Numerical Models of the Onset of Yield Strength in Crystal^Melt Suspensions

Earth and Planetary Science Letters 187 (2001) 367^379 www.elsevier.com/locate/epsl

Numerical models of the onset of yield strength in crystal^melt suspensions

Martin O. Saar *, Michael Manga, Katharine V. Cashman, Sean Fremouw

Department of Geological Sciences, University of Oregon, Eugene, OR 97403, USA

Received 15 November 2000; received in revised form 15 February 2001; accepted 19 February 2001

Abstract

The formation of a continuous crystal network in magmas and lavas can provide finite yield strength, dy, and can thus cause a change from Newtonian to Bingham rheology. The rheology of crystal^melt suspensions affects geological processes, such as ascent of magma through volcanic conduits, flow of lava across the Earth's surface, melt extraction from crystal mushes under compression, convection in magmatic bodies, and shear wave propagation through partial melting zones. Here, three-dimensional numerical models are used to investigate the onset of `static' yield strength in a zero-shear environment. Crystals are positioned randomly in space and can be approximated as convex polyhedra of any shape, size and orientation. We determine the critical crystal volume fraction, Pc, at which a crystal network first forms. The value of Pc is a function of object shape and orientation distribution, and decreases with increasing randomness in object orientation and increasing shape anisotropy. For example, while parallel-aligned convex objects yield Pc = 0.29, randomly oriented cubes exhibit a maximum Pc of 0.22. Approximations of plagioclase crystals as randomly oriented elongated and flattened prisms (tablets) with aspect ratios between 1:4:16 and 1:1:2 yield 0.08 6 Pc 6 0.20, respectively. The dependence of Pc on particle orientation implies that the flow regime and resulting particle ordering may affect the onset of yield strength. Pc in zero-shear environments is a lower bound for Pc. Finally the average total excluded volume is used, within its limitation of being a `quasi-invariant', to develop a scaling relation between dy and P for suspensions of different particle shapes. ß 2001 Elsevier Science B.V. All rights reserved.

Keywords: yield strength; crystals; lava £ows; £ow mechanism; rheology; magmas

1. Introduction tal-free in some volcanic eruptions, to melt-free in portions of the Earth's mantle. Moreover, the Magmas and lavas typically contain crystals, concentration of suspended crystals often changes and often bubbles, suspended in a liquid. These with time due to cooling or degassing, which crystal^melt suspensions vary considerably in causes crystallization, or due to heating, adiabatic crystal volume fraction, P, and range from crys- decompression, or hydration, which causes melting. Such variations in P can cause continuous or abrupt modi¢cations in rheological properties such as yield strength, viscosity, and £uid^solid * Corresponding author. Fax: +1-541-346-4692; transitions. Thus, the mechanical properties of E-mail: [email protected] geologic materials vary as a result of the large

EPSL 5805 26-4-01 Cyaan Magenta Geel Zwart 368 M.O. Saar et al. / Earth and Planetary Science Letters 187 (2001) 367^379 range of crystal fractions present in geologic sys- suspensions of di¡erent particle shapes. Our nu- tems. The rheology of crystal^melt suspensions merical models complement the experimental a¡ects geological processes, such as ascent of studies presented in a companion paper [10]. magma through volcanic conduits, £ow of lava across the Earth's surface, convection in magmatic reservoirs, and shear wave propagation 2. Rheology of magmatic suspensions through zones of partial melting. One example of a change in rheological proper- In this section we provide an overview of the ties that may in£uence a number of magmatic conceptual framework in which we interpret our processes is the onset of yield strength in a sus- results. Fig. 1 illustrates schematically the rela- pension once P exceeds some critical value, Pc. tionship between e¡ective shear viscosity, Weff , The onset of yield strength has been proposed yield strength, dy, and particle volume fraction, as a possible cause for morphological transitions P. We make a distinction between £uid and solid in surface textures of basaltic lava £ows [1]. Yield (regions A and B, respectively, in Fig. 1). The strength development may also be a necessary subcategories AP and AQ are used for suspensions condition for melt extraction from crystal mushes with dy = 0 (Newtonian) and dy s 0 (Bingham), re- under compression, for example in £ood basalts spectively. We assume here that the suspensions or in the partial melting zone beneath mid-ocean are not in£uenced by non-hydrodynamic (e.g. col- ridges [2]. Furthermore, an increase of viscosity loidal) forces, Brownian motion, or bubbles. First and development of yield strength in magmatic we discuss viscosity (region A) then yield strength suspensions may cause volcanic conduit plug for- (subregion AQ). mation and the transition from e¡usive to explosive volcanism [3]. Indeed, Philpotts et al. [2] state that the development of a load-bearing crystalline network is ``one of the most important steps in the solidi¢cation of magma''. In this paper we employ three-dimensional (3D) numerical models of crystal^melt suspensions to investigate the onset of yield strength, dy. Yield strength development due to vesicles [4,5] is not considered in this study. The objective is to under- stand the geometrical properties of the crystals in suspension that determine the critical crystal volume fraction, Pc, at which a crystal network ¢rst forms. Our simulations suggest that the onset of yield strength in crystal^melt suspensions may occur at crystal fractions that are lower than the 0.35^0.5 commonly assumed [6^8] in static (zero- shear) environments. We demonstrate that yield strength can develop at signi¢cantly lower P when crystals have high shape anisotropy and are randomly oriented, and that an upper bound Fig. 1. Sketch of the development of e¡ective shear viscosity, should be given by Pc = 0.29 for parallel-aligned Weff , and yield strength, dy, as a function of crystal fraction, objects. Because particle orientation is a function P. The critical crystal fractions Pc and Pm depend on the to- of the stress tensor, we expect increasing particle tal shear stress, d, and particle attributes, h, such as particle shape, size, and orientation distribution. A ¢rst minimum alignment with increasing shear stress and perfect yield strength develops at PcrPc(h, dy = 0) and increases with alignment in pure shear only [9]. Furthermore, we increasing P. The ¢elds A, AP, AQ, and B are explained in the suggest a scaling relation between dy and P for text.

