The effect of particle size on the rheology of liquid-solid mixtures with application to lava flows: Results from analogue experiments



[1] We investigate the effect of crystal size on the rheology of basaltic magmas by means of a rheometer and suspensions of silicon oil with natural magmatic crystals of variable size (from 63 to 0.5 mm) and volume fraction ϕ (from 0.03 to 0.6). At constant ϕ, finer suspensions display higher viscosities than coarser ones. Shear thinning (flow index n < 1) occurs at ϕ > 0.1–0.2 and is more pronounced (stronger departure from the Newtonian behavior) in finer suspensions. Maximum packing and average crystal size displays a nonlinear, positive correlation, while yield stress develops at ϕ > 0.2–0.3 irrespective of the crystal size. We incorporate our results into physical models for flow of lava and show that, with respect to lava flows containing coarser crystals, those with smaller crystals are expected to: 1) flow at lower velocity, 2) have a lower velocity gradient, and 3) be more prone to develop a region of plug flow. Our experimental results explain the observation that phenocryst-bearing and microlite-bearing lavas at Etna volcano (Italy) show smooth pahoehoe and rough aa' surfaces, respectively.


[2] Pure silicate melts at high temperature usually display a Newtonian rheological behavior, while crystal-bearing magmas and lava flows can exhibit a non-Newtonian behavior (e.g., yield stress or shear thinning) mainly depending on the amount of solid fraction and shear rate. Studies on the rheology of crystal-bearing silicate melts [Lejeune and Richet, 1995; Thies and Deubener, 2002; Sato, 2005; Caricchi et al., 2007; Ishibashi and Sato, 2007; Vetere et al., 2010; Vona et al., 2011] and analogue materials [Soule and Cashman, 2005; Mueller et al., 2010; Cimarelli et al., 2011] have explored the effect of variable solid fraction and particle shape on the rheology of suspensions. However, the rheological effect of particle size remains poorly known. Using suspensions of spherical particles, Farris [1968] showed that a bimodal suspension including coarse and fine particles has, at the same solid fraction, a lower viscosity with respect to a unimodal suspension including fine particles only. Results from numerical models on zero-shear suspensions [Saar et al., 2001] suggest that particle size plays a negligible role in controlling suspension rheology, e.g., the onset of yield stress. Zhou et al. [1999] suggest that the size distribution of solids in a suspension can significantly affect its yield stress.

[3] Here we perform rheological measurements of suspensions of silicon oil and different volume fractions of natural crystals of varying size. Our aim is to clarify the effect of crystal populations with a different range of size on the rheological behavior of crystal-melt suspensions in lava flows, and, more in general, of other natural flows consisting of Newtonian fluids loaded with solids. The experiments have been conducted at controlled shear (strain) rates in the same range of those expected in many natural flows, including basaltic lava flows.

2. Starting Material and Experimental Technique

2.1. Material

[4] Suspensions of crystals of four different equivalent diameters (<63, 63–125, 125–250, and 250–500 µm, Figure 1) immersed in silicone oil (Cannon standard oil N15000 with density=1870 kg/m3 and viscosity η=41.32 Pa·s at 25°C) are used as analogues of basaltic magma. The use of N15000 standard oil prevents electroviscous effects, and ensures a Newtonian behavior at experimental strain rates (≤ 500 s−1). The viscosity of oil at 25°C is comparable to that of crystal-free basalts at 1200–1300°C [e.g., Giordano and Dingwell, 2003] and sand-water mixtures from debris-flows [Scotto di Santolo et al., 2010].

Figure 1.

(a) Images from field emission scanning electron microscopy of the used crystals in the class 250–500 µm (top) and <63 µm (63–20 µm) (bottom). Box plots summarizing the (b) size, (c) axial ratio, and (d) Form factor (FF) of the crystal populations used in the experiments. The averaged axial ratio among the different populations is 1.69±0.14, and the averaged form-factor is 0.63±0.08.

