Universality and criticality of a second-order granular solid-liquid-like phase transition

10.5 MB·English

by  Gustavo Castillo

Universality and criticality of a second-order granular solid-liquid-like phase transition
Gustavo Castillo,1,2 Nicol´as Mujica,1,∗ and Rodrigo Soto1
1Departamento de F´ısica, Facultad de Ciencias F´ısicas y Matem´aticas,
Universidad de Chile, Avenida Blanco Encalada 2008, Santiago, Chile
2Laboratoire de Physique Statistique, Ecole Normale Sup´erieure,
UMR CNRS 8550, 24 Rue Lhomond, 75005 Paris, France
(Dated: January 22, 2015)
Weexperimentallystudythecriticalpropertiesofthenon-equilibriumsolid-liquid-liketransition
thattakesplaceinvibratedgranularmatter. Thecriticaldynamicsischaracterizedbythecoupling
5 of the density field with the bond-orientational order parameter Q , which measures the degree of
4
1 local crystallization. Two setups are compared, which present the transition at different critical
0 accelerations as a a result of modifying the energy dissipation parameters. In both setups five
2
independentcriticalexponentsaremeasured,associatedtodifferentpropertiesofQ : thecorrelation
4

n length,relaxationtime,vanishingwavenumberlimit(staticsusceptibility),thehydrodynamicregime
a of the pair correlation function, and the amplitude of the order parameter. The respective critical
J exponentsagreeinbothsetupsandaregivenbyν =1,ν =2,γ =1,η≈0.6−0.67,andβ =1/2,
⊥ (cid:107)

0 whereas the dynamical critical exponent is z =ν(cid:107)/ν⊥ =2. The agreement on five exponents is an
2 exigenttestfortheuniversalityofthetransition. Thus,whiledissipationisstrictlynecessarytoform
the crystal, the path the system undergoes towards the phase separation is part of a well defined

] universalityclass. Infact,thelocalordershowscriticalpropertieswhiledensitydoesnot. Beingthe
h
laterconserved,theappropriatemodelthatcouplesbothismodelCintheHohenbergandHalperin
c
classification. The measured exponents are in accord with the non-equilibrium extension to model
e
C if we assume that α, the exponent associated in equilibrium to the specific heat divergence but
m
with no counterpart in this non-equilibrium experiment, vanishes.
-
t
a PACSnumbers: 64.60.Ht,64.70.qj05.40.-a,45.70.-n,
t
s
.
at I. INTRODUCTION to be quite general, allowing other systems to be com-
m pared with them. Some of these are, for example, the
- Thermodynamics and equilibrium statistical mechan- directed percolation process, the Kardar-Parisi-Zhang
d ics have shown an extraordinary success in describing modelforsurfacegrowthandtheSwift-Hohenbergmodel
n phase transitions. It has been possible to classify them, [13–17]. Thecriticalphenomenamethodologyinequilib-
o
predicttheirproperties,andunderstandthecriticalphe- rium has been extended to these cases, where again we
c
[ nomena. Also, they provide a framework that can be find critical exponents, universality classes, and scaling
usedtobuildefficientnumericaltoolsastheMonteCarlo functions, with the dynamic renormalization group as a
1 simulations or to design and analyze experiments, for usefulapproach. But,stillmuchunderstandingisneeded
v
example exploiting the concept of universality classes. to reach a state of knowledge comparable to the equilib-
2
0 From a theoretical viewpoint the critical phase transi- rium case.
0 tions were classified in universality classes in terms of Granularmatter,duetothestrongdissipationpresent
5 symmetry and conservation properties [1, 2]. at the particle interactions and the corresponding need
0 In several order-disorder phase transitions the dynam- forcontinuousenergyinjectiontosustaindynamicstates,
1. ics of the transition needs the interplay of two or more is an excellent candidate for studying out of equilibrium
0 order parameters [3–9]. One particularly interesting phase transitions. Important progress has been made
5 case is the so called model C in the Hohenberg and along this direction [18–27], but a wider view in the con-
1 Halperinclassification,whereanon-conservedcriticalor- text of dynamical phase transitions is still lacking. Re-
v: der parameter couples to the conserved noncritical den- cently, we presented an experimental study of a granular
i sity [1, 9, 10]. Examples where this model can be ap- liquid-solid-like phase transition in a vibrated quasi-two-
X
plied are varied, comprising liquid-liquid phase transi- dimensionalgranularsystem[28]. There,weshowedthat
r tions [5, 7, 8, 11], binary alloys [6], and anisotropic mag- the transition is characterized not only by the density
a
nets [12]. field but also by a bond-orientational order parameter,
In out of equilibrium conditions there is no such sys- which is described by the model C. To our understand-
tematicframeworkthatcanbeusedtoanalyzeorclassify ing, this is the first non-equilibrium realization of this
phase transitions. There are, nevertheless, some notori- class of phase transition.
ous examples of prototype models that have been shown In this manuscript we extend the results found previ-
ously by focusing now on the universality of the transi-
tion. That is, we aim to verify if by varying some ex-
perimental parameters, the same critical exponents are
∗ Correspondingauthor: nmujica@dfi.uchile.cl found. We remark that in Ref. [28] five independent
2
critical exponents were found which, therefore, put an
exigent test the universality condition. (b)
!
