Nucleation in sheared granular matter

TL;DR

This study demonstrates a first order phase transition in sheared granular spheres from disordered to ordered states, driven by homogeneous nucleation, with critical nuclei forming at a specific volume fraction.

Contribution

It provides experimental evidence of nucleation-driven crystallization in granular matter under shear, highlighting the transition mechanism and critical nucleus size.

Findings

Transition occurs at 64.5% volume fraction.

Nucleation involves dissolution of small nuclei and growth of critical nuclei.

Crystallites exhibit FCC and HCP symmetries.

Abstract

We present an experiment on crystallization of packings of macroscopic granular spheres. This system is often considered to be a model for thermally driven atomic or colloidal systems. Cyclically shearing a packing of frictional spheres, we observe a first order phase transition from a disordered to an ordered state. The ordered state consists of crystallites of mixed FCC and HCP symmetry that coexist with the amorphous bulk. The transition, initiated by homogeneous nucleation, overcomes a barrier at 64.5% volume fraction. Nucleation consists predominantly of the dissolving of small nuclei and the growth of nuclei that have reached a critical size of about ten spheres.

Full text

Packings of spheres show interesting features such as phase transitions between disordered and ordered states, and can be useful in the study of amorphous atomic configurations Finney2013 . Examples include thermal colloidal packings Pusey1986 ; Gasser2001 ; Liber2013 , packings of macroscopic granular spheres Scott1960_Scott1969_Finney1970 ; Finney2013 ; Bernal1964_Nicolas2000_Mueggenburg2005 ; Panaitescu2012 ; Francois2013 ; Hanifpour2014_Hanifpour2015_Saadatfar2017 ; Tsai2003_Daniels2006 ; Chen2006_Slotterback2008 ; Knight1995_Richard2005 ; Schroeter2005 ; Berryman1983 ; Komatsu2015 , and simulations of the mathematical hard sphere model Royer2015 ; Kapfer2012 ; Mickel2013 ; Anikeenko2007_Baranau2014 ; Jin2010 ; Wood1957_Alder1957_Hoover1968 ; Auer2004 ; Sanz2011_Valeriani2012 ; Yanagishima2017 ; Sanz2014 . The behavior of such systems is determined by the fraction of space filled by the spheres, their packing fraction $\phi$.

The hard sphere model exhibits an entropically driven first order phase transition. Disordered fluid states are observed below the freezing density of $\phi=0.495$ and crystalline ordered states appear above the melting density of $\phi=0.545$, with coexistence of the two phases for intermediate densities Wood1957_Alder1957_Hoover1968 .

Granular spheres can also be packed in disordered and ordered states. In contrast to colloidal packings and the mathematical hard sphere model, granular packings are characterized by the existence of permanent contacts between the particles. Most experimental protocols for increasing the bulk volume fraction, such as vertical shaking Knight1995_Richard2005 , centrifugation Liber2013 , thermal cycling Chen2006_Slotterback2008 and sedimentation Schroeter2005 , do not achieve an ordered state from an initial disordered state. The “random close packed state” (RCP) is used operationally to describe the highest density state achieved by these methods. The RCP volume fraction is in the range $0.635<\phi<0.655$ Scott1960_Scott1969_Finney1970 ; Berryman1983 ; Anikeenko2007_Baranau2014 ; Kapfer2012 , about $15\%$ lower than the densest possible packing of ordered face centered cubic (FCC) or hexagonal close packing (HCP), which each have a volume fraction $\phi=\pi/\sqrt{18}\approx 0.74$ Hales2012 .

Ordered clusters of granular spheres have been obtained with a system density $\phi_{global}$ in the range 0.64–0.74 by multidimensional shaking Francois2013 ; Hanifpour2014_Hanifpour2015_Saadatfar2017 , cyclic shear Bernal1964_Nicolas2000_Mueggenburg2005 ; Panaitescu2012 , and shear in a Couette cell Tsai2003_Daniels2006 . By analogy with the freezing-melting transition in the hard sphere model, the emergence of growing crystallites found in our granular experiment can be interpreted in terms of nucleation and a first order phase transition Radin2008_Aristoff2010 ; Jin2010 .

In our experiment we compact a granular packing by shearing. With increasing shear cycles, a well-defined plateau emerges at phase transition density $\phi_{global}=0.645$, which is in the range of densities associated with RCP Scott1960_Scott1969_Finney1970 ; Berryman1983 ; Anikeenko2007_Baranau2014 ; Kapfer2012 . Such a plateau was not reported in a previous experiment that used a setup similar to ours, and the growth rate of nuclei that we find differs qualitatively from that found in the previous experiment Panaitescu2012 .