[5] Sanidine and pyroxene crystals from pyroclastic deposits of the Stenzano Eruptive Unit [Latera caldera, Vulsini Volcanic District, Italy; Taddeucci and Palladino, 2002] were separated from other components by sieving and hand picking under binocular microscope, and then rinsed in distilled water and dried at 110°C. Image analysis on images from field emission scanning electron microscopy [Jeol JSM-6500F] on 212–456 crystals per size class was conducted following Vetere et al. [2010]. Analysis shows that crystals are parallelepiped-like shape. Distribution of major axis, axial ratio (AS) and form factor [FF = 4π surface/perimeter2; Cox and Budhu, 2008] in the four different sizes is reported in Figure 1. Crystal size varies according to the respective granulometric class, and the variation range of shape and size is similar in the different classes.

[6] In order to produce suspensions with a defined volume fraction of particles, a weighted amount of crystals was added to a defined volume of oil. The mixture was left in a furnace at 60–70°C for 3 h to exclude the formation of bubbles during the free sinking of crystals in the oil. Then, the mixture was carefully mixed and then put in the furnace for 1 h. Finally, the suspension was carefully and slowly hand mixed using a spoon to prevent the possible breakage of crystals. This procedure was used for all suspensions. The density of the crystals, measured by the immersion method, is 2770±250 kg/m3. Densities are used to determine the volume fraction of solid ϕ in the different suspensions, which, in the experiments, varies from 0.03 to 0.6. Although the density of silicon oil is lower than that of particles, settling experiments performed on a mixture of oil with 10 vol.% of crystals with size 125–250 µm reveal settling velocities lower than 0.06 mm/min. As a result, the short duration of experiments (few minutes) negates the potential effects of crystals sinking to create concentration differences. The difference between the density of crystals and that of oil N15000 is comparable with that reported in parallel-plate experiments by Mueller et al. [2010] and Cimarelli et al. [2011], where sinking effects were also excluded.

2.2. Experimental Apparatus and Measurement Technique

[7] Suspension rheology was measured using an Anton Paar Physica MCR301 rheometer. Flow curves inline image and inline image are determined by rotational rheometry using a parallel plate geometry (plate diameter of 2.5 cm and fixed distance between the two measuring plates of 1.0, 1.5 and 2.5 mm) equipped with rough surfaces in controlled shear-rate mode [Mezger, 2006] at 25°C, where τ is the applied shear rate, ηsusp the viscosity of the suspension and inline image the strain rate. In all the experiments, inline image increases from 0 to 1 s−1 and decreases from 1 to 0 s−1. We performed a second run where inline image was increased from 0 to 500 s−1 and reversal from 500 to 0 s−1 to investigate the onset of the non-Newtonian behavior at very dilute suspensions. We also performed preshear tests on suspensions with different crystal concentration or size to check possible effects of a preshear treatment on the measurement. Results show that the preshear treatment does not influence the rheological measurements. Flow curves for dilute suspensions are well reproduced during the first and the second runs over the inline image ramps of 0–1 s−1. Hysteresis of the flow curve is observed only in suspensions showing shear thinning. In these suspensions, at a given τ, values of inline image are systematically lower on the down ramp than on the up ramp. Since we consider the averaged up-down values of viscosity related to Newtonian plateau, we do not account for the hysteresis effect with the fitting of the data.

[8] Figure 2 shows viscosity measurements performed with three different distance (1.0, 1.5, and 2.5 mm) between the measuring parallel plates, for both finer and coarser suspensions with different solid fraction. The observed difference in viscosity values is less than 0.1 log units, well within the measurement error, suggesting a negligible effect of the distance between the plates within the 1.0–2.5 mm.

Figure 2.

The effect of changing the gap (1.0, 1.5, and 2.5 mm) between the two parallel plates of the rheometer on the measured viscosity versus solid fraction behavior for suspension with particles of size <63 and 250–500 µm.

[9] We test the experimental reproducibility by repeating the measurements on separately prepared samples having the same crystal content and range of sizes. We have also varied the strain rate interval and data point time interval recording. The flow curves are reproducible within 0.1 log units.

3. Results

[10] We fit the inline image flow curve with the three-parameter model by Herschel and Bulkley [1926] (HB):

display math(1)

where K is the consistency, το is the yield stress, and n is the flow index. According to Castruccio et al. [2010], this model well fits the flow curves of crystal-bearing lavas. Our measurements, which are summarized in Table 1 and Figures 3 and 4, are qualitatively consistent with results from previous experimental studies [e.g., Hoover et al., 2001], which show a decrease in flow index with increasing solid fraction (Figure 3), and (b) the onset of yield stress at solid fraction ≥0.2–0.3 (Figure 4).