Thearticleisorganizedasfollows. SectionIIdescribes
the liquid-solid-like transition that takes place in con-
fined quasi two dimensional granular systems. In Sec.
IIItheexperimentalsetupandproceduresareexplained,
describingthetwoconfigurationsthatareusedtotestthe
(2)
universality of the critical exponents. The experimental
resultsandthedeterminationofthecriticalaccelerations (4)
and exponents are presented in Sec. IV. Finally, conclu-
sions are given in Sec. V.
II. THE LIQUID-SOLID-LIKE TRANSITION IN
QUASI TWO DIMENSIONAL GRANULAR
SYSTEMS
FIG.1. Schematicoftheexperimentalsetup. (a)Topviewof
Granular matter is ubiquitous in our daily life, still
thequasi-2Dcell,withL =L =100d. (b)Sideviewofthe
a fully understanding of its dynamical behavior has re- x y
setup. The vertical height in the cell is L = 1.94d±0.02d.
mainedratherelusiveforseveralyears[29,30]. Drygran- z
The cell is illuminated from below with a 2D array of light
ular matter is a collection of athermal macroscopic par-
emitting diodes, which light is diffused with a white acrylic
ticles that interact mainly through hard core-like dissi-
sheet placed between the array and the cell. (1) camera, (2)
pative collisions, and depending on external conditions quasi-2Dcell,(3)electromechanicalshaker,(4)accelerometer.
such as pressure or packing fraction it may behave as a
solid, a liquid or even a gas.
Energy injection is needed to compensate the energy
dissipatedingrain-grainandgrain-wallcollisions. Vibra- uous. In the later case critical phenomena develop, with
tionsareanefficientmethodtoperformthistaskinadis- staticanddynamicalpropertiesthatshowpowerlawbe-
tributed and controlled way, being possible to reach sta- haviorintermsofthedistancetothecriticalacceleration
ble states. Among the different possibilities to perform [28].
the vibrations, the quasi-two dimensional (Q2D) geom-
etry has gained attention because it allows the granular
system to be followed in detail at the individual grain III. EXPERIMENTAL SETUP AND
scale together with the collective dynamics. In Q2D sys- PROCEDURES
tems, grains are placed in shallow box, with a height
that is between one and two particle diameters, while it In this paper we extend the analysis of the previously
islargeinthehorizontaldirections. Theboxisvertically published study [28]. Two sets of experiments are used
vibrated with maximal accelerations larger than gravity to test the universality hypothesis. They differ in the
such that grains acquire vertical energy that is trans- dissipationcoefficients,whichwillbelabeledexperiments
ferred via collisions to the horizontal degrees of freedom. A and B, with larger and lower dissipation respectively.
Inthereducedtwo-dimensionaldynamics,thesystemre- The granular system is composed of N 104 stainless
sembles a liquid but with the important difference that steel spherical particles of mass m = 4.45∼ 10−3 g, and
×
the collisions are not conservative. d=1mmdiameter. TheQ2Dboxhaslateraldimensions
For fixed density and vibration frequency, by increas- L = L L = 100d. The box consists of two 10 mm
x y
≡
ing the maximum acceleration the system presents a thick glass plates separated by a square metallic frame.
phasetransitionwheregrains clusterinsolid-likeregions Each inner glass surface has an indium tin oxide (ITO)
with crystalline order [18, 25]. The instability is origi- coating, which dissipates electrostatic charges generated
nated in the development of an effective negative com- by collisions of particles with the walls. The box is fixed
pressibility [26, 31] and the transient dynamics is gov- to a base by four posts placed at each corner of the cell.
erned by waves [26]. Once the system is segregated The base supports an array of high intensity light emit-
the solid-like cluster show fluctuations in the interface ting diodes. A piezoelectric accelerometer is fixed to the
that are well described by the capillary theory, allowing base, allowing the measurement of the imposed forcing
us to extract an effective surface tension that has non- accelerationwitharesolutionof0.01g. Themainadvan-
equilibrium origin [32]. tage of this setup is that particles are illuminated from
In our previous work we studied experimentally the below. Theyarethenvisualizedasdarkdisksonawhite
solid-liquid-likephasetransitioninthevibratedQ2Dge- background. Thisallowstodetectparticlesindenseclus-
ometry. Depending on the filling fraction and height of ters, evenwhenparticlesarepartiallymountedontopof
thecell,thetransitionwaseitherdiscontinuousorcontin- each other. A scheme of the setup is shown in Fig. 1. A
3
cal and electric properties depend on fabrication details
such as thickness, annealing, substrate, feed gas, among
others[33]. So,itisexpectedthatthedissipationparam-
eters,namelyinelasticandfrictioncoefficients,shouldbe
different for the two sets of ITO coated glass plates. As
willbeshownbelow,ourresultsindeeddemonstratethat
the dissipation is stronger for the thinner ITO coated
glass plates (type A experiments).