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2.1. Fluid behavior: region A as on total applied stress, d, e.g. Pm = Pm(h, d). The dependence of Pc and Pm on d results from hydro- Einstein [11] found an analytical solution, dynamic forces that can break and orient the crys- Weff = Ws(1+2.5P), describing the e¡ective shear vis- tal network and transform it into a more ordered cosity, Weff , of a dilute (P 9 0.03) suspension (left state of denser packing. Here we de¢ne PcrPc(h, hand side of ¢eld A) of spheres for very low Rey- d = 0). As dy approaches zero, the minimum crit- nolds numbers (Stokes £ow). The viscosity of the ical crystal fraction, Pc, is obtained for a given h. suspending liquid is Ws. The particle concentra- The development of yield strength, dy, may thus tion, P, has to be su¤ciently low that hydrody- be described by: namic interactions of the particles can be ne- 1=p glected. d y P P =P c31= 13P =P m d co 2 At higher P (right hand side of ¢eld A) the so- called Einstein^Roscoe equation [12^14] is often where dco re£ects the total interparticulate cohe- employed to incorporate e¡ects of hydrodynami- sion resisting hydrodynamic forces and p may re- cally interacting, non-colloidal particles: £ect the response of the aggregate state to shear- ing [16^18]. 32:5 W r W eff =W s 13P =P m 1 In this study we investigate the in£uence of h on Pc as dC0, with the implication that the onset where Pm is the maximum packing fraction. The of zero-shear yield strength is a lower bound for form of Eq. 1 allows the relative viscosity Wr to the onset of yield strength in shear environments. diverge as PCPm. Eq. 1 is commonly used to Thus, Pc may be viewed as the lowest crystal vol- calculate the viscosity of magmas [7]. However, ume fraction at which dy could possibly form the value of Pm, which determines the transition under the given assumptions (e.g. no non-hydro- to a solid, is uncertain; suggested values range dynamic forces, bubbles, or Brownian motion). from Pm = 0.74 [13] to Pm = 0.60 [15]. Pm =0.74 corresponds to the maximum packing fraction of uniform spheres, so Pm might be expected to be 3. Methods di¡erent if non-spherical particles are considered [7,15]. An additional complication is the tendency In contrast to natural and analog experiments, of crystals to form aggregates or networks. Je¡rey computer models permit investigation of the for- and Acrivos [14] argue that a suspension that mation of a continuous crystal network at low P forms aggregates can be viewed as a suspension under static conditions (zero strain rate). As ex- of single particles of new shapes (and sizes) and perimental and in situ measurements of yield thus possessing di¡erent rheological properties. strength may disrupt the fragile network that ¢rst forms at Pc, it has been argued [6,19] that incor- 2.2. Onset and development of yield strength: rect extrapolation of stress versus strain rate mea- region AQ surements towards zero strain rate can lead to ¢ctitious yield strength values when in actuality Suspensions may have a range of yield the suspension is shear-thinning. Moreover, simu- strengths dy (subregion AQ) [16]. At Pc a ¢rst sam- lations allow the intergrowth of crystals, which is ple-spanning crystal network forms to provide important for systems where the crystal growth some minimum yield strength (dyC0). For P v Pc rate, G, is signi¢cantly larger than the shear yield strength increases with increasing P. The £u- rate, Q_ , as is the case in some natural systems id^solid transition occurs at the maximum pack- [2,20]. In this study we assume a zero-shear envi- ing fraction, Pm. ronment, thus: In general we expect Pc and Pm to depend on Q_ particle attributes (denoted h), such as particle ! 0 3 shape, size, and orientation distribution, as well G

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We employ continuum percolation models to study the possible development of dy as a function of P and h. Percolation theory describes the inter- connectivity of individual elements in disordered (random) systems and suggests a power law relationship of the form [21]: 0 P 6P c d yV R 4 P 3P c P vP c where Pc is the percolation threshold which is reached when crystals ¢rst form a continuous phase across the suspending £uid. The exponent Fig. 3. Critical crystal volume fraction, Pc, versus ratio of cube side length, l , over bounding box side length, l =1, R describes the development of dy for P v Pc close c b to P . Although phenomenological, we determine for parallel-aligned cubes. Inset: Pc versus number of resolu- c tion grains, a, per distance from the crystal center to the far- a geometrical percolation threshold, pc, and as- thest vertex, d, for randomly oriented cubes. Symbols are the sume that it is related to Pc [22]. With this ap- mean and shaded areas indicate standard deviations. Circle proach, we can investigate the dependence of pc (light gray area): lc/lb = 0.1; square (medium gray area): on crystal shape, size, and orientation distribution lc/lb = 0.05; diamond (dark gray area): lc/lb = 0.025. The total number of grains is a W(a/d)3, where for a cube d =(3l2)1=2. and draw conclusions about the e¡ects of these t c geometric properties on the development of yield strength. overlapping crystals of a ¢nite size may simulate The crystals in our simulations can be approxi- crystal intergrowth in a zero-shear environment. mated as convex polyhedra of any shape, size, Crystal orientation distribution is pre-assigned so and orientation distribution in 3D space. Crystals that, for example, parallel-aligned or randomly are positioned randomly and interpenetrate each oriented distributions can be simulated. Crystals other (soft-core continuum percolation). In soft- are positioned inside and outside a bounding unit core percolation the concept of maximum packing cube (Fig. 2b). To avoid ¢nite size e¡ects, the fraction, Pm, does not apply. Fig. 2a shows that maximum length of the largest crystal in a given simulation is never longer than 1/10, and typically 1/20 to 1/40, of the bounding cube side length. Fig. 3 shows the decrease in standard deviation and mean of Pc for decreasing particle side length for parallel-aligned cubes. We determine crystal overlaps analytically. Vol- ume fractions, P, are determined numerically by discretizing the bounding cube into subcubes (grains). The inset of Fig. 3 illustrates that while crystal size in£uences the standard deviation of Pc (gray shaded areas), `grain resolution', a, is rela- Fig. 2. (a) Simulation of crystal intergrowth. (I) Crystals tively uncritical for calculations of P. This insen- touch; (IIa) in natural systems under high growth rate to sitivity could be due to the angular crystal shapes shear rate ratios, touching crystals grow together; (IIb) in that may be approximated well by a few angular the computer model crystals overlap; (III) same end result. grains. We also determine P by using the number (b) Schematic 2D illustration of crystals forming clusters. of crystals per unit volume, n, and the volume of One cluster connects opposite sides of the bounding box and a crystal, v,inP =13exp(3nv) [23,24]. forms the backbone. It is necessary that some crystals are positioned outside the bounding box, so that they can pro- Crystals that overlap are part of a `cluster'. trude into it. Actual simulations are 3D (Fig. 4). Overlapping crystals of di¡erent clusters cause

