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Nucleation of athermal hard spheres

Frank Rietz,1, 2, 3, 4, ∗ Charles Radin,5 Harry L. Swinney,6 and Matthias Schroter1, 2, †

1Max-Planck-Institute for Dynamics and Self-Organization Gottingen, 37077 Gottingen, Germany2Institute for Multiscale Simulation, Friedrich-Alexander-Universitat Erlangen-Nurnberg (FAU), 91052 Erlangen, Germany

3Department of Nonlinear Phenomena, University Magdeburg, Universitatsplatz 2, 39106 Magdeburg, Germany4Department of Pattern Formation, University Magdeburg, Universitatsplatz 2, 39106 Magdeburg, Germany

5Department of Mathematics, University of Texas at Austin, Austin, TX, USA6Center for Nonlinear Dynamics and Department of Physics,

University of Texas at Austin, Austin, TX, USA(Dated: June 8, 2017)

We present an experiment with packings of macroscopic spheres that model disordered and par-ticulate systems and provide insight into the less accessible atomistic world. Cyclically shearinga packing of frictional spheres, we observe a first order phase transition from a disordered to anordered state. The ordered state consists of crystallites of mixed FCC and HCP symmetry thatcoexist with the amorphous bulk. The transition, initiated by homogeneous nucleation, overcomesa barrier at 64.5% volume fraction. Nucleation consists predominantly of the dissolving of smallnuclei and the growth of nuclei that have reached a critical size of about ten spheres. Our resultsindicate that classical nucleation theory is not applicable if the nucleation barrier is identified witha depleted region at the nucleus interface.

PACS numbers: 64.70.ps, 61.43.Gt, 64.60.Q-

Packings of spheres show interesting features such asphase transitions between disordered and ordered states,and can be useful in the study of amorphous atomicconfigurations [1]. Examples include thermal colloidalpackings [2–5], macroscopic athermal packings of hardspheres [1, 6–18], and simulations of the mathematicalhard sphere model [19–27]. The behavior of such sys-tems is determined by the fraction of space filled by thespheres, their packing fraction φ.

The hard sphere model exhibits an entropically drivenfirst order phase transition. Disordered fluid states areobserved below the freezing density of φ = 0.495 andcrystalline ordered states appear above the melting den-sity of φ = 0.545, with coexistence of the two phases forintermediate densities [23].

Athermal spheres can also be packed in disordered andordered states. Most experimental bulk protocols for in-creasing their volume fraction such as vertical shaking[14], centrifugation [4], thermal cycling [12, 13] and sed-imentation [15] do not achieve an ordered state from aninitial disordered state. The “random close packed state”(RCP) is informally used to describe the highest densitystate achieved by these methods, with a packing frac-tion in the range 0.635 < φ < 0.655 [18–21], about 15%lower than the densest possible packing of ordered facecentered cubic (FCC) or hexagonal close packing (HCP),which both have a volume fraction φ = π/

√18 ≈ 0.74

[28].

Ordered clusters of spheres have been obtained witha system density φglobal in the range 0.64–0.74 by mul-tidimensional shaking [8–10], cyclic shear [6, 7], andshear in a Couette cell [11, 16]. By analogy with thefreezing-melting transition in the hard sphere model, theemergence of growing crystallites exhibited by athermal

spheres can be described in terms of nucleation and afirst order phase transition [22, 29]. The homogeneousnucleation and growth of ordered clusters which we re-port support such a description.

The description of phase transitions in equilibrium sta-tistical mechanics is often enriched by classical nucleationtheory (CNT) [30]. The free energy cost required to cre-ate an interface between coexisting amorphous and or-dered phases gives rise to a free energy barrier. The bar-rier is a function of the critical nucleus size and consists ofa positive quadratic interfacial free energy and a negativethird order free energy phase difference. For small clus-ters the surface energy is significant; small nuclei tend toshrink while large nuclei tend to grow.

