The effect of particle shape on suspension viscosity and implications for magmatic flows

Authors


Abstract

[1] The rheology of crystal-bearing magma and lava depends on both the shape and volume fraction of the suspended crystals. We present the results of analogue rheometric experiments on monodisperse suspensions of solid particles in a Newtonian liquid, in which particle volume fraction ϕ and aspect ratio rp are varied systematically. We find that the effect of ϕ on viscosity is well captured by the Maron–Pierce model, and that this model is valid across the range of particle aspect ratios investigated (0.04 ≤ rp ≤ 22, i.e., strongly-oblate to strongly-prolate) when the maximum packing fraction ϕm is treated as a fitting parameter. The value of ϕm derived from fitting to our experimental data depends strongly and systematically on particle aspect ratio; hence, ϕm represents an effective proxy for the influence of particle shape on suspension rheology. We present a simple relationship for ϕm (rp) which allows the viscosity of a suspension to be calculated as a function of ϕ and rp only. We investigate the impact of accounting for crystal shape when modelling volcanic flows by simulating the eruption of magma carrying crystals of different aspect ratio, and conclude that the effect of crystal shape should not be neglected.

1. Introduction and Theoretical Background

[2] The flow of particle suspensions is central to volcanic and magmatic processes throughout the volcanic system; examples include the mingling of phenocryst-bearing magmas, the eruption of microlite-bearing magma, and the flow of crystal-bearing lava down a volcano's flank. Accurate modelling of such flows depends on detailed knowledge of the suspension's rheology. It is well-known from laboratory rheometry of magmatic suspensions that the volume fraction of suspended crystals exerts a first-order control on the effective viscosity of a magma [e.g., Lejeune and Richet, 1995; Caricchi et al., 2007; Ishibashi and Sato, 2007; Lavallee et al., 2007], hence on eruption dynamics [Melnik and Sparks, 2005]. In this article, we demonstrate that the influence of crystal shape is similarly strong, and should not be neglected. We perform rheometric experiments using pre-sheared analogue particle suspensions; whilst analogue experiments do not capture processes such as crystal breakup and pressure solution, they have the advantage over experiments with magmatic suspensions that they are better characterized and allow more thorough exploration of parameter space. Based on our data, we define a relationship between the shape of suspended particles, expressed as their mean aspect ratio, rp, and the microstructural properties of the suspension, encapsulated by the maximum packing fraction of particles, ϕm. Since the average aspect ratio of the dominant crystal phase in a magma is often known, or can be estimated on the basis of petrological models, the relation proposed here will facilitate the use of rheological models in which ϕm is a critical parameter and will improve their accuracy by accounting for crystal shape.

[3] The rheology of a material is usually described by a constitutive equation which relates the driving stress τ to the resulting strain rate equation image. For a Newtonian liquid, this equation is τ = μequation image, where μ is the Newtonian viscosity, which is independent of strain rate. Particle-bearing liquids typically develop a more complex, non-Newtonian rheology as particle concentration increases, becoming first shear-thinning at moderate volume fractions, then developing a yield stress at high volume fractions [Mueller et al., 2010], though we note that yield stress is not universally observed in experiments on natural samples [e.g., Caricchi et al., 2007]. This non-Newtonian behaviour is well-described by the Herschel–Bulkley model [Herschel and Bulkley, 1926]:

equation image

where the consistency K is cognate with viscosity and has the units Pa sn, the flow index n defines the degree of non-Newtonian behaviour (n < 1 for shear thinning behaviour), and τ0 is the yield strength. For non-Newtonian liquids, the viscosity is commonly expressed as an ‘apparent viscosity’ η = τ/equation image at a specified strain rate; expressed in these terms, the Herschel–Bulkley model becomes:

equation image

For dilute and moderately-concentrated suspensions, where yield stress is negligible [Mueller et al., 2010], it can be seen that K is equivalent to η evaluated at equation image = 1 s−1. For weakly shear-thinning materials, i.e., when n is close to 1, K can often be sufficiently well approximated by η. For suspensions, η is commonly non-dimensionalized by the Newtonian viscosity of the suspending liquid μ0 to give the relative viscosity ηr = η/μ0; by analogy a relative consistency can be defined as Kr = K/μ0.

[4] Particles affect the rheology of a suspension because additional energy is dissipated due to fluid–particle and particle–particle interactions. As particle volume fraction ϕ increases, these interactions become more common, leading to a dramatic, non-linear increase in relative viscosity until a maximum packing fraction ϕm is reached [e.g., Einstein, 1906, 1911; Roscoe, 1952; Maron and Pierce, 1956], at which point the suspension becomes jammed. If the particles are perfectly rigid, flow ceases when ϕm is reached; in magmatic suspensions, however, processes such as plastic deformation or creep may take place at ϕ > ϕm [e.g., Lavallee et al., 2007; Caricchi et al., 2007].

