PHYSICAL REVIEW E 88, 052203 (2013)

Effect of aspect ratio on the development of order in vibrated granular rods

Vikrant Yadav,* Jean-Yonnel Chastaing,† and Arshad Kudrolli‡

Department of Physics, Clark University, Worcester, Massachusetts 01610, USA(Received 5 September 2013; published 21 November 2013)

We investigate ordering of granular rods in a container subject to vibrations in a gravitational field as a functionof number density of the rods. We study rods with three different length to diameter aspect ratios Ar = 5, 10, and15. The measurements are performed in three dimensions using x-ray computer tomography to visualize the rodsin the entire container. We first discuss a method to extract the position and orientation of the rods from the scanswhich enables us to obtain statistical measures of the degree of order in the packing. We find that the rods withAr = 5 phase separate into domains with vertical and horizontal orientation as the number density of the rods isincreased, whereas, for Ar = 10 and 15 the rods are predominately oriented vertically in layers. By calculatingtwo-point spatial correlation functions, we further show that long range hexagonal order occurs within a layerwhen the rods are oriented along the vertical axis. Thus, our experiments find that long range order increasesrapidly in granular rods with growing anisotropy.

DOI: 10.1103/PhysRevE.88.052203 PACS number(s): 45.70.Qj, 81.05.Rm

I. INTRODUCTION

The shape of granular materials is well known to havea significant effect on their packing properties. Sphericalgranular materials compact when vibrated in a containerbut remain disordered even when subjected to prolongedexcitations [1]. By contrast, significant ordering is observed tooccur with granular rods [2–4]. Development of order above acritical concentration and aspect ratio is well known in thermalrods and occurs because of maximization of configurationalentropy [5]. Although these principles have been considered topredict the nature of packing of rods in the absence of gravity[6], and in vibrated rods confined to quasi-two-dimensions[7], they do not apply in general to athermal granularmaterials.

In the presence of gravity, experimental studies with rods ina narrow tall vertically vibrated container have found that rodswith vertical orientation grow in from the boundaries [3]. Itwas proposed that shearing by the vertical oscillating containersidewalls was important to the observed vertical ordering.Later, rods vibrated in wide containers were shown to formdomains of vertically oriented rods above a critical numberdensity of rods when the container was vibrated verticallyor horizontally [4]. These domains were observed to formeverywhere within the container and not just at the boundaries.Therefore, a void filling mechanism similar to that used tounderstand size separation in bidisperse granular materials [8]was proposed to explain the appearance of vertically orientedrods [4]. This mechanism makes the assumption that smallvoids are more likely to occur than larger ones. Therefore, it ismore likely that a vertical rod can fall into a void in the presenceof gravity than a horizontal one. While subsequent work hasfurther examined ordering in rods in a vertical plane [9],using simulations [10,11], and x-ray tomography [12], a fullunderstanding of the conditions under which granular rods

*VYadav@clarku.edu†Present address: Laboratoire de Physique, Ecole Normale

Supérieure de Lyon, F-69007 Lyon. France.‡AKudrolli@clarku.edu

order, still remains far from clear. In particular, no study existswhich investigates the effect of degree of anisotropy on thedevelopment and nature of order.

In this paper, we discuss the emergence of order in vibratedrods in three dimensions (3D) enabled by a microfocus x-ray computer tomography (CT) instrument. This instrumentallows us not only to nondestructively visualize the nature ofthe internal packing of rods but also to obtain quantitativeinformation on the degree of order and defects enabled bytracking the position and orientation of all the rods in thepacking. This data is then used to calculate mean orientationorder parameter and the spatial pair correlation function as afunction of the number of the rods in the container. We varythe aspect ratio of the rods by changing the length of the rodsto investigate the impact of the degree of anisotropy on theappearance of order. As we discuss in the following, we findthat only short range order is observed for the smallest aspectratio investigated in a large container, and vertically orientedrods arranged in layers with hexagonal order are observed forhigher aspect ratios.