The ITO coating works very well for many hours of
experimental runs. Eventually, it does however get dam-
aged by particle collisions. The precise time of repro-
ducible runs is probably function of the ITO coating
thickness. Thisissupportedbythethicknessdependence
ofthecrackonsetstraininITOthinfilms,whichislower
forthickercoatings[33]. Alldatapresentedinthispaper
correspond to reproducible runs for which no important
damage was noticeable. In fact, a surface damaged ITO
coating is manifested in important changes of the mea-
suredquantities–suchasthedensityandorderstructure
factors–withrespecttothoseobtainedforanewpairof
ITO coated glass plates. We conjecture that the damage
occurs because of erosion of the ITO coating, which in
FIG.2. Atypicalrawimageofthecompletesystem,fortype turn affects particle interactions by an increase of elec-
BexperimentandΓ=4.05±0.01. Particlesarevisualizedas trostatic forces and contamination of the system by dust
black disks on a white background. formed from the ITO coating. In order to insure repro-
ducibility, glass plates were changed periodically during
the time duration of all the experimental runs, and all
partsoftheexperimentsarecleanedinanultrasonicbath
typical image of the system is shown in Fig. 2. before mounting the experimental cell again, including
The whole setup is forced sinusoidally with an elec- the particles.
tromechanical shaker, with vertical displacement z(t) = Two important issues are the top and bottom plates
Asin(ωt). Top view images are obtained with a camera flatnessandhomogeneityofLz throughtheexperimental
at 10 fps. The images acquired have a typical resolu- cell. With the cell empty of particles the height is mea-
tion of 1020 1020 pix2, thus we obtain particles of 10 sured at nine different positions on a regular spaced grid
×
pix diameter approximately. Particle positions are de- with an optical microscope (see supplementary material
termined at sub-pixel accuracy. Results have been ob- of [28]). Within experimental errors the vertical height
tained by fixing the particle number N, cell height Lz is homogeneous, Lz = 1.94d 0.02d. However, the ho-
±
and driving frequency f = ω/2π = 1/T = 80 Hz. The mogeneity of Lz does not insure that the glass plates
dimensionless acceleration Γ = Aω2/g is varied in the are both flat. Because of the mechanical constrains and
range 1 6. In this present work we focus on the config- stressesexertedontheplates,somesmallresidualcurva-
−
uration C2 of reference [28], namely L = 1.94d 0.02d ture could exist.
z
±
and N = 11504, which implies the filling fraction to be In Fig. 3(a) we show an average of 3270 images for
φ = Nπd2/4L2 = 0.904. This filling fraction value is Γ=2.05, well below the solid-liquid transition (for type
possible because the system is Q2D, which allows to ac- Bexperiments,Γ 4.5). Onlyasmallpartofthesetup
c
commodate more particles than for a pure 2D system. is shown, of size 15≈d 15d. Viewed from above it corre-
×
As was demonstrated in [28], under this particular sponds to the left side wall at almost half of the cell. It
configuration the system undergoes a second-order type isclearthatnearthesidewallparticlestendtoformlay-
phase transition. The aim of this paper is to present a ers of thickness d, within which particles barely move
∼
more detailed study of the transition itself. Addition- compared with the rest of the system. This layering,
ally, we are particularly interested if there is universality observed at all side walls, is induced by the extra dissi-
in this granular out-of-equilibrium phase transition. In pation given by the side wall collisions, similar to what
ordertostudytheuniversalbehaviorofthecriticalexpo- has been shown in sheared granular matter [23]. How-
nents,wechangethedissipationparametersofthesystem ever, for a distance (cid:38) 5d from a side wall, the system
by changing both the bottom and top lids. We use two is nearly homogeneous. The inset of Fig. 3(a) shows an
pairsofITO(indiumtinoxide)coatedglassplates. Inthe average intensity plot as a function of the position (time
experimentsoftypeAthecoatingis25nmthick,whereas andy averaged),showingthelayersclosetotheedgeand
for type B experiments it is 750 nm thick. It has been then saturating at a constant value. Also, by means of a
shown that ITO films mechanical, fracture, morphologi- time average coarse-graining procedure we can compute
4
b) c)
180
160 2.5 1.01
Intensity111024000 2
80 N0 N0 0.99
600 1 2 3 4 5 6 7 8 N/i 1.5 N/i
x/d
= =
ρi 1 ρi 0.97
0.5
0.95
0 1 2 3 4 5 6 7 8 10 20 30 40 50 60 70 80 90
x/d x/d
FIG.3. (a)Averageimageobtainedfrom3270rawimages. Onlyasectionofthesetupnearawallisshown(ofsize15d×15d).
