Tutorials and Reviews

International Journal of Bifurcation and Chaos, Vol. 9, No. 8 (1999) 1467–1484c© World Scientific Publishing Company

MIXING OF GRANULAR MATERIALS:A TEST-BED DYNAMICAL SYSTEM

FOR PATTERN FORMATION

K. M. HILL, J. F. GILCHRIST and J. M. OTTINO∗

Department of Chemical Engineering,Northwestern University, Evanston, IL 60208, USA

D. V. KHAKHARDepartment of Chemical Engineering,

Indian Institute of Technology Bombay,Powai, Mumbai, 400076, India

J. J. McCARTHYDepartment of Chemical and Petroleum Engineering,

University of Pittsburgh, PA 15261, USA

Received April 1, 1999

Mixing of granular materials provides fascinating examples of pattern formation and self-organization. More mixing action — for example, increasing the forcing with more vigorousshaking or faster tumbling — does not guarantee a better-mixed final system. This is becausegranular mixtures of just barely different materials segregate according to density and size; infact, the very same forcing used to mix may unmix. Self-organization results from two com-peting effects: chaotic advection or chaotic mixing, as in the case of fluids, and flow-inducedsegregation, a phenomenon without parallel in fluids. The rich array of behaviors is ideallysuited for nonlinear-dynamics-based inspection. Moreover, the interplay with experiments isimmediate. In fact, these systems may constitute the simplest example of coexistence betweenchaos and self-organization that can be studied in the laboratory. We present a concise summaryof the necessary theoretical background and central physical ideas accompanied by illustrativeexperimental results to aid the reader in exploring this fascinating new area.

1. Introduction

Granular materials display a rich variety of dy-namical phenomena that have far-reaching impli-cations for industrial and natural processes. Oneof these phenomena is mixing, especially when itis tied to its counterpart, flow-induced segregation.Industrial examples of mixing and segregation ap-pear in the pharmaceutical, food, chemical, ce-ramic, metallurgical, and construction industries[Bridgwater, 1995; Fan et al., 1990; Nienow et al.,

1985]. However, the understanding of the funda-mentals of granular mixing remains incomplete, es-pecially when compared with that of fluid mixing[Ottino, 1989a]. The picture is changing though.Granular materials have attracted considerable re-cent attention in the physics community [Jaeger &Nagel, 1992; Behringer, 1993; Jaeger et al., 1996a,1996b]. Notably for granular mixing is that chaos-inspired ideas — which have been central in ad-vancing the understanding of fluid mixing [Aref,

∗E-mail: ottino@chem-eng.nwu.edu

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1468 K. M. Hill et al.

Fig. 1. The result of flow-induced segregation of mixtures of different-sized [(a) and (b)] and different density (c) granularmaterials placed in a rotating cylinder. The system quickly evolves, in a revolution or so, from a mixed state (a) to asegregated state [(b) and (c)]. (d) shows the equilibrium segregation structure corresponding to a mixture of particles withdifferent densities using the computational model described in the text.

1990; Ottino, 1990] — have been found applicableto granular mixing as well [Khakhar et al., 1999a].Granular materials, however, introduce additionaldifficulties. The main difference among these isflow-induced segregation (for a general review see[Rosato, 1999]). Often, granular systems evolvequickly through complex dynamics into a state ofself-organization. For example, in a 2D (short) tum-bled cylinder this may lead to radial segregation,a segregated core of smaller or denser particles, asshown in Fig. 1 [Ristow, 1994; Cantalaub & Bideau,1995; Clement et al., 1995; Khakhar et al., 1997b;Dury & Ristow, 1997]. In long cylinders axial band-ing may follow, according to certain particle prop-erties [Fig. 2] [Nakagawa, 1994; Hill & Kakalios,1995]. The 2D case is the focus of this paper.

Recently it was shown that chaotic advection,which can be used as a tool to improve fluid mix-ing, may also improve granular mixing [Khakharet al., 1999a]. This concept was investigated in sys-tems of granular materials in rotating pseudo 2Ddrum mixers of different shapes. Consider the sys-tem depicted in Fig. 3. Under suitable conditions —easy to achieve in the laboratory — the flow of non-cohesive granular materials in 2D rotating circularcontainers achieves a continuous flow, the so-calledrolling regime. A sketch of the flow is shown inFig. 3. The flow is confined to the top free surfacein the form of a thin shear-like layer whereas the restof the material moves in solid-like rotation with themixer walls. Material is fed into the flowing layerfor x < 0 and leaves the layer for x > 0. When the

Mixing of Granular Materials 1469

Fig. 2. Axial segregation of different-sized particles in a long rotating drum mixer.

Fig. 3. Schematic view of the continuous flow regime in a rotating cylinder. The dotted curve denotes the interface betweenthe continuously flowing layer and the region of solid body rotation. The mixer is rotated with angular velocity, ω, and thevelocity profile within the layer, vx, is nearly simple shear. The vy component is not shown. A typical particle trajectory isdepicted.

