Table Of ContentParticle-scale reversibility in athermal particulate media below jamming
Carl F. Schreck1,2, Robert S. Hoy3, Mark D. Shattuck4,1, and Corey S. O’Hern1,2,5
1Department of Mechanical Engineering & Materials Science,
Yale University, New Haven, Connecticut 06520-8260, USA
2Department of Physics, Yale University, New Haven, Connecticut 06520-8120, USA
3Department of Physics, University of South Florida, Tampa, Florida 33620, USA
4Benjamin Levich Institute and Physics Department,
The City College of the City University of New York, New York, New York 10031, USA and
5Department of Applied Physics, Yale University, New Haven, Connecticut 06520-8120, USA
We perform numerical simulations of athermal repulsive frictionless disks and spheres in two
3
1 andthreespatialdimensionsundergoingcyclicquasi-staticsimplesheartoinvestigateparticle-scale
0 reversible motion. We identify three classes of steady-state dynamics as a function of packing
2 fraction φ and maximum strain amplitude per cycle γmax. Point-reversible states, where particles
n do not collide and exactly retrace their intra-cycle trajectories, occur at low φ and γmax. Particles
in loop-reversible states undergo numerous collisions and execute complex trajectories, but return
a
totheirinitial positions at theendof each cycle. Loop-reversible dynamicsrepresents anovelform
J
of self-organization that enables reliable preparation of configurations with specified structural and
1
mechanical properties over a broad range of φ from contact percolation to jamming onset at φJ.
3
Forsufficiently large φ and γ , systems display irreversible dynamicswith nonzero self-diffusion.
max
]
t PACSnumbers: 83.80.Fg61.43.-j,63.50.Lm,62.20.-x
f
o
s
. In equilibrium thermal systems, reversible processes cyclic simple shear [9] over a wide range of packing frac-
t
a areimaginedastransitionsfromonethermodynamicmi- tionφ andshear strainamplitude γ . We identify two
max
m crostate to another with no change in free energy. For classes of grain-scale reversible motion, point and loop
- example,thermalfluctuationsinhigh-temperaturefluids (which are stable to finite perturbations). For point-
d give rise to particle motions that yield only very small reversible dynamics, particles do not collide during the
n
changes in entropy. However, finite deformations of su- forwardcycle,andthus they exactlyretracetheirtrajec-
o
percooled liquids and amorphous solids, upon reversal toriesuponreversal. Incontrast,particlecollisionsoccur
c
[ of the strain, can produce microscopically irreversible frequently during loop-reversible dynamics, but the sys-
motion, such as collective particle rearrangements [1], tem self-organizes so that particles return to the same
1
anelasticity, and plastic flow [2]. The identification of positions at the beginning of each cycle, despite com-
v
topological defects in crystalline materials is straight- plex particle motion during the cycle. We map out the
2
9 forward, whereas it is much more difficult to identify ‘dynamical phase diagram’ as a function of φ and γmax.
4 particle-scalemotionthatgivesrisetoplasticityinamor- Thesystemtransitionsfrompoint-toloop-reversibleand
7 phous materials [3]. then from loop-reversible to irreversible (with nonzero
. self-diffusion) dynamics with increasing φ and γ . We
1 Granularmaterials, foams, andother athermalpartic- max
0 ulatemediaarehighlydissipative,andthereforemustbe show that the time evolution toward steady-state point-
3 andloop-reversiblebehaviorcanbecollapsedontoauni-
driventoinduceparticlemotion. Experimentalstudiesof
1 versal scaling function with power-law scaling at short
granularmediahaveshownmacroscalereversibilityofthe
: and intermediate times, and exponential decay at long
v packingfractionduringcyclicshear[4]andvibration[5].
Xi Sincethesesystemsarefarfromthermalequilibrium,one times.
WeperformnumericalstudiesofN athermaldisksand
might assume that they do not display microscale re-
r spheres undergoing quasi-static, cyclic simple shear in
a versiblemotionwhensubjectedtocyclicdriving. Experi-
2D and 3D at constant φ using shear-periodic boundary
mentalandcomputationalstudiesof2Dfoamshaveiden-
conditionsinsquareorcubic cells [11]. Particlesinteract
tified both reversible and irreversible T1 bubble neigh-
via the pairwise, purely repulsive linear spring potential
bor switching events during cyclic shear [6]. Researchers
havealsoshownthatmotionofindividualparticlestran- 2
ǫ r
sitions from reversible to irreversible beyond a density- V(r )= 1− ij Θ(σ −r ), (1)
ij ij ij
2(cid:18) σ (cid:19)
dependentcriticalstrain,whichdecreaseswithincreasing ij
packing fraction, in cyclically sheared suspensions [7, 8].