Regions in dark grey (low intensity) results from more particles in these zones. The vertical stripes at the left account for
layers of particles that move less and spend more time near the wall because of the extra dissipation at the side wall. The
inset shows the intensity of this image as a function of position, showing that for x (cid:38) 5d the system is almost homogeneous.
(b) Time averaged coarse-grained particle density obtained directly from particle detection as a function of x, with d/4 as bin
width. It also shows oscillations due to side wall layering and confirms the homogeneity for x(cid:38)5d at this scale. (c) The same
coarse-grained density at a different scale, showing that the density at the center is about 2% higher than close to the edges.
Here the bin width is d/2. For both (b) and (c) N is the average number of particles detected in a vertical bin spanning
i
y =[0,100d] and x =[(i−1)w ,iw ), where w =d/4 or d/2 is the bin width. N is defined as the number of particles that
i b b b 0
would fit into a bin with an homogeneous density ρ =N/L2.
o
the density of particles as a function of distance from a direction. For example, S(k ,k ) shows the characteris-
x y
wall. This is shown in Figs. 3(b) and 3(c). Close to the tic symmetric circular annular shape at kd = 2π, where
wallwenoticethesameparticlelayeringandanapparent k = (cid:126)k . A second verification is given by the even sym-
homogenization for x (cid:38) 5d (Fig. 3(b)). The asymptotic metr|y|of the coarse-grained density with respect to the
limit is slightly less than expected to compensate for the cell center line, as can be observed from the data pre-
particle excess at the side walls. At a larger scale, as sented in Fig. 3(c), which is indeed symmetric with re-
shown in Fig. 3(c), we observe that there is a small spect to x/d=50.
density gradient that leads to an excess of particles at
the cell’s center compared to the edges: density is about
2% higher at the center that near the side walls. This
can be the result of three effects, which are probably all
The particle detection is done by using a modified
present. Firstly, eventhoughthecell’sheightishomoge-
open source Matlab code which uses a least-square al-
neouswithinexperimentalerrors, theremightbeasmall
gorithm [34]. Our modified version in C++ & CUDA
residualconcavity, whichmakestheparticlestoaccumu-
allows faster computation for large number of particles
late near the center. A second factor could be that the
[35, 36]. The algorithm allows to detect both layers of
verticalaccelerationmaynotbeconstantthroughoutthe
particlesinadensesolidcluster,wherethetoplayerpar-
cell. In other words, there could be some “flapping” of
ticles are placed in the valleys that the bottom particles
thesystem. Indeed,wehavemeasuredtheaccelerationof
form. Typical experimental runs consist of at least 30
thecellatdifferentpositionsandwefindthatitisabout
video acquisitions, one for each A, of about 3300 images
0.2% higher near the borders that at the center. Finally,
each. Therefore, the complete number of images to ana-
the presence of dissipative side walls could induce such
lyze for a single experimental run is about 105.
large scale inhomogeneity.
Another important issue is the mechanical leveling of
the whole setup (for details, see supplementary material
of Ref. [28]). The cell’s horizontality, and thus the sys- Finally, for fixed geometry, particle density and vibra-
tem’s isotropy, can be checked by two ways: First, it tion frequency we perform vibration amplitude ramps,
is verified through the static structure factor S((cid:126)k) (de- from Γ (cid:38) 1 in the liquid phase, increasing A going
finedbelow)andtheaveragedensityFouriercomponents through the solid-liquid transition that is reached at
ρ((cid:126)k,t) (for now we can say that S((cid:126)k) is a measure of Γ=Γ . In order to check that the transition is continu-
c
(cid:104) (cid:105)
densityfluctuationsinFourierspace). Whenthesequan- ous, for some runs we also perform decreasing amplitude
titiesareplottedversusk andk thereisnopreferential ramps starting above Γ .
x y c
5
d)
1.5
) 1
k
(
S
0.5
0
0 10 20 30 40
kd
FIG. 4. (Color online) Color plots of log ((cid:104)ρ((cid:126)k,t)ρ((cid:126)k,t)∗(cid:105)/N) (a), log ((cid:104)ρ((cid:126)k,t)(cid:105)(cid:104)ρ((cid:126)k,t)∗(cid:105)/N) (b) and the structure factor
10 10
function S((cid:126)k) (c) as functions of k and k . From S((cid:126)k) the isotropy of the fluctuations is clear from the circular halo shape of
x y
(cid:112)
the pre-peak. (d) S(k) for a wide range of kd, obtained from S(k ,k ) by simply plotting as a function of k= k2+k2. For
x y x y
all these figures Γ=3.75<Γ and data was obtained from type B experiment.