In our experiment we compact spheres by shearing.With increasing shear cycles, a well-defined plateauemerges at nucleation/phase transition density φglobal =0.645; such a plateau was not reported in a previous ex-periment [6] that used a setup similar to ours. The ob-served plateau density is in the range of densities asso-ciated with RCP [18–21]. Our observations also yield anuclei growth rate that differs qualitatively from a pre-vious experiment [6]. We cannot identify a nucleationbarrier at the nucleus surface in the framework of CNT.

The shear cell is cubical with side length 10.5 cm(Fig. 1(a)). Nucleation from the side walls is suppressedby half-spheres that are glued to the walls at randompositions (glue: Vitralit 7562, Panacol). The cell isfilled with 49,400 precision BK7 glass spheres of diame-ter 3±0.0025 mm and density 2.51 g/cm3 (size tolerancegiven by manufacturer, Worf Glaskugeln GmbH).

The top plate applies a pressure of 2.1 kPa on thebed, and the pressure increases downward to 3.1 kPa atthe bottom. The cell is sheared by sinusoidally tilting

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opposite vertical shear walls by ±0.6◦ about axes indi-cated by red circles about half-way up opposite side walls(Fig. 1(a)); the total peak-to-peak oscillation displace-ment at the bottom of the cell is about one-third of asphere diameter.

The glass spheres are index matched [31] to a mixtureof phthalate esters (Cargille Laboratories, identifier code1160, ρ = 1.1 g/cm3, ν = 41 cSt) with a dissolved fluores-cent dye (Oxazine 750 perchlorate, c = 10 mg/l). Hydro-dynamic interactions are negligible for our small ampli-tude shear with a period of 2 s, and sphere deformationis also negligible. The spheres, half-spheres, and shearcell walls are made of BK-7 optical glass and are indexmatched with the liquid mixture in a temperature con-trolled environment (n = 1.5198± 0.0001 at λ = 589 nmand T = 22.9± 0.1◦C).

The shearing process is periodically stopped to mea-sure the packing using an horizontally translated ver-tical laser light sheet (λ=658 nm, P=75 mW). Fluo-rescent light is imaged simultaneously by the camera(Figs. 1(b,c)). The image slices are combined to forma three-dimensional volume. The measurements pre-sented here are from 618 scans made during a run with1.955× 106 cycles.

In the course of the two month long run 10% of thespheres escaped from the sample cell through a gap be-tween one of the shear walls and a side wall. This un-planned draining of spheres may have increased the mo-bility of spheres farther inside the packing, thereby en-hancing their crystallization.

Positions of the 20,000 spheres at least 3 diame-

Figure 1. Precision spheres are contained in a cubical volumewith opposite side walls (lime green) that produce oscillatingshear. The cell is filled with a liquid that is index matchedto the beads. The top plate is levered to the side walls andis mounted on a piston that is constrained to move vertically.Shear is produced by periodically oscillating the cell bottom inthe z direction with a period of 2 s. A laser sheet illuminatesa slice in the x− y plane, as illustrated by contrast enhancedimages in (b) and (c); (b) shows a layer near the camera, and(c) shows a layer much further from the camera. Note thatthe image quality in both layers is comparable.

ters from any wall are detected by convolution witha gradient-based template [32]. Then the peak of thecorrelation map in the (x,y,z) directions is determinedwith (37,37,30) pixel/diameter resolution using a 3-pointGauss estimator [33]. This yields the positions of all par-ticles. From the pair correlation function we can deter-mine the accuracy to better than 2% of a sphere diameterfor 99% of the particles (see Supplemental Material [34]).

For each sphere there is a Voronoi cell consisting of allpoints closer to that sphere than to any other sphere inthe sample (see insets in Fig. 2(a)). φlocal is then the ratioof the sphere volume to the volume of its Voronoi cell.The mean volume fraction of the whole sample φglobal isgiven by the harmonic mean of all φlocal values [35].