1470 K. M. Hill et al.

flow is steady (i.e. it is not a function of time) asis the case for a circular mixer the “streamlines” orcirculation paths are closed and time-invariant. Atwo-dimensional flow can be derived from a stream-function Ψ, such that vx = ∂Ψ/∂y, vy = −∂Ψ/∂xwhere Ψ = Ψ(x, y). The structure of the steadyflow is Hamiltonian with one-degree of freedom, andtherefore it cannot be chaotic (see [Ottino, 1990,pp. 215, 216]). However, if the container is non-circular the situation is very different; the flowinglayer grows and shrinks in time with a frequencyno less than twice that of the mixer rotation, andΨ = Ψ(x, y, t). Adding time-periodicity — the sys-tem is now referred to as having one-and-a-half de-grees of freedom — makes chaos possible (for de-tails see [Ottino, 1989]). Thus, if chaotic advectionimproves mixing, noncircular mixers should mixmore efficiently than circular mixers. Experimen-tal studies using colored tracer particles in other-wise mono-disperse granular materials confirm thatincreased mixing rates occur in noncircular contain-ers [Khakhar et al., 1999a].

Differences in particle properties make the sys-tem’s behavior even more interesting from a dy-namical systems viewpoint. Small differences in sizeand/or density of the grains of the granular mate-rial lead to flow-induced demixing or segregation, aphenomenon without parallel in fluid systems.Granular flow is almost invariably accompanied bysegregation, a major impediment to mixing that in-teracts nontrivially with chaotic advection.

Much of the interesting dynamics for segrega-tion and mixing of granular materials can be tracedback to the flowing layer; it is only here that thegrains are free to move relative to their neighbors.For the rest of the bed the grains are locked intoposition relative to each other in solid body rota-tion until they are fed into the layer. However, ad-ditional dynamical details governed by the shapeof the container are also important, as will be dis-cussed below. Consider now a model that can beused to investigate the behavior of these kinds ofsystems.

2. The Basic Model

Figure 3 depicts the essential details of a model de-veloped by Khakhar et al. [1997a]. This modeldescribes the flow of identical particles in the flowregime where the flowing layer is steady and thin,and the free surface is nearly flat. The particlesoutside of this layer move in solid body rotation.

Metcalfe et al. [1995] addressed the case of slowerflow, where this flow becomes time periodic andconsists of discrete avalanches; the case correspond-ing to cohesive particles, leading also to a time-periodic flow is considered by Shinbrot et al. [1999].For the purposes of this tutorial description, we fo-cus solely on the regime corresponding to steadyflow.

Our initial remarks are restricted to a half-full circular container. In this case the free sur-face has length 2L (constant) and the flowing layerhas thickness δ(x) that varies along the length butwhich is otherwise independent of time. A theoreti-cal analysis indicates that δ(x) can be approximatedby a parabola [Khakhar et al., 1997a]

δ = δ0(1− (x/L)2) , (1)

where δ0 is the thickness at x = 0. The velocity pro-file within the flowing layer can be approximated by

dx

dt= vx = 2u(1 + y/δ) (2)

dy

dt= vy = −ωx(y/δ)2 (3)

where u is the average velocity at the center of theflowing layer and is given by u = ωL2/(2δ0). Thevy-component [Eq. (3)] is chosen to satisfy massconservation. This model captures the essence offlow in the layer. Additional computations indicatethat different velocity fields, for example vx ∼ y3/2,vy ∼ xy5/2, give essentially the same mixing pat-terns, demonstrating that macroscopic geometricaleffects (i.e. the shape of the container) control theimportant details of the physics.

This model, coupled with solid-body rotationin the bed, is sufficient to compute a Poincare sec-tion for the circular mixer, shown in Fig. 4(a). As isstandard in fluid mechanics, the flow is interpretedin a continuum sense and collisional diffusion — therandom-like motion of individual particles as theycollide with other particles — is ignored in the par-ticle trajectory calculations. The plot is trivial forthe circular mixer; the system is regular; particlescan cross streamlines only by collisional diffusion,and the system exhibits slow radial mixing.

To adapt the model for noncircular mixers, oneneeds only to make L time dependent thus chang-ing the model from a 2D model in x and y to a3D model in x, y and t. This transforms the equa-tions for δ(x), vx and vy, to interact nonlinearly.These equations, without further modification, are

Mixing of Granular Materials 1471

adequate for mixers that are rotationally 180◦ sym-metric providing a stationary origin and convexboundaries to preserve the layer shape. For exam-ple, the dimensionless length of the flowing layerL(t) for a square mixer is expressed as

L(t) =

1

| cos θ| if θ <π

4or |π − θ| < π

4or θ <

7π

4

1

| sin θ| otherwise

(4)

Consider one final element to complete the model.Experiments show that the layer maintains geo-metric similarity, so that δ0(t)/L(t) is a constant

[Khakhar, 1997a]. This implies that u ∼ L(t); thusthe speed u changes with mixer orientation; thelonger the layer, the faster and deeper the flow.