c
IV. EXPERIMENTAL RESULTS The static structure factor S((cid:126)k) measures the intensity
of density fluctuations in Fourier space:
A. Static structure function. ρ((cid:126)k,t) ρ((cid:126)k,t) 2
S((cid:126)k)= (cid:104)| −(cid:104) (cid:105)| (cid:105), (2)
N
Particle positions (cid:126)rj(t) in the plane (x,y) are deter- ρ((cid:126)k,t)ρ((cid:126)k,t)∗ ρ((cid:126)k,t) ρ((cid:126)k,t)∗
minedforeachtimet. Experimentally,thereisnoaccess S((cid:126)k)= (cid:104) (cid:105)−(cid:104) (cid:105)(cid:104) (cid:105), (3)
N
to the z coordinate. Thus, the 2D microscopic density
field Fourier components are where denotes time averaging. In general
(cid:104) (cid:105)
ρ((cid:126)k,t) = 0,duetoinhomogeneitiesinducedbybound-
(cid:104) (cid:105) (cid:54)
ary conditions, as those shown in Fig. 3. The wave
(cid:90) N
ρ((cid:126)k,t)= d2(cid:126)rei(cid:126)r·(cid:126)kρ((cid:126)r,t)=(cid:88)ei(cid:126)k·(cid:126)rj(t). (1) vectors are computed from (cid:126)k = π(nxˆı+nyˆ)/L, where
n ,n N, and k = (cid:126)k .
j=1 x y
∈ | |
6
a) b) 1.2
1
0.6
0.8
(k) 0.4 Γ max 0.6
S S
0.4
0.2
0.2
A
B
0 0
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.5 2 2.5 3 3.5 4 4.5 5 5.5 6 6.5
kd Γ
c) 35
d) 0.9
0.8
30
0.7
25
0.6
/d 20 λd0.5
ξ
0.4
15
0.3
0.2
10
A A
B 0.1 B
5 2 3 4 5 6 7 8 9 10
1.5 2 2.5 3 3.5 4 4.5 5 5.5 6 6.5
Γ (ξ/d)−1 10−2
×
FIG. 5. (Color online) (a) S(k) in the large wavelength limit for different accelerations Γ. The pre-peak grows and shifts
towards lower k for increasing Γ. We define the pre-peak’s maximum S that is obtained at k = k∗. The width λ of the
max
pre-peakisdefinedasthewidthathalfheight. (b,c)S andξ/d=π/k∗ asfunctionsofΓforbothtypesofexperiments. (d)
max
λd versus d/ξ. A linear dependence is observed for ξ(cid:38)15d (d/ξ(cid:46)0.07).
InFigs. 4(a)and4(b)wepresentcolorplotsofthetwo 3(c). Thus, itsFourierdecompositionyieldsthatthenot
terms that are used for the computation of S(k ,k ), pure odd harmonics should vanish, having very low val-
x y
namely ρ((cid:126)k,t)ρ((cid:126)k,t)∗ /N and ρ((cid:126)k,t) ρ((cid:126)k,t)∗ /N as uesinpractice. InFig.4(c), thesubtractionofthesetwo
functions(cid:104) of kx and ky(cid:105) (both in(cid:104)log10 (cid:105)s(cid:104)cale). I(cid:105)t turns termsisshown,whichdefinesS(kx,ky). Thesmoothness
outthatbothquantitiesarestronglynon-monotonicand oftheresultingfunction,withnodiscriminationbetween
aredifferentbyseveralordersofmagnitude. Inthelower even or odd modes, indicates that the density fluctua-
wavenumber range that is plotted these quantities show tions are governed by long wavelength dynamics and not
asetofpeaksplacedonaregulargridontopofasmooth by the static density profile.
background,takinglargevalueswhenbothn andn are
x y
Thestructurefactorpresentsanotoriouspre-peakcen-
odd, whereas the other modes are much lower. This is
tered at kd 0.2, which corresponds to a large wave-
understood by the even symmetry that the density field
∼
length structure of size 15d (the pre-peak term refers
has with respect to the cell’s center, as shown in Fig.
∼
tothepeakbeingatlowerwavenumberthantheonecor-
7
responding to the first coordination shell at kd=2π, see d
below). The associated density fluctuations are indeed
visiblebysimplevisualinspection(seeFig.2). Fig. 4(d)
shows S(k) for a wider range of kd, up to kd 40. It
has the usual form expected for liquids with sho∼rt range d/p2
order, presenting a maximum at kd = 2π related to
the first coordination shell as well as the pre-peak dis-
cussed before. This pre-peak can be located in the range
kd=0.1 0.3 depending on the exact value of Γ. Simi-
−
lardensityfluctuationshavebeenobservedinamorphous
materials[37,38],whichhavebeenconsistentlyrelatedto
the existence of medium-range-crystalline-order. These
density fluctuations have also been related to the pres-
ence of a pre-peak in the structure factor. In fact, in the
amorphous literature the pre-peak is known as the First FIG. 6. Schematic representation of two square interlaced
layers for which particles are closed packed. The center of
SharpDiffractionPeak(FSDP)becauseitappearsatlow
particlesinthebottom(top)layerareshownwithsolid(open)
k and is obtained from X-ray diffraction measurements.