The angular order between spheres sharing a face oftheir Voronoi cells is characterized by a weighted versionof the order parameter, q6 [27]. A sphere is called crys-talline if it is densely packed and its neighbors are or-dered, i.e., φlocal > 0.72 and q6 is either in the rangeq6(FCC)=0.575±0.02 or q6(HCP)=0.485±0.02. Otherauthors suggest different definitions for local order, butour results do not depend qualitatively on the choice ofdefinition or threshold (see Supplemental Material [34]).Crystalline spheres that share a Voronoi face are consid-ered to be part of the same nucleus. The distance of anucleus to another single sphere or nucleus is given bythe shortest distance between the sphere centers.

During densification we observe three distinct regions,as can be seen in Fig. 2(a). The packing starts from adisordered state and compacts approximately logarithmi-cally with time for about 20,000 cycles. The compactionthen slows to a stop, and the second region, a plateau(φglobal = 0.645), emerges and persists for about 50,000cycles. Then the first growing nucleus appears, indicatinga first order phase transition to the third distinct region.The volume fraction slowly begins to increase as shear-ing continues, and nuclei increase in number and size buthave no preferred orientation of their hexagonal layers;an intermediate state of the system is shown in Fig. 3(a).The first growing nucleus is shown in Fig. 3(b), which il-lustrates that nuclei fluctuate in shape, size, and crystalsymmetry as they grow, as can be seen in the movie inthe Supplemental Material [34].

All nuclei start their growth at least 10 sphere diam-eters distance away from any wall. By the end of theexperiment, after two million shear cycles, nuclei with upto ∼ 600 spheres are present, and 9% of all spheres are incrystallites, which have FCC or HCP symmetry with ap-proximately equal probability; no icosahedral symmetrywas observed.

Histograms of local densities for amorphous packingshave a single peak and are approximately symmetricalabout that peak, as Fig. 2(b) illustrates. During com-paction the peak narrows slightly and shifts to higherdensities. Beyond the first order phase transition a sec-ond peak emerges at φlocal = 0.74, the density of densest

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packed arrangements.

The end of the densification plateau is identified bythe emergence of a nucleus in the interior region; sub-sequently all nuclei are tracked for each successive scan.Nuclei with fewer than about ten spheres are found toshrink more often than grow, while growth graduallybecomes more probable for nuclei with more than tenspheres, that is, ∆ρ = (ρgrow − ρshrink) > 0, where ρis the probability to grow or shrink, respectively [3, 6].q6 that defines a nucleus does not shift the critical sizeat which the difference of the probabilities ∆ρ becomespositive (see Supplemental Material [34]).

Our main result is the observation of homogeneous nu-

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Figure 2. The formation of the crystalline phase showsfeatures of a first order phase transition. (a) Starting from aloose packing, the global volume fraction φglobal increases log-arithmically with the number of shear cycles until it reachesa plateau at the density φglobal = 0.645. After approximately70,000 shear cycles, φglobal increases again due to the for-mation of crystalline regions inside the sample. The 2D dia-grams on the upper left show spheres (blue) and their Voronoicells (yellow) for cases with Voronoi neighbors that are dis-ordered loosely packed (left) and symmetric densely packed(right). (b) Histograms of local volume fractions reveal thecoexistence of crystalline and amorphous regions inside thesample. Below the transition there is only a single peakeddistribution, which shifts towards higher densities until theplateau is reached. After the onset of crystallization the pre-vious peak population diminishes and a new peak appears atφlocal = 0.74. The histogram for the plateau is the averageof 32 consecutive scans; elsewhere the histogram is the aver-age of 10 scans. Colors in (b) correspond to the colored datapoints in (a).

cleation in a system of athermal spheres. The emer-gence of a plateau at a well-defined packing fraction,φglobal = 0.645, indicates the onset of a first order phasetransition. A previous experiment used a setup similarto ours and observed inhomogeneous nucleation at thecell walls but no nucleation in the interior of the cell [6].Another experiment used shaking rather than shearingto study compaction, but the evolution of the nucleationdynamics was not analyzed ([8, 9, 36] and SupplementaryMaterial in [8]).

