Quasi-One Dimensional Models for Glassy Dynamics

1.9 MB·

by  Prasanta Pal

Quasi-One Dimensional Models for Glassy Dynamics
Prasanta Pal,1 Jerzy Blawzdziewicz,2 and Corey S. O’Hern3,4,5
1 Department of Diagnostic Radiology, Yale University School of Medicine, New Haven, CT, 06520-8042
2Department of Mechanical Engineering, Texas Tech University, Lubbock, TX 79409-1021
3Department of Mechanical Engineering & Materials Science, Yale University, New Haven, CT 06520-8286
4Department of Applied Physics, Yale University, New Haven, CT 06520-8267
5Department of Physics, Yale University, New Haven, CT 06520-8120
(Dated: January 7, 2014)
Wedescribenumericalsimulationsandanalysesofaquasi-one-dimensional(Q1D)modelofglassy
dynamics. Inthismodel,hardrodsundergoBrowniandynamicsthroughaseriesofnarrowchannels
4 connected by J intersections. We do not allow the rods to turn at the intersections, and thus
1
there is a single, continuous route through the system. This Q1D model displays caging behavior,
0
collectiveparticlerearrangements,andrapidgrowthofthestructuralrelaxationtime,whicharealso
2
found in supercooled liquids and glasses. The mean-square displacement Σ(t) for this Q1D model

n displaysseveraldynamicalregimes: 1)short-timediffusionΣ(t)∼t,2)aplateauinthemean-square
a displacement caused by caging behavior, 3) single-file diffusion characterized by anomalous scaling
J Σ(t) ∼ t0.5 at intermediate times, and 4) a crossover to long-time diffusion Σ(t) ∼ t for times t

6 that grow with the complexity of the circuit. We develop a general procedure for determining the
structuralrelaxationtimetD,beyondwhichthesystemundergoeslong-timediffusion,asafunction
] of the packing fraction φ and system topology. This procedure involves several steps: 1) define a
t
f set of distinct microstates in configuration space of the system, 2) construct a directed network of
o microstatesandtransitionsbetweenthem,3)identifyminimal,closedloopsinthenetworkthatgive
s rise to structural relaxation, 4) determine the frequencies of ‘bottleneck’ microstates that control
.
t theslowdynamicsandtimerequiredtotransitionoutofthem,and5)usethemicrostatefrequencies
ma and lifetimes to deduce tD(φ). We find that tD obeys power-law scaling, tD ∼ (φ∗−φ)−α, where
both φ∗ (signaling complete kineticarrest) and α>0 dependon thesystem topology.
-
d
PACSnumbers: 64.70.kj,61.43.Fs,82.70.Dd
n
o
c
I. INTRODUCTION compressed rapidly or are sufficiently polydisperse) [10].
[
Random close-packed states are amorphous, mechan-

1 ically stable sphere packings with φ ≈ 0.64 for
Developingafundamentalunderstandingofglasstran- rcp
v
monodisperse spheres [6, 11]. In Fig. 1, we show the
0 sitions in amorphous materials is one of the remain-
mean-square displacement Σ(t) (MSD) versus time t
6 ing grand challenges in condensed matter physics [1–
9 3]. Glass transitions occur in myriad systems including over a range of φ from 0.50 to 0.62 from molecular
0 atomic, magnetic, polymer, and colloidal systems. Hall- dynamics (MD) simulations of polydisperse [12], elas-
. tic hard spheres with ballistic (not Brownian) short-
1 marks of the glass transition include a stupendous in-
time dynamics. This data was obtained from studies
0 creaseinthestructuralandstressrelaxationtimes[4]and
4 a concomitantdramatic decrease in the mobility over an by M. Tokuyama and Y. Terada, and is similar to re-
1 extremelynarrowrangeoftemperatureordensity,broad sults in Refs. [13–15]. For relatively dilute systems, the
: MSD crosses over from ballistic (Σ(t) ∼ t2) to diffusive
v distributions of particle motions that are spatially and
(Σ(t)∼t) when it reaches ≈0.1σ2, where σ is the aver-
i temporally heterogeneous, and aging behavior in which
X age particle diameter. The formation of a plateau in the
thesystembecomesprogressivelymoreviscouswithtime
r after it has been quenched to the glassy state [5]. MSD (for φ&0.57)signalsthe onsetof cagingbehavior,
a
whereparticlesaretrappedbyneighboringparticlesthat
Glasstransitionsinliquidsshowmarkedsimilaritiesto
surround them. The height and length of the plateau
jamming transitions in athermal systems such as granu-
characterizethecagesizeandthetimeoverwhichcaging
lar media, foams, and emulsions that do not thermally
persists. The appearance of the plateau and two-stage
fluctuate[6]. Athermalsystemstypicallyjam,ordevelop
relaxationin the MSD leads to dramaticincreasesin the
a nonzero static shear modulus, at sufficiently largeden-
structuralandstressrelaxationtimesasshowninFig.2.