black circles. The 2D projected square lattice has a unit cell
√
Fig. 5(a) presents S(k) for small wavenumbers at dif- length d/ 2.
ferent accelerations Γ below Γ . From this figure it is
c
clear how the pre-peak evolves as the system is driven
towards the transition by increasing its acceleration to-
wards Γc: its height increases and its position shifts to In what follows, medium range order will be analyzed
lower k. This implies that density fluctuations become withanappropriatebond-orientationalorderparameter,
larger in size and stronger as Γ increases. We character- which does indeed present critical behavior.
ize the pre-peak by its maximum value S S(k∗),
max
and the associated characteristic length scale ξ≡= π/k∗,
where k∗ is the position of the pre-peak. These quanti-
B. Bond-orientational order parameter.
ties are plotted in Figs. 5(b) and 5(c) as functions of Γ
for increasing amplitude ramps and for both experiment
In the vicinity of the transition, fluctuations of high
types A and B. Although the data points are scattered
density present the same square symmetry as the solid
andthattheydonotreallyoverlap,speciallyathigherΓ,
phase. In the quasi-2D geometry the solid phase con-
both quantities show similar trends for each experimen-
sistsoftwosquareinterlacedlayersinsteadofthehexag-
tal type. Both S and ξ/d increase as the transition
max
onal layer that is characteristic of 2D systems [25]. The
is approached, although the former seems to saturate at
local order can be characterized through a 4-fold bond-
larger Γ. We observe no great differences between the
orientationalorderparameter. Thisisstillvalidinquasi-
twoITOcoatings,beingtheirfinalvalues(nearthetran-
2Dgeometrybecausetheinterlacedtwo-layersquarelat-
sition) very similar, S 0.6 0.1 and ξ/d 20 35.
max
≈ − ≈ − tices (with unit cell length d in each plane) result also
WealsopresentinFig. 5(d)thewidthofthepre-peak,λ,
in a square lattice when projected in 2D, with unit cell
definedasitswidthathalfS . Thisquantityisamea-
max length d/√2 when the grains are close packed, as shown
sure of the dispersion around the characteristic length ξ.
inFig. 6. The4-foldbond-orientationalorderparameter
The collapse and scatter of the data are improved with
per particle is defined
respect to the other quantities, with no dependence on
the different dissipation parameters.
By observing visually the persistence of the solid clus- Qj = 1 (cid:88)Nj e4iαjs, (4)
ters we conclude that for the A type experiment, the 4 Nj
s=1
transition is located at Γ 5.1, whereas for type B
c
∼
it is found to be Γ 4.5. Given that we are dealing where N is the number of nearest neighbors of particle
c j
withacontinuouspha∼setransitionthesearejustapprox- j and αj is the angle between the neighbor s of parti-
s
imatedvalues. However,neitherS norξ showevident cle j and the x axis. For a particle in a square lattice,
max
changes at these values. Qj =1andthecomplexphasemeasuresthesquarelat-
| 4|
As a conclusion to this first part we can say that den- tice orientation respect to the x axis. For details on the
sity fluctuations do not show critical behavior, but they computationofQj4 wereferthereadertothesupplemen-
are needed to create regions of high order. This is also tary information of Ref. [28].
evidentfromvisualinspection;higherdensitypatchesare Representative maps of Qj are shown in Fig. 7 for
| 4|
indeed more ordered, as can be verified in Fig. 2. Den- three accelerations, Γ < Γ , Γ Γ and Γ > Γ . In
c c c
≈
sityisaconservedfield. Itsfluctuationsarehoweverlim- this case the maps are obtained from images for exper-
ited by the system’s vertical geometrical constrain and iment type A (Γ 5.1). Below the transition the or-
c
∼
the fact that the particles are in practice hard spheres. deredpatches,orcrystallites,arefirstsmall,moreorless
8
./0 ./0
!") 1 !") 1
23%2141!"!1(cid:239)1!"$ 23%2141!"!1(cid:239)1!"$
23%2141!"$1(cid:239)1!"' 23%2141!"$1(cid:239)1!"'
23%2141!"'1(cid:239)1!") 23%2141!"'1(cid:239)1!")
23%2141!")1(cid:239)1!"5 23%2141!")1(cid:239)1!"5
!"( 23%2141!"51(cid:239)16"! !"( 23%2141!"51(cid:239)16"!
!"' !"'
- -
-+, !"& -+, !"&
!"% !"%
!"$ !"$
!"#1 !"#1
!"# !"$ !"% !"& !"' !"( !") !"# !"$ !"% !"& !"' !"( !")