Our experiment uses a cubical shear cell with side length 10.5 cm (Fig. 1(a)). Nucleation from the side walls is suppressed using half-spheres glued to the walls at random positions (glue: Vitralit 7562, Panacol). The cell is filled with 49,400 precision BK7 glass spheres of diameter 3 $\pm 0.0025$ mm and density 2.51 g/cm3 (size tolerance given by manufacturer, Worf Glaskugeln GmbH). The top plate applies a pressure of 2.1 kPa on the bed, and the pressure increases downward to 3.1 kPa at the bottom. The cell is sheared by sinusoidally tilting opposite vertical shear walls by $\pm 0.6^{\circ}$ about axes indicated by red circles about half-way up opposite side walls (Fig. 1(a)); the total peak-to-peak oscillation displacement at the bottom of the cell is about one-third of a sphere diameter.

The glass spheres are index matched Dijksman2012 to a mixture of phthalate esters (Cargille Laboratories, identifier code 1160, $\rho=1.1$ g/cm3, $\nu=41$ cSt) with a dissolved fluorescent dye (oxazine 750 perchlorate, $c=10$ mg/l). Hydrodynamic interactions are negligible for our small amplitude shear with a period of 2 s, and sphere deformation is also negligible. The spheres, half-spheres, and shear cell walls are made of BK7 optical glass and are index matched with the liquid mixture in a temperature controlled environment ($n=1.5198\pm 0.0001$ at $\lambda=589$ nm and $T=22.9\pm 0.1^{\circ}$C).

The shearing process is periodically stopped to measure the packing using an horizontally translated vertical laser light sheet ($\lambda$=658 nm, $P$=75 mW). Fluorescent light is imaged simultaneously by the camera (Figs. 1(b) and 1(c)). The image slices are combined to form a three-dimensional volume. The measurements presented here are from 618 scans made during a run with $1.955\times 10^{6}$ cycles. In the course of the two month long run 10% of the spheres escaped from the sample cell through a gap between one of the shear walls and a side wall, and this possibly increased the mobility of spheres farther inside the packing, thereby enhancing their crystallization.

Positions of the 20,000 spheres at least 3 diameters from any wall are detected by convolution with a template. Then the peak of the correlation map in the ($x$,$y$,$z$) directions is determined with (37,37,30) pixel/diameter resolution using a 3-point Gauss estimator Raffel2007 . This yields the position of each particle. From the pair correlation function we can determine, for more than 99% of the particles, the position of each particle to less than 2% of a sphere diameter (see Supplemental Material supp ).

For each sphere there is a Voronoi cell consisting of all points closer to that sphere than to any other sphere in the sample (see insets in Fig. 2(a)). $\phi_{local}$ is then the ratio of the sphere volume to the volume of its Voronoi cell. The mean volume fraction of the whole sample, $\phi_{global}$, is given by the harmonic mean of all $\phi_{local}$ values Weis2017 .

The angular order between spheres sharing a face of their Voronoi cells is characterized by a weighted version of the order parameter $q_{6}$ Mickel2013 . A sphere is called crystalline if it is densely packed and its neighbors are ordered, i.e., $\phi_{local}>0.72$ and $q_{6}$ is either in the range $q_{6}$(FCC)=0.575$\pm$0.020 or $q_{6}$(HCP)=0.485$\pm$0.020. Other spheres are called amorphous. A nucleus is a connected set of crystalline spheres, each sharing at least one Voronoi face with another sphere in the set. Some authors suggest different definitions for local order Kapfer2012 ; Faken1994_Bargiel2001_Stukowski2012_Tanaka2012_Leocmach2013 , but our results do not depend qualitatively on the choice of definition or threshold (see Supplemental Material supp and a recent review Reinhart2017 ). The distance of a nucleus to another single sphere or nucleus is given by the 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 a disordered state and compacts approximately logarithmically with time for about 20,000 cycles. The compaction then slows to a stop, and the second region, a plateau ($\phi_{global}=0.645$), emerges and persists for about 50,000 cycles. Then the first growing nucleus appears, indicating a first order phase transition to the third distinct region. The volume fraction slowly begins to increase as shearing continues, and nuclei increase in number and size but have 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 illustrates that nuclei fluctuate in shape, size, and crystal symmetry as they grow, as can be seen in the movie in the Supplemental Material supp .

All nuclei start their growth at least 10 sphere diameters distance away from any wall. By the end of the experiment, after two million shear cycles, nuclei with up to $\sim 600$ spheres are present, and 9% of all spheres in the analyzed volume are in crystallites, which have FCC or HCP symmetry with approximately equal probability; no icosahedral symmetry was observed (see Supplemental Material supp ).

Histograms of local densities for amorphous packings have a single peak and are approximately symmetrical about that peak, as Fig. 2(b) illustrates. During compaction the peak narrows slightly and shifts to higher densities. Beyond the first order phase transition a second peak emerges at $\phi_{local}=0.74$, the density of densest packed arrangements.

The end of the densification plateau is identified by the emergence of a nucleus in the interior region; subsequently all nuclei are tracked for each successive scan. Nuclei with fewer than about ten spheres are found to shrink more often than grow, while growth gradually becomes more probable for nuclei with more than ten spheres, that is, $\Delta\rho=(\rho_{grow}-\rho_{shrink})>0$, where $\rho$ is the probability to grow or shrink, respectively Gasser2001 ; Panaitescu2012 (Fig. 4(a)). The critical size at which the difference of the probabilities $\Delta\rho$ becomes positive does not depend on the definition of local crystallinity (see Supplemental Material supp ).