sities or confining pressures, and remain jammed for ap-
In this figure, we demonstrate that the structural relax-
plied shear stresses below the yield stress. Similarities
ation time t (time beyond which the MSD scales as
between jammed and glassy systems include highly co- D
MSD ∼ D t) grows by nearly four orders of magnitude
operative and heterogeneous particle motion in response L
over a small range in packing fraction.
to perturbations [7, 8] and extremely slow relaxation [9]
as a system approaches the glass or jamming transition. Because of the rapid rise in relaxation times and the
Dense colloidal suspensions undergo a glass transi- factthatdensecolloidalsystemscanonlybeequilibrated
tion when they are compressed to packing fractions φ at packing fractions well below random close-packing, it
approaching random close packing (provided they are is difficult to accurately measure the precise form of the
2
6 5
4
4
2
) 3
(t D
Σg 10 0 g t102
lo −2 lo
1
−4
−6 0
−3 −2 −1 0 1 2 3 4 5 6 0.48 0.5 0.52 0.54 0.56 0.58 0.6 0.62
log t φ
10
FIG.1: Mean-squaredisplacementΣ(t)versustimetforelas-
FIG. 2: Structural relaxation time tD required for the hard
tichard-spheresystemswithuniform15%polydispersityover
sphere systems in Fig. 1 to reach the long-time diffusive
arangeof packingfractions φ=0.50 (◦),0.53 (△),0.54 (✁),
0.55 (▽),0.56(✄),0.57(+),0.58 (×),0.59 (∗),0.60 (•),0.61 regime (Σ(t)∼DLt) as a function of packing fraction φ.
(✷),and0.62(⋄)fromtoptobottom. Thesolid,dashed,and standing of the particle-scale origins of dynamical het-
dottedlineshaveslopes 2,1, and0.5, respectively. Thisdata
erogeneities, cage formation and relaxation, and struc-
was obtained from studies by M. Tokuyama and Y. Terada,
tural rearrangements that give rise to subdiffusive be-
and is similar to theirresults in Ref. [13].
havioris lacking. Severalfactorshavecontributedto the
divergence of the relaxation times [16]. In particular, slowprogress. First,itiswell-knownthatitisdifficultto
there is current vigorous debate concerning the packing predictdynamicalquantitiesfromstaticstructuralprop-
fractionatwhichcompletedynamicalarrestoccurs—isit erties. Thus, even though one can visualize all colloidal
before random close packing or does dynamic arrest co- particles in 3D, it is difficult to determine in advance
incidewithrandomclosepacking[17]? Ifitistheformer, which particles will move cooperatively. Further, it has
it is possible that these systems undergo an ideal glass proveddifficulttoidentifyandsampletheraretransition
transition to a static, but not mechanically stable state. states that allow the system to move from one region in
Furtheropenquestionsincludedeterminingthecollective configuration space to another.
particle motions that are responsible for subdiffusive be- We have developed a quasi-one-dimensional (Q1D)
haviorandtheonsetofsuper-Arrheniusdynamics,which model, where hard rods diffuse through a series of con-
occur well below random close packing. nected loops and junctions (or intersections) [29] as
There have been a number of theoretical and compu- shown in Fig. 3. There are a number of advantages for
tational studies aimed at understanding slow dynamics employing this model to explore slow dynamics in col-
in dense colloidal suspensions and related glassy sys- loidal and other glassy systems. First, this model dis-
tems [18–21]. These include the application of mode plays many features of glassy behavior including caging,
coupling theory to colloidal systems [22] and the devel- heterogeneousandcollectivedynamics, anda divergence
opment of coarse-grained facilitated [23] and kinetically ofthe structuralrelaxationtime t . Second, the formof
D
constrainedlatticemodels[24,25]. Modecouplingtheory thedivergenceoft withincreasingpackingfractioncan
D
hasbeensuccessfulinpredictingtheformofthetwo-step be determined analytically. Third, simulations and ex-
relaxationof the intermediate scattering function, but it periments of the colloidal glass transition show evidence
predicts an ergodicity-breakingtransitionwell-abovethe for quasi-one-dimensional behavior such as correlated
experimentallydeterminedcolloidalglasstransition. Re- string-likemotion of the fastestmoving particles [30, 31]
latedtheoriesareabletopredictactivateddynamics,but and subdiffusive behavior with MSD ∼t0.5 that is char-
thelocationofthedivergenceinthestructuralandstress acteristic of single-file diffusion in quasi-one-dimensional
relaxationtimesisstillaninputparameter,notapredic- systems [32, 33]. For example, possible single-file sub-
tion [26]. Models of dynamic facilitation, in which mo- diffusive behavior occurs in simulations of polydisperse,
bile regions increase the probability that nearby regions elastichard-spheresystemsforpackingfractionsφ=0.61
will also become mobile, are able to explain important and 0.62 as shown in Fig. 1.