Table 1. Summary of the Rheological Parameters in the Different Suspensionsa
<63 µm63–125 µm125–250 µm250–500 µm
  1. a

    φ is the particle volume fraction, ηr is the relative viscosity, τ0 is the yield stress, and n is the flow index.

Figure 3.

Flow index n versus volume fraction for suspensions of variable particle size. Error bar is the standard deviation.

Figure 4.

Determined yield stress of suspensions of variable particle size at different volume fractions. Error bar is the standard deviation.

[11] A shear thinning behavior (n<1) is observed in suspensions with crystals size ≤125 µm at ϕ larger than about 0.1 (Figure 3). Suspensions with crystals size >125 µm exhibit shear thinning at ϕ >0.2. The deviation from Newtonian behavior is more pronounced with finer crystal sizes. Our results on suspensions with crystals in the 125–250 µm size class match well those of Mueller et al. [2010], which also found n < 1 at ϕ > 0.2 for samples bearing particles of comparable size (angular silicon carbide grit with size between 126 and 312 µm).

[12] Suspensions with finer (<63–125 µm) and coarser (>125 µm) crystals exhibit a negligible το at ϕ less than about 0.3 while, το steeply rises at larger ϕ (Figure 4), indicating that το is dependent on the solid fraction but not on the crystal size of the suspension.

[13] The viscosity of suspension as function of inline image is plotted in Figure 5. Dilute suspensions (0.03–0.06 vol. % of crystals) display Newtonian behavior at inline image and show viscosity in the range 43–58 Pa·s. At ϕ=0.30−0.31, coarser suspensions do not show a strain rate-dependent viscosity in the range 0–1 s−1, whereas the finest suspension (<63 µm) displays a shear-thinning behavior at inline image. On the whole, shear thinning develops at lower strain rates in finer suspensions than in coarser ones.

Figure 5.

Viscosity versus strain rate of suspensions with different crystal size and volume fraction.

[14] We fitted the relative viscosity versus solid fraction (ϕ) data (Table 1 and Figure 6) using the Krieger and Dougherty [1959] (KD) equation:

display math(2)

where B is the Einstein coefficient (also known as intrinsic viscosity); B is 2.5 for spheres, while it is an adjustable parameter for non spherical particles, as in our case; ϕm is the maximum packing of solid particles (crystals in this study). An evident nonlinear dependence of ηr on crystal size at ϕ >0.1–0.2 (Figure 6) is observed. At the same solid fraction, suspensions with finer crystals display higher ηr than those with coarser ones. Using the KD equation, we calculate the maximum packing ϕm for each sample by fitting experimental data (Figure 7). ϕm increases with increasing crystal size from 0.35 (crystals <63µm) to 0.71 (250–500 µm crystals). The adjustable parameter ϕm is thus strongly dependent on particle size. This result is consistent with that of numerical models on the packing of populations of spheres [Shi and Zhang, 2008; Qi and Tanner, 2012]. These models show that populations with a larger fraction of coarser particles have larger density packing. Our suspension with crystals of size 125–250 µm has a calculated maximum packing (ϕm=0.63) equal to that of random close packing (RCP=0.63) of spheres [Stickel and Powel, 2005], i.e., highest volume fraction of spheres packed to retain a random configuration. Suspensions with crystals between 125 and 500 µm have calculated ϕm values spanning between the random loose packing (RLP=0.56), which represents the minimum packing of spheres, and RCP=0.74 of Weitz [2004] in monomodal spheres. The suspension with crystals of size <63 µm has ϕm = 0.34, a value consistent with that found by Mueller et al. [2010] on high-aspect ratio wollastonite crystals (ϕm = 0.316) and by Cimarelli et al. [2011] on high aspect ratio wollastonite and silicon carbid mixtures (ϕm = 0.36). This value is also consistent with that found in suspensions of starch granules with irregular shape and AR<2, which have ϕm between 0.37 (rice) and 0.34 (amaranth) [Willet, 2001]. Our results summarized in Figure 7 and the above comparison with results from other studies on particle suspensions show that the maximum packing of suspensions is a complex function of the particle size and shape.

Figure 6.

Relation between relative viscosity and solid fraction for different suspensions. Viscosity is determined averaging the Newtonian plateau (see Figure 5). Continuous lines are the Krieger and Dougherty [1959] fitting curves. Error bar is the standard deviation.