Classical nucleation theory (CNT) has been success-fully used to interpret packings of thermal hard colloids,where the size of critical nuclei was measured to be 60–160 spheres [3]. We find that a similar approach does notwork for our results for athermal hard spheres. Our mainresult with respect to the CNT framework is an interfacedensity larger than the bulk density for nuclei smallerthan 20 spheres (Figs. 4(b,c)); hence it is energetically fa-vorable for a nucleus to grow. Therefore, the free energyhypothesis of CNT is in contradiction with the observednegative growth probability of small nuclei (Fig. 4(a)).We conclude that CNT seems to be the wrong frame-work to describe nucleation in athermal sphere packings.Further, we found no qualitative differences if the inter-faces of growing and shrinking nuclei were analyzed sepa-

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Figure 3. (a) This cross section of the cell after 106 cyclesshows crystalline regions in the cell interior; the amorphousphase is not shown. The colors of the nucleated spheres in-dicate crystal type. The wire frame indicates the inner partof the shear cell that is used for evaluation. (b) The nucleusthat started to grow first (encircled in (a)) fluctuates in shapeand size until a stable seed was reached at about 3× 105 cy-cles. (c) Time evolution of the nucleus in (b). A movie in theSupplemental Material [34] illustrates the nucleation in theshear cell.

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rately (see Supplemental Material [34]). We also checkedfor two-step nucleation as recently described for colloidalsystems [5], but we found this did not fit our observationsof growing nuclei with initial FCC or HCP symmetrywithout any intermediate phase (Fig. 3).

In conclusion, the present experiment revealed an onsetof nucleation growth at 70,000 shear cycles in a systemof 49,400 spheres, which indicates that future numericalsimulations of sheared packings should extend well be-yond a recent study of 2,000 frictional grains cyclicallysheared for 2,000 cycles [17].

We thank Markus Benderoth, Thomas Eggers, TiloFinger, Kristian Hantke, Wolf Keiderling, Udo Krafft,

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Figure 4. (a) The growth (shrinking) probability of a nucleusdepends on its size: nuclei with about ten or more spherespredominately grow (∆ρ > 0). (b) Local volume fraction asa function of distance from the surface of a nucleus. Largenuclei (red) show an initial dip [37], but there is no dip forsubcritical nuclei (black), and very small nuclei have a minoreffect on the surroundings (dashed). (c) The packing densitynear the surface of a nucleus – in the range in (b) between thevertical dotted lines – is higher for the small nuclei that tendto dissolve than for the large nuclei that tend to grow. Thehorizontal dashed lines in (b) and (c) indicate the amorphousplateau far away from any nucleus. The symbol size in (a)and (c) indicates the number of examples observed.

and Vishnu Natchu for technical support and discus-sions. The project was financed by grants from GermanAcademic Exchange Service (DAAD), German ResearchFoundation (DFG) STA 425/24 and Cluster of ExcellenceEngineering of Advanced Materials, and in part by NSFDMS-1509088.

∗ email: [email protected]† email: [email protected]

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[34] See Supplemental Material at [URL will be inserted bypublisher] for a movie and additional evaluations.

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[36] For a characterization of the melting process of a crys-talline packing using persistent homology see also [10].

[37] Alexey V. Anikeenko and Nikolai N. Medvedev (Labora-tory of Molecular Dynamics and Structures, Novosibirsk,Russia) have observed depletion around large nuclei inathermal packings that were generated by Lubachevsky-Stillinger [26] and Jodrey-Tory [24] densification algo-rithms (private communication).