aspects of dynamical heterogeneities and non-Arrhenius In our previous studies of quasi-one dimensional mod-
relaxation times. However, these models have been im- els,wefocusedonthe‘figure-8’systemwithasinglejunc-
plemented using either coarse-grainedor lattice descrip- tion (or intersection) [29] and N hard rods. We found
tions, not particle-scale, continuum models. that the structural relaxation time diverges as a power-
Even though researchers have been able to visualize law with increasing packing fraction,
the motions of colloidal particles in 3D using confocal
microscopy for more than a decade [27, 28], an under- t ∼(φ∗−φ)−α, (1)
D
3
FIG. 3: (a) The Q1D channel consists of two different types of lobes, middle and end, with lengths Lm and Le = 2Lm +l,
respectively. The width of the channel, l, is the same as the length of the intersection. Particles move on the closed loop
(green dashed line) with an origin that is in the center of one of the end lobes. The plus and minus directions are indicated.
A close-up of the top intersection is shown in (b). When traversing the circuit, particles move in direction I for the first half
of the circuit and then in direction II for the second half. Directions I and II alternate between the NE/SW and NW/SE
directions for systems with multiple junctions. Q1D channels are pictured with (c) J = 1, (d) 2, and (e) 3 junctions. The
directions of motion I and II in thejunction are labeled.
∗
where α = N/2−1 and φ = N/(N +4) is the packing The residence time for the squeezed configurationin the
fractionatwhichkinetic arrestoccurs. At kinetic arrest, figure-8 model τ ∼ (φ∗ −φ)2 tends to zero in the limit
r
a plateau in the MSD persists for t → ∞. Near φ∗, the φ∗−φ→0, and thus t ∼(φ∗−φ)−α as in Eq. 1.
D
mostlikelyconfigurationsarethosewithN/2particlesin
both the top and bottom end lobes, and no particles in
the junction. tD is controlled by rare ‘junction-crossing’ II. MODEL DESCRIPTION
events, in which a particle from the bottom (top) lobe,
crosses the junction, enters the top (bottom) lobe from
A. System geometry
onesideofthejunction,andanotherparticleexitsthetop
(bottom)lobeandentersthebottom(top)lobefromthe
We consider the collective dynamics of N non-
other side of the junction. Thus, to undergo structural
overlapping Brownian particles in a quasi-one-
relaxation, the system transitions from a relaxed config-
dimensional (Q1D) channel that forms a closed loop
uration with half of the particles in each lobe to a rare,
with multiple intersections, as illustrated in Fig. 3. The
squeezedconfigurationwithanextraparticleinoneofthe
particlesmovethroughtheintersectionsinmodeI inthe
lobes, andback to a relaxedconfigurationthat is similar
first half of the circuit and mode II in the second half.
to the initial one but with particle labels shifted forward
For systems with multiple intersections, modes I and
orbackwardbyone. Thefrequencyf ofjunction-crossing
II alternate betweenthe northeast/southwest(NE/SW)
eventsisdeterminedbytheprobabilityP forasqueezed
S
and northwest/southeast (NW/SE) directions. The
configurationtooccurdividedbytheresidencetimethat
particles can move in both the forward and backward
the system spends in the squeezed configuration τ ,
r
directions, but they cannot turn at the intersections.
P Thus, to switch the traffic mode at a given intersection,
S
f = . (2)
particles in one mode must vacate the junction to allow
τ
r
particles in the other mode to enter.
