi-ments; (f) Axial ratio distribution of crystals in the two populations in Figure 1e.

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from 1 to 0 s21. Because of the variability in thegrain size of the crystals, the gap h, distancebetween the two plates, has been fixed at three dif-ferent distances (h 5 1.0, 1.5, and 2.5 mm).According to Del Gaudio et al. [2013], the differ-ence in viscosity values is <0.1 log units, wellwithin the measurement error.

3. Results

[8] The recorded data points (s, _c, gsusp) from therheological measurements are fitted with theparameters of the Herschel and Bulkey’s [1926]equation (HB):

s5s01K _cn (1)

where K is the consistency, so is the yield stress,and n is the flow index. By this relationship, n andso for each suspension are determined. The Newto-nian behavior is represented by so5 0 and n 5 1.As concerns the rheology of the populations A andB, a deviation from the Newtonian behavior (shearthinning; n< 1) has been observed as the solidfraction increases, as expected. Figure 2 shows therelationship between n and / in the suspensions Aand B (data from Del Gaudio et al. [2013]) andthose with mixed A and B studied in this work.Population A departs from the Newtonian behaviorat /� 0.1, and B shows a shear-thinning behaviorat /� 0.25 (Figure 2). The extent of the deviationfrom the Newtonian behavior varies with the crys-tal size and becomes more pronounced as /increases. In the two mixtures 0.2A/B and 0.2B/A,

this deviation occurs at a total volume fraction ofcrystals /� 0.25–0.30. The suspensions A, B, and0.2A/B show a decrease of n as / increases. Thesuspension 0.2B/A is characterized by a morecomplex tendency, with an increase of n at/> 0.25–0.30 and a decrease for /> 0.45.

[9] Figure 3 shows that the suspensions A, B, and0.2B/A exhibit a negligible or null so at /� 0.35;at higher / values, so of A and B steeply rises. Forthe suspension 0.2B/A, an abrupt increase of so isalso observed at / 5 0.4, but a decrease of so

occurs at higher /. The suspension 0.2A/B dis-plays a steeply increase of so at / 50.25.

[10] Figure 4 summarizes the variation of gr as afunction of /. The data are fitted according to theKrieger and Dougherty [1959] model and theCosta et al. [2009] parameterization (see Table 1).The Krieger and Dougherty [1959] relation KDis:

gr5 12/=/mð Þ2E/m (2)

where E is the Einstein coefficient (also known asintrinsic viscosity) and /m is the maximumpacking.

[11] The semiempirical parameterization of Costaet al. [2009] is:

gr /ð Þ5 11 /=/�ð Þdh i

= 12F /=/�; n; cð Þ½ �E/� (3a)

where F is

F5 12nð Þerf p1=2=2 12nð Þ/=/� 11/c=/�cð Þ

h i(3b)

Figure 2. Flow index n of suspensions A and B and mixturesof the two size populations; error bar is the standarddeviation.

Figure 3. Yield stress s0 of suspensions A and B and mix-tures of the two size populations; error bar is the standarddeviation.

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where /* is the critical solid fraction at which therheological transition from a dominant liquidphase regime evolve to a crystals dominatedregime. c is a measure of the rapidity of viscosityincrease with / as this parameters approaches /*;d controls gr at />/*. d is equal at 13-c, and ndetermines gr at / 5 /*.

[12] Despite the KD’s model is known to be work-ing well for dilute suspensions even for nonequantparticles [Cimarelli et al., 2011], the curves in Fig-ure 4 evidence that the KD model produces curveswell fitting the data of the suspensions A and B,but it lacks the matching with data of 0.2B/A and0.2A/B. The semiempirical parameterization fol-lowing Costa et al. [2009] model, which is specifi-cally designed to fit the gr-/ relationships inmagmas, better describes this paper data on0.2B/A and 0.2A/B.

[13] In Figure 4, the suspension B has lower vis-cosity values of A and a weaker dependence on /with respect to A. At /� 0.4, gr of the suspensions0.2A/B and 0.2B/A do not significantly variesfrom that of A and B. At /> 0.4, gr of 0.2A/B and0.2B/A become larger than that of A and B, and gr

of 0.2B/A reaches the highest values. An outcomeof the above reported results is that in bimodal sus-

pensions like those analyzed here, the addition offiner crystals to a suspension of coarser ones pro-duces a more pronounced increase of viscositywith respect to suspensions constituted by coarseor fine crystals alone, or by finer crystals withadded coarse ones. The higher viscosity ofbimodal suspensions with respect to that of theunimodal ones at a fixed / has been observed inslurries [Ancey, 2007], but not in suspensions ofspheres [Chong et al., 1971].