where r is the center-to-centerseparationbetweenpar-
ij
An important open question is whether athermal par- ticles i and j, Θ(x) is the Heaviside step function, σ =
ij
ticulate media can undergo completely reversible mo- (σ +σ )/2, and σ is the diameter of particle i. We fo-
i j i
tionduetointer-graincollisionswhensubjectedtocyclic cus on bidisperse particle size distributions, i.e. 50-50
loading. We address this question by performing nu- mixtures by number with diameter ratio σ /σ =1.4, to
l s
merical simulations of frictionless granular materials in frustrate crystallization during shear [12]. We consider
two and three spatial dimensions undergoing quasistatic system sizes from N = 32 to 512 to assess finite size ef-
2
fects for packing fractions below and near the onset of (a) (d)
jamming (φ ∼0.84 [10] in 2D and ∼0.65 [13] in 3D).
J
The particles are initially placed randomly in the sim-
ulation cell at packing fraction φ and then relaxed using
conjugate gradient energy minimization [12]. We apply
simple shear strain by shifting each particle horizontally
xi =xi +∆γyi , (2)
n,k+1 n,k n,k
in increments of ∆γ = 10−3, where xi and yi are
n,k n,k
coordinates of particle i at the kth step of strain cycle (b) (e)
n [14]. After each strain step, we minimize the total
potential energy at fixed shear strain, γ = k∆γ for the
k
forward or γ =2γ −k∆γ for the reverse part of the
k max
cycle. This process is repeated for up to n=106 cycles.
During the simulations, we measure the single cycle
mean-square displacement (at step k =0 for each cycle)
∆r2(n) = (Nσ2)−1 (Xi −Xi )2
1 s n,0 n+1,0
Xi (cid:16)
(c) (f)
+ (Yi −Yi )2+(Zi −Zi )2 (3)
n,0 n+1,0 n,0 n+1,0
(cid:17)
and arc length
L2(n) = (Nσ2)−1 (Xi −Xi )2 (4)
s (cid:18) n,k+1 n,k
Xi Xk (cid:2)
2
+ (Yi −Yi )2+(Zi −Zi )2 1/2
n,k+1 n,k n,k+1 n,k (cid:19)
(cid:3)
versus n, where Xi = xi −γ y , Yi = yi , and
n,k n,k k n,k n,k n,k
FIG. 1: (color online) Particle tracks (solid lines) in a small
Zi = zi are the non-affine displacements of parti-
n,k n,k window of a N = 128 system of bidisperse disks undergoing
cle i after subtracting off the affine contribution. The
(a) point-reversible, (b) loop-reversible, and (c) irreversible
long-time dynamics are either reversible or irreversible
behavior during cyclic simple shear. Panels (d), (e), and (f)
depending on φ and γmax. We quantify the steady-state showthedisks’tracksinpanels(a),(b),and(c),respectively,
behavior by measuring ∆r2(n) and L2(n) of the intra- after subtracting off the affine motion. The systems in (a)
1
cycle particle trajectories. and (d) correspond to φ = 0.64 and γ = 0.8, (b) and (e)
max
Particles in point-reversible systems organize to avoid to φ = 0.8 and γmax = 0.8, (c) and (f) to φ = 0.82 and
collisions. At long times, no collisions take place, and γmax = 0.8. Pluses, circles, and crosses mark the beginning,
L(n)=∆r (n)=0, or more aptly, they fall below small middle, and end of the particle tracks, and particle outlines
1
numerical thresholds, e.g. ∆r (n) < τ = 5×10−4 and correspond to thebeginning of thecycle.
1 r
L(n)<τ =10−8. The values of τ and τ do not qual-
L r L
itatively affect our results as long as they aresufficiently
small. Particle motions for point-reversible systems are
elongatedinthedirectionofaffinedisplacement,whereas
affine and in the direction of the imposed affine shear
nonaffine displacements are more compact.