The structural relaxation time is the inverse of this fre- The topology of the system is characterized by the
quency,andthust =f−1 =τ /P . Ifweassumeergod- number of junctions J. Each channel has two end lobes,
D r S
icity, P can be calculated from configuration integrals, and for a given J there are 2(J −1) symmetric middle
S
and for the figure-8 model, P ∼ (φ∗ −φ)N/2+1, where lobes. Thechannelgeometryisdescribedbythreelength
S
N/2+1 is the number of particles in the squeezed lobe. parameters: the channel width l (which also determines
4
FIG. 4: (a) Q1D system with N = 6, J = 2, and K = 1. (b) Mean-square displacement (MSD) Σ(t) versus time t for the
system in (a) from φ = 0.444 (filled circles) to ≈ 0.500 (open downward triangles) from left to right. The numbers 1, 2, and
3 indicate short-time diffusive, plateau, and long-time diffusive behavior of the MSD, respectively. (Inset) The time scale tD
beyond which the system displays diffusive behavior is obtained by setting Σ(tD) = 1 (long-dashed line). The dotted and
solid lines have slope 1 corresponding to short- and long-time diffusive behavior, respectively. (b) The timescale tD versus
packingfraction φfortheQ1Dmodelin (a). Theslopeofthelong-dashed linein theinsetis−1. tD increases asapowerlaw,
tD ∼(φ∗−φ)−1 with φ∗ =0.5.
the length of the intersection), andthe length of the end diffusivemotionatlongtime scales. Inthis work,wean-
and middle lobes L and L . alyze the slow dynamics of Q1D systems as the packing
e m
To reduce the number of independent parameters we fraction is increased by changing the lobe length L at
m
focus on a model with constant particle number N.
L =2L +l. (3)
e m
B. System dynamics
We also assume that the particle size d is equal to the
channel width,
In our model, each particle undergoes Q1D Brownian
d=l. (4) motion[34]alongthechannellength. ThisBrownianmo-
tion is implemented numerically using a Monte Carlo al-
With these assumptions, exactly K particles fit into a gorithm [35–37] with random single-particle moves and
middle lobe and 2K + 1 particles fit into an end lobe the step size chosen from a Gaussian distribution [38].
when L =Kl, where K is an integer. Thestandard-deviationoftheBrownianstepdistribution
m
With the lengths of the middle and end lobes related σ is chosen small enoughto accurately representBrown-
by Eq. (3), the total length of the channel is iandynamicsofnon-overlappingparticleswithshorttime
diffusioncoefficientD ∝σ2. Atlowpackingfractionswe
s
L=2(J +1)(L +l), (5) use σ =0.1∆,where∆=(L−Nl)/N is the averagegap
m
size between particles. For large packing fraction, we re-
where the length of each intersection is counted both in ducedthestandarddeviationtoσ ∝(L −(2K+1)l)/N ,
e e
the NE and NW directions. The packing fraction of the whereN isthenumberofparticlesinthemostoccupied
e
particles in the channel is given by endlobe, to ensurethatrareconfigurationsaresampled.
Nl
φ= . (6)
L
C. Close-packing and kinetic arrest
In our intersecting-channel model, particles moving
through an intersection in one direction block the mo- It is important to emphasize that KA states are dis-
tion of particles in the perpendicular direction. Thus, tinct fromclose-packedconfigurations. In a close-packed
at high packing fractions the system undergoes kinetic configuration, some or all particles in the system cannot
arrest. In a kinetically arrested (KA) configuration, the move, and the system size L cannot be reduced in a
m
particles can perform local movements, but the system continuous manner without creating particle overlap. In
cannot undergo collective rearrangements that lead to KA states, local particle motions are possible (and L
m
5
FIG. 5: Illustration of the bottleneck event that causes slow dynamics in the Q1D system with N = 6, J = 2, K = 1, and
M = 2 pictured in Fig. 4. For a rearrangement event to occur, particle 6 must migrate into the middle lobe ((a) and (b)),
reside there until other particles (2 and 3) pass from the upper to the lower part of the channel ((c) and (d)), and then pass
through the lower intersection intothe lower end lobe (e).
can be reduced), but the particles are blocked at the in- reach the long-time diffusive regime (where Σ(t)∼D t)
L
∗
tersections, and no particles are able to complete a full forL slightlyabovethekinetic-arrestthresholdL =l
m m
circuit around the channel. or packing fraction slightly below the critical packing
∗
There are two possible types of behavior for systems fraction (φ =0.5).
withmiddle lobe lengths thatareslightly abovethe crit-
ical value L = L∗ for kinetic arrest. After passing TheresultsinFig.4(b)showthatneartheKAthresh-
m m ∗
through the geometrical bottleneck associated with ki- old φ = φ the Q1D model displays slow dynamics that
neticarrest,thesystemeitherarrivesatanunconstrained resembles the dynamics observed in glass-forming sys-
state where the particles diffuse around the circuit on a tems. We find three dynamical regimes: (a) short-time
timescale of the order of τ = L2/D , or remains con- diffusion, (b) the formation of a plateau, where Σ(t) re-
0 s
strained by a sequence of bottlenecks that need to be mains nearly constant, and (c) long-time diffusion. As
cleared to complete a circuit. shown in Fig. 4 (c), the long-time diffusive motion is
∗
arrested at φ = φ . A cursory examination of the sys-
We are interested here in the kinetic arrest of the sec-
temdepictedinFig.4(a)isinsufficienttodetermine the
ond kind, where not only the initial escape from the
mechanismthatcausestherapidgrowthofthe timescale
nearlyKAstate occursona divergenttimescale,but the
requiredtoreachthelong-timediffusiveregimeasshown
timescaleforthe subsequentlong-timediffusive behavior
∗ in Fig. 4 (b) and (c).
also diverges at L = L . In what follows, the term
m m
kinetic arrest refers only to the second-kind behavior.