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Fig. 4. Visualizations of a simulation of randomly oriented elongated rectangular biaxial soft-core prisms of aspect ratio 5:1, at the percolation threshold; (a) only backbone crystals, connecting four of the six cube faces, (b) all crystals, (c) all crystals that protrude out of the bounding cube. (d), (e), and (f) are magni¢cations of a region in (a), (b), and (c), respectively. The total crystal volume fraction in this simulation is Pc = 0.132 (backbone volume fraction: 0.016). The total number of crystals is 28 373 of which 3400 are part of the backbone. the two clusters to merge into one. The percola- low crystal numbers to con¢rm calculations of tion threshold is reached when a continuous crys- simulation parameters (Fig. 4). tal chain (hereafter referred to as the `backbone') exists, connecting one face of the bounding cube with the opposing face (Figs. 2b and 4a). A back- 4. Results bone in this study includes the `dead-end branches' which may not be considered part of In this section we present our simulation results the backbone in other studies [21] (Fig. 2b). and reserve interpretation for the discussion sec- All simulations are repeated at least 10 times to tion. Fig. 5 shows Pc for biaxial rectangular 32 determine a mean and a standard deviation of Pc. prisms with aspect ratios ranging from 10 to 2 0 Standard deviations of Pc are about 0.01. The 10 , where 10 indicates a cube, and negative typical number of crystals in each simulation and positive exponents indicate oblate and prolate ranges from 103 to 3U105, depending on particle prisms, respectively. All crystals are soft-core ob- attributes h. jects of uniform size and are oriented randomly. We test our computational method by running Standard deviations for Pc are shown with vertical simulations for which percolation theory results bars. A maximum Pc = 0.22 þ 0.01 is reached for are well-established [25^28], such as for parallel- cubes with decreasing values of Pc for less equant aligned cubes, where Pc = 0.29. Furthermore, we shapes. The dashed curve in Fig. 5 shows results use visualizations of crystal con¢gurations at from Garboczi et al. [22] for overlapping, ran-

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Fig. 5. Simulation results of Pc versus aspect ratio for randomly oriented biaxial soft-core rectangular prisms (solid curve, this study), compared with values for randomly oriented rotational soft-core ellipsoids (dashed curve) determined by [22]. Maximum values of Pc = 0.22 þ 0.01 and P = 0.285 are reached for the most equant shapes, i.e. for Fig. 6. Simulation results of Pc versus aspect ratio for ran- c domly oriented triaxial soft-core rectangular prisms. A maxi- cubes and spheres, respectively. The number of crystals per mum value of P = 0.22 þ 0.01 is reached for the most equant simulation ranges from 4U102 to 8U104 (extreme prolate c shape (cube). The gray shaded area indicates the range of as- case). The standard deviation is shown with vertical bars. pect ratios for typical tabular plagioclase crystals [2,10,35]. Two simulations, with 10 repetitions each, are performed for cubes, ¢rst with 401 þ 49 cubes and second with 2597 þ 128 cubes. Both simulations yield the same result, indicating that a large crystal is Vlarge =8Vsmall, where Vsmall is the the larger cube size is su¤ciently small to avoid ¢nite size ef- volume of a small crystal. For both parallel- fects. The large and small shaded areas represent experimental results from Hoover et al. [10] and Philpotts et al. [2], re- aligned (solid line) and randomly oriented (dashed spectively. line) crystals, Pc is invariant (to within the stan- domly placed and randomly oriented, rotational ellipsoids. The general form of the two curves in Fig. 5 is the same, but the curves are o¡set for the more equant shapes. Both curves converge at the extreme prolate and oblate limits. Results for triaxial soft-core rectangular prisms at random positions and orientations are shown in Fig. 6. Aspect ratios are given as short over medium and as long over medium axis for oblate and prolate prisms, respectively. Again, the uniaxial limit and thus most equant shape (cube) provides the maximum value of Pc = 0.22 þ 0.01. Deviation from an equant shape by either elonga- Fig. 7. Critical crystal volume fraction, Pc, versus occurrence tion or £attening causes a decrease in Pc, with the fraction of large crystals in a bimodal size distribution. Crys- largest combined decrease when all three axes tals are parallel-aligned (solid line) and randomly oriented have di¡erent lengths. (dashed line) soft-core cubes. Open circles and error bars in- The e¡ect of crystal size on the percolation dicate numerical crystal volume fraction calculations using a threshold was examined by simulations involving space discretization method. Also shown are calculations of crystal volume fraction using the number of crystals, n, and bimodal size distributions. Fig. 7 shows Pc versus the mean volume of the crystals, Vm,inP =13exp(3nVm) occurrence fraction of large crystals in a bimodal (¢lled squares). Error bars for the latter calculation of P are size distribution of soft-core cubes. The volume of comparable in size to the ones shown.

