Liquid-solid phase transition in a confined granular suspension

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by  Nariaki Sakaï

Liquid-solid phase transition in a confined granular suspension
Nariaki Saka¨ı,∗ S´ebastien Moulinet, and Fr´ed´eric Lechenault
Laboratoire de Physique Statistique, Ecole Normale Sup´erieure,
PSL Research University, UPMC, CNRS, 75005 Paris, France
Mokhtar Adda-Bedia
Univ. Lyon, ENS de Lyon, Univ. Claude Bernard,
CNRS, Laboratoire de Physique, F-69342 Lyon, France
(Dated: 10 janvier 2017)
We experimentally investigate the phase behaviour of a model two-dimensional granular system
undergoingstationarysedimentation.Buoyantcylindricalparticlesarerotatedinliquid-filleddrum,
thusconfinedinaharmoniccentripetalpotentialwithtunablecurvature,whichcompeteswithgra-
7
vity to produce various thermal-like states : the system can be driven from liquid-like to solid-like
1
configurationsastherotationrateisincreased.Westudythestatisticsofthelocalhexagonalorder
0
2 across this transition, which exhibits large fluctuations and heterogeneities as the dense phase of
the system develops spatial ordering. In particular, the spontaneous appearance of locally crystal-
n
lized regions in the disordered phase points to the existence of an underlying critical point : size
a
distribution displays scale invariance when approaching the transition, which in turn occurs when
J
thepackingfractioncrossestherandomclosepacking.Thissuggeststhattheorderingtransitionis
9 only due to geometrical effects, in contrast with other experiments on liquid-solid phase transition

ofgranularmedia.Finally,asimpleforcebalanceexplainswhentheperfectcrystallinestateoccurs
]
t yielding an upper bound for the critical frequency of the transition.
f
o
PACSnumbers: 83.80.Hj,47.57.E,47.57.Gc
s
.
t
a
m Statistical approaches to the phase behavior of granu- equilibrium situations [9, 19–22].
lar matter have flourished during the past twenty years,
- On the one hand, most of the experimental studies on
d fromkinetictheoriestoEdwardshypothesis,culminating
dense granular media and their liquid-solid phase tran-
n with the jamming paradigm [1]. However, no unifying
sition have been carried out by controlling the packing
o
framework has emerged that captures the physics of this
c fraction,e.g.bychangingthenumberofparticlesinagi-
class of systems the way thermal statistics do for mo-
[ ven fixed volume with hard walls [4]. On the other hand,
lecular systems. Energy dissipation and athermality are simple granular sedimentation experiments cannot be
1
two defining features of granular matter : since thermal
v sustainedwithindynamicalsteady-statesforlongenough
fluctuations are irrelevant for millimeter-sized particles,
9 to decipher the resulting statistical ensemble they evolve
0 achievingadynamicalsteadystaterequiresacontinuous in. Here we present an experimental situation where a
2 energy injection to compensate for the dissipative pro-
buoyantgranularsuspensionisinsteadconfinedinahar-
2 cesses.Thisusuallytakestheformofamechanicalagita-
monic trap with tunable curvature and maintained in a
0
tion which plays the role of the thermal bath. When this
. continuous sedimentation state. In addition to gravity
1 is achieved, particles behave like a fluid, and for 2D mo-
effects induced by the mismatch in density between the
0 nodisperse systems, the granular fluid crystallizes when
fluid and the grains, a centripetal confining pressure is
7
the density of particles is increased [2–6]. The transition
1 adjusted by changing the rotation rate of the system.
alsooccursbycoolingthesystemi.e.decreasingthemean
: Thecompetitionbetweenthisconfinementandbuoyancy
v
kinetic energy by decreasing the injected power [5, 7–9].
allows to select the density profile of the assembly and
i
X Particles can also interact with each other through elec-
thus to explore various packing states. Furthermore, ins-
r trostatic [10] or magnetostatic [11, 12] interactions, in tead of solid friction, the particles are coupled through
a which case it is possible to obtain a transition by tu-
hydrodynamic interactions. Altogether, the system can
ning the strength of the interaction. These observations
be continuously driven from a dispersed to a crystalline
are very similar to what is observed in simulations of
state, undergoing in particular a disordered to ordered
harddiskswithelasticcollisions[13,14,17],orforcolloi-
phase transition.