II. EXPERIMENTAL APPARATUS

The rods used in the experiments consist of hollow plasticcylinders and their dimensions are listed in Table I. We usehollow rods because it is easier to write algorithms to trackthem in 3D. The rod packings are studied within a cylindricalPlexiglas container with a diameter dc = 120 mm and heighthc = 150 mm. The container filled with rods is mounted ona Labworks ET-139 electromagnetic shaker which vibratessinusoidally in the vertical direction driven with a prescribedacceleration amplitude � and frequency f using an Agilent33120A function generator. In the experiments discussed in thefollowing � = 3g, where g is the gravitational acceleration,and f = 15 Hz, unless otherwise stated.

Because of the gravitational field, the rods are not uniformlydistributed along the vertical axis even under the highestexcitations possible in our system. Therefore, we find itconvenient to use a number fraction nf to parametrize the

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http://dx.doi.org/10.1103/PhysRevE.88.052203

YADAV, CHASTAING, AND KUDROLLI PHYSICAL REVIEW E 88, 052203 (2013)

TABLE I. The aspect ratio Ar and the corresponding length l anddiameter d , of the hollow plastic (polypropylene) rods used in theexperiments. The interrod coefficient of friction is 0.52.

Ar d (mm) l (mm)

5 3.02 14.9910 3.02 30.5315 3.02 45.27

system [4] which is given by

nf = nnmax

, (1)

where n is the total number of rods in the container, and nmax isthe maximum number of rods which can be packed verticallyinside the container in one layer. In the container with dc =120 mm, nmax = 1450, corresponding to packing rods in avertical layer with hexagonal order.

The internal imaging of the packing structure is carriedout with a microfocus x-ray CT instrument manufacturedby Varian Medical Systems. This instrument can image amaximum cylindrical volume with a 15 cm diameter and 15 cmheight. This sets the upper limit of the sample size that wecan examine. In this technique, radiograms are taken frommultiple viewpoints, and a reconstruction of the 3D spatialmap of the x-ray absorption by the medium is performed bythe instrument. Spatial resolution of voxels with dimensionsless than 100 μm can be obtained with scans lasting 242 susing 5700 views. Even higher resolutions can be obtainedwith longer scans. The x-ray absorption is proportional tothe amount of material in the voxel and the reconstructionalgorithm yields intensity with 16-bit depth. These spatialmaps can be then used to track individual rods as discussed inthe next section.

Because of the time taken by the CT technique to completethe scans needed to resolve the rods inside the packing, westop the excitation during the scan. While rods at very lowconcentrations (where they do not collide with each other)move rapidly under the vibration amplitudes we use, at highdensities the rods do not move rapidly as they interact withmultiple neighbors. Thus the rods appear to be essentially incontact and slide past each other as they rearrange. In thisregime, which is the focus of our study, we find that the rodsrapidly come to rest in place due to dissipation when the shakeris stopped. Therefore, the scans we analyze subsequently canbe viewed as snapshots of the packing dynamics. We visuallychecked that turning off the excitations to obtain the scansdoes not affect the nature of the subsequent evolution of thepacking by comparing to packings where excitations were notturned off.

III. TRACKING THE RODS IN THREE DIMENSIONS

The various possible cross sections that a hollow tube canhave are shown in Fig. 1. A typical cross section of the rodpacking after the intensity map is converted to binary data,corresponding to the presence of materials (white) and void(black/blue), is shown in Figs. 2(a) and 2(b). Then we invertthis image so that all rod boundaries appear dark, as shown

FIG. 1. (Color online) The intersection of a plane with a hollowrod can produce ellipses as in (a), (b), and (c); elliptical arcs (d); andlines as in (e).

in Fig. 2(c). A label matrix is created from this image whereall connected pixels of a particular signature are denoted bythe same label. As the dimensions of the rods are known, wedefine an upper and lower cutoff limit in terms of the pixel sizethat the rod cross section can have. Any size above or belowthese limits is discarded [see Fig. 2(d)]. To remove any falsepositives that are not the interior of rods we run a convexitycheck on the signatures. The convex area of the space exteriorto rods is always larger than their true area due to the presenceof kinks in their boundaries. Removing these, we are left withonly the true closed signatures of the rods [see Fig. 2(e)].