EPSL 5805 26-4-01 Cyaan Magenta Geel Zwart M.O. Saar et al. / Earth and Planetary Science Letters 187 (2001) 367^379 373 dard deviation) of bimodal size distribution. Fig. 7 5.1. Onset of yield strength also shows good agreement between calculations of crystal volume fraction using the dis- For randomly oriented biaxial soft-core prisms cretization approach (open circles) and using we obtain 0.01 6 Pc 6 0.22 for oblate crystals with P =13exp(3nVm) [23,24], where n is the number aspect ratios ranging from 0.01 to 1, and and Vm the mean volume of the crystals (solid 0.006 6 Pc 6 0.22 for prolate crystals with aspect squares). ratios of 100 to 1, respectively (Fig. 5). Deviation from the uniaxial shape, a sphere for ellipsoids, or a cube for rectangular prisms, leads to a decrease 5. Discussion of Pc (Fig. 5). The percolation threshold for spheres and parallel-aligned convex objects of The onset of yield strength, dy, can be related to any shape is PcW0.29 [25^28]. When anisotropic the formation of a continuous particle (or bubble) particles are not perfectly aligned, Pc depends on network that provides some resistance to applied the orientation distribution of the objects [33,34]. stress [6,16]. This particle network ¢rst forms at Randomly oriented cubes yield Pc = 0.22 þ 0.01 in the percolation threshold, Pc. No yield strength is our simulations, a result that supports the predic- expected to exist for crystal volume fractions of tions of Balberg et al. [33,34]. Thus, the onset of P 6 Pc. Transitions in magmatic processes con- yield strength is a function of both shape (Fig. 5) trolled by dy may thus not be expected to occur and the degree of randomness in the particle ori- before P has reached or exceeded Pc. Therefore, entation, with dy occurring at lower P for more the percolation threshold, Pc, may be a crucial elongated or £attened shapes, as well as for more parameter in understanding the occurrence of randomly oriented objects. While it is well-estab- transitions in magmatic £ow and emplacement lished that size heterogeneity of non-overlapping behavior. particles, for example in sediments, increases the Gue¨guen et al. [29] emphasize a necessary dis- maximum packing fraction, size heterogeneity of tinction between mechanical and transport perco- overlapping objects does not appear to in£uence lation properties. They suggest that the e¡ective Pc, as shown in Fig. 7. Therefore, onset of yield elastic moduli of a material that contains pores strength should not change with size variations of and cracks is explained by `mechanical percola- crystals. tion'. In contrast, elastic moduli for media that Philpotts et al. [2,20] observe a crystal network contain particles with bond-bending interparticle in the Holyoke £ood basalt at total crystal vol- forces are probably described by the same perco- ume fractions of about 0.25. Based on 3D imag- lation models that describe transport properties ing of the crystals using CT scan data, they con- (permeability, conductivity) [29,30]. Therefore, clude that plagioclase (aspect ratio 5:1) forms the while rheological properties at critical melt frac- crystal network, although it comprises only half tions [31] probably belong to mechanical perco- of the total crystal volume fraction (W0.13). Our lation that describes solid behavior, the networks results show that the formation of a continuous of crystals that form solid bonds, investigated in network of randomly oriented plagioclase crystals this study, may be a transport percolation prob- of aspect ratio 5:1 at PcW0.13 is expected on a lem. purely geometrical basis and thus show good In suspensions, dy may be created by friction, agreement with the experimental results (dashed lubrication forces, or electrostatic repulsion be- line at aspect ratio 5 in Fig. 5). Because the elon- tween individual particles [32]. In addition, crys- gated plagioclase crystals form a network at lower tal^melt suspensions may provide dy by solid con- Pc than the more equant (cube-like) pyroxenes, it nections of intergrown crystals. We expect the may be reasonable, as a ¢rst approximation, to latter to occur at lower P and provide larger dy model crystal network formation in this plagio- than friction. Therefore, we consider only the con- clase^pyroxene system with plagioclase crystals tribution of crystal network formation to dy. only.