3. Advection and Poincare Sections

Figures 4(b) and 4(c) show the Poincare sec-tions for a half-filled elliptical and square mix-ers respectively, illustrating both regular regions(KAM islands) where particles can be trappednear elliptic points (marked in red) and also re-gions where chaotic trajectories exist near hyper-bolic points. The Poincare sections shown weregenerated by plotting the location of advected

(a) (b)

(c)

Fig. 4. Poincare sections corresponding to (a) a half-filled circle, (b) ellipse and (c) square. The positions of the particles aremarked after every half revolution for the circle and ellipse and every quarter revolution for the square.

1472 K. M. Hill et al.

particles every half rotation of the mixer, exceptthose for the square which were plotted every quar-ter rotation. Figures 5(a) and 5(b) show thePoincare sections for the square mixer when filled5% less and 5% greater than half, respectively, il-lustrating that the degree of filling is critical. Themixer is no longer 180◦ rotationally symmetric fromthe perspective of the rotating particles. The cir-culation time in solid body rotation increases withfill level, and thus the circulation of the particlesis no longer synchronous with the rotation of themixer. We focus primarily on the half-full case forpurposes of this tutorial.

4. Collisional Diffusion

Consider now the addition of collisional diffusion.Savage [1993] proposed the following scaling rela-tion for the diffusion coefficient based on hard par-ticle dynamics simulations

Dcoll = f(η)d2 dvxdy

, (5)

where dvx/dy is the velocity gradient across thelayer, and d is the particle diameter. The prefactorf(η) is a function of the solids volume fraction, η;in our simulation we take f = 0.025.

How does collisional diffusion enter into themixing picture? Consider what happens to a groupof localized tracer particles in the flowing layer. Fig-ure 6 shows a computational simulation of the timeevolution of a blob of tracer particles during a typ-ical mixing experiment. The left column shows theevolution without collisional diffusion, and the rightcolumn includes diffusion. Thus, in a typical exper-iment the blob is deformed into a filament by theshear flow and blurred by collisional diffusion un-til particles exit the layer. In mixers with circularcross-sections, this is the only form of mixing. Par-ticles then execute a solid body rotation in the bed,re-enter the layer, and the process repeats.

The diffusional mixing process can be simulatedby adding a noise term to the particle advectionEqs. (2) and (3) to mimic diffusion, a Lagrangianapproach. Thus the dynamical system is:

dx

dt= vx (6)

dy

dt= vy + S (7)

where S is a white noise term which upon inte-gration over a time interval (∆t) gives a Gaussian

< 1/2 full > 1/2 full

Fig. 5. Illustration of the variation of the Poincare sections for the square when the filling level is changed about 50%. Thelocations of the elliptic and hyperbolic points, are very sensitive to the degree of filling about the half-full level. (a) When thefill level drops to 45%, both the hyperbolic points and the elliptical points move significantly away from the center of rotationradially, and there is a string of high period islands surrounding the elliptical points. By contrast, (b) when the fill levelis increased to 55%, both the hyperbolic point and elliptical points approach the center and the overall circulation patternbecomes dominated by unbroken tori (not unlike that for the circular mixer).

Mixing of Granular Materials 1473

Fig. 6. Computational results depicting the time evolution of a “blob” of tracer particles as it moves through the flowinglayer. The left-hand column shows the effect of advection; the right-hand column includes collisional diffusion.

(a) (b) (c)

Fig. 7. Computational results for a system of unmixed particles differing only in color. The initial conditions are shown inthe inset of (a), and both advection and collisional diffusion are included during the calculation of the mixing process. Theintensity of segregation is shown for large systems PeL = 104 and small systems PeS = 102. The states of the larger systemsafter 20 rotations are shown in Figs. 7(b) and 7(c).

random number with variance 2Dcoll∆t. Note thatthe diffusional effect is incorporated only in the y-direction. Diffusion along the layer (x-direction) isneglected, since diffusional effects are masked byconvection (i.e. particles move much faster thanthey diffuse). This can be put on a quantitativebasis by computing the Peclet number, a dimen-sionless ratio that measures the importance of con-vection versus diffusion. The Peclet number forflow in the x direction is Pe = uL/Dcoll, rangingfrom 102 in laboratory mixers to 104 in industrialmixers, whereas the Peclet number in the y di-rection is much smaller (by a factor of (δ0/L)2 ∼

0.0025). At this stage the model gives us a com-plete model of mixing of monodisperse sphericalparticles in a mixer of noncircular, 180◦ symmet-ric cross-sectional shape.

While diffusion works approximately the samefor all mixers, the combination of diffusion with ad-vection introduced by unsteady flow increases mix-ing efficiency as shown in Fig. 7. For two systems ofidentical particles differing only in color with initialunmixed conditions shown in the inset of Fig. 7(a),the mixing rate is significantly higher for the squaremixer than the circular mixer as shown in thegraph in Fig. 7(a). The state of the systems after

1474 K. M. Hill et al.

20 rotations are shown in Figs. 7(b) and 7(c) to fur-ther illustrate the difference in the mixing efficiencyof the two mixers.