[5] Particle shape affects rheology by changing the nature of fluid–particle and particle–particle interactions, introducing two additional considerations: 1) anisometric particles are orientable, so their influence depends on their orientation with respect to the flow; 2) anisometry enhances particle–particle interactions because the orbit of a rotating non-spherical particle encloses a greater volume for potential interactions than that of a spherical particle. Previous theoretical and analogue experimental work on the rheology of suspensions of anisometric particles is less extensive than for spherical particles, and focusses primarily on dilute suspensions [e.g., Jeffery, 1922; Brenner, 1974; Powell, 1991; Pabst et al., 2006]. An exception is our earlier study [Mueller et al., 2010] in which we consider suspensions of anisometric particles with volume fractions spanning the range 0 ≤ ϕ ≲ ϕm; we found that the following relationship holds across that range:

equation image

and that ϕm varies with particle aspect ratio. Equation (3) is based on the widely-applied relationship of Maron and Pierce [1956] (with Kr substituted for ηr).

[6] For monodisperse spherical particles, the theoretical maximum packing fraction ϕm ≈ 0.74 for an ordered packing and ϕm ≈ 0.64 for a random (i.e., disordered) packing [Bernal and Mason, 1960]. The effect of particle shape on maximum packing fraction in the absence of hydrodynamic shearing has been investigated experimentally [e.g., Milewski, 1973; Parkhouse and Kelly, 1995; Rahli et al., 1999], geometrically [Evans and Gibson, 1986], and numerically [e.g., Williams and Philipse, 2003; Gan et al., 2004; Donev et al., 2004]. In general, these studies show that ϕm tends to decrease with increasing particle anisometry. There have been comparatively few studies of ϕm for anisometric particles in shearing flows; we are aware of just two, Kitano et al. [1981] and Pabst et al. [2006], who both infer ϕm from rheological measurements of sheared suspensions and find that ϕm decreases for increasingly prolate particles.

[7] In the present study we define the aspect ratio rp = la/lb, where la is the particle's axis of rotational symmetry and lb is its maximum diameter perpendicular to that axis. Various empirical and semi-empirical models have been proposed to describe the dependence of ϕm on rp: Evans and Gibson [1986], Rahli et al. [1999] and Mueller et al. [2010] propose inverse relationships between the two parameters; Parkhouse and Kelly [1995] propose a logarithmic relationship; and Kitano et al. [1981] and Pabst et al. [2006] suggest a linear decrease in ϕm with rp. In this work, we use high-resolution rheometry to investigate the relationship between rp and ϕm for shearing flows, and compare existing models with our experimental data.

2. Experiments

[8] Rheometric measurements were performed on suspensions of particles in silicone oil (Cannon Viscosity Standard N15000. A variety of particles were used to cover the range of aspect ratios 0.04 ≤ rp ≤ 22 (i.e., the range most relevant to magmatic suspensions): biotite and polyacrylic glitter (oblate); glass beads (spherical); glass and carbon fibres (prolate). Examples are shown in Figure 1. Suspensions were prepared by mixing a known mass of particles with a known mass of silicone oil. Samples were centrifuged to remove trapped gas bubbles, and particle volume fractions were calculated using the oil and particle densities, to an estimated accuracy of 3%. Table 1 lists the main characteristics of the particles used in this study.

Figure 1.

Examples of particles used in this study: (a) oblate polyacrylic glitter, (b) spherical glass beads, (c) prolate glass fibres. Length of scale bar is 1 mm.

Table 1. Particle Characteristics
ParticlesMaterialMean rpaσ (rp)bϕmc
  • a

    Determined from optical microscopy.

  • b

    σ (rp) is the standard deviation of the particles' aspect ratio.

  • c

    Obtained from equation (3) (see section 3 for description of procedure).

Btbiotite flakes0.040.0180.220
Gl1polyacrylic glitter0.140.0140.540
Gl2polyacrylic glitter0.160.0160.550
spheresglass spheres1.00-0.633
F1glass fibres2.501.120.573
F2glass fibres3.501.460.558
F3glass fibres5.752.770.538
F4glass fibres5.903.030.538
F5carbon fibres9.056.530.430
F6glass fibres10.65.740.404
F7glass fibres12.63.990.359
F8glass fibres22.07.360.323

[9] The rheology of the homogenized suspensions was determined using a ThermoHaake Mars II rotational rheometer, in parallel plate arrangement with 35 mm sensor diameter and 1.5 mm gap width. In order to avoid wall-slip effects with high viscosity samples, the surfaces of both the upper and lower sensor were modified by attaching sandpaper with grit size of the order of the particle sizes. Flow curves τ (equation image) were obtained by running a 20-step ‘up-ramp’ of incrementally increasing shear stress, followed by a 20-step ‘down ramp’. At each step the strain rate was recorded when equilibrium stress conditions were reached. The maximum shear stress was usually set between 300 and 500 Pa. In order to eliminate the transient effects observed during flow initiation by Mueller et al. [2010], each flow-curve determination was preceded by a pre-shear treatment in which the suspension was sheared to large strain at constant strain rate.