A detailed analysis (cf. Sec. III) reveals that the bot-
tleneck causing the slow diffusion correspondsto the dy-
namical event depicted in Fig. 5. During this event one
D. Critical dynamics near kinetic-arrest threshold
of the particles (particle 6 in the example considered)
needs to migrate into the middle lobe and reside there
The characteristic dynamics in the system for L ap- until other particles pass from the upper to the lower
m
∗ ∗
proaching the critical value L is illustrated in Fig. 4. part of the channel. In the limit L → L = l, the
m m m
AsdepictedinFig.4(a),the systemhasJ =2junctions particleresidinginthemiddlelobedoesnothaveenough
andcontainsN =6particles. Fig.4(b)showsthemean- room to move, which results in a low probability of this
square displacement (MSD) Σ(t) of the particles versus squeezed configuration and implies that the correspond-
timet,andFig.4(c)depictsthetimethesystemneedsto ing bottleneck-crossing event is rare.
6
E. Critical packing fractions
Our numerical simulations indicate that kinetic arrest
withdivergenttimescalesrequiredtoreachthelong-time
diffusive regime occurs for critical lobe lengths equal to
integer multiples of the intersection or particle length,
∗
L =Kl. (7)
m
Since each of the J junctions can be filled by at most
a single particle, the maximal number of particles in a
∗
system with L =L is
m m
N =2(J +1)(K+1)−J. (8)
cp
The corresponding close-packing fraction is
N
φcp(J,K)= cp . (9)
N +J
cp
According to our analysis presented in Sec. III, a sys-
tem near the KA threshold (7) requires at least two
particle-size vacancies to allow long-time diffusive mo-
tion. One vacancy is needed to empty an intersection,
and the other to allow a particle moving in the other
direction to completely cross the intersection.
Infact,asystemwithJ junctionsandlobeoccupation
number K exhibits critical scaling of t in the presence
D FIG. 6: Each Q1D configuration can be
of 2+M voids, i.e., for
mapped to one of the discrete microstates S =
N =Ncp−2−M (10) {EbJ1Mr1Ml1...JJ−1MrJ−1MlJ−1JJEt}, which is a set
of integers that represents the occupancy of the lobes and
particles, where intersections of the system. The integer Eb (Et) is the
number of particles in the bottom (top) end lobe and Mr
0≤M ≤M (11) i
max andMl givethenumbersofparticlesintheithrightandleft
i
and middle lobes. The integer Ji represents thestate of junction
i definedby Eq. 15.
M =2(J +1)K (12)
max
is the maximum number of particle size voids in the sys-
A. Definition of Microstates
tem such that when M → M the system still under-
max
goeskineticarrest. Ifonemoreparticlesizevoidisadded
to the system (or conversely a particle is taken out), the We represent each discrete microstate by the occu-
systemnolongerrequiresa‘squeezed’,bottleneckconfig- pancy variable
uarrraetsiotnisto relax. Thus, the packing fraction for kinetic S ={EbJ1,Mr1,Ml1...,JJ−1,MrJ−1,MlJ−1,JJ,Et},
(14)
N −M −2
∗ cp which is the set of integers that represents the states of
φ (J,K,M)= . (13)
Ncp+J lobes and intersections (as illustrated in Fig. 6). The
integer Eb (Et) is the number of particles in the bottom
(top) end lobe and Mr (Ml), i = 1...J, is the number
i i
III. DISCRETE MICROSTATES AND of particles in the ith right(left) middle lobe. A particle
CONSTRUCTION OF MICROSTATE NETWORK is assumed to reside in the lobe if its entire length is
contained within the lobe length.
If any portion of a particle enters an intersection, the
To describe the structural relaxation mechanisms in
particleisassignedtothisintersection. Sincetheparticle
the Q1D-channel model, it is convenient to map all of
lengthisthesameastheintersectionlength(Eq.(4)),the
theconfigurationsofthesystemontoasetofdiscretemi-
maximal number of particles that can reside in a given
crostates. The microstates correspond to configuration-
space regions defined by: (i) the number of particles intersection is two. The state of intersection i, which is
residing in each lobe; and (ii) the number of particles occupied by ki particles, is given by
presentineachintersectionandtheirdirectionofmotion.