5. Segregation

Consider now the effects of segregation. Granularmixtures of even slightly dissimilar materials willsegregate in many situations, including when theyare shaken or tumbled. One of the better-knownexamples of this behavior is the “Brazil-nut” effect[Williams, 1963; Rosato et al., 1987], where largeparticles rise to the top of a shaken container ofmixed nuts. This may be due to a “shadowing”effect. That is, while all particles are jostled upduring a shake, there is a greater probability thata space will be made for a smaller particle to slipbelow a larger one than vice versa. In this way thesmaller particles gradually work their way down tothe bottom of the container, and the larger particlesdrift up. In certain situations, this segregation ef-fect has been shown to be driven by convective mo-tion in the granular materials [Ehrichs et al., 1995].

Another example of segregation involves seg-regation driven by the flow of granular materials.When granular mixtures are poured in a thin spacebetween two vertical walls (like a Hele–Shaw cell),the components may separate into stratified lay-ers. The pattern resembles the stratification pat-tern found in sedimentary rocks and may be re-sponsible for this phenomenon in certain situations[Makse et al., 1997; Koeppe et al., 1998]. Anotherexample of flow-induced segregation is radial segre-gation in a rotating drum mixer, where the differentcomponents separate in the direction perpendicularto the axis of rotation [shown in Figs. 1(b)–1(d)]. Interms of the streamfunction, ∇Ψ, and the gradientof concentration, ∇c, are collinear. Thus, a radi-ally segregated structure, in a circle, is an invariantstructure.

For a physical picture of flow-induced segrega-tion, consider a mixture of particles of different sizesin a steady chute flow. As first observed over a cen-tury ago [Reynolds, 1895], when granular materialsflow, they dilate and as the material dilates, voidsare created. Small particles can squeeze into smallvoids below a large particle, but the reverse is muchless likely to occur resulting in a net segregating fluxof the smaller particles downward, away from thefree surface. The experimental work of Nityanandet al. [1986] illustrates the typical behavior of sys-tems with size segregation. Similarly, when mix-

tures of particles differing in density move througha shear layer, the denser particles are more likelyto sink into a lower layer. This will be described inmore detail below.

In long rotating cylinders, radial segregationis often followed by axial segregation [Donald &Roseman, 1962; Das Gupta et al., 1991; Hill &Kakalios, 1995]. After continued rotation of granu-lar materials in a long mixer, certain mixtures willfurther segregate into relatively pure, single compo-nent bands along the axis of rotation. [See Fig. 2]Here the mechanism for the case of different sizeparticles is believed to be due to differences in an-gles of repose of the two materials which producesmall differential axial flows for the two materials[Das Gupta et al., 1991; Hill & Kakalios, 1995].This is a more complex phenomenon than simpleradial segregation. Some mixtures do not axiallysegregate, whereas other systems of granular mate-rials exhibit reversible phenomena when they seg-regate at higher speeds and remix at lower speedsof rotation. Some mixtures evolve from the simpleinitial banding pattern into further pattern devel-opment. This is discussed in more detail in otherworks [Hill et al., 1997; Choo et al., 1998]. Many ofthese issues remain imperfectly understood. Radialsegregation in the cross-section of the circular mixermay play a role in the axial segregation, and thusthe cross-sectional shape of the container may havea strong influence in the results [Hill et al., 1999].

How is segregation added to the advection pic-ture? A model for segregation based on densitydifference can be based on an “effective buoyancyforce” [Khakhar et al., 1997b]. The effects of seg-regation can be accounted for in terms of drift ve-locities with respect to the mean mass velocity. Asusual the effects are significant only in the directionnormal to the flow. The segregation velocity for themore dense particles (labeled “1”) can be written as

vy1 = −2β(1 − ρ)Dcoll(1− f)

d(8)

and for the less dense particles (labeled “2”) as

vy2 =2β(1 − ρ)Dcollf

d(9)

Here, as before, Dcoll, is the collisional diffusion,β is the so-called dimensionless segregation veloc-ity which is determined by fitting Monte Carlo and“soft-particle” computations; f is the number frac-tion of the more dense particles, ρ < 1 is the

Mixing of Granular Materials 1475

density ratio, and d is the particle diameter. Simi-lar expressions can be obtained for the case of sys-tems differing in size, at low solids volume fractions[Khakhar et al., 1999b]. In our simulations we takeβ = 2.