(Fig. 1 (a)). Thus, the non-affine tracks of each particle
are zero (Fig. 1 (d)). Particles in systems undergoing irreversible dynamics
In loop-reversible systems, particle collisionsoccur fre- do notreturnto their originalpositions atthe beginning
quently, but the system self-organizes so that particles of each new cycle, e.g. Xni+1,0 6= Xni,0 and Yni+1,0 6=
return to the same positions at the start of each cy- Yi in 2D. (Figs. 1 (c) and (f).) Irreversible systems
n,0
cle. Since collisions between particles occur, L(n) > 0, have nonzero ∆r1(n) and L(n) (i.e. L(n) > τL and
but ∆r1(n) = 0 (i.e. below τr). Particle trajectories ∆r1(n) > τr). Systems can be “transient” irreversible
(Fig. 1 (b)) and non-affine displacements (Fig. 1 (e)) intime andevolveintopoint- orloop-reversiblesystems,
trace out complex paths, yet all particles end up in the orsteady-stateirreversibleandremainirreversibleinthe
same locations at the beginning of each new cycle, i.e. large-cycle limit with nonzero self-diffusion.
Xi = Xi and Yi = Yi in 2D. Thus, particle The steady-state“dynamicalphasediagram”inFig.2
n+1,0 n,0 n+1,0 n,0
trajectoriesformclosedloopsinconfigurationspace. We (a) for cyclically sheared athermal disks shows point-
focus on period one loop-reversible systems, but multi- and loop-reversible, as well as irreversible regimes ver-
period dynamics are also found. Particletrajectoriesare sus φ and γ [14]. Point-reversible systems occur at
max
3
and 2.2±0.2 for φ ≪ φ . Over a finite number of cy-
J
cles (i.e. n < 104 in Fig. 2 (b)), transient irreversible
dynamics can occur, but these systems become point-
reversible, loop-reversible, or steady-state irreversible as
n → ∞. Point-reversible systems tend to form ordered,
size-segregated layers, in which particles cannot collide
during simple shear. Further, the loop-reversibleto irre-
versible transition in steady-state φ (γ ) is bounded
R max
in the large-γ limit by the highest packing fraction
max
φ at which systems can form ordered, size-segregated
R
layers; φ (γ → ∞) = φ = π/4 ≃ 0.785 in 2D and
I max R
0.605 in 3D in the N →∞ limit.
InFig.2(b),scatterplotsofL(n)versus∆r (n)for2D
1
systems illustrate the evolution of the dynamics with in-
creasingn. Thepointsformseveralwell-definedclusters:
point-reversible (P) with L < τ and ∆r < τ , loop-
L 1 r
reversible (L) with nonzero L (L > τ ) and ∆r < τ ,
L 1 r
and irreversible (I) with nonzero L (L > τ ) and ∆r
L 1
(∆r > τ ). The P, L, and I clusters are separated by
1 r
more than 3 orders of magnitude in ∆r or L. For re-
1
gion L, we also mandate L > 30∆r since systems with
1
L<30∆r typically relax to point-reversible states. We
1
also enforce ∆r > 0.3σ to define region I since sys-
1 s
tems with ∆r <0.3σ typically relax to point- or loop-
1 s
reversible states. Systems that do not fall within the
bounds defining regions P, L, and I are categorized as
transientirreversible(T). As n increases,the fractionF
t
ofsystemsinthetransientregimevanishesasapower-law
FIG. 2: (color online) (a) “Dynamical phase diagram” n−α (where α=0.56±0.01),while the fractionofpoint-
for N = 128 bidisperse disks after n = 104 cycles show-
reversible, loop-reversible, and steady-state irreversible
ing point-reversible (circles), period-one (pluses) or multi-
systems saturates near 104 cycles (inset to Fig. 2 (b)).
period (crosses) loop-reversible, and transient (triangles) or
We also characterized the dynamics of these systems
steady-state (squares) irreversible dynamics versus γ and
max
as they approach steady-state point- and loop-reversible
φ. ThesolidanddashedlinesindicateφL(γmax)andφI(γmax),
the boundaries between point- and loop-reversible dynamics states (Fig. 3 (a)). We find that the single cycle mean-
and between loop-reversible and irreversible dynamics, re- square displacement can be described by a function that
spectively. The vertical dotted and dot-dashed lines define interpolates between power-law and exponential decays
φR = 0.785 and φJ ≈ 0.84. (b) Intra-cycle mean-square at short and long times, respectively:
displacement ∆r versus arc-length L after n = 10 (blue),
3×102 (red), an1d 104 (black) cycles. The regions labeled ∆r1f(n)=f+(n)(n/nc)−α+f−(n)e−β(n−nc), (5)
P, L, T, and I define point-reversible, loop-reversible, tran-
sient irreversible, and steady-state irreversible dynamics, re- where f±(n)= (1+e±γ(n−nc))−1, γ ∼ 1, nc is the cycle
spectively. The P-region extends from L = ∆r = 10−6 to number at which the decay changes from power-law to
1
10−16(notshown). Thedashedboundariesindicate∆r1=τr, exponentialbehavior, αis apower-lawscalingexponent,
∆r1 = 0.3σs, L = τL, L = 30∆r1, and L = ∆r1 discussed and β characterizes the exponential decay.