4. Discussions and Conclusions

[14] The rheological behavior of bimodal or multi-modal suspensions has been investigated by Farris[1968], Chong et al. [1971], Shapiro et al. [1992],Barnes [2003], Pishvaei et al. [2006], and Qi andTanner [2012]. Most of these studies have ana-lyzed (a) bimodal suspensions in which the mainsolid phase is represented by spheres of a coarsesize and (b) the effect of particle size ratio on theviscosity. The results evidence that, for a constant/, the relative viscosity of a unimodal suspensionof spherical particles is higher than that of bimodalsuspensions with an added finer fraction [see alsoCimarelli et al., 2011]. This effect seems to ceasewhen the particle size ratio (small to large spheres)become less than 1/10 [Chong et al., 1971]. In thatcase, the bimodal suspension behaves as a unimo-dal ones made by a solid phase (larger spheres)immersed in a ‘‘fluid’’-like material, i.e., the liquidplus the finer spheres. According to Pishvaei et al.[2006], Ancey [2007], Del Gaudio et al. [2013],and previously reported, suspensions charged by asolid phase of small size have a viscosity higherthan that of suspensions with solids of larger size.Data in Figure 4 further indicate that once finerparticles of a suspension become predominantwith respect to the coarser ones, the relative vis-cosity increases more than that of suspensions inwhich the coarse phase becomes predominant withrespect to the finer one. Results of this study andthose of the above cited studies on nonsphericalparticles do not corroborate the results of Chong

Figure 4. The figure shows the relationship between relativeviscosity (viscosity of suspensions divided by viscosity of sili-cone oil) and total solid fraction for suspensions A and B andthe suspensions 0.2A/B and 0.2B/A (with / 5 0.05, 0.10,0.15, 0.20, 0.25, and 0.30); error bar is the standard deviation.Black curves are KD curves for suspensions A and B, graycurves the KD’s for suspensions 0.2A/B and 0.2B/A, blueand green curves are Costa et al. [2009] predictions for 0.2A/B and 0.2B/A, respectively, continuous and dashed redcurves are Costa et al. [2009] predictions for suspensions Aand B, respectively.

Table 1. Fit Parameters for Curves in Figure 4a

Costa ‘09 KD

n /� c d E /m

A 0.0007 0.50 0.21 12.8 8.0 0.56B 0.0025 0.70 0.60 12.4 6.1 0.700.2A/B 0.0008 0.44 1.01 12.0 8.7 0.550.2B/A 0.0001 0.44 4.02 9.0 4.5 0.50

aBest fit parameters of KD equation and Costa et al. [2009] model.

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et al. [1971] and Cimarelli et al. [2011] onspheres. These latter authors report that, at con-stant /, unimodal suspensions of spheres have aviscosity higher than that of bimodal suspensions.Figure 4 shows that the suspension A has viscosityvalues comparable to those of 0.2B/A at/� 0.35–0.40. This discrepancy could be due tothe nonspherical shape of crystals, which haveaxial ratios 5 1.52–1.53 and FF 5 0.68–0.70. It issuggested that, at /� 0.35–0.40, the addition offiner crystals to a population of larger ones allowsthe filling of the available rooms between thelarger particles and, within such spaces, the addedparticles do not interact together or with the largerparticles. According to this interpretation and datafrom Figure 3, yield stress and significant shear-thinning behavior are not recorded at /� 0.35–0.40. When the fraction of finer crystals equals oris higher than that of the coarser ones, i.e., at/� 0.35–0.40, a steep rise of gr occurs. It is pro-posed that when the volume of finer crystals ishigher than that of the coarser ones, the finer solidsstart to interact together and a cluster-like structureforms within the rooms among the coarser crys-tals. As result, and a yield stress should appear.This is consistent with the data from Figure 3,where such rheological behavior is recorded.

[15] Figure 4 shows the viscosities of a unimodalsuspension of fine crystals and of bimodal suspen-sions of fine and added coarser crystals are compa-rable at /� 0.35–0.40. In that case, the addition oflarger particles to a suspension of smaller onesmay produce a confinement of both the fluid phaseand smaller particles in the spaces among thelarger ones. Data from Figure 3 reveal that theappearance of yield stress in the bimodal suspen-sion 0.2A/B occurs at /� 0.25, at a lower / withrespect to A, B, and 0.2B/A. Such a behavior couldreflect an early formation of crystal networks offiner particles, which, with the addition of thelarger ones, tend to be confined in progressivelysmaller space.

[16] As concerns the departure from a Newtonianbehavior, suspensions of finer particles develop anearly shear-thinning behavior with respect to thebimodal suspensions with the addition of largercrystals (Figure 2) because, according to Del Gau-dio et al. [2013], finer particles reduce the averagesize of clusters earlier than coarser particles. Con-cerning the suspension 0.2A/B, reduction of theclusters is developed later since smaller particlesare spatially confined by the larger ones. For / >0.4, gr values of the suspension A are lower thanthose of the suspension with added larger crystals.