Figure 7.

Maximum packing calculated by fitting the relative viscosity-solid fraction curve with the Krieger and Dougherty [1959] equation. The solid line allows the eye.

4. Discussion and Conclusions

[15] We have shown that crystal size has a strong effect on values of Newtonian viscosity and the onset of non-Newtonian (e.g., shear thinning) behavior of suspensions, but it does not play a relevant role on the onset of yield stress (Figure 4). This latter result accords well with experimental results of Ancey and Jorrot [2001] on well-graded, unimodal, sand-water suspensions, and contradicts the results from numerical models of Saar et al. [2001]. This discrepancy could be explained by the fact that the models is based on zero-shear environment and soft-core continuum percolation, where the crystals interpenetrate each other and the maximum packing has no physical significance). Concerning the relations between τοand ϕ, Figure 4 shows a good agreement between our data and the experiments on subliquidus basalts [Hoover et al., 2001], which set the onset of το at ϕ between 0.2 and 0.35.

[16] In our experimental conditions, the onset of non-Newtonian behavior (shear thinning; n < 1) is reached at ϕ between 0.1 and 0.2 (Figure 3) and finer suspensions develop shear thinning at lower ϕ than the coarser suspensions. The shear thinning effect could be due to different processes including shear heating, particle rotation along the flow direction, and cluster reduction. According to Deubelbeiss et al. [2011], we exclude an important effect of shear heating due to the low values of viscosity of the fluid and suspensions and shear rate. We propose a possible effect of crystal rotation and interaction. According to Chang and Powell [1993], the reduction of viscosity during shearing is related to the influence of particle size on the average number of particles in clusters. Our results suggest that, at a fixed volume fraction and increasing strain rate, finer particles tend to reduce the average size of clusters earlier than coarser particles, particularly at larger ϕ (Figure 5). This reduction in the number of clusters may also favor the development of a preferred orientation along the flow direction, as experimentally observed by Ildefonse et al. [1992], so promoting a decrease in viscosity as the strain rate increases.

[17] The slight discrepancy between our results, i.e., n<1 at ϕ ≥0.1–0.2, and those of Castruccio et al. [2010], n<1 at ϕ ≥0.3, could be due to the different shape of the particles used in the experiments. Our experiments featured relatively elongated particles, whereas the cubic particles used by Castruccio et al. [2010] have a lower capability to develop an along-flow preferred orientation and, as a consequence, to disrupt and reduce the number of clusters via rotation and dispersion [Iezzi and Ventura, 2002; Kawabata et al., 2013].

[18] The observed higher viscosity of finer suspensions with respect to coarser ones at constant ϕ (Figure 6) is likely related to the tendency of smaller particles to physically interact forming crystal networks (clusters) earlier than larger particles. This tendency is due to the smaller particle-to-particle distance in finer suspensions. The onset of an early clustering of finer suspensions with respect to coarser ones is also suggested by the inverse relationship between crystal size and ϕm (Figure 7). The dependence of relative viscosity on crystal size described above may be explained by the physical model of Tordesillas et al. [2009] and Kawabata et al. [2013]. Following that model, clusters developed in a suspension evolve according to a “buckling-breakup”–mode. A consequence of the evolution of clusters in the suspension is the increase of viscosity with increasing contacting particles.

[19] Figure 8 shows a comparison of viscosities of this work with the KD curves with the parameters fitting the data by Cimarelli et al. [2011] (wollastonite crystals of size 72.26 µm with AR=8.5; ϕm = 0.33), Vona et al. [2012] (Etna basalt with crystals of 140 µm and AS = 4.6; ϕm = 0.4), and prediction of Einstein-Roscoe equation using maximum packing for uniform unimodal spheres (ϕm =0.74; McBirney and Murase, 1984) and (ϕm =0.6; Marsh, 1981]. Our data span between the curve for unimodal, ϕm=0.6 suspensions of spheres, and that for suspensions with elongated crystals [Cimarelli et al., 2011]. None of the model curves (Figure 8) can fit all of the experimental data, except where ϕ<0.2. For those values of ϕ, the KD curve by Cimarelli et al [2011] overlaps our data for suspensions with crystals ≤125 µm, while Einstein-Roscoe curve overlaps those for suspensions with crystals of size between 125 and 500 µm. This match can be due to the similar values of ϕm resulting by fitting data using KD equation. Therefore, in dilute suspensions, the viscosity data are reproducible within 1 log unit. The lack of acceptable fits for ϕ>0.2 could be due to a more relevant effect of crystal size and clusters development on ϕm and shape. Plotting ηr versus ϕ/ϕm (Figure 9), relative viscosities can be reasonably fitted just at ϕ/ϕm <0.2 (i.e., dilute suspensions) by the empirical equation for suspensions of spheres with different size by Chong et al. [1971]:

display math(3)
Figure 8.