*+, *+,
* *
./0
!") 1
23%2141!"!1(cid:239)1!"$
23%2141!"$1(cid:239)1!"'
23%2141!"'1(cid:239)1!")
23%2141!")1(cid:239)1!"5
!"( 23%2141!"51(cid:239)16"!
!"'
-
,
-+ !"&
!"%
!"$
!"#1
!"# !"$ !"% !"& !"' !"( !")
*+,
*
FIG.7. (Coloronline)Colormapsoftheabsolutevalueofthe4-foldbond-orientationalorderparameterinrealspace(foreach
particle we plot |Qj|) for Γ=4.18 (a), 5.10 (b) and 5.42 (c) (Experiment type A, Γ ≈5.1). In each figure only a part of the
4 c
systemisshown,from0.2Lto0.8Lineachhorizontaldimension. Thecoloringisdetailedinthelegend,withthemostordered
particles in red (|Q |=0.9−1). Solid (open) circles correspond to particles classified as particles in the solid (liquid) phase.
4
This classification is performed by measuring the area of the Voronoi area of each particle (see supplementary document of
[28]).
homogeneously distributed in space and are of short life- As discussed before, the local density can not change
time. They increase in size and live for longer time as stronglyatthetransitionbecauseofthesystem’svertical
Γ approaches Γ . Also, they tend appear more near the geometricalconfinement. Thus,thecorrectorderparam-
c
center than at the sidewalls, which we relate to the large eter must be the local 4-fold symmetry order, measured
scale small density inhomogeneity discussed before. The through the orientational parameter Qj. With this in
4
quantitative study of crystallites size and lifetime is pre- mind,wedefineitsglobalaverage,inspacefirstandthen
sented below.
9
0.5
b)
0.48 0.03
0.46
i
| 4
Q4 0.44 Q
∆ 0.01
|
h
0.42
0.4 A(increasingΓramp)
B(increasingΓramp) A
B(decreasingΓramp) 0.003 B
0.38
2 2.5 3 3.5 4 4.5 5 5.5 6 0.001 0.01 0.1
ε
Γ
FIG. 8. (Color online) (a) Global average of 4-fold orientational order parameter (cid:104)|Q |(cid:105) versus Γ for the two ITO coatings:
4
Experiment type A (thin ITO) with increasing Γ ramps ((cid:3)) and experiment type B (thick ITO) with increasing (◦) and
decreasing(•)Γramps. Continuouslinesshowthelineartrendfitfor2.5<Γ<Γ andthesupercriticaldeviationforΓ(cid:62)Γ .
c c
TheadjustedcriticalaccelerationsareΓ =5.12±0.01(typeA)andΓ =4.48±0.03(typeB).(b)∆Q =(cid:104)|Q |(cid:105)−QL versus
c c 4 4 4
ε = (Γ−Γ )/Γ in log-log scale for each ITO coating thickness (type A: (cid:3); type B ◦ for both increasing and decreasing Γ
c c
ramps), where QL = aΓ+b is obtained from the linear trend below Γ . For the thin ITO coating, a = 0.011±0.001 and
4 c
b=0.380±0.002. ForthethickITOcoating,a=0.016±0.001andb=0.359±0.002. Insakeofclarityjustonerepresentative
errorbarisshownforeachcase. Thestraightlinesarepowerlawswithexponentsequalto0.4(dash-dotted),1/2(continuous)
and 0.6 (dashed), shown as guides to the eye.
in time, 102
(cid:42) N (cid:43)
1 (cid:88)
Q = Qj , (5)
(cid:104)| 4|(cid:105) N | 4|
j=1
which measures the fraction of particles in the ordered
phase. ) 101
k
(
Inordertoshowthatthetransitionisindeedofsecond 4
S
order for both experiment types, we present in Fig. 8a
theglobalaverage Q versusΓforthethinandthicker
4
(cid:104)| |(cid:105)
ITO coated plates (type A and B respectively). For the
latter, both increasing and decreasing Γ ramps are pre-
sented, showing good reproducibility. This figure indeed 100
demonstrates that both configurations present a second
order type transition, continuous and with no hysteresis.
0 0.2 0.4 0.6 0.8 1
The qualitative behavior is the same for both experi-
kd
ment types. First, Q has a linear dependence on Γ
4
(cid:104)| |(cid:105)
below the transition. This reflects the fact that the frac-
FIG.9. (Coloronline)4-foldbond-orientationalstructurefac-
tion of particles that form crystallites with square fold
tor S (k) for several Γ for experiment type A (the results for
4
symmetry, even if it is transiently, increases when the
B are basically the same). The vertical axis is in log scale.