3. Results and Analysis

[10] Between four and eight flow-curve determinations were performed for each particle aspect ratio, at varying particle concentrations ϕ. We used the Herschel–Bulkley model (equation (1)) to fit the flow curve from each experiment. A full table of results is included in the auxiliary material. Measurements revealing yield stress behaviour were discarded, hence τ0 = 0 for all reported experiments. Our data show that shear thinning is weak (0.9 ≲ n ≤ 1) for all but the most strongly anisometric particles used, and that the degree of shear thinning is very sensitive to slight changes to the suspension composition; hence, no simple correlation between n, ϕ and rp could be found. By contrast, Kr and ϕ are strongly correlated and are plotted in Figure 2 for suspensions of prolate (Figure 2a) and oblate (Figure 2b) particles. The error in Kr is calculated by combining the standard error obtained from the fitting procedure with an estimate of the measurement error (3%). Following Mueller et al. [2010], we determine a value of ϕm for each aspect ratio, by fitting equation (3) to each Kr dataset. The fitting process was done on log10 (Kr) to avoid biasing the fit to large values of Kr.

Figure 2.

Relative consistency Kr versus particle volume fraction ϕ for suspensions of (a) prolate and (b) oblate particles. Kr (ϕ) has been fitted using the Maron–Pierce model (equation (3)) for each dataset (grey lines) to determine ϕm (values in grey boxes). Error bars are shown unless they are smaller than the symbol.

[11] The values of ϕm determined from this fitting procedure are listed in Table 1, and are plotted in Figure 3 against the mean aspect ratio of the suspended particles for each dataset. The data are approximately symmetrically-distributed around rp = 1 and are well-described by a log-Gaussian function:

equation image

where image is the maximum packing fraction for particles with rp = 1 and b is a fitting parameter (b2 is equivalent to the variance of the log-Gaussian function). For our data, the best fit is achieved for image = 0.656 and b = 1.08. This value of image is in excellent agreement with previously reported values for sheared suspensions of spheres [Rutgers, 1962]. Note that alternative functional forms were tried, including log-logistic and log-Cauchy functions, but these yielded worse fits to data.

Figure 3.

The maximum packing fraction of particles ϕm as a function of the particle aspect ratio rp. The data have been fitted with a log-Gaussian function (equation (4); R2 = 0.905). Error bars in rp correspond to 1 σ of the la-distribution of the particles, assuming constant lb. The errors in ϕm in Figure 2 are determined by fitting equation (3) to the data in Figure 2, but with Kr ± error in (Kr) substituted for Kr.

[12] Figure 3 also compares published ϕm(rp) models with our experimental results and with equation (4). Agreement between the data and previous models is generally very poor, and is restricted to small intervals of rp in the prolate range. Furthermore, most of the previous models produce unphysical results (such as ϕm > 1 or ϕm < 0) for certain aspect ratios. By contrast, the model that we propose is in good agreement with the data across a wide range of oblate and prolate aspect ratios, is continuous at rp = 1, and the predicted value of ϕm tends asymptotically towards zero for extreme aspect ratios.

4. Practical Applications

[13] Equations (3) and (4) can be combined so that suspension consistency can be computed as a function of particle aspect ratio and particle volume fraction; however, it is the apparent viscosity that is more commonly sought for practical applications. In the absence of yield stress, the consistency K is identical to apparent viscosity when n = 1 or equation image = 1 s−1 (equation (2)). In this case, the non-dimensionalization to form Kr is robust, indicating that our results scale to arbitrary melt viscosity. If the model is applied to situations in which both n ≠ 1 and equation image ≠ 1 s−1, the analogy between Kr and ηr is inexact and the true suspension viscosity can only be calculated if the value of flow index n is known (which it usually is not). From equation (2), we can see that the fractional difference between Kr and ηr is given by:

equation image

hence, for values of n and equation image close to unity, this error is small. Our experiments indicate that typically 0.9 < n ≤ 1 for ϕ/ϕm < 0.7 so the flow index only deviates substantially from unity at high particle volume fractions. Consequently, Kr can be used as a proxy for ηr at intermediate strain rates for all but the most concentrated particle suspensions. For suspensions that are only weakly shear thinning, consistency and apparent viscosity are approximately equivalent for all situations of practical interest.