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Typical plagioclase shapes may be approximated as triaxial polyhedra that are elongated and £attened (tablets). Fig. 6 shows that, for random orientations, the combined e¡ect of three independent axis lengths in rectangular prisms reduces Pc further with respect to biaxial prisms. The shaded area in Fig. 6 indicates the range of rectangular prism shapes that best approximate typical plagioclase crystals [2,10,35]. These plagioclase tablets of aspect ratios (short:medium:long) 1:4:16 to 1:1:2 yield 0.08 6 Pc 6 0.20, respectively, in our simulations. Therefore, under the condition of random crystal orientation, i.e. in a zero-shear environment, we expect typical plagioclase tablets to form a ¢rst fragile crystal network Fig. 8. Backscattered electron image of a pahoehoe sample from Hawaii close to the percolation threshold after partial at crystal volume fractions as low as 0.08^0.20, melting experiments [10]. An image-spanning crystal pathway where the particular values depend on the crystal (arrows) formed of plagioclase (black) and pyroxene (gray) aspect ratios. crystals can be observed. Our results also agree reasonably well with Hoover et al.'s [10] analog experiments with are about equal). For this aspect ratio and ran- prismatic ¢bers in corn syrup, in which dom orientation, neglecting clustering e¡ects visi- 0.10 6 Pc 6 0.20 for aspect ratios 3^4 (gray shaded ble in the experiments, our simulations suggest area in Fig. 5). However, the particles in the ex- PcW0.18 (Fig. 6). periment are non-overlapping, not soft-core as in Hoover et al. [10] also carried out melting ex- our simulations, and thus we expect some devia- periments with 'a'a samples from Lava Butte, OR, tion from the numerical results. In the same USA, containing mainly plagioclase, with only study, Hoover et al. [10] conduct partial melting minor pyroxene (volume fraction 6 0.05). The experiments with pahoehoe and 'a'a samples from plagioclase crystals show some local alignment Hawaii and Lava Butte, OR, USA, respectively. and exhibit a small dy for a plagioclase volume Partially melted pahoehoe samples with subequal fraction of 0.31 at 1142³C, where the sample amounts of plagioclase and pyroxene show some shape is preserved, but not at 0.26 at 1150³C, ¢nite yield strength, and thus the sample main- where the sample collapses. The plagioclase aspect tains its cubical shape, at volume fractions of ratio is about 1:2:5 for which our simulations 0.35 at a temperature of 1155³C. At 1160³C and suggest PcW0.16. However, as discussed above, volume fractions of 0.18, the sample collapses, alignment of crystals causes an increase of Pc, indicating that the yield strength dropped below where the upper bound Pc = 0.29 is reached for the total stress of about 5U102 Pa applied by parallel-aligned objects of any convex shape. gravitational forces. Backscattered electron im- The general trend of our simulations and other ages indicate that plagioclase and pyroxene form percolation threshold studies appears to be con- local clusters and a sample-spanning cluster net- sistent with experiments presented by Hoover et work at 1155³C (Fig. 8). Cluster formation sug- al. [10] and agrees well with experiments presented gests that crystal con¢gurations in the pahoehoe by Philpotts et al. [2,10] (Fig. 5). The drastic de- sample may be dominated by nucleation site ef- crease in Pc we observe with increasing particle fects, possibly due to rapid cooling [36]. If we take shape anisotropy and increasing randomness in the average of the reported median plagioclase particle orientation is consistent with other nu- aspect ratio of about 1:2:5 and the estimated me- merical, experimental, and theoretical studies dian pyroxene aspect ratio of 1:1:2, we obtain an [23,29,37^39] that investigate the formation of aspect ratio of about 1:1.5:3.5 (volume fractions continuous object networks.

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5.2. Scaling relation for dy(P) curves of di¡ering for systems containing interpenetrating objects of particle shapes, and other generalizations any convex shape. Eq. 6 and Bc = GVexf = 2.8 for spheres agree well with the established percolation To develop general rules that explain the de- threshold pc = 0.29 for soft-core spheres [27]. Soft- pendence of the geometrical percolation thresh- core parallel-aligned convex shapes also yield old, pc, on object geometries we consider two per- pc = 0.29 [28]. Because the ratio of v/Gvexf is con- colation theory concepts. First is the average stant for objects of di¡erent sizes and identical critical number of bonds per site at pc, Bc. Second shape and orientation distributions, pc may be is the excluded volume, vex, which is de¢ned as the expected to be invariant if only the size distribu- volume around an object in which the center of tion changes. This expectation is con¢rmed in our another such object cannot be placed without simulations (Fig. 7). overlap. If the objects have an orientation or Generalization of a relationship between pc, size distribution, the average excluded volume of GVexf, nc, and v to include variations in object an object is averaged over these distributions and shapes have been less successful. Garbozci et al. denoted by Gvexf and the average critical total ex- [22] investigate a number of possible shape func- cluded volume is given by GVexf = ncGvexf [33], tionals for randomly oriented soft-core rotational where nc is the number of soft-core particles at ellipsoids, including GVexf. They ¢nd that the percolation threshold. Balberg et al. [33] ncGvexfW1.5 and 3.0 for extremely prolate and found that Bc is equal to GVexf, i.e.: oblate rotational ellipsoids, respectively, which agrees reasonably well with the range of 1.4 (ran- Bc ncGvexf GV exf 5 domly oriented widthless sticks) to 2.8 (spheres) suggested by Balberg [34]. Similar results for the and suggested that GVexf is invariant for a given other shape functionals lead Garboczi et al. [22] shape and orientation distribution and thus inde- to conclude that simple shape functionals do not pendent of size distribution [34]. Values of GVexf produce invariants and thus cannot predict pc for are highest for spheres and parallel-aligned con- overlapping rotational ellipsoids. vex objects of any shape, where GVexf = 2.8, lowest Drory [41] also concluded that the total average for orthogonally aligned (macroscopically iso- excluded volume is not truly universal, but is in- tropic) widthless sticks, where GVexf = 0.7, and in- £uenced instead by the shape of objects ``in a termediate for randomly oriented cylinders, for complicated way''. However, Drory et al. [42] which GVexf = 1.4 [34]. In natural systems particles point out that GVexf ``seemed to be relatively in- can be oriented between random and parallel, de- sensitive to the shape of particles'' when com- pending on the shear stress tensor and particle pared with pc and that ``it was therefore consid- shapes [9,40], and thus we expect GVexf to fall ered an approximately universal quantity''. A within the bounds of 1.4 and 2.8. Therefore, more rigorous theoretical model is developed to GVexf probably varies by a factor of 2 in natural explain values of Bc = GVexf for interacting objects systems. of di¡erent shapes [41^43]. While this theory pre- Shante and Kirkpatrick [25] found dicts Bc well it does not provide an alternative to pc =13exp(3Bc/8) for parallel-aligned convex ob- Bc that would be a true invariant. jects in continuum percolation. Using Eq. 5 and Based on the above studies, we have to content Gvexf =8v for spheres (and parallel-aligned convex ourselves with GVexf as an approximate invariant objects in general), where v is the volume of the that varies by a factor of 2. Predictions of Pc to sphere, it is possible to make further generaliza- within a factor of 2 should thus be possible for a tions. At the percolation threshold, where n = nc, large range of particle shapes [38]. In geological B = Bc, and P = pc, Eq. 5 is substituted into applications, a factor of 2 uncertainty in Pc may P =13exp(3nv) [23,24] to yield: be too large to interpret some observations, but may still be a valuable constraint for developing pc 13exp 3Bcv=Gvexf 6 models and predictions.