10
transition is approached. For both experiments A and Curves obtained for Γ < Γ are in blue (dark gray) and for
c
B there is a clear deviation from this linear trend above Γ>Γ areinred(lightgray). ForbothITOthicknessesand
c
a given threshold. The critical acceleration –defined for forΓ<Γ ,allcurvesshowanOrnstein-Zernike-likebehavior
c
now as the value where the qualitative change occurs– in the limit kd (cid:28) 1, S4(k) ≈ S4(0)/[1+(ξ4k)2]. For Γ> Γc
for the thicker ITO coating is lower (Γ 4.5) than the the curves tend to collapse together.
c
∼
one for the thin ITO coating (Γ 5.1). Also, the initial
c
∼
linear slope below the transition is larger for the thick
ITO coating case. Both facts, the lower critical value
10
b)
a) A
A 101 B
B
101
) d
0 /
( 4
4 ξ
S
100
100
10−2 10−1 100 10−2 10−1 100
ε ε
FIG.10. (Coloronline)S (0)(a)andξ /d(b)versusεforexperimentstypeA((cid:3))andtypeB(◦). Continuouslinesarecritical
4 4
powerlaws,withexponentsγ =ν =1forS (0)andξ ,shownasguidestotheeye. ThefittedcriticalaccelerationsforS (0)
⊥ 4 4 4
andξ /dare: Γ =5.09±0.07andΓ =5.24±0.08respectivelyforexperimenttypeA;Γ =4.43±0.06andΓ =4.58±0.06
4 c c c c
respectively for experiment type B. More details are provided in Table I.
and larger slope for the thicker ITO coating, are con- orientational structure factor
sistent with a lower effective dissipation at the top and
Q ((cid:126)k,t) Q ((cid:126)k,t) 2
bottom walls. Indeed, in this case the transition occurs S ((cid:126)k)= (cid:104)| 4 −(cid:104) 4 (cid:105)| (cid:105). (7)
4
at a lower amplitude, thus at lower energy injection and N
dissipation rates, and transient crystals form and grow This structure factor is shown in Fig. 9 for several ac-
more easily as Γ increases. In this figure, the continuous celerationsandforexperimenttypeA.Resultsforexper-
linescorrespondtofitsofalineardependenceforΓ<Γc iments type B are basically the same and are not shown.
and a supercritical deviation for Γ > Γc (details in the For both experiment types and for Γ < Γc, S4(k) shows
figure caption). an Ornstein-Zernike-like behavior in the limit kd 1,
The deviation from the linear trend observed for Γ < (cid:28)
fiΓnceids adsefitnheedeaxstr∆apQo4lat=ion(cid:104)|Qof4|t(cid:105)h−e QlinL4e,arwhtreernedQoL4veirs tdhee- S4(k)≈ 1+S4((ξ0)k)2, (8)
4
complete range of Γ. Fig. 8(b) presents ∆Q versus
4
ε = (Γ Γc)/Γc in log-log scale for each ITO coating whereξ4 andS4(0)arethe4-foldbond-orientationalcor-
thicknes−s. The continuous line shows the supercritical relation length and static susceptibility respectively.
law ∆Q4 √Γ Γc as a guide to the eye. In Fig. 10 we present both S4(0) and ξ4/d for Γ<Γc,
The resu∝lts of F−ig. 8(a) are fitted to the supercritical obtained from the fits of the Ornstein-Zernike behavior
law ∆Q =c√Γ Γ . The adjusted parameters are c= for both ITO coatings. Both quantities are plotted as
4 c
0.029 0.002(typ−eA)andc=0.024 0.002(typeB).We functions of the reduced acceleration ε = (Γc Γ)/Γc,
conjec±turethatthedifferentadjusted±cvaluesalsoreflect where Γc is obtained from a specially adapte−d fitting
the difference of dissipation parameters that control the procedure [28]. The correlation length and susceptibil-
particle-wall collisions. ityvarystronglyasthetransitionisapproached. Forthe
Next, in order to characterize quantitatively the or- two ITO coatings these quantities obey critical power
deredpatchesshowninFig. 7, inparticulartheirtypical laws. In the limit ε 0 they both saturate, presumably
lengthandtimescales,weanalyzetheorientationalorder duetothesystem’sfi→nitesize. Forε(cid:46)3 10−2 theysat-
×
parameter in momentum space. Its Fourier components uratetoS4(0) 10 20andξ4/d 10respectively. The
≈ − ≈
are critical-like behavior is fitted with both free and fixed
exponents. As discussed in our previous work [28], the
N
Q ((cid:126)k,t)=(cid:88)Qjei(cid:126)k·(cid:126)rj(t). (6) precise measurement of Γc and the critical exponents is
4 4 far from trivial. Initial fits give Γ in the range 5.1 5.6
c
j=1 −
and exponents that can vary up to a factor of almost 2.
Then, local order can also be analyzed through its fluc- Additionally, thefittedΓ canbequitedifferentdepend-
c
tuations in Fourier space by means of the 4-fold bond- ing from which quantity they are obtained. The lack of