Our main result is the observation of nucleation centers throughout the interior of a granular packing. The emergence of a plateau at a well-defined packing fraction, $\phi_{global}=0.645$, indicates the onset of a first order phase transition. A previous experiment used a setup similar to ours and observed heterogenous nucleation at the cell walls but no nucleation in the interior of the cell Panaitescu2012 . Another experiment used shaking in a spherical container which may or may not have suppressed heterogeneous nucleation; however, the spatiotemporal evolution of the nuclei was not determined in this study (see Francois2013 ; Hanifpour2014_Hanifpour2015_Saadatfar2017 and Supplementary Material in Francois2013 ).

Our experiment analyzes static granular matter between episodes of cyclic shearing, which increase the density. Our data are snapshots, measurements of static packings. When the snapshots are viewed consecutively, as in a movie, nuclei emerge, grow, and shrink until after sufficient shearing cycles, nuclei of critical size are created that then grow indefinitely under further shearing (see movie in Supplementary Material supp ). A nucleation theory should explain why shearing at low amplitude and low frequency, in the presence of pressure and gravity, leads to such a sequence of static granular configurations. This is distinct from the usual nucleation theory, which shows that a thermal or Brownian dynamics, starting from a supercooled state, leads to the creation, growth, and shrinking of nuclei.

The observed compaction of a granular bed under gravity and pressure can be described by a simple mechanical picture. A bed of frictional hard spheres increases in density if subjected to repeated small disturbances that are introduced, for example, by shearing, shaking, or fluidization. The repeated shear cycles in our experiment break some of the force chains that form a skeleton supporting the bed, and this leads to compaction Yanagishima2017 . However, this simple picture does not address the homogeneous nucleation seen in our experiment (Fig. 4(a)); very dense crystallite clusters ($\phi=0.74$) form, grow, and dissolve without the help of a flat wall. Figure 4(b) shows that near a small nucleus the volume fraction is slightly higher than the background volume fraction, which suggests from a mechanical argument that under a confining pressure small nuclei should grow. However, Fig. 4(a) reveals that this does not happen. Any theory for nucleation in our system would have to account for these results in Fig. 4.

Our results are qualitatively different from the crystallization dynamics observed in constant volume molecular dynamics simulations of supercooled hard spheres: the existence of a nucleation barrier in our system disagrees with the autocatalytic growth of crystalline regions observed in Sanz2011_Valeriani2012 . Further, we do not observe burst-like growth events as seen in mature glasses Yanagishima2017 ; Sanz2014 . These differences indicate that nucleation in our sheared granular system is governed by its own dynamics.

In conclusion, we have found that small amplitude cyclic shear of a bed of spherical particles under gravity and pressure leads to compaction until a well-defined random close packed volume fraction is reached. Then after many more cycles (50,000 in our experiment), clusters with HCP and FCC symmetry emerge, and these crystallites grow if they contain about ten or more particles. Previous experiments Bernal1964_Nicolas2000_Mueggenburg2005 ; Tsai2003_Daniels2006 ; Panaitescu2012 ; Francois2013 ; Hanifpour2014_Hanifpour2015_Saadatfar2017 had observed nucleation centers on flat confining walls, but our experiment shows nucleation throughout the interior of the cell. Further, the present experiment on 49,400 spheres indicates that future numerical simulations of sheared packings should extend well beyond a recent study of 2,000 frictional grains cyclically sheared for 2,000 cycles Royer2015 .

We thank Markus Benderoth, Thomas Eggers, Tilo Finger, Kristian Hantke, Wolf Keiderling, Udo Krafft, and Vishnu Natchu for technical support and discussions. The project was financed by grants from German Academic Exchange Service (DAAD), German Research Foundation (DFG) STA 425/24, Cluster of Excellence Engineering of Advanced Materials, and in part by NSF DMS-1208941, NSF DMS-1509088 and Robert A. Welch Foundation Grant F-0805.

Appendix A Supplemental Material: Nucleation in sheared granular matter

Frank Rietz, Charles Radin, Harry L. Swinney, and Matthias Schröter

Appendix B A. movie

Link to movie: https://youtu.be/_oV-WwtW4Xo

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

Appendix C B. Precision of sphere coordinates

Appendix D C. Absence of icosahedral symmetry

Appendix E D. No qualitative influence of different thresholds for crystal classification

Appendix F E. No qualitative influence of a different crystal definition 1:

Geometry of Voronoi cells

Appendix G F. No qualitative influence of a different crystal definition 2:

Common neighbor analysis

Appendix H G. No qualitative influence of a different crystal definition 3:

Bond angle analysis

Appendix I H. Influence of bin size on interface behavior

Appendix J I. No qualitative difference in the comparison of growing and shrinking nuclei