A phase-separation perspective on dynamic heterogeneities in glass-forming liquids

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by  C. Cammarota

A phase-separation perspective on dynamic heterogeneities in glass-forming liquids
C. Cammarota∗,1,2 A. Cavagna,2,3 I. Giardina,2,3 G.
Gradenigo†,4,5 T. S. Grigera,6 G. Parisi,1,2 and P. Verrocchio4,5,7
1Dipartimento di Fisica, Universita` di Roma “Sapienza”, p.le Aldo Moro 5, 00185, Roma, Italy.
2Centre for Statistical Mechanics and Complexity (SMC), CNR-INFM.
3Istituto Sistemi Complessi (ISC), CNR, Via dei Taurini 19, 00185 Roma, Italy.
4Dipartimento di Fisica, Universita` di Trento, via Sommarive 14, 38050 Povo, Trento, Italy.
5INFM CRS-SOFT, c/o Universita` di Roma “Sapienza”, 00185, Roma, Italy.
6Instituto de Investigaciones Fisicoqu´ımicas Te´oricas y Aplicadas (INIFTA) and Departamento de F´ısica,
0
Facultad de Ciencias Exactas, Universidad Nacional de La Plata, and CCT La Plata,
1
Consejo Nacional de Investigaciones Cient´ıficas y T´ecnicas, c.c. 16, suc. 4, 1900 La Plata, Argentina.
0
7Instituto de Biocomputaci´on y F´ısica de Sistemas Complejos (BIFI), Spain.
2

n Westudydynamicheterogeneities inamodelglass-former whoseoverlap withareferenceconfig-
a uration is constrained to a fixed value. The system phase-separates into regions of small and large
J overlap, so that dynamical correlations remain strong even for asymptotic times. We calculate an

4 appropriatethermodynamicpotentialandfindevidenceofaMaxwell’sconstructionconsistentwith
1 a spinodal decomposition of two phases. Our results suggest that dynamic heterogeneities are the
expression of an ephemeral phase-separating regime ruled bya finite surface tension.