To add this element into the advection model,assume first that the mean flow is still the same as ifall particles were identical, so that Eqs. (6) and (7)still apply. Note also that the effects of segregation,as well as those of collisional diffusivity describedearlier, are significant only in the direction normalto the flow (apparent, again, when the Peclet num-bers in each direction are considered) so Eq. (6)remains unchanged. Thus to compute the effect ofsegregation on the earlier models for the circulationpatterns in the different mixers, one needs only torewrite Eq. (7) with the segregation terms, Eqs. (8)and (9). Substituting the form for vy from Eq. (3),Eq. (8) takes the following form for the more denseparticles:

dy1

dt= −ωx

(y1

δ

)2

+ S − 2β(1 − ρ)Dcoll(1− f)

d,

(10)

and for the less dense particles, Eq. (9) becomes:

dy2

dt= −ωx

(y2

δ

)2

+ S +2β(1 − ρ)Dcollf

d. (11)

Thus we have a complete model for mixing and seg-regation for both circular and noncircular rotating2D drum mixers. Results are presented in Figs. 8(g)and 8(h), and will be discussed in Sec. 7. The casecorresponding to size segregation is more complex[Khakhar et al., 1999b].

6. Experiments

Typical experiments in this area are fast, inexpen-sive and reproducible. They can be conducted us-ing a variety of noncohesive particles from seeds tospherical beads. We caution the reader that whenthe particle size is less than 1 mm in diameter, ef-fects such as clumping due to moisture or attractiveand repulsive forces due to static electricity becomemore prominent. Thus, it is recommended thatbead sizes not smaller than 0.5 mm in diameter areused. For reducing the effects of static electricity,antistatic spray is also recommended. The particleswe used for the results presented below are sphericalbeads (Quackenbush Co.) with sizes ranging from0.8 to 2 mm and densities of 2.5 and 7.8 g/cm3 (glass

and steel, respectively). To minimize axial segrega-tion effects, it is recommended that the depth of thetumblers used be limited to a few particles, say 5–7.The design of the mixers is simple. The outer shapemay be cut from anything from thick poster-boardto Plexiglas and sandwiched between two plates.The results presented in this paper were obtainedusing three different quasi-2D mixers: a circularmixer, an elliptical mixer, and a square mixer, all ofapproximately the same depth (∼ 6 mm) and sur-face area (∼ 600 cm2). The faceplate of the mix-ers is made of Plexiglas for ease in observation anddata acquisition while the rear plate is fashioned ofaluminum and can be grounded to minimize elec-trostatic effects.

Two types of experiments are possible forvisualizing the chaotic flow, one that is more diffi-cult, another that is surprisingly easy and efficient.In the first case, one might attempt to capture thecirculation patterns in a mixer by seeding the bulkgranular materials with, for example, darker butotherwise identical particles in certain regions of themixer. This is what was done for the blob exper-iment shown in Fig. 9. In previous work we haveshown this to be useful in determining the mixingefficiency for the systems of identical particles. Onemight also use this type of experiment to depictthe general mixing pattern, although, in generalit is not effective in identifying the details of thePoincare sections, or even distinguishing regular is-lands from the chaotic regions. Collisional diffusioneffects in laboratory-size experiments are large andblur the details. Furthermore, this experiment isdifficult to execute and its effectiveness is stronglydependent on the initial placement of the tracerparticles.

On the other hand, segregation experiments areeasy to set up. In general, for these experiments, westart with a (roughly) well-mixed system of gran-ular materials in the mixer and rotate the mixerat a constant rate until an equilibrium state is ob-tained for a specific orientation of the mixer. (Thecomparison should be made at equal angles of ori-entation; the pattern becomes a periodic functionof time with a frequency no less than twice that ofthe mixer rotation.) The equilibrium segregationstructure obtained is independent of the initial ar-rangement of the particles in the mixer and stronglyresembles the Poincare section. As we are primar-ily interested in visualizing the rough details of thePoincare section, only the latter (segregation) ex-periments are discussed here.

Fig. 8. Results from segregation experiments for quasi two-dimensional tumbling mixers filled halfway with mixtures of granular materials. “E” denotes experimentalresults and “C” denotes computational results. (Compare with the segregation results for the circular mixer in Fig. 1 and the Poincare sections in Fig. 3.) Images aretaken while the mixer is rotated, though the images of the square are rotated counter-clockwise by ∼ 30◦ to maximize the use of space. The competition between mixingand segregation was studied using ternary mixtures of particles differing in size (0.8 mm blue, 1.2 mm clear, and 2.0 mm red glass spheres) and binary mixtures ofparticles differing only in density (2 mm glass and steel spheres). The volume fraction of dense beads in the glass and steel bead system is 0.25, and the volume fractionof the smaller beads used in the mixture of different-sized glass beads is 0.25 — equal fractions of medium and larger beads are used. Computational results are shownonly for binary mixtures of particles differing in density. Equilibrium segregation structures are shown for two different orientations for the systems of different-sizedspheres (a)–(d), showing that, unlike the systems segregated in the circle, the shape of the segregated structures depend on the mixer orientation. Results from systemsof granular materials differing only in density are similar, both in experiment (e) and (f) and computations (g) and (h).

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For the segregation experiments, it is helpful tostart with a well-mixed system. Any imperfect ini-tial mixing leads to a longer time for the system toresemble its final equilibrium state of segregation.The mixers described above are small enough to al-low one to accomplish a thoroughly mixed initialcondition without too much difficulty. We simplymix the materials first by hand, and then shake androtate the mixers randomly once they are sealed toachieve a well-mixed system.