J =2(1−δ )(1−δ )+k , (15)
Such a discrete mapping allows us to employ graphical i qi0 ki0 i
techniques to identify bottleneck states that control the where q =0 and 1 for directions of motion I and II (as
i
slow dynamics of the system. defined in Fig. 3). Several examples of microstates and
7
FIG. 7: (Color online) Illustration of microstates 505, 514, 424, 534, 444, and 406 for Q1D systems with N = 10 and J = 1.
Red and black shaded particles occupy the top and bottom end lobes, respectively. Gray and purple shaded particles occupy
thejunction (or intersection) in directions I and II,respectively.
theircorrespondingoccupancyvariablesS areillustrated φ∗(J,K,M),wedecomposeallmicrostatesintotwomain
inFig.7forafigure-8modelwithN =10andJ =1. The categories: the sets of unsqueezed (U) and squeezed
examples show all five states of the intersection, J = (Q) microstates. For unsqueezed microstates, none of
1
0,...,4, which is the middle integer in the microstate the lobes is completely filled with particles at φ =
∗
label. φ (J,K,M). In contrast, for squeezed microstates at
The number of microstates N (φ) allowed by the least one lobe becomes completely filled (i.e. squeezed
S
∗
excluded-volume constraints is maximal at φ → 0 and or compressed) when φ → φ (J,K,M). Hence, the
decreasesasthepackingfractionapproachesφ∗. InFig.8 configuration-space volume corresponding to squeezed
∗
(a)and(b)weshowthatforafixedtopology,thenumber microstates vanishes when φ = φ , whereas the vol-
of microstates N (φ∗) at kinetic arrest does not depend ume corresponding to unsqueezed microstates remains
S
or only weakly depends on the number of particles. In nonzero.
contrast, the number of microstates grows exponentially A squeezed microstate can contain one or more com-
when the number of particles N and intersections J is pressed regions (CRs). For the channel geometry de-
increased simultaneously, as depicted in Fig. 8 (c). scribed in Sec. IIA, there are three types of CRs. First,
a simple CR consists of a single compressed lobe, e.g.
a compressed top end or left middle lobe as shown in
B. Classification of states: Squeezed and trapped Fig. 9 (a) and (b). According to Eq. 3 and the notation
microstates introducedinSec.IIE,compressedmiddleandendlobes
containNc =K andNc =2K+1particles,respectively.
Second,acomposite CR(showninFig.9(c))is acon-
As discussed in Secs. I and IID, the system dynam-
∗ tiguous region that consists of compressed lobes and in-
ics near the KA threshold φ→φ (J,K,M) is controlled
tersectionsthatconnectthem. Eachconnectingintersec-
bylow-probabilitybottleneck microstatesthroughwhich
tion contains a single particle that is moving in a direc-
the system must pass to continuously move the parti-
tion that will connect the lobes (without causing a turn
cles around the channel. The bottleneck microstates oc-
at the intersection). The number of particles in a com-
cur with low probability, P , because they correspond
S
positeCRthatincludesk compressedmiddlelobesand
to a vanishingly small portion of the configurationspace m
k compressed end lobes is
when the system approaches kinetic arrest. Thus, the e
infrequentsamplingofthe bottleneckmicrostatesresults
Nc =k (K+1)+2k (K+1)−1. (16)
in the rapidly growing timescale required to reach the m e
∗
long-time diffusive regime as φ→φ .
A redistributed CR, as shown in Fig. 9 (d), is a region
that can be obtained from a composite CR by moving
someparticlesfromthecompressedlobestotheadjacent
1. Types of squeezed states connecting intersections. The number of particles in a
redistributed CR is the same as the number of particles
To facilitate the identification of the bottleneck states in the corresponding composite CR given by Eq. (16).
and analysis of the scaling behavior of the struc- We define a simple squeezed microstate to be one that
tural relaxation time t near the KA packing fraction containsonlysimpleCRs. Squeezedmicrostatesthatare
D
8
∗
FIG. 8: The number of microstates NS as a function of φ/φ for a Q1D model with (a) J =1, N =6 (circles), 20 (squares),
100 (diamonds), and (b) J = 2, K = 1, and N = 4 (M = 6; circles), 5 (M = 5; squares), 6 (M = 4; diamonds), 7 (M = 3;
∗
upward triangles), and 8 (M = 2; leftward triangles). (c) Number of microstates at φ = φ for a Q1D system with K = 1,
M =J, and N =Ncp−2−M.