dal systems [15–18], despite the fact that granular fluids
In this Letter, we study the transition of the system
are out of equilibrium. However, the depth of this ana-
from a loose liquid-like state towards an ordered crystal
logy remains elusive, partly due to dissipative processes
as a function of the confinement. We first focus on the
likesolidfriction,e.g.theso-calledgranulartemperature
spatial arrangement of particles in the harmonic trap,
doesnotequilibratebetweenphaseswhenthereiscoexis-
and characterize the transition by the so-called bond-
tence [7]. This leads to question to what extent concepts
orientational order parameter. We find that the mean
from thermodynamics can be exported to these out of
size of monocrystalline regions grows sublinearly beyond
2
acriticalrotationrateatwhichthespatialdistributionof The phenomenology of this system is quite rich : at
the structure is maximally heterogeneous. The size dis- very low rotation rate, the grains float up to the top of
tributions of these regions display scale invariance in the the cell and avalanche similarly to what occurs in a par-
vicinityofthisrotationrate.Furthermore,thecut-offsize tially filled rotating drum [24], a regime we do not study
of these distributions scales with the distance to random here. Three pictures of the experiment at different rota-
closepacking,confirmingthecriticalnatureofthetransi- ting rates are shown in Fig. 1; they correspond respec-
tion. Finally, we discuss the mechanism of the transition tively to the fully dispersed state where the interparticle
as the onset and equilibration of internal forces and in- distanceislargerthanthegrainsdiameter,thestatenear
terpret the value of the critical packing fraction as the thecriticalpointwhereadense,disorderedregionhasin-
random close packing. vadedasignificantcentralregion,andfinallyastatewith
alargeorderedcrystalinthecenterofthecellsurrounded
by a gas-like ring. An illustrative movie of the different
a) b)
phases is provided in [25].
Let us now detail the physical ingredients at play wi-
thin the system. The rotation of the cell creates a ra-
dially linear outward pressure gradient for the buoyant
particles, resulting in a centripetal force. On top of this
rotationally symmetric potential, gravity pushes on the
particlesupwards.Interestingly,thecouplingtothefluid
0.34 Hz 0.59 Hz in this Hele-Shaw geometry yields a strong drag on the
c) d)1 (cid:1) 0.34 grains, which in the regimes of interest tend to co-rotate
C
0.45 withthecell.Thusintherotatingframe,gravityisaspin-
0.8 (cid:1) 0.55
RCP 0.64 ning force that tends to disperse the particles outwards,
(cid:1)(r)0.6 00..7857 competing with the confining potential and injecting po-
0.4 wer into the system. Changing the rotation rate shifts
0.2 thebalancebetweentheseforces.Inthepolarcoordinate
system((cid:126)e ,(cid:126)e )associatedwiththelaboratoryframe,the
0 r θ
0 20 40 60 equation of motion of a single grain reads :
r/d
0.87 Hz
d(cid:126)v
ρV =µ((cid:126)v−ωr(cid:126)e )+(ρ−ρ )V(cid:126)g−ρ Vω2r(cid:126)e , (1)
Figure 1. Snapshots of the system at different rotation fre- dt θ l l r
quencies:(a)0.34Hz,(b)0.59Hz and(c)0.87Hz.Notethat
for the latter, the system is composed of a monocrystal sur- where ω =2πf is the rotation rate, µ is the viscous drag
rounded by a gaseous phase. (d) Mean packing fraction pro- coefficient measured to be 1.1×10−3Nm−1s in the ex-
files at 6 values of rotation rates (in Hz). The dashed hori- periment, V is the volume of the particle, r is the radial
zontal line corresponds to the random close packing and the positionofitscenterofmassawayfromthecenterofthe
bold line to the crystal packing fraction.