Once the systems are mixed, we use a com-puter-controlled stepper motor (Compumotorr) torotate the mixers at approximately 1 revolution perminute so that a steady flow with a flat free surfaceis developed. (The Froude number F = ω2L/g ≈2 × 10−4 for the circle.) Preliminary experimentsare recorded by means of Polaroidr pictures un-til suitable conditions are identified. More de-tailed data was taken from images obtained using aKodakr CCD camera, ideal for quantitative imageanalysis.

Direct measurement of the concentration dis-tribution of particles in a segregating system is alsopossible by injecting a setting fluid into the particlebed and freezing the particle configuration. Sam-ples are then taken from different regions by cut-ting the bed and concentration may be obtained byseparating the particles by sieving (different-sizedparticles) or using a magnet (if one of the compo-nents is magnetic).

7. Typical Results

Consider now examples of the results produced bythese systems. Equilibrium segregation structuresfor half-filled circular, elliptical, and square mixersare shown in Figs. 1 and 8 for systems of spheresdiffering in size [Figs. 1(b), 8(b) and 8(d)] anddensity [Figs. 1(c), 1(d), 8(e) and 8(f)]. Com-putational results based on the incorporation ofall details (Secs. 4 and 5) into the basic model(Sec. 2) are shown in Figs. 1(d), 8(g) and 8(h).As seen in Fig. 8, the shape of the container notonly has a strong influence on the Poincare sec-tions (Fig. 4), but it plays a dominant role indetermining the equilibrium segregation structuresfor the different systems as well. The segregatedregions for half-filled systems are similar to theregular islands in the Poincare sections. Thus,segregation has an unanticipated benefit: Heavierparticles tag the regular regions in the Poincaresections.

The classic radially segregated structure for thecircular mixers is shown in Figs. 1(b)–1(d). In thiscase, the dynamics in the flowing layer cause thesmaller (more dense) particles to move to lowerlevels in the layer leading to a segregated time-invariant (rotationally symmetric) core region thatcoincides with the streamlines. However, the equi-librium segregation structures obtained in noncircu-lar mixers do not follow this simple rule, as shownfor the half-filled elliptical [Figs. 8(a), 8(c), 8(e) and8(g)] and square [Figs. 8(b), 8(d), 8(f) and 8(h)]mixers. In this case the region with a high con-centration of the small (or more dense) particlesis located away from the center of rotation and isnearly separated into two distinct regions betweenthe corners of the square and the center of rotation.Unlike the circular mixer, the precise shapes of thesegregated regions are periodic in time and dependon the instantaneous orientation of the mixer, asillustrated by comparing Figs. 8(a) and 8(b) withFigs. 8(c) and 8(d), respectively.

How do these complicated structures arise frominitially well-mixed systems? In all cases shown, ini-tially well-mixed systems segregate rather quickly— apparent within a half rotation, all systems seg-regate into a state resembling radial segregation.For the circular mixer, this pattern remains rela-tively stable and becomes the final state of segre-gation. Since the shape of the mixer is rotationallyinvariant, the important dynamics lie solely in thedifferential flow. While diffusion in the flowing layermay tend to jostle the particles away from the cen-ter of rotation, an equilibrium, radially segregatedstructure, is eventually reached.

This initial radial structure may not be in-variant though. Instead, in the case of the half-filled noncircular mixers, the initial radial segrega-tion gradually evolves into the more complicatedsegregation patterns, as shown for the square mix-ers in Fig. 10. (The pattern development for theellipse follows the same course.) The heavier orsmaller particles that sink into the regular regionsor islands are more likely to be mapped back tothe same region than the particles in the chaoticregion. Hence, as the system evolves, the segre-gated islands around the elliptical points grow andbecome more distinct. This argument is furthersupported by the strong agreement between exper-iment and computational predictions [Figs. 1(e),8(g) and 8(h)] based only on the mixer shape, dif-fusion, and the density segregation model describedabove.

1478 K. M. Hill et al.

Fig. 9. Images from mixing experiments and computations using tracer particles. Deformation and mixing of a “blob” dueto chaotic advection and diffusion are clearly apparent, though due to the low Peclet number for typical experimental mixersthe mixing patterns are blurred due to collisional diffusion.

Mixing of Granular Materials 1479

Fig. 10. Evolution of binary mixture of 2 mm glass and steel beads as the system is rotated at 1 rpm in a half-filled squaremixer. The number of rotations completed is noted in the upper right-hand corner of each image as the system evolves froman initially well mixed state (denoted as IC) to the final (steady) segregation pattern.