We use a coordinate system with the x axis and the y axisoriented along the horizontal, and the z axis oriented alongthe vertical. The matrix with ellipses in a cross section canbe easily tracked using a standard image processing algorithmto determine their centers in the horizontal x-y plane. Wethen repeat this process over all the cross-sectional imagesalong the z axis. The cross sections belonging to the same rodin different planes are identified by proximity and are usedto calculate the center of mass and orientation of each rod.Because we detect only ellipses (or closed signatures) alongthe one axis, we rerun the whole process for cross sectionsorthogonal to the x and the y axes. This enables us to trackthe rods which have an open signature in the x-y plane andwhich would be missed otherwise. Thus, we generate a list ofthe position and orientation of each rod in each packing forsubsequent analysis.

FIG. 2. (Color online) (a) A typical cross section of the packingobtained with x-ray CT scanning. The steps used to detect the rodsare illustrated in (b), (c), (d), and (e) (see text). The center of a rodcross section is indicated by a dot in (e).

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IV. EVOLUTION OF ORDER

We now discuss experiments which illustrate the evolutionof order when the rods are added randomly in a containerand then vibrated. Figures 3(a)–3(f) show an example of theevolution when the container is vibrated over 20 min fornf = 1. It can be noted that the rods rapidly begin to rotatetoward the vertical direction everywhere within the container.Regions with hexagonal order begin to appear after only afew minutes of vibrations, and with slow healing of defectsand grain boundaries occurring over time. However, we finda few defects persist even after prolonged vibrations. Afterinspecting the entire packing, we find that these defects arecreated by a few rods that get trapped at the bottom of the

(g)1.0

0.8

0.6

0.4

0.2

0.0

n v

6000400020000t (sec)

5 deg. 10 deg. 15 deg.

Eqn. (2)

θcθcθc

(f)

(d)

(e)

(c)

===

(b)(a)

FIG. 3. (Color online) The evolution of the rod packing inside thevibrated container for Ar = 10 and nf = 1. The images correspondto a horizontal cross section at a height of l/2 from the bottomxbrkof the container at (a) t = 0 s, (b) t = 30 s, (c) t = 120 s, (d) t =240 s, (e) t = 840 s, and (f) t = 1200 s. (g) The number fractionof rods oriented along the vertical axis. The evolution assuming thecorresponding cutoff tilt angles θc are also shown.

container due to the pressure of rods which have packedvertically overhead.

In order to examine the development of ordering quantita-tively, we plot the ratio of the number of vertical rods nv andthe total number of rods n as a function of time in Fig. 3(g). Ifθ is the angle that the rod makes with the vertical z axis, wedefine a cutoff tilt angle θc, so that a rod is considered verticalfor θ < θc. As a guide, we also plot the expected trend if theinitial distribution were isotropic and if the rate of growth ofvertical rods was proportional to the number of vertical rods,and independent of each other [4]:

∂nv

∂t= αnv(β − nv), (2)

where α is the rate constant, and β is the saturation fraction.While the final number fraction of vertical rods can be observedto change with the choice of θc, the ordering is well describedby Eq. (2) in each case. We find α ≈ 0.22 independent ofchoice of θc. And β = 0.69, 0.76, and 0.83 for θc = 5,10,and 15, respectively. The rods which are not perfectly aligned(tilted at an angle greater than θc) are present in the bulk asdefects, and at the top or bottom of the packing. The observedphenomena and quantitative analysis enabled by x-ray CTtechnique is consistent with those reported by visual inspectionby Blair et al. [4]. Those experiments were performed withan order of magnitude larger number of solid brass rods andin a wider container with a container diameter to rod lengthratio dc/ l which was three times larger. But, we find similarbehavior while comparing across the same number fraction nf .Thus, the smaller system size and materials used here to aidscanning does not have a qualitative influence on the observeddevelopment of order.