cell and (cid:126)v is its velocity. Interestingly, this equation ad-
mits a fixed point solution (r∗,θ∗) given by r∗ =l sinθ∗
c
Theexperimentalsetupisinspiredfrom[23].Itconsists andtanθ∗ =−ρlω,withl = (ρl−ρ)g beingdefinedasthe
µV c ρlω2
inputtingamonolayerofN ≈3800cylindricalparticles confinement length. Thus the trajectory of a single par-
0
of diameter d = 4mm, height h = 2mm and density ρ = ticleinaninfinitesystemwouldspiraltowardsthispoint
1.01×103kg·m−3 inatwo-dimensionalcylindricaldrum for all finite rotation rates. However, due to the collec-
of radius R = 250mm. The two parallel plates of the tive nature of the system, the hydrodynamic coupling
0
drumareseparatedbyadistance1.5handthecellisfilled between particles at the intermediate Reynolds numbers
with a solution of cesium chloride of density ρ =1.55× at hand tends to spread them apart, as verified indepen-
l
103kg·m−3, making the liquid denser than the particles. dentlythroughadditionaltwo-particlesedimentationex-
The axis of the cell is horizontal, so particles feel the periments,resultingintheobservedstationarydispersed
effect of gravity, and the system is rotated with a motor states.
(ABB Motors) at a frequency f ranging from 0.35Hz to To characterize the spatial distribution of particles in
0.95Hz. A high resolution camera (SVS Vistek) is placed the cell, we first compute the mean packing fraction
in front of the cell and triggered by means of an optical φ(r) = < dA(r)/2πrdr > where dA(r) is the cumula-
fork once every cycle. We focus on the statistics of the ted area of the particles located in a ring of width dr at
packingcomputedoveraseriesof2000picturespervalue distance r from the instantaneous centre of mass of the
of the rotation rate. Particles are hollow cylinders made grains,whichdoesnotcoincidewiththecenterofthecell.
of white polystyrene and filled with a green silicon core, ThisquantityisdisplayedinFig.1(d).Atlowfrequency,
in order to ease their detection : error on the positions is this profile is roughly parabolic. However, at f ≈0.4Hz,
of the order of 20µm. the packing fraction starts to develop a plateau at the
3
a) (cid:1)10-3 b) andhighrotationrates,butsharplyriseandpassthrough
1 2.5
<m> a maximum at f ≈0.64Hz, much like in the vicinity of
0.8 Var(m) 2 c
a critical point.
0.6 1.5 If two particles belong to the same monocrystal, the
0.4 1 orientationsoftheirψ arethesame.Thus,inordertobe
6
0.2 0.5 able to distinguish between neighboring ordered regions
with different orientations, we define for every particle
0 0
0.4 0.6 0.8 1 k the first-neighbor correlation C(Ψk) = ψ(cid:126)k. < ψ(cid:126)l >
f (Hz) 6 6 6 l
c) 0.34 d) which is the scalar product of the bond-orientational or-
0.39 der parameter of the particle k and its first neighbors -
1n(s)00 1 -2 0000....45555157 s10c2 1/(cid:3) = 2.2 twheeeaxvperreasgseth<e.co>rlreilsattaioknenbyovtearkianlgl fithrsetvneecitgohrbiaolrfsorlmanodf
the complex number ψ . A snapshot of the particles dis-
6
10-4 100 1 tributionatthetransitionisdisplayedinFig.2(b).When
the system approaches this ordering transition from the
100 101s 102 103 100 1(cid:1)0(cid:2)-1 10-2 disordered state, many locally crystallized clusters ap-
pear in the liquid phase. These clusters are intermit-
Figure2.(a)Timeaverageandvarianceoftheorderparame- tent and grow in size when the system approaches the
ter m=|1/N (cid:80)N0 ψk|. (b) Instantaneous configurations of transition. Above the transition frequency, they perco-
0 k=1 6
thesystematf =0.59Hz,whereparticlesarepatchedwitha late to form a unique connected region, but heterogenei-
colorcorrespondingtothevalueofthefirstneighborscorrela- tiescontinuetobelarge,asembodiedbytherugosityand
tionC(ψ ).Particleswhicharecrystallizedareyellow(bright)
6 fluctuationsoftheinterfacebetweenthemonocrystaland
incontrastwithparticlesinadisorderedphasewhichareblue
its surroundings. When the system is driven away from
(dark).(c)Sizedistributionsofmonocrystalsfordifferentva-
the transition point, this rugosity decreases and the mo-
lues of the rotation frequency. (d) Characteristic cut-off size
of these distributions as a function of the distance to φ . nocrystal becomes circular, where interface fluctuations
RCP
are only due to adsorption and desorption of particles at
the interface.