Note that both the Poincare section and theequilibrium segregation structures depend stronglyon the degree of filling about the half-full level, asshown in Figs. 5 and 11, respectively. Figures 11(a)and 11(b) show that for mixtures of spheres differ-ing only in density, and systems at just below (45%)and just above (55%) the half-full level, the com-plicated segregation patterns disappear. In thesecases, the patterns revert to the simple radial struc-ture one might expect from earlier experiments withthe circular mixers. For mixtures of spheres differ-ing in size, the results are the same for a mixer filledless than half full [Fig. 11(c)] but are more compli-cated when just over half full [Fig. 11(d)], wherea striping pattern dominates. This phenomenon,unrelated to the Poincare sections, is outside thescope of this paper. For filling fractions of 25%and 75%, the circulation patterns (not shown) in-dicate a return of regular regions partway betweenthe corners of the square and the center of the flow-ing layer. This is mirrored in the equilibrium seg-regation structures where the asymmetric islandsreturn. For an example, see the cover of this issue.

8. Methods of Analysis

The fact that the data are taken in the form of dig-ital images lends itself to a range of digital analysistechniques, many of which may be known to readersof this journal. We restrict ourselves to two, thoughin fact, many others are possible.

The most common measure of mixing is thestandard deviation of the concentration fluctua-tions about the mean concentration — the so-called intensity of segregation [Dankwerts, 1952].For the systems in this work, this measure ofsegregation captures the initial radial segrega-tion but, unfortunately, indicates that all systemsconsidered segregate nearly equally. Thus, thismeasure fails to capture any of the aspects aris-ing from spatial structure. (See Fig. 12(a), forexample.)

Figure 12(b) demonstrates that a differentmethod of analysis picks up these differences. Cer-tain aspects of the structure can be quantified bymeasuring the perimeter length p and the area, a,

1480 K. M. Hill et al.

> 1

/2 fu

ll<

1/2

full

Fig. 11. Variation in segregation patterns in the square mixer when the filling level is changed to about 50%. (Comparewith the Poincare sections in Fig. 4.) Figures 11(a) and 11(b) show that for systems of different density (equal size) granularmaterials, the complicated segregation patterns disappear, and the equilibrium state of segregation resembles that of a circularmixer (simple radial segregation, Fig. 1). For different-sized particles (equal density), the 45% case results [Fig. 11(c)] aresimilar to those in Fig. 11(a). However, due to instabilities in the flow, the 55% system never reaches a final segregatedpattern, but produces instead an ever changing pattern of streaks. The corresponding patterns in the circular mixer remainradially segregated state for all filling levels examined.

of the segregated region, where p corresponds toan iso-concentration line c(x, y) = K, where Kis a suitable threshold. (We have done this forK = 0.75, i.e. for regions of segregation contain-ing 75% or greater concentration of dense beads.)Thus, p2/a is a nondimensional measure of this seg-regated structure. For a perfect semi-circle thisvalue is p2/a = (2/π)(2 + π)2 ≈ 17. Any devia-tion from straight radial segregation will increasethe value of p2/a. As shown in Fig. 12(b), thismeasure gives a dramatically different result for the

50% full and 45% full square. The result for thesquare filled slightly less than half full remains con-stant, between 20 and 25. On the other hand, thevalue for a half-filled square rises up slowly over thefirst 3–5 revolutions while the final stable structureis first forming (see Fig. 8), to nearly twice thatvalue. It remains nearly constant for any particularorientation for over forty revolutions. The value ofthis measure oscillates depending on the orientationof the mixer; the measurements in this graph weredone with the orientation of the mixer shown in

Mixing of Granular Materials 1481

Fig. 12. Measurements of the segregation pattern for differ-ent fill levels of the square mixer. In Fig. 12(a) the standarddeviation from the mean concentration quickly rises with theinitial radial segregation for the 7/16 full and the 1/2 fullmixers. It then shows little change (if any) when the finalmore complicated pattern forms over the next four rotationsfor the half-full square mixer. In contrast, Fig. 12(b) demon-strates that a nondimensional measure of the pattern com-plexity — p2/a described in the text — captures the changes.For the graph in Fig. 12(b), the measurements were taken forthe region of the segregated structure of 75% or greater densebeads. The inset shows the change in this value as the thresh-old value is changed from 50% to 80%, demonstrating thatthe value is nearly constant, though it increases somewhat forincreasing threshold value. The error bars in the main graphrepresent the variance of p2/a from its mean value once thesegregated structure had reached its equilibrium state (afterfive full rotations).

Fig. 10. The p2/a measure captures the differencein segregation structure for the different fill levels ofthe square and the gradual evolution for the half-filled square mixer from one of radial segregation toa more complex pattern of petals.

9. Recommendations

The experiments described in this paper providea test-bed for the study of nonlinear behavior. Un-doubtedly, there are several avenues that may bepursued to fully elucidate the physics governingthese systems; the experimental and computationalphenomena are quite rich, and there are many ques-tions that have yet to be answered regarding theuniversality of the experimental results. For exam-ple, we noted that the details of the patterns ob-served are sensitive to the fill level of the mixer,particularly when it is near 50%. The reasons forthis behavior, however, are not completely under-stood though the underlying Poincare section seemsto play an important role. To better understandhow the structure of the Poincare section interplayswith segregation phenomena one might need to con-sider variations in the type of mixture (studyingsegregation by size, density, etc.); the ratio of therelevant values for the components (size, density,etc.); the speed of rotation, and the mixer size.All these may need to be supplemented by com-panion computational studies; these may focus oneffects hard to capture experimentally. A listingmay include varying the depth of the flowing layer,the velocity profile, and changing the Peclet num-ber (effectively, the relative value of the diffusioncoefficient).