Relative viscosity versus solid fraction in different suspensions (data from Figure 6). Black lines are the Krieger and Dougherty [1959] fitting curves using KD parameters of suspensions by Cimarelli et al. [2011] (wollastonite crystals; length 72.2 µm; axial ratio 8.5), solid curve, and by Vona et al. [2011] (plagioclase and pyroxene crystals; length 140 µm; axial ratio 4.6), dashed curve. Gray curves are the Einstein Roscoe fitwith ϕm=0.36 (solid curve) and 0.374 (dashed curve).

Figure 9.

The relative viscosity at varying normalized solid volume fraction ϕ/ϕm for different suspensions. Solid line is the Chong et al. [1971] fit for suspensions with spheres of different size.

[20] Chong et al. [1971] discovered an independent relation between relative viscosity of suspensions of spherical particles and particle size. Our data clearly show that, for nonspherical crystals, the relative viscosity of suspension is dependent on size, in particular for crystal suspensions with size > 63 µm or ϕ/ϕm <0.2.

[21] Our study has immediate implications for the rheology of magmatic suspensions like lava flows. As a working example, we consider a basaltic lava flow containing crystals with the same size populations as for analogue experiments we performed on silicon oil. Following the velocity profile equation by Dragoni et al. [1986] for a lava of thickness h=1 m flowing on a plane with slope =2° and, assuming a magma (liquid+crystal), density of 2700 kg/m3, we determine the velocity profile for three ideal lavas characterized by the same crystal content (ϕ=0.3) but three different sizes of crystal (<63, 125–250, and 250–500 µm) and the corresponding three different apparent viscosities calculated on the basis of the experimental results of Figure 6. Results show that the lava with the finer crystals has a maximum velocity about one order of magnitude smaller than that of the lava containing coarser crystals (Figure 10a). This implies that, for a constant crystal content, the effect of crystal size on the velocity, and, as a result, fluidity of lava flows may be remarkable. This result may explain the different morphology observed on lava flows of differing crystal size distributions at Etna volcano (Italy). Here lava flows displaying relatively fluidal morphology of pahoehoe to transitional types are characterized by the presence of 16–18 vol. % of phenocrysts of size up to 4 cm [Hughes et al., 1990], while lava flows with the same bulk chemical composition and crystal content, but with crystals of smaller maximum size (up to 2.4 cm) and a higher amount of micron-sized microphenocrystals, display nonfluidal, aa'-type morphology [Hughes et al., 1990].

Figure 10.

Calculated velocity profiles (a) and velocity gradient (b) for basaltic lava flows of thickness h=1 m on a slope=2° in the region τ>τo with ϕ=0.3 and three crystal populations of size 63, 63–125, and 250–500 µm. The apparent viscosity has been estimated on the basis of the relative viscosity values from Figure 6.

[22] In addition, our results show that the velocity gradient is higher for flows with coarser crystals (Figure 10b). This suggests that lava flows with finer crystals (e.g., microlites) may develop more easily a plug flow with respect to flows with crystals of larger size (e.g., phenocrystals), which are expected to be less viscous and more fluidal.

[23] In summary, results of our study show that the particle (crystal) size has an important effect on the rheology of magmatic suspensions and other natural liquid-solid mixtures (e.g., debris-flows) that must be considered in numerical or physical models of emplacement. In addition, further studies need to be carried out to analyze the effect(s) of polymodal suspensions on the rheology of natural flows.


[24] We thank FIRB-MIUR “Research and Development of New Technologies for Protection and Defense of Territory from Natural Risks” for funding the project. We also thank Kelly Russell and Corrado Cimarelli for the critical and constructive reviews of the manuscript, and James Tyburczy for the editorial handling.