centerasfrequencyisincreased,correspondingtoadense In order to quantify these heterogeneities, we mea-
region where particles are in contact. At high rotation sure the size s and perimeter p of the crystalline clus-
rates f > fcrys ≈ 0.8Hz, the value of the plat√eau tends ters. They are defined as ensembles of Voronoi neigh-
to the packing fraction of the crystal φc = π/ 12 ≈ 0.9 bors that all have C(ψ6) larger than a given threshold
as the assembly fully orders. fixed at 0.5. We have checked that the results are in-
The characterization of the state of the granular as- sensitive to a variation of a thresholding around this va-
sembly is monitored using the bond-orientational or- lue. Size and perimeter correspond respectively to the
der parameter ψk [14]. For each particle k, one defines number of particles in the cluster and on its boundary.
6
ψ6k =1/Nk(cid:80)Nl=k1e6iθkl, where the sum runs over all first First, we find that approaching the transition from the
neighboring particles l and θ is the angle between the disordered side, the size distribution of these clusters
kl
line from particles k to l and a globally fixed arbitrary displays a power-law regime stretching further towards
axis - the neighbors are found by computing the Voronoi larger sizes as the transition is approached, as can be
diagram from the particles positions. For particles in a seen in Fig.2(c). This scale invariance developping in a
hexagonally ordered neighborhood, |ψ | = 1. Moreover, finiteregionaroundthetransitionisaclearindicationof
6
this quantity is constructed such that its phase is the criticality. Furthermore, we find that in this parameter
same for all particles in the same monocrystal. In order range, the relation between the size and perimeter of the
tocapturetheglobalstateofthesystem,wedefineamo- clusters is also a power law with a non-trivial roughness
nocrystalline order parameter as m = |1/N (cid:80)N0 ψk|, exponent s ∝ p1.2 (Fig. 3(a)). As the rotation rate is
0 k=1 6
where the sum runs over all particles in the system. The further increased, the unique monocristalline region that
time average and variance of this quantity are shown crosses the system continues to grow in size, but its per-
in Fig. 2(a). At low rotation rates, the time average imeternowdecrease,resultinginavanishingroughness:
is constant and not significantly different from zero. at high rotation rates, sizes and perimeters are just rela-
It starts rising around f = 0.4Hz and tends towards ted by s=p2/4π which corresponds to a perfect circular
m ≈ 0.8 at high rotation rates. Notice that if all the shape, as emphasized in Fig. 3(a). In the same spirit, we
grainswereinacrystallinestate,oneshouldhavem≈1, have computed the radius of gyration of each cluster s
however even for the highest frequency we probed one is asgivenbyR2(s)=1/s(cid:80)s ((cid:126)r −<(cid:126)r(s)>)2.Fig.3(b)
g k=1 k
still left with a thin layer of grains in the liquid state shows that the cluster size is a power-law function of R
g
whichmakestheorderparameterm<1.Moreover,fluc- with an exponent d ≈ 1.8, as one would expect from a
f
tuations of this order parameter are close to zero at low percolation-like transition.