An alternative to physical experimentation forprobing these systems is Particle Dynamic Simu-lations, and a few words in this regard should bementioned here. These techniques are based on thesame premise as Molecular Dynamic Simulations —i.e. particles (molecules) are treated as distinct en-tities. The macroscopic behavior of the particlesis developed from either conservation of linear andangular momentum for each binary collision (hardparticle approach) [Campbell, 1989] or the simul-taneous solution of Newton’s equation of motionfor each particle (soft particle approach) [Cundall& Strack, 1979; Walton, 1993]. These two meth-ods are primarily used for low-density, fast-flow orhigh-density, slow-flow, respectively. The flow in atumbler consists of a spectrum which spans theseextremes — low-density, fast-flow in the surfacelayer, and high-density, slow-flow or in some casenearly “stagnant” regions in the bulk of the bed.For this reason, “hybrid” techniques have been de-veloped to model this type of device [McCarthy &Ottino, 1998a]. The advantage of these techniques

1482 K. M. Hill et al.

is that they allow measurements of quantities thatwould be difficult or impossible to measure in aphysical experiment — such as velocity and con-centration profiles, segregation rates and other de-tails. Also, computational experiments allow closecontrol of particle properties (size, shape, density,frictional characteristics), experimental conditions(gravity, wall friction, etc.), and additional interac-tion forces like liquid-bridging and van der Waalsforces [Thornton & Yin, 1991]. Using these tech-niques, therefore, one might more easily probe suchcomplexities as: 3D effects, the role of friction, sizesegregation, the role of cohesive forces [Shinbrotet al., 1999; McCarthy & Ottino, 1998b], etc. onthe current model.

In the context of the ideas presented here, Par-ticle Dynamic Simulations should be regarded asa tool that allows for the elucidation of constitu-tive equations (size and density segregation) thatcan be incorporated into continuum descriptions ofmixing and segregation in granular flows. (For acomplete review see [Ottino & Khakhar, 2000]). Infact, the track advocated here — one focusing on adynamical system viewpoint — considers the con-tinuum description as the starting point. In its sim-plest form the continuum model outlined in the ear-lier sections gives a good description of mixing andsegregation of granular materials in rotating cylin-ders of different cross-sectional shapes and for equalsized particles [Khakhar et al., 1999a]. Acceptingthis base-model as essentially correct, there are sev-eral questions that need to be investigated as addi-tional physical effects are added to it. Equations (6)and (7) define a conservative chaotic dynamical sys-tem with stochastic forcing due to particle diffusion.An analysis of how diffusion destroys the invariantstructures (KAM surfaces, etc.) present in the un-forced system, i.e. without diffusion, and to whatextent diffuse versions of these structures remain,is important for understanding mixing in the sys-tem. The issue of diffusion is more important inthe case of granular materials than in the case offluids. (See Fig. 7.)

The inclusion of segregation fluxes in the model[Eqs. (10) and (11)] changes the character ofthe system. The system is now dissipative, andparticle trajectories of a component in the mix-ture converge to an attractor for that componentin the absence of diffusion. In the case of a circularmixer, the boundary of the attractor correspondsto a streamline of the flow. When the shape of thecontainers is changed from the base-case of a circle

the situation becomes more complex; the system isnow chaotic. The results presented above indicatethat attractors survive in the presence of diffusionand equilibrium segregation structures correspondroughly with the Poincare sections for the base-flow(i.e. with no diffusion or segregation). These as-pects need to be studied in more detail and somedegree of mathematical investigation should be pos-sible. The situation resembles the case of a fluidwith suspended solid particles but in many respectsis simpler.

Finally consider extensions of the model thatmay bring new concepts from a dynamical systemsviewpoint. For example consider the extension ofthe model for the case of different-sized particles.In this case the velocity gradient must become com-position dependent — higher gradients for a largerfraction of the smaller particles. While a rigorousapproach to this problem appears formidable andwould require simultaneous solution of the massand momentum balance equations together with theconvective diffusion equation, a simpler option, ac-ceptable on physical grounds, may be to considerthe average velocity (u) and layer thickness (δ) tobe functions of composition chosen so as to satisfycontinuity and mimic the actual flow. Such a modelcould reveal the implications of the coupling be-tween velocity and composition on the mixing andsegregation and pattern formation in the system.

It is apparent that granular mixing and seg-regation experiments can benefit from an infusionof dynamical systems thinking. The enrichmentis not entirely one-sided though; granular flow ex-periments serve to pictorially illustrate importantconcepts in nonlinear physics and dynamical sys-tems and the balance between disorder and self-organization.

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