not simple form a composite squeezed-microstate cluster, • kinetically arresting if the directionof particle mo-
such as that shown in Fig. 9 (e), which is a set that tion in the terminal intersection is orthogonal to
contains (i) a givensqueezed microstate Q that includes the compressed lobe, and the intersection is a part
only simple and composite CRs and (ii) all microstates of a compressed region.
thatcanbeobtainedfromQbyreplacingcompositeCRs
with the corresponding redistributed CRs. Asqueezedmicrostate(consistingofoneormoreCRs)
The particles contained in CRs of a given squeezed is untrapped if at least one CR end is free. In a trapped
microstate are termed compressed particles. The total state there are no free CR ends, but at least one end is
number ofcompressedparticles in a squeezedmicrostate trapping. IfallCRendsarekineticallyarresting,themi-
that has k CRs is crostate is KA, and the evolution is constrained to this
microstateortheassociatedmicrostatecluster. Sincethe
k microstate occupancy variable (15) specifies the number
Nc =XNic, (17) ofparticlesineachlobeaswellasboththenumberofpar-
i=1 ticles and the direction of motion for each intersection,
the untrapped, trapped, and KA CRs can be identified
where Nic is the number of particles in the ith CR. As by analyzing the sequence of integers in S. As further
will be discussed in Secs. IIIB2 and IIIB3, the number
discussed in Sec. IIIB3, trapped, squeezed microstates
of squeezed particles Nc, combined with the effects of
relaxmoreslowlythanuntrapped,squeezedmicrostates,
trapping on the ends of the CRs, determine the scaling
which influences the frequency of rare microstate sam-
∗
(withφ −φ)ofthefrequencyf withwhichacompressed
pling.
microstate (or microstate cluster) is sampled as the sys-
In addition to the three basic types of squeezed mi-
tem evolves at long times.
crostates described above, we consider a special case of
a compressed pair of left and right middle lobes for a
system with lobe size K = 1 in the middle panel of Fig.
2. Untrapped, trapped, and KA squeezed microstates
10. AsdiscussedinSec.IIIB3,the crossingfrequencyof
such microstates is not controlled by particle dynamics
Toestimatehowlong,onaverage,thesystemresidesin within the CRs, but by particle motion in the neighbor-
agivensqueezedmicrostate,we introducethe conceptof hoodofthe CRs. Seethe leftandrightpanelsofFig.10.
microstate trapping. To this end, we first establish three We will refer to these systems as semitrapped.
types of boundaries of a CR as shown in Fig. 11. The
boundary (i.e., the intersection that terminates the first
or last lobe in a CR) is:
3. Relaxation timescales
• free if the terminal intersection is empty or the
direction of particle motion in this intersection is By the arguments leading to Eq. (2), the frequency of
along the line passing through the compressed ter- crossing a bottleneck microstate S (or an associatedmi-
minal lobe; crostate cluster) depends on the microstate probability
P and on the time τ the system spends in microstate
S r
• trapping if the direction of particle motion in the (cluster) S during a crossing event. Due to system er-
terminal intersection is orthogonal to the com- godicity,theprobabilityP isproportionaltothefraction
S
pressedlobe,andthe intersectionis nota partofa ofthe configurational-spacevolumethe microstate(clus-
compressed region; ter) occupies. For a squeezed microstate (cluster) with
9
FIG.9: (Color online) Illustration of thetypesofcompressed regions (CRs) for aQ1D system with N =7,J =2, K =1,and
M =1: simple CRs (red) with compressed (a) top end and (b) left middle lobes; (c) a composite CR (gray) with compressed
topendandleftmiddlelobeswithparticle4movingindirectionI;(d)redistributedCR(blue)withparticles3and4occupying
thetopintersection movingindirection I,and(e)acomposite squeezedstateclustercomposed oftheredistributedCR (blue)
from (d) and simple CR (red) in thebottom end lobe.
because the particles blocking the intersections need to
diffuse an O(1) distance to release the trapped particles.
A trapped composite CR can relax to an associated re-
distributed CR on the fast timescale (19); however, the
system remains in the cluster of trapped microstates for
the longer time interval (20).
The scaling of the frequency (Eq. (2)) for crossing a
microstate corresponding to a simple or composite CR
neara KAtransitionis obtainedby combining (18) with
(19)foruntrappedmicrostatesandwith(20)fortrapped
microstates. Thus, the bottleneck crossing frequency is
f ∼(φ∗−φ)α, (21)
FIG. 10: (Color online) Illustration of semitrapped middle
lobes(S:green)inmicrostate101101. Whenthesystemtran- where
sitionstomicrostate130011,particle4ispreventedfrommov-
ing downward and particle 2 is prevented from moving up- α=Nc−2 (22a)
ward. A similar effect occurs when the system transitions to
110031. and
α=Nc (22b)
Nc compressed particles, the probability scales as
are the crossing-frequency exponents for untrapped
PS ∼Ω∼(φ∗−φ)Nc, (18) states with Nc >2 and for trapped states, respectively.