V. NUMBER FRACTION AND ASPECTRATIO DEPENDENCE

We now examine the effect of rod aspect ratio on theobserved structure at steady state as a function of the number ofrods in the container. To illustrate qualitative differences whicharise with aspect ratio Ar , we show snapshots of the packingsobserved with Ar = 5,10, and 15 for nf = 1 in Fig. 4 [13].Here, the rods are rendered in POV-RAY using different shadeswhich depends on the rod tilt angle θ from the vertical. It canbe noted that the rods with Ar = 5 form domains of horizontaland vertical rods. The rods with larger aspect ratio (Ar = 10)have fewer horizontal rods, and are predominately orientedalong the vertical axis, whereas rods with Ar = 15 are seen tobe all oriented vertically. Thus, we find a qualitative change inthe nature of order which we characterize quantitatively next.

A. Orientation statistics

We plot the distribution of θ in Fig. 5 to discuss thesystematic evolution of the packing as a function of nf foreach Ar . For small nf = 0.1 for each Ar , we observe thatthe distribution is peaked around 90◦. This is consistent withthe fact that the rods are essentially horizontal in the containerdue to the effect of gravity. We examined the distribution of theazimuthal orientation of the rods and found it to be isotropicbecause of the symmetry of the container. However, as nf isincreased the packing changes depending on Ar . In the case

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YADAV, CHASTAING, AND KUDROLLI PHYSICAL REVIEW E 88, 052203 (2013)

FIG. 4. (Color online) The steady state packing of rods with (a) Ar = 5, (b) Ar = 10, and (c) Ar = 15 (nf = 1). The 3D rendering of thepacking from position and orientations obtained from the x-ray CT scans. The rods with vertical and horizontal orientation are shaded red(gray) and yellow (light gray), respectively.

of Ar = 5, we observe that the distribution becomes bimodalwith the peak at 90◦, decreasing in amplitude, and a secondarypeak developing around θ = 0◦. Thus we find domains ofrods which are oriented vertically and horizontally in domains

(a)

(b)

(c)

1.0

0.8

0.6

0.4

0.2

0.0

P(

9060300

= nf 0.1 nf 0.3 nf 0.7 nf 1.0

Ar = 5

1.0

0.8

0.6

0.4

0.2

0.0

P(

9060300

nf 0.1 nf 0.3 nf 0.7 nf 1.0

Ar = 10

1.0

0.8

0.6

0.4

0.2

0.0

P(

9060300

nf 0.1 nf 0.3 nf 0.7 nf 1.0

Ar = 15

= = =

= = =

=

= =

= =

(deg.)

FIG. 5. (Color online) Distribution of the orientation angle of therods with respect to the vertical for various number density and aspectratio (a) Ar = 5, (b) Ar = 10, and (c) Ar = 15. The distribution forAr = 5 becomes bimodal with peaks near θ = 0◦,90◦, whereas, inthe case of Ar = 10 and 15, a single peak at θ = 90◦ develops fornf = 1.

with few rods which are in fact at an intermediate angle. Atetratic phase has been observed in two-dimensional vibratedgranular rods in the horizontal plane [14]. But the phase seenin Fig. 4(a) is somewhat different, because the orthogonalalignment of domains is in the vertical plane, but the domainsappear to be randomly aligned in the horizontal plane. Bycontrast, a second peak is observed to develop in the case ofAr = 10 at an angle which starts to decrease with nf . Thispeak shows that a significant number of rods are tilted at thatangle. Correlated with the appearance of this peak, we observevortex-like motion consistent with previous observations byBlair et al. [4] with rods with similar aspect ratios.

As nf approaches 1, and the tilt decreases, the vortex motionslows down and stops. This is consistent with the model formotion of rods by Volfson et al. [15], who showed that theconvective motion arises from collision of a tilted rod withthe vibrating frictional substrate. At nf = 1, we still observea bimodal distribution of rods but the extent of horizontaldefects is smaller when compared to the rod of Ar = 5. Thetrend toward vertical orientation with nf is even stronger inthe case of Ar = 15. The P (θ ) is observed to show a singlepeak which moves from θ ≈ 90◦ to θ ≈ 0◦, and very few rodsare oriented horizontally.

To have a simple measure of the global vertical order of thesystem, we define a parameter

S = 〈cos θ〉, (3)where, 〈·〉 represents an average over all particles in thepacking, and a value of S close to 1 implies predominantalignment of rods along the vertical axis.