[14] We now present a concrete example to illustrate the importance of accounting for particle-shape-dependent rheology when considering the fluid dynamics of crystal-bearing magma. Over the past few decades, numerical models of conduit flow have been used extensively to investigate the physical controls on eruption style [e.g., Sahagian, 2005]. Such models typically find that magma rheology exerts a first-order control. Following Llewellin and Manga [2005], we use the open-source conduit flow model CONFLOW [Mastin, 2002] as a test-bench to explore the impact of using a rheological model that accounts for crystal shape. We use the standard ‘Pinatubo white pumice’ parameters supplied with CONFLOW, and the fragmentation criterion of Papale [1999]. CONFLOW uses a version of the Einstein–Roscoe viscosity model (ηr = (1 − ϕ/ϕm)−2.5) to account for crystal volume fraction. We adapt the source code to use equation (3) in its place, introducing the assumption that consistency and viscosity are equivalent and limiting validity to ϕ/ϕm ≲ 0.7 as described above (note that computed strain rate is in the range 0.1 < equation image ≤ 2 for all simulations, hence the error in viscosity due to this assumption, given by equation (5), is never greater than 20% and is usually much smaller). Dependence of viscosity on aspect ratio is included through equation (4) for ϕm.

[15] Results, plotted in Figure 4, demonstrate the variations in predicted eruption parameters that arise as crystal aspect ratio is varied from rp = 1 (equant) to rp = 10 (highly prolate). The range of aspect ratios considered is representative of the range found in nature, considering both phenocryst and microlite phases: for phenocryst phases rp < 5 is typical [e.g., Mock and Jerram, 2005]; microlites are commonly found with 5 < rp < 15 [Castro et al., 2003; Couch et al., 2003].

Figure 4.

Gas volume fraction, pressure and vertical velocity against depth in a volcanic conduit, during eruption, calculated using CONFLOW [Mastin, 2002] for the standard ‘Pinatubo white pumice’ parameters. In each case ϕ = 0.3 and rp takes the values 1, 2, 5 and 10.

[16] Figure 4 shows that the predicted fragmentation depth (indicated by the inflection point in each curve) increases from ∼1.25 km for equant crystals to ∼2.6 km for the most prolate crystals. To achieve the same change without varying aspect ratio, the volume fraction of equant crystals would have to be increased from ϕ = 0.3 to ϕ = 0.47. Whilst we recognize that magma at depth is unlikely to carry significant volume fractions of very high aspect ratio crystals, it seems very likely that large vertical variations in average aspect ratio are common [Melnik and Sparks, 2005]. This modelling example is not, therefore, intended to represent any particular volcanic eruption, but to demonstrate the great sensitivity of eruption models to crystal aspect ratio through its impact on rheology. It is likely that strongly non-linear eruption models would show even greater sensitivity.

5. Conclusions

[17] We have performed laboratory analogue experiments to determine the rheology of monodisperse suspensions of particles for a range of particle aspect ratios from highly-oblate to highly-prolate. We have found that the impact of particle shape on rheology can be captured through its effect on the maximum packing fraction of particles ϕm. We find that the maximum fraction of particles that a suspension can hold before jamming is highest for equant particles, and decreases systematically for particles that are increasingly prolate or oblate; consequently, a suspension of equant particles will have a lower viscosity than a suspension with the same volume fraction of prolate or oblate particles.

[18] We have developed a method for calculating the viscosity of a particle suspension if particle volume fraction ϕ and aspect ratio rp are known: first, maximum packing fraction ϕm is calculated using equation (4); second, relative consistency Kr is calculated using equation (3). Relative consistency is approximately equivalent to relative apparent viscosity ηr for a suspension which shows only mild shear thinning; practically, this puts the following, conservative, constraints on the validity of our approach: 0.04 ≤ rp ≤ 22; ϕ/ϕm ≲ 0.7.

[19] Finally, we have used the practical example of the numerical modelling of a volcanic eruption to demonstrate the importance of accounting for crystal shape; in our example, changing crystal aspect ratio from rp = 1 to rp = 10 has a similar impact to changing particle volume fraction from ϕ = 0.3 to ϕ = 0.47. It is crucial, therefore, that modellers of volcanic and magmatic flows ask not only what volume fraction of crystals is in the flow, but also what shape the crystals have.

Acknowledgments

[20] S.M. is supported by NERC Research Fellowship NE/G014426/1. We thank M. Mangan and Y. Lavallee for their reviews, which helped to improve the manuscript, and L. Mastin for supplying the source code for CONFLOW and facilitating our modifications. Dataset F5 has been collected by C. Cimarelli.

[21] The Editor thanks Yan Lavallée and Margaret Mangan for their assistance in evaluating this paper.

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