For a semitrapped microstate illustrated in Fig. 10
where Ω is the configurational-space region occupied by
there is no geometrical trapping (i.e., the particles are
the bottleneck microstate (cluster).
free to leave the CR). Since Nc = 2, Eq. (22a) predicts
Theanalysisdescribedinourpreviousstudy[29]shows
α = 0 in this case. However, our numerical simulations
that the residence time τ for an untrapped, squeezed
r indicate that, instead, the crossingfrequency scales with
microstate scales as
the exponent
τ ∼(φ∗−φ)2. (19)
r
α=1. (22c)
The residence time (19) is the timescale for an end par-
∗ This anomalous behavior indicates that the crossing fre-
ticle in a CR to diffuse a distance proportionalto φ −φ
quencyis notcontrolledbythe CRitself, butby particle
to the CR border. In contrast, for a trapped simple CR,
dynamicsinitsneighborhoodduringtheapproachtoand
τ ∼O(1), (20) subsequent separation from the CR.
r
10
FIG. 11: (Color online) Illustration of the three types of ends of compressed regions (CRs) using a Q1D system with N =7,
J =2, K =1, and M =2. (a) The redistributed CR formed by particles 1, 2, 3, 4, and 5 has kinetically arrested (particle 1;
blue) and free (particle 5; pink) ends. (b) The redistributed CR formed by particles 1, 2, 3, 4, and 5 has kinetically arrested
(particle 1; blue) and trapped (particle 5; violet) ends. (c) The simple CR formed by particle 3 (red) has two trapping ends
(particles 6 and 7).
Relation (22a) for untrapped CRs can be derived us- C. Diffusion through the microstate network
ing an alternative first-passage time argument. Accord-
ingly, we treat the boundary of an untrapped CR in Forsmallsystems suchas the figure-8withJ =1 con-
Nc–dimensionalconfigurationspaceasanabsorbingsur- sidered in Ref. [29], the bottleneck microstates can be
face,andconsiderastationaryprobabilitydistributionρ determined by inspection. However, when the number
that tends to the constant equilibrium value at infinity. of particles and intersections is increased, the number
BysolvingtheNc–dimensionalLaplaceequationforthis of microstates grows exponentially (Fig. 8 (c)), and the
boundary-value problem, we find that the perturbation system becomes rapidly too complex for a simple analy-
δρ of the probability distribution due to the presence of sis. Tofacilitateanautomatedanalysis,werepresentthe
the absorbing boundary scales as system evolution as a diffusive process on a network of
connected microstates. Key features of the network are
δρ∼(R/r)Nc−2, (23) determined using graph-theoreticaltechniques.
∗
where R ∼ φ −φ is the characteristic dimension of the
CR,andristhedistancefromtheCRregion. Integrating 1. States, transitions, and graphs representing the
the corresponding probability flux density microstate network
j ∼r−1(R/r)Nc−2 (24) In our approach, the set of microstates and transi-
ρ
tions between them (for a given φ near kinetic arrest)
over the (Nc−1)–dimensional CR surface yields are represented by a directed graph. The microstates
correspond to nodes of the graph, and the transitions
between states correspond to the edges connecting the
J ∼RNc−2, (25)
ρ nodes. This graphical representation is illustrated in
Figs. 12 and 13 for systems with a single and two in-
consistent with Eq. (22a). tersections, respectively.
We note that the above argument does not apply to a Transitionsbetweentwomicrostatesoccurwhenapar-
semitrappedCR,becausethesolutionofthecorrespond- ticle crosses a border between a lobe and an intersection
ing 2D Laplace equation for the probability density ρ (see Fig. 14). Since the particle can cross the border in
diverges logarithmically at infinity. This logarithmic di- a positive or negative direction (cf. the definition in Fig.
vergence may suggest that the crossing frequency f de- 3), the transitions are represented by directed edges de-
cayslogarithmicallywhenpackingfractionφtendstothe picted as arrows (the arrow orientation corresponds the
KA value; however, our numerical simulations yield the positive direction of particle motion).
power-lawbehavior(22c). Resolvingthisdiscrepancyre- Our goal is to identify bottleneck microstates (mi-
quires further study. crostate clusters) that control the slow dynamics of the