Evolution of S as a function of number fraction for the rodswith the three different Ar is plotted in Fig. 6. We observe thatas nf is increased from 0 to 1, rods with Ar = 10 and 15 gofrom an initially disordered state to an ordered state, whereasrods with Ar = 5 do not order as significantly. We also notethat the critical nf at which ordering begins, decreases with anincrease in Ar . A broad peak is seen for packing of particleswith Ar = 5 at low nf . However, this does not signify actualordering. It can be attributed to the fact that when particleswith Ar = 5 are vibrated at an acceleration of 3g, even at nfclose to 0.5 most of the rods are in a rare-gas-like state. Whenthe shaker was turned off, this state quickly quenches to a staticstate where particles get trapped in various orientations givingan overall nonzero S.

In the case of rods with Ar = 10 and 15, we observe a dipin value of S as nf increases from 0.9 to 1. From the scans,

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EFFECT OF ASPECT RATIO ON THE DEVELOPMENT OF . . . PHYSICAL REVIEW E 88, 052203 (2013)

1.0

0.8

0.6

0.4

0.2

0.0

S

1.00.80.60.40.2nf

A =r 5 10 15

A =r A =r

FIG. 6. (Color online) Evolution of the global vertical orderparameter S as a function of nf for rods with various Ar .

it is clear that this dip in the case of rods with Ar = 10 isbecause of the presence of defects in the bulk of packing. Inthe case of rods of Ar = 15, the dip is due to the accumulationof free horizontal rods on the top of packing. As nf tends to1, the dynamics of the system slows down and the probabilitythat a void can be filled by a free vertical rod near the top ofthe packing decreases drastically. This is because significantenergy is required to rotate the rod vertically so that it can fallinto a narrow void. Thus, these free rods create a dip in thevalue of S.

B. Spatial order

As we can note from Fig. 4, the rods appear hexagonallyordered when oriented vertically. Therefore, we calculate thepair correlation function g(r) for different values of nf toanalyze the spatial ordering of rods within a layer. For n rodsinside a volume V , g(r) as a function of distance r is given by

g(r) = Vn2

〈∑i

∑j �=i

δ(�r − �rij )〉

, (4)

where �rij corresponds to the distance between particles labeledi and j , and 〈·〉 indicates averaging over the rods in thevolume V .

In Fig. 7, we plot g(r) for various values of nf and Ar .For rods with Ar = 5, there is just one primary peak atr/d = 1 because of the hard core repulsion potential betweenrods, and the high density of the packing. This peak becomesprominent with an increase in nf , but no further peaks areobserved. This implies a lack of local ordering in rods withAr = 5 as measured by g(r). For rods with Ar = 10, we notethat there is little to no ordering at low number fractions. Atnf = 0.7, we see the emergence of a dominant first peak anda weak secondary peak. As nf is increased further to 1.0, thesecondary features of g(r) become prominent. The existence ofpeaks at r = d, √3d, 2d, √7d, and 3d indicated by the verticaldashed line in Fig. 7 correspond to a hexagonal lattice. Thus,the existence of a nonzero value of S that is close to 1, andthe presence of a hexagonal lattice, indicates the presence of acrystalline phase. Similar signatures are observed for particleswith Ar = 15. In this case additional peaks corresponding to a

80

60

40

20

0

g(r)

54321

nf = 0.1 nf = 0.4 nf = 0.7 nf = 1.0

80

60

40

20

0g(

r)54321

nf = 0.1 nf = 0.4 nf = 0.7 nf = 1.0

80

60

40

20

0

g(r)

54321

r/d

nf = 0.1 nf = 0.4 nf = 0.7 nf = 1.0

(a)

(c)

(b)

FIG. 7. (Color online) The evolution of pair correlation functiong(r) as a function of distance scaled by the diameter of the rod forrods with (a) Ar = 5, (b) Ar = 10, and (c) Ar = 15. The vertical linescorrespond to a hexagonal lattice.

hexagonal lattice appear in g(r) which shows that spatial orderexists at even longer spatial scales.