The above finding could be due to the occurrence,in 0.2A/B, of nearly locked (dynamically) clustersof smaller particles filling the available roomamong larger particles.

[17] The results on the effect of the crystal size onthe rheology of bimodal suspensions reportedhere, along with those from previous studies onunimodal suspensions [Lejeune and Richet, 1995;Ishibashi and Sato, 2007; Mueller et al., 2010;Del Gaudio et al., 2013], have some general impli-cations for the flow of magmas in conduits and onthe Earth surface. However, some cautions mustbe used in the full extrapolation of these experi-mental data to real natural cases because of theconsiderable lower variability in crystal size andshape of used crystal populations with respect tothat observed in natural magmas. Taking intoaccount such a lower variability and specifyingthat these results may be extrapolated only to mag-mas with a nearly unimodal or bimodal size distri-bution and not to magmas with a seriate-likefabric, some inferences on the crystal size-dependent viscosity variations in magmatic sus-pensions may be done.

[18] At / smaller than a threshold (our data sug-gest / 5 0.30–0.35), magmas having same /and crystals of different size, have comparableviscosity values. Below this threshold, crystalsbreakage can occur in a flowing magma withcrystallinity of about 30 vol % producing 5% offiner fragments of the whole crystal population[Forien et al., 2011]. Following the results ofthis study, crystals breakage does not producesignificant viscosity variations. Over this thresh-old and at constant /, the viscosity of magmascontaining growing phenocrystals, or incorporat-ing large xenocrystals, may be lower than that ofmagmas containing not growing phenocrystalsbut crystallizing microlites by degassing, decom-pression or cooling [Cashman and Blundy, 2000;Couch et al., 2003; Armienti, 2008; Hastieet al., 2013]. The viscosity of a microlite-poormagma including coarse xenocrystals during itsascent [e.g., Michaud, 1995] is lower than thatof a microlites dominated magma with the sametotal crystal content. A shear-thinning behavioris expected to develop earlier in a microlitesdominated magma rather than in a magma con-taining exclusively phenocrystals or phenocrys-tals and growing microlites. The above reportedpossible effects of the crystals size on the viscos-ity of magmatic suspensions have immediatefalls on the ascent velocity. Here, it is modeledthe ascent velocity u of magmas flowing in a

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laminar regime in a dike using this paper dataand by using the relation [Lister and Kerr,1991]:

u5 ½w2=ð3gsuspÞ�Dqg (4)

where w is the half width of the dike, Dq is the dif-ference between the density of the country rockand that of the magma, and g is the gravity (9.81m/s2). The viscosity of oil used in these experi-ments is 41.32 Pa s, a value within the range ofbasaltic magmas at subliquidus temperature[McBirney and Murase, 1984]. It is assumed thatthe melt viscosity does not change significantlyduring crystallization, and therefore, the velocityestimates have the significance of ‘‘minimumvelocities’’ because of, in natural systems, the vis-cosity of the residual melt increases as crystalliza-tion proceeds. In the following, it is assumedw 5 10 m, Dq 5 0.2, and use the gsusp values forthe following suspensions: A, B, 0.2A/B, and0.2B/A with / 5 0.3, 0.4, and 0.5. Results aresummarized in Figure 5 and show how a magmahaving an initial / 5 0.3 decreases its velocity of1 order of magnitude as / approaches 0.4. This isdue only to an increase of / and it is irrespectiveof the size distribution of the crystal populations.As the crystal content further increases (/A> 0.4,/B> 0.4, /A> 0.4 2 /B and /B> 0.4 2 /A), thevelocity of the magma (a) remains nearly constantif the early formed crystals grow without new-formed microlites, (b) decrease of 1 order of mag-nitude if only microlites form during the ascent,

and three to fourth orders of magnitude if a magmawith phenocrystals starts to form microlites.Therefore, the size of crystal population, which isa function of cooling, decompression and degass-ing processes within a conduit, becomes relevantin controlling the ascent velocity when a critical/-threshold is reached.

Acknowledgments

[19] This study was found by FIRB-MIUR ‘‘Research andDevelopment of New Technologies for Protection and Defenseof Territory from Natural Risks.’’ The author thanks an anon-ymous reviewer and Mattia Pistone for the critical and con-structive reviews of the manuscript, James Tyburczy for theeditorial handling, Guido Ventura for constructive help anddiscussions, and Andrea Cavallo for taking field emissionscanning electron microscopy of the starting material.

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DEL GAUDIO: RHEOLOGY OF BIMODAL CRYSTALS SUSPENSIONS 10.1002/2013GC005078

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