4
a) b) confinement is sufficiently loose to enable the buoyancy
0.34 0.34
103 00..4555 00..4555 to disperse the grains across the cell, whereas at higher
0.64 0.64
0.75 f 102 rotation rates it becomes strong enough to bring some
s 0.87 dr = 1.2 s df = 1.8 of the grains to contact (cf. Eq. 1). When the grains are
1 1 1 packed in a perfect crystalline state, there is equilibra-
10
0 tion of the two forces at the boundary of the crystal,
10
which starts being able to sustain internal stress. This
0 1 2 0 1
10 10 10 10 10
p R /d transition occurs typically when the confinement length
g
c) d) l = (ρl−ρ)g becomes comparable to the typical radius
101 Rc ≈(d/ρl2ω)2(cid:112)N /φ ofthecrystallinephase.Thisleadsto
2 0 c
10
aanestimatefest =0.82Hzofthefrequencyatwhicha
> d crys
<s (cid:3)/ (cid:4) = 1 crystal is formed, which is reasonably close to the mea-
(cid:3) = 1.5
sured value f ≈ 0.8Hz and leads to an upper bound
100 1 crys
1 for the critical frequency f ≈0.64Hz.
0 c
10
0 -1 -2 0 -1 -2
10 10 10 10 10 10
(cid:1)(cid:2) (cid:1)(cid:2) In summary, we report an experimental study of a
liquid-solid-like phase transition in a two dimensional
Figure3.(a)Scatterplotofsizesvs.perimeterpoftheclus- granular suspension in which the state of the system
ters for rotation frequencies ranging from 0.34Hz to 0.9Hz.
is controlled by applying an adaptive confinement. The
Each point correspond to one cluster and each color corres-
transition is controlled by the competition between the
pond to a given rotation rate. When the rotation rate is in-
buoyancy and the centripetal force which allows us to
creased(arrows),sizesandperimetersofclustersintheliquid
phase grow according to a power law with a non-trivial ex- relate the critical rotation rate with the parameters of
ponentd ≈1.2.Afterthetransition,thesystemiscomposed the experiment. The behavior of assembly is well des-
r
of a unique crystallized region. Its roughness decreases and cribed by the global bond-orientational order parame-
departsfromthes∝p1.2 law(solidline)toreachs=p2/4π, ter m = |1/N (cid:80)N0 ψk| which time average is zero
0 k=1 6
(dottedline)whichcorrespondstoacircularcompactregion.
in the liquid-like phase and non-zero for the solid, and
(b) Scatter plot of the cluster size versus the radius of gyra-
which fluctuations peak around a transition frequency
tion of the clusters. (c) Average cluster size < s > and (d)
f = 0.64Hz. This means that near the transition, large
correlationlengthζ versusthepackingfractiondifferenceδφ. c
heterogeneities develop within the system through the
appearance of many small monocrystals. In this disor-
dered phase, sizes and perimeters of these clusters obey
Interestingly, at the transition frequency f ≈0.64Hz,
c geometricalscalingrelations,anddisplayacriticalbeha-
the packing fraction at the center of mass of the granu-
viour with respect the distance to random close packing,
lar suspension reaches φ = 0.82 ≈ φ , the random
RCP which thus appears as a critical point. Future work in
close packing density, a value that is much larger than
deciphering this system’s statistics will focus on its dy-
the solidus point in both thermal [13, 14] and ather-
namics,sinceitappearstobeundergoingacoupledshear
mal [4] hard disks. In order to further assess the critical
flowpattern.Largedeviationsofthepackingfractionand
nature of the transition at hand, we computed additio-
kinetic energy are also under scrutiny, in the hope to ex-
nal quantities that usually scale with the distance of the
tract relevant state functions related to the aforementio-
control parameter to the critical point. First, Fig. 2(d)
ned criticality.
shows the cut-off size s when the cluster size distribu-
c
tions n(s) is fitted by n(s) = s−τexp(− s ). Though s
sc c
doesn’t exhibit any particular behaviour when plotted
against other quantities, it appears to scale with the dis-
tanceofthepackingfractiontotherandomclosepacking, ∗ [email protected]
δφ ≡ φ −φ(r = 0), with an exponent close to 2.2. [1] A. J. Liu and S. R. Nagel, Annu. Rev. Condens. Matter
RCP
Phys. 1, 347 (2010).
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