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For the case of randomly oriented rods with hemispherical caps of length L+W and width W [33]:

3 2 2 Gvexf 4Z=3W 2ZW L Z=2WL 7

For LEW, Eq. 7 reduces to that of Onsager [44]:

2 GvexfW Z=2WL 8

Because the calculation of an average excluded volume for randomly oriented rectangular prisms 2 is di¤cult, we de¢ne a volume v* = (4/3)Zr1r2 that scales in a similar manner to Eq. 8. Here, r1 is the Fig. 10. Scaling of an experimental dy(P) curve for ¢bers of distance from the center to the closest edge and r2 aspect ratio 3.4:1 (open symbols along solid line) to the ex- is the distance to a vertex (Fig. 9). When we nor- perimental dy(P) curve for spheres (¢lled symbols); data are malize v* by the critical number of crystals per from [10]. Dashed lines indicate scaled curves for the ¢bers using the scaling constants 1.5 and 2.2. unit volume, nc, we obtain a quasi-invariant, ncv*. Between the asymptotic limits of large and small aspect ratios ncv* varies by about a factor object (here a sphere) by: of 3 (Fig. 9). The relatively small variations of n v* compared with P suggests that n v* may P g Gvexfg c c c P eq 9 be considered a reasonable estimate of a charac- 8V g teristic normalized volume. If nGvexf were truly invariant, and could be ap- where the subscript g denotes values for general plied to conditions of PgPc and non-overlapping particle shapes. For Eq. 9, we use P = nV (for (hard-core) particles, we could relate the volume hard-core particles). Similarly, for soft-core par- fraction for particles of general shape and volume, ticles, where P =13exp(3nV), we obtain the rela- V, to the volume fraction, Peq, of an equivalent tionship:

Gvexfg=8V g P eq 13 13P g 10

Because ncGvexf varies by a factor of about 2, scaling in Eqs. 9 and 10 is accurate to a factor of about 2. As an example, we use Eqs. 7 and 9, and L = 2.5W for hard-core ¢bers of mean aspect ratio 3.4:1 [10] to calculate that Peq = 1.5Pg.Wemay then use this scaling constant of 1.5 to scale ex- perimentally measured dy(P) curves for these ¢bers to the values of the equivalent object (sphere). The two curves shown in Fig. 10 can be super- imposed using Peq = 2.2Pg. The factor 2.2 is great- er than our estimated value of 1.5 by a factor of 1.5, and within the anticipated uncertainty range

2 of a factor of 2. Fig. 9. The volume v* = (4/3)Zr1r2 scales in a similar manner to the average excluded volume (Eq. 8). We normalize v*by An equivalent way of interpreting the experi- the number of particles, nc, at the percolation threshold. mental results is to determine the average ex-

EPSL 5805 26-4-01 Cyaan Magenta Geel Zwart M.O. Saar et al. / Earth and Planetary Science Letters 187 (2001) 367^379 377 cluded volume, Gvexf. Our result of Peq =1.5Pg im- dy. Our results indicate that Pc is a function of plies that Gvexf =12Vg for the hard-core ¢bers. We both shape and the degree of randomness in par- now use Gvexf =12Vg together with the computed ticle orientations. In general, the onset of dy number of crystals at the percolation threshold, should occur at lower crystal volume fractions nc. For biaxial randomly oriented prisms of aspect for larger shape anisotropy, as well as for more ratio 3:1 and a normalized particle volume of randomly oriented objects. 36 3 1.7U10 , simulations yield nc = 113U10 which Formation of a sample-spanning network of results in ncGvexf = 2.3. The average total excluded randomly oriented plagioclase crystals of aspect volume thus falls within the expected range of 1.4 ratio 5:1 is expected at PcW0.13 and thus con- to 2.8 for particle shape and orientation distribu- ¢rms experimental observations [2,20]. In general, tions in natural systems. under random orientations, for typical plagioclase aspect ratios ranging from 1:4:16 to 1:1:2 we 5.3. Implications expect 0.08 6 Pc 6 0.20, respectively. Randomly oriented crystals that have larger aspect ratios Possible implications of the development of may exhibit yield strength at even lower crystal yield strength in crystal^melt suspensions include volume fractions. Therefore, the development of transitions in surface textures of basaltic lava yield strength may occur at lower crystal volume £ows [1], melt extraction from crystal mushes fractions than the 0.35^0.5 commonly assumed [2], transitions from e¡usive to explosive volcan- [6^8], provided the crystals are anisotropic in ism [3], and propagation of shear waves through shape and exhibit random orientations, as can zones of partial melting [45]. Furthermore, the be the case in low-shear environments. characteristics of mineral textures and the spatial For overlapping (soft-core) particles, Pc in- distribution of ore deposits in komatiites are creases with the degree of alignment and reaches sometimes explained by £uid dynamical models a maximum of Pc = 0.29, the value for spheres, for that suggest post-emplacement convection parallel-aligned particles of any convex shape. [46,47]. Aggregate and network formation of den- This dependence of Pc on the particle orientation dritic olivine crystals in komatiites increase viscos- distribution suggests that the £ow regime, and ity and may provide yield strength, which would therefore the resulting particle ordering, of a sus- reduce the convective vigour and may potentially pension is an important parameter in de¢ning Pc suppress convection. In general, the development and hence the onset of yield strength. In contrast, of yield strength is likely to a¡ect convective pro- crystal size distributions are not expected to in£u- cesses in a variety of geologic settings including ence Pc unless crystal overlap is inhibited. The lava lakes, £ood basalts, and magma chambers. presented results of Pc in a zero-shear environ- Finally, loss of a rigid crystal framework may ment may be viewed as a lower bound for the be a necessary condition for some geologic pro- onset of yield strength in shear environments. cesses to occur. For example, diapiric ascent of Phenocryst volume fractions larger than 0.30 magma through the crust [48] or thorough mixing [50] and even 0.50 [15] have been reported for of partially solidi¢ed material into an intruding dikes. Our results suggest that a ¢rst minimum melt [49] may require loss of continuous crystal yield strength (dyC0) may develop at much lower networks. volume fractions and potentially impede magma £ow. Suppression of £ow under high-shear stresses, however, requires P larger than about 0.5 [51]. 6. Conclusions Thus, it should be emphasized that even for P s Pc, £ow can occur if stresses exceed the yield We employ numerical simulations for soft-core strength. (overlapping) rectangular prisms to determine the Finally, we con¢rm that the average total ex- critical crystal volume fraction, Pc, at which a cluded volume nGvexf is a quasi-invariant that suspension may develop a ¢nite yield strength, varies by only a factor of about 2 over a large