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Supplemental Material: Nucleation of athermal hard spheres

Frank Rietz, Charles Radin, Harry L. Swinney, Matthias Schroter

movie-spheres-nucleate-Rietz.avi

The movie shows the nucleation of spheres in the shear cell. For visualization only the spheres that are in a crystallinestate are shown in a rotating side view. Color indicates the crystal symmetry. The current state is quantified in thetop right by global packing fraction, shear cycle, and the number of crystalline spheres. The end of the plateau inthe packing fraction is marked by the advent of the first growing nucleus, which is encircled in the rotating animationand depicted at a constant viewing perspective on the lower right. The wire frame indicates the inner part of thecell, while the green and brown frames are parallel to the shear walls. The bottom of the interrogation volume is cutnon-orthogonally because of optical accessibility.

Precision of sphere coordinates

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Figure S1. (a) The accuracy of the detected coordinates of the spheres with diameter 3± 0.0025 mm is derived from the widthof the first peak of the radial distribution function [35]. For ideal monodisperse spheres and known coordinates there wouldbe a step function at a distance of one particle diameter. Under experimental conditions the maximum of the peak is used asthe mean particle diameter and the width of the left shoulder characterizes the precision of the sphere coordinates. (b) Theoverlap length for a given percentile of overlapping contacts. Before counting the number of overlaps, spheres are sorted inincreasing order, i.e., the smallest overlaps are first. To exclude outliers the width of the left tail is defined by the largest of the99% smallest overlaps, the quantile q99. An accuracy of the sphere positions of 2% of a diameter is deduced. The blue rangein (a) indicates the overlap by the smallest spheres if only the manufacturer’s size tolerance is considered, i.e., the maximumshoulder width if the coordinates were precisely known. The coordinate error due to size tolerance is negligible compared tothe inaccuracy of the detection method. The evaluation includes 2.45× 106 calculated sphere coordinates.

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No qualitative influence of different thresholds for crystal classification

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Figure S2. Combinations of different thresholds for q6 and φlocal do not change the general picture of preferred dissolutionof nuclei with fewer than approximately ten spheres. In this paper spheres are classified as crystalline if φlocal > 0.72 and∆q6 < 0.02, i.e., if q6 is either in the range q6(FCC)=0.575±0.02 or q6(HCP)=0.485±0.02 (circle symbol). The rotationallyinvariant parameter q6 is used in the literature to characterize the local order of a sphere by considering the relative positions ofits surrounding particles [27]. The form of q6 we use weighs neighbors by the area of the Voronoi faces that they share with thecentral sphere [27]; thus close neighbors are more influential than distant ones. Symbol size indicates number of observations.

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Figure S3. Combinations of different thresholds do not change the general picture of an absence of depletion zones aroundsubcritical nuclei. The red solid line corresponds to the parameter used in the paper.

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No qualitative influence of a different crystal definition

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Figure S4. The results do not depend on the selected definition of local crystallinity as shown here. In addition to q6crystallinity can also be geometrically characterized by comparison of a given Voronoi cell with the Voronoi cells for FCC andHCP [19]. For comparison a rotationally invariant fingerprint is calculated from the face normals of a single Voronoi cell. Asphere has either FCC or HCP symmetry if the difference of their fingerprints is below a threshold, ∆FCC,HCP. Symbol sizeindicates number of observations.

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Figure S5. Selection of a different definition for local order checked for two thresholds does not yield in a qualitative changeof properties at the nucleus interface.

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Influence of bin size on interface behavior

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Figure S6. Influence of bin size on φlocal at the nucleus interface. Left column: bin size 0.5 diameter, as used in paper; rightcolumn: bin size 0.01 diameter. The bin step is always 0.005 diameter. The number density at higher resolution is a variantof the radial distribution function. The pronounced peaks at

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No qualitative difference in the comparison of growing and shrinking nuclei

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Figure S7. The nuclei are split in groups of growing and shrinking nuclei. There is no qualitative difference in φlocal at thenucleus interface.

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