in the main text. The inset shows the fraction Ft of systems In Fig. 3 (b), we plot the best fit ∆rf(n) versus
in (a) categorized as point-reversible (solid), loop-reversible 1
∆r (n) at each γ and φ (averaged over 16 initial
1 max
(dashed),transientirreversible(dotted),andirreversible(dot-
conditions) for the 2D systems in Fig. 2 that evolve to
dashed) dynamics versus n. Similar data for 3D systems is
point-reversiblestates. ThescalingfunctioninEq.5col-
shown in SupplementaryMaterial.
lapses more than 60% of point-reversibile systems with
deviations ∆= h(log ∆rf(n)−log ∆r (n))2i <0.18.
10 1 10 1
The top and bottom insets in Fig. 3 (b) show the
power-law scaling and exponential decay of ∆r (n) sep-
1
low φ and γ , whereas irreversible systems occur for arately. We findsimilarscalingforthe approachtoloop-
max
φ & φ . At intermediate packing fractions between reversible states. However, the exponential decay for
J
roughlycontactpercolation[10]and jamming onset, e.g. loop-reversible systems is difficult to differentiate from
0.6.φ.0.84 in 2D, loop-reversible systems are found. numericalerrorbecausethelong-timedynamicsoccursat
The boundary between point- and loop-reversible sys- larger n and smaller ∆r than that for point-reversible
c 1
tems is γ ∼ A(φ)(φ − φ)λΘ(φ − φ), where A(φ) systems. In Fig. 3 (c), we show the power-law decay
max J J
depends weakly on φ and λ ∼ 1.2 ± 0.1 for φ → φ for all systems that evolve to loop-reversible dynamical
J
4
>2.0
2 1.5
1.5 1.0
γ max 1 0.5
0.0
0.5
(a)
0
0.4 0.6 φ 0.8 1
>104
2 103
1.5 102
γ max 1 10
1
0.5
(b)
0
0.4 0.6 φ 0.8 1
FIG.4: (coloronline)(a)Contourplotofthepower-lawexpo-
nentα(Eq.5)versusφandγ for2Dsystemsthatevolveto
max
either point- (circles) or loop-reversible (crosses) states. (b)
Contourplotofthecyclenumbernc (Eq.5)thatcontrolsthe
crossover from power-law to exponential decay versus φ and
γ for thepoint-reversible systems in (a).
max
crossover from point- to loop-reversible behavior, which
FIG. 3: (color online) (a) Single cycle mean-square displace-
suggeststhattheoriginoftheslowdynamicsisrelatedto
ment ∆r versus n for a point-reversible system at φ = 0.64
1
contact or “collision” percolation. In contrast, α>1 for
and γ = 0.5 (circles) and a loop-reversible system at
max
φ = 0.8 and γ = 0.5 (pluses) with best fits to ∆rf point-reversiblesystemsatlowφandγmax. InFig.4(b),
max 1 weplotn forpoint-reversiblesystemsversusφandγ .
(Eq. 5) indicated by solid and dashed lines. (b) Compari- c max
son of ∆r1(n) (averaged over 16 initial conditions for each We find that nc increases with φ and γmax, and appears
γ and φ) to ∆rf(n) (black dots), where n is the cycle to be diverging as the system approaches the transition
max 1
number,forpoint-reversiblesystemsinFig.2with∆<0.18. from point- to loop-reversibility.
∆r1f(n)=∆r1(n)isindicatedbythedashedline. Thetopleft To test the stability of steady-state loop-reversible
iTnhseetdsahsohweds lloinge10i∆ndri1c(ant)e/sa∆vre1r(snu)s/αal=og1(0nn/n/nc)c−(αb.laTckhedobtost)-. satmatpelsit,uwdeeδpeinrtruarnbdoamll pdairretcictlieosnsat[14st].raWineγth=en0pebryforamn
tom rightinset showslog10∆r1(n)/a versusβ(n−nc) (black cyclicsimpleshearontheperturbedsystemandmeasure
dots). ∆r (n)/a=e−β(n−nc) isindicated bythedashed line.