EPSL 5805 26-4-01 Cyaan Magenta Geel Zwart 378 M.O. Saar et al. / Earth and Planetary Science Letters 187 (2001) 367^379 range of shapes and we suggest that it provides a [9] M. Manga, Orientation distribution of microlites in obsi- reasonable means to de¢ne an equivalent volume dian, J. Volcanol. Geotherm. Res. 86 (1998) 107^115. [10] S. Hoover, K.V. Cashman, M. Manga, The yield strength fraction, Peq. Peq captures the characteristic par- of subliquidus basalts: experimental results, J. Volcanol. ticle geometry that determines onset of Pc and Geotherm. Res. (2001), in press. may also be applied to PgPc. Using Peq for all P [11] A. Einstein, Eine neue Bestimmung der Moleku«ldimensio- nen, Ann. Phys. 19 (1906) 289^306; English translation appears to allow scaling of experimental dy(P) curves [10] for suspensions of di¡erent particle in: Investigation on the Theory of Brownian Motion, Dover, New York, 1956. shapes to within a factor of 2. [12] R. Roscoe, Suspensions, in: J.J. Hermans (Ed.), Flow Properties of Disperse Systems, North Holland, Amster- dam, 1953, pp. 1^38. Acknowledgements [13] H.R. Shaw, Comments on viscosity, crystal settling, and convection in granitic magmas, Am. J. Sci. 263 (1965) 120^152. This work was supported by NSF (EAR [14] D.J. Je¡rey, A. Acrivos, The rheological properties of 0003303 and EAR 9902851), and a GSA student suspensions of rigid particles, AIChE J. 22 (1976) 417^ research grant. Acknowledgement is made to the 432. Donors of the Petroleum Research Fund, admin- [15] B. Marsh, On the crystallinity, probability of occurence, istered by the American Chemical Society, for and rheology of lava and magma, Contrib. Mineral. Pe- trol. 78 (1981) 85^98. partial support of this research. We also thank [16] Z.Q. Zhou, F. Tunman, G. Luo, P.H.T. Uhlherr, Yield John Conery and Wolfram Arnold for computa- stress and maximum packing fraction of concentrated sus- tional advice and Bruce Marsh, Ross Kerr, and pensions, Rheol. Acta 34 (1995) 544^561. Jean Vigneresse for their comments and careful [17] C.R. Wildemuth, M.C. Williams, Viscosity of suspensions reviews of the manuscript. James Kauahikaua modeled with a shear-dependent maximum packing fraction, Rheol. Acta 23 (1984) 627^635. and the Hawaiian Volcano Observatory are [18] C.R. Wildemuth, M.C. Williams, A new interpretation of thanked for their support.[SK] viscosity and yield stress in dense slurries: coal and other irregular particles, Rheol. Acta 24 (1985) 75^91. [19] H.A. Barnes, K. Walters, The yield strength myth, Rheol. References Acta 24 (1985) 323^326. [20] A.R. Philpotts, L.D. Dickson, The formation of plagio- [1] K.V. Cashman, C. Thornber, J.P. Kauahikaua, Cooling clase chains during convective transfer in basaltic magma, and crystallization of lava in open channels, and the tran- Nature 406 (2000) 59^61. sition of pahoehoe lava to 'a'a, Bull. Volcanol. 61 (1999) [21] D. Stau¡er, A. Aharony, Introduction to Percolation 306^323. Theory, Taylor and Francis, London, 1992. [2] A.R. Philpotts, J. Shi, C. Brustman, Role of plagioclase [22] E.J. Garboczi, K.A. Snyder, J.F. Douglas, M.F. Thorpe, crystal chains in the di¡erentiation of partly crystallized Geometrical percolation threshold of overlapping ellip- basaltic magma, Nature 395 (1998) 343^346. soids, Phys. Rev. E 52 (1995) 819^828. [3] D.B. Dingwell, Volcanic dilemma: £ow or blow?, Science [23] I. Balberg, Excluded-volume explanation of Archie's law, 273 (1996) 1054^1055. Phys. Rev. B 33 (1986) 3618^3620. [4] F.J. Ryerson, H.C. Weed, A.J. Piwinskii, Rheology of [24] E.J. Garboczi, M.F. Thorpe, M. De Vries, A.R. Day, subliquidus magmas, 1: picritic compositions, J. Geophys. Universal conductivity curve for a plane containing ran- Res. 93 (1988) 3421^3436. dom holes, Phys. Rev. A 43 (1991) 6473^6482. [5] D.J. Stein, F.J. Spera, Rheology and microstructure of [25] K.S. Shante, S. Kirkpatrick, An introduction to percola- magmatic emulsions: theory and experiments, J. Volca- tion theory, Adv. Phys. 20 (1971) 325^367. nol. Geotherm. Res. 49 (1992) 157^174. [26] A.S. Skal, B.I. Shklovskii, In£uence of the impurity con- [6] R.C. Kerr, J.R. Lister, The e¡ects of shape on crystal centration on the hopping conduction in semiconductors, settling and on the rheology of magmas, J. Geol. 99 Sov. Phys. Semicond. 7 (1974) 1058^1061. (1991) 457^467. [27] G.E. Pike, C.H. Seager, Percolation and conductivity: a [7] H. Pinkerton, R.J. Stevenson, Methods of determining the computer study. I, Phys. Rev. B 10 (1974) 1421^1434. rheological properties of magmas at sub-liquidus temper- [28] S.W. Haan, R. Zwanzig, Series expansion in a continuum atures, J. Volcanol. Geotherm. Res. 56 (1992) 108^120. percolation problem, J. Phys. A 10 (1977) 1547^1555. [8] A.-M. Lejeune, P. Richet, Rheology of crystal-bearing [29] Y. Gue¨guen, T. Chelidze, M. Le Ravalec, Microstruc- silicate melts: an experimental study at high viscosities, tures, percolation thresholds, and rock physical proper- J. Geophys. Res. 100 (1995) 4215^4229. ties, Tectonophys. 279 (1997) 23^35.