VI. MULTIPLE LAYERS

We next extend our study to the case when nf is increasedabove 1, and multiple layers of rods are poured into thecontainer. In Fig. 8(a), we show the number density of thecenter of the rods P (z) as a function of height z obtainedin steady state by vibrating rods with Ar = 5 and nf = 3.We observe sharp peaks at around l/2, 3l/2, and 5l/2signifying the development of vertical layering with rodsoriented predominately along the z axis. Further, because theamplitude of the peaks decreases with z, and the backgroundrises, we gather that the layer at the bottom is most ordered,and the packing becomes progressively more disordered as afunction of height. To further characterize the order with this

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YADAV, CHASTAING, AND KUDROLLI PHYSICAL REVIEW E 88, 052203 (2013)

(a)

(b)

0.20

0.15

0.10

0.05

0.00

P(z

)

43210z/l

0 < z/l < 1

40

30

20

10

0

g(r)

4321 r/d

2 < z/l < 3 1 < z/l < 2

FIG. 8. (Color online) (a) Probability of finding a particle at aheight z from the bottom of the container normalized with rod lengthl. Peaks at l/2, 3l/2, and 4l/2 indicate layering. Inset: A POV-RAYrendering of the packing observed with nf = 3. The different shadescorrespond to different θ orientations as in Fig. 4. (b) g(r) within thelayer shows the decreasing hexagonal order with height.

layer, we plot g(r) corresponding to each layer in Fig. 8(b).The layer at the bottom is clearly most ordered with peakscorresponding to the hexagonal lattice. However, disordergrows with height.

The greater order in the lower layers occurs because of atleast two factors. First, the voids that are created in the firstlayer due to vibrations are filled more efficiently due to verticalrods falling from top layers into the bottom layer. Further,horizontally oriented rods are more likely to be blocked fromfalling through the gaps between the rods by other rods.This leads to an increase in the number of horizontal rodswith height. Secondly, energy is input into the system due tocollisions of the rods with the vibrating bottom of the container.But this energy is lost because of inelastic collisions amongrods as it reaches the upper layer. Because of this decrease in

energy, the effective rods fluctuations (granular temperature)decreases with height. Thus, the frequency of creating voidsdecreases with height, and there is less energy available tocause a horizontal rod to rotate to a more vertical orientation.Therefore, defects that are initially present may not evolvesignificantly, and the packing remains in a disordered stateeven after being subject to prolonged vibrations.

VII. CONCLUSION

In summary, we investigated the evolution of the orderingtransition in vibrated granular rods as a function of theirnumber fraction and aspect ratio with x-ray tomography. Wediscussed a method to track the rods in 3D which is then usedto calculate the microstructural properties of the system. Wefind that the degree of ordering increases with the aspect ratioof rods. From the distribution of the vertical component oforientation vector of rods, we showed that rod packings withall aspect ratio have some ordering as their number fraction isincreased. Using mean orientation data we could also showthat rods with larger aspect ratio were more ordered. Paircorrelation functions are used to show that rods tend to arrangethemselves in a hexagonal lattice in a layer, and further thatrange of ordering increases with aspect ratio of the rods. Wefurther found that the ordering extended into multiple layersas nf was increased above 1. However, the ordering decreasesfrom the bottom to the top layer because of decreased energytransfer to higher layers from the container bottom due to thedissipative nature of the system.

The experiments and the quantitative analysis are overallconsistent with the void filling mechanism for development ofvertical order. However, it is unclear why the rods with Ar = 5phase separate into horizontal and vertical rod domains, andsimply do not form a more broader angle distribution centeredaround the vertical. The development of hexagonal order witha layer of vertically oriented rods is consistent with the factthat the maximum packing of rods lowers the center of massof the entire packing. However, analytical results on the effectof rod aspect ratio on the degree of order are not available tofurther analyze the system. It is hoped that this experimentalinvestigation will enable further development of a theory ofordering in athermal granular rods in a gravitational field.

ACKNOWLEDGMENT

This work was supported by the National Science Founda-tion Grants No. CBET-0853943 and No. DMR-0959066.

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