(c) log10∆1r1(n)/a versus αlog10n (black dots) for systems the deviation, ∆r = q(Nσs2)−1 i ~rni,0−~rni,,p0 2, where
i∆n F<ig0..024t.h∆atrev(nol)v/eat=olno−opα-riesvienrdsiibcaleteddynbaymthicealdsatsahteeds wliniteh. ~rni,,p0 are the coordinates of the Ppert(cid:12)(cid:12)urbed syst(cid:12)(cid:12)em after
1 t cycles required to reach steady steady at each φ and
Theinset shows ∆r /a versusnfor threeindependentinitial
1 γ . We find that loop-reversible systems are stable
conditions (squares, triangles, and pluses) at φ = 0.76 and max
(∆ < τ ) for perturbations δ < δ ≃ 10−1, i.e. per-
γ =0.8. Exponential fits to the large n regime are shown r r c
max
assolid,dashed,anddottedlineswithslopesβ=0.029,0.026, turbations as large as one-tenth of a particle diameter,
and 0.016, respectively. where δc is relatively insensitive to φ for γmax .1.
Inconclusion,we studied the extent to whichparticle-
scalemotioncanbereversibleinathermalsystemsunder-
going cyclic quasistatic loading. We identified two types
states. In the inset, we also show several systems for of reversible behavior. For point-reversible states, par-
which we captured the long-time exponential decay. ticles do not collide and therefore trivially retrace their
In Fig. 4 (a), we show the power-law scaling exponent paths. For loop-reversible states, all particles undergo
αfor2Dsystemsthatevolvetopoint-andloop-reversible multiple collisions and have complex trajectories, yet all
statesversusφandγ . Wefindthatα.1forallloop- particlesreturntothelocationstheywereinatthebegin-
max
reversible systems and point-reversible systems near the ning of the cycle. We determined the regions of packing
5
fractionandstrainamplitudesin2Dand3Dwherethese withparticularstructuralandrheologicalpropertiesover
dynamical states are stable. In particular, we find that a broad range of packing fractions.
loop-reversiblestates occur over a range of packing frac-
tionfromcontactpercolation[10]tojamming onset,and We acknowledge support from NSF grant numbers
thus our results emphasize that complex spatiotemporal NSF MRSEC DMR-1119826 (CS), DMR-1006537(RH),
dynamics are found well below φ . Loop-reversible dy- CBET-0968013(MS),andCBET-0967262(CO).Wealso
J
namicalstatesrepresentanovelformofself-organization thank S. Papanikolaou, S. S. Ashwin, and T. Bertrand
that will enable reliable preparation of configurations for helpful discussions.
[1] P. Schall, D. A. Weitz, and F. Spaepen, Science 318 [9] S.Slotterback,M.Mailman,K.Ronaszegi,M.vanHecke,
(2007) 1895. M. Girvan, and W. Losert, Phys. Rev. E 85 (2012)
[2] J. S. Harmon, M. D. Demetriou, W. L. Johnson, and K. 021309.
Samwer, Phys. Rev. Lett. 99 (2007) 135502. [10] T. Shen, C. S. O’Hern, and M. D. Shattuck, Phys. Rev.
[3] M. L. Falk and J. S. Langer, Ann. Rev. Condens. Matt. E 85 (2012) 011308.
Physics 2(2011) 353. [11] M.P.AllenandD.J.Tildesley, Computer Simulationof
[4] M. Nicolas, P. Duru,and O. Pouliquen, Eur. Phys. J. E Liquids (Oxford UniversityPress, New York,1987).
3(2000) 309. [12] G.-J. Gao, J. Blawzdziewicz, and C. S. O’Hern, Phys.
[5] E. R. Nowak, J. B. Knight, E. Ben-Naim, H. M. Jaeger, Rev. E 80 (2009) 061303.
and S. R.Nagel, Phys. Rev. E 57 (1998) 1971. [13] N. Xu and C. S. O’Hern, Phys. Rev. Lett. 94 (2005)
[6] M. Lundberg, K. Krishan, N. Xu,C. S. O’Hern, and M. 055701.
Dennin,Phys. Rev. E 77 (2008) 041505. [14] See Supplementary Materials, which include results for
[7] D.J.Pine,J.P.Gollub,J.F.Brady,andA.M.Leshan- 2Dand3DsystemsasafunctionofsystemsizeN,shear
sky,Nature 438 (2005) 997. strain increment∆γ,andenergyminimization threshold
[8] Laurent Corte, P. M. Chaikin, J. P. Gollub, and D. J. V .
th
Pine, Nature Physics 4(2008) 420.