EPSL 5805 26-4-01 Cyaan Magenta Geel Zwart M.O. Saar et al. / Earth and Planetary Science Letters 187 (2001) 367^379 379

[30] M. Sahimi, S. Arabi, Mechanics of disorderd solids, II, [42] A. Drory, B. Berkowitz, G. Parisi, I. Balberg, Theory of Phys. Rev. B 47 (1993) 703^712. continuum percolation. III. Low-density expansion, Phys. [31] J. Renner, B. Evans, G. Hirth, On the rheologically crit- Rev. E 56 (1997) 1379^1395. ical melt fraction, Earth Planet. Sci. Lett. 181 (2000) 585^ [43] A. Drory, Theory of continuum percolation. II. Mean 594. ¢eld theory, Phys. Rev. E 54 (1996) 6003^6013. [32] R.G. Larson, The Structure and Rheology of Complex [44] L. Onsager, The e¡ects of shapes on the interactions of Fluids, Oxford University Press, Oxford, 1999. colloidal particles, Ann. N.Y. Acad. Sci. 51 (1949) 627^ [33] I. Balberg, C.H. Anderson, S. Alexander, N. Wagner, 659. Excluded volume and its relation to the onset of percola- [45] G.A. Barth, M.C. Kleinrock, R.T. Helz, The magma tion, Phys. Rev. B 30 (1984) 3933^3943. body at Kilauea Iki lava lake: potential insights into [34] I. Balberg, `Universal' percolation-threshold limits in the mid-ocean ridge magma chambers, J. Geophys. Res. 99 continuum, Phys. Rev. B 31 (1985) 4053^4055. (1994) 7199^7217. [35] T.L. Wright, R.T. Okamura, Cooling and Crystallization [46] H.E. Huppert, R.S.J. Sparks, J.S. Turner, N.T. Arndt, of Tholeiitic Basalt, 1965 Makaopuhi Lava Lake, Hawaii, Emplacement and cooling of komatiite lavas, Nature US Geological Survey Prof. Paper 1004, 1977. 309 (1984) 19^22. [36] L. Keszthelyi, R. Denlinger, The initial cooling of pahoe- [47] J.S. Turner, H.E. Huppert, R.S.J. Sparks, Komatiites II: hoe £ow lobes, Bull. Volcanol. 58 (1996) 5^18. experimental and theoretical investigations of post-em- [37] S.H. Munson-McGee, Estimation of the critical concen- placement cooling and crystallization, J. Petrol. 27 tration in an anisotropic percolation network, Phys. Rev. (1986) 397^437. B 43 (1991) 3331^3336. [48] B.D. Marsh, On the mechanism of igneous diapirism, [38] A. Celzard, E. McRae, C. Deleuze, M. Dufort, G. Furdin, stoping, and zone melting, Am. J. Sci. 282 (1982) 808^ J.F. Mareªche¨, Critical concentration in percolating sys- 855. tems containing a high-aspect-ratio ¢ller, Phys. Rev. B [49] M.D. Murphy, R.S.J. Sparks, J. Barclay, M.R. Carroll, 53 (1996) 6209^6214. T.S. Brewer, Remobilization of andesite magma by in- [39] M.O. Saar, M. Manga, Permeability^porosity relationship trusion of ma¢c magma at the Soufriere Hills volcano, in vesicular basalts, Geophys. Res. Lett. 26 (1999) 111^ Montserrat, West Indies, J. Petrol. 41 (2000) 21^42. 114. [50] Y. Wada, On the relationship between dike width and [40] Y. Wada, Magma £ow directions inferred from preferred magma viscosity, J. Geophys. Res. 99 (1994) 17,743^ orientations of phenocrysts in a composite feeder dike, 17,755. Mikaye-Jima, Japan, J. Volcanol. Geotherm. Res. 49 [51] R.C. Kerr, J.R. Lister, Comment on ``On the relationship (1992) 119^126. between dike width and magma viscosity'' by Yutaka [41] A. Drory, Theory of continuum percolation. I. General Wada, J. Geophys. Res. 100 (1995) 15541. formalism, Phys. Rev. E 54 (1996) 5992^6002.

EPSL 5805 26-4-01 Cyaan Magenta Geel Zwart