]
n
n
The conspicuous lack of a growing correlation length, of dynamic heterogeneities. In so doing, we establish a
-
s contrastingwiththeverysteepincreaseoftherelaxation much-needed further link between dynamic and thermo-
di time,hasbeenapuzzleinthephysicsofstructuralglasses dynamic relaxation in glass-forming liquids.
. for quite a long time. Arguably, the first breakthrough Let us start with the standard measurement of the
t
a hasbeenthediscoveryofdynamicheterogeneities[1],and dynamic correlation length ξ . Our glass-former is the
d
m
the detection of a growing dynamical correlationlength, well-known soft-sphere model in 3-d [26]. A useful tool
- ξ [2, 3]. If we take two snapshots of the system sepa- to measure ξ is the overlap,which quantifies how much
d d
d
ratedbya time lagcomparableto the αrelaxationtime, a configuration at time t is similar to the reference con-
n
o τα,theparticledisplacementsvaryenormouslyacrossthe figuration at t = 0. If we partition the system in small
c system,andthe typicalsize ξd ofthe mobility-correlated cubicboxesandletnibethenumberofparticlesinboxi,
[ regions increases on lowering the temperature. thelocaloverlapisdefinedasq(r ,t)≡n (t)n (0),where
i i i
1 Morerecently,bystudyingthethermodynamicsofsys- ri refers to the centre of cell i [27]. The spatial map of
v temssubjecttoamorphousboundaryconditions[4,5],an the local overlap tells us how much different regions of
9 entirely different, fully static, correlation length ξ has the system have decorrelated (respect to the initial con-
3 s
been discovered [6, 7]. ξ also grows upon cooling, even figuration) over a time t. In Fig. 1 (top) we show two
5 s
2 though its surge occurs at lower temperatures than ξd. snapshots of the overlap field. We see that at t = τα
. The static correlation length has a natural interpreta- there are large heterogeneous regions, which eventually
1
tion as the size of the cooperatively rearranging regions fade away for longer times. To quantify their size we
0
0 [8], and within the random first-order theory [9] it is de- must compute the overlap correlationfunction,
1 termined by the balance between a surface tension cost G(r,t)≡(cid:10)q(0,t)q(r,t)(cid:11)−(cid:10)q(0,t)i(cid:10)q(r,t)(cid:11), (1)
v: and a configurationalentropy gain of a rearrangement.
i Clearly,itwouldbedesirabletounifythedynamicand or its Fourier transform S(k,t) (Fig. 2, left). In general,
X
the thermodynamic frameworks,so as to understandthe given a correlation function in Fourier space, it is well-
r
a interplay between the two correlation lengths. Although establishedpractice[12]toextractthe correlationlength
the static-dynamic connection is clear in mean-field sys- ξ from the small-k linear interpolation of S−1 vs. k2,
tems [10] and some progresses have been made in more
S(k,t)−1 =A+Bk2, (2)
realistic systems [11], we are quite far from a unifying
picture in real glass-formers. Here we show that surface from which the correlation length is obtained as ξ(t)2 =
tension, which is a crucial ingredient of the thermody- B/A. ThevalidityofEq.2isshownintheinsetofFig.2,
namic framework, also plays a key role in the formation right. [28]. The time-dependent correlation lenght ξ(t)
represents the size of the dynamical heterogeneities at
time t. This lengthscale grows as the time approaches
τ and the heterogeneities become more extended (inset
α
∗Present address: CEA, Institut de Physique Theorique, Saclay,
of Fig. 2, left). The largest value of ξ(t) (reached at the
F-91191Gif-sur-Yvette,France
†Present address: SMC-INFM and Dipartimento di Fisica, Uni- τα) defines the so-called dynamical correlation length,
vesita´diRoma“Sapienza”, p.leAldoMoro5,00185,Roma,Italy. ξd ≡ξ(τα) [2, 3, 13].
2
t=40000 MCsteps t=1000000 MCsteps QQQQQQ======000000......000000 QQQQQQ======000000......222222555555
3 tttttt======111111......000000⋅⋅⋅⋅⋅⋅111111000000444444
Q0 =0.0 ξ(t)1 ttttttttt=========225252525.........000000000⋅⋅⋅⋅⋅⋅⋅⋅⋅111111111000000000444444444
ttt===111...000⋅⋅⋅111000555
111111 104 105 106 111111 tt==22..55⋅⋅110055
t t=5.5⋅105
k,t)k,t)k,t)k,t)k,t)k,t)
S(S(S(S(S(S(
SS((kk,,tt))--11
Q0 =0.25 000000......111111 000000......111111
kk22
111111 111111000000 111111 111111000000
kkkkkk kkkkkk
FIG. 1: Fluctuations of the overlap field, δq(r,t) = q(r,t)− FIG. 2: S(k,t) at different times for the unconstrained (left)
hq(t)i for a 2-d slice of the system. Upper panels: uncon- and constrained (right, (Qˆ = 0.25) cases. Left inset: cor-
strained system. Lower panels: constrained system (Qˆ = relation length ξ(t) as extracted from Eq. (2). Right inset:
0.25). Leftpanels: t=τα. Rightpanels: largetimes. L=16. S(k,t)−1 vs. k2. T =TMC and L=16.
What happens beyond τ ? The memory of the initial if there is a nonzero surface tension between high and
α
configuration is gradually lost, so that the correlation low overlap regions, such a scenario is not what we ex-
function S(k,t) decays sharply, Fig. 2 (left). What hap- pect: the surface tension would force different domains
pens to ξ(t) is less clear,because the vanishing of S(k,t) to merge, driving the system towardsa phase-separated,
makesithardtofitareliablevalueofξ(t)throughEq.2. highly correlated state [15]. Hence, the second hypoth-
Although this point is debated [14], our results indicate esis is that the correlation does not decay and that the
that ξ(t) decreases beyond τα (left inset of Fig. 2), in dynamic correlation length ξ(t) grows beyond ξd, up to
linewithotherstudies[13]. However,whatisimportant, an asymptotic value of the order of the system’s size L.
and definitely out of question, is that the correlation de- The stark difference between these two hypotheses sug-
creases for large times, irrespective of its spatial range. gests that the constrained experiment may clarify the
Heterogeneities blur as q(r,t) becomes zero everywhere, mechanisms of formation of dynamical heterogeneities.
and their size ξ(t) becomes somewhat ill-defined. Inspection of the overlapfield in the constrained case,
Let us now make a different experiment. We want to Fig. 1 (bottom), is quite telling: for large t the system
impose a constraint on the dynamics, so that the sys- phase separates into high and low overlapregions, form-
tem cannot entirely lose memory of its initial configu- ing stable dynamical heterogeneities of the order of the
ration. This can be implemented by imposing a lower system size. From a quantitative point of view the sit-
bound on the global overlap, Q(t)=1/V R dr q(r,t). In uation is equally clear: in contrast with the free case,
the unconstrainedcase Q(t) goesasymptotically to zero, the constrained correlation function does not go to zero
as the memory of the initial configurationfades [29]. On for large times, but saturates at a finite value, Fig. 2
the other hand, if we run the dynamics with the con- (right). Hence, even in the late time regime dynamic
straint Q(t)≥Qˆ things change [30]. Initially the system heterogeneities remain strongly correlated.
does notfeel the constraint: the globaloverlapdecreases The study of the correlation length in the constrained
from its t = 0 value, Q = 1, and everything proceeds case confirms this scenario [31]. In an infinite system
as described above, including the growth of the hetero- undergoing phase separation, or at a critical point, the
geneities. However, at later times Q(t) hits its lower intercept A in Eq. 2 vanishes while the slope B remains
bound Qˆ and it cannot decrease further. What happens finite, so that ξ grows indefinitely [12]. On the other
to the dynamical heterogeneities in this case? hand, in a finite system phase separation means that
Therearetwoalternativehypotheses. First,thecorre- ξ becomes comparable with system size L. As a con-
lation S(k,t) and its spatial range ξ(t) decay to zero for sequence, the finite-size (periodic) real space correlation
largetasinthefreecase. Duetotheconstraint,however, functionG(r,t)ceasestobeasimpleexponentialforlarge
such endgame cannot happen in the same manner as in r(smallk). ThisimpliesthattheinterceptAinEq.2can
the free case, i.e. with q(r,t) becoming zero everywhere. go below zero and take small negative values (O(1/L2)).
Heterogeneities must thus become very small, forming a In a finite-size system it is therefore convenient to com-
salt-and-pepper configuration of the field q(r,t), so that pare A/B = ξ−2 vs. L−2 to check whether or not phase
the total integral of the field stays equal to Qˆ. Yet, separation occurs. From Fig. 3 (left) we see that in the
3
33
11
22..55
0.01
00..88
22
-2-2ξξ 00..66 11..55 QQ==00..2255 ∆E(t) constraifnreeed
00..44 QQ==00 t-1/3
11
00..22
00..55 0.001
00 LL--22 00 LL--22 1000 10000 100t000 1e+06 1e+07
110033 110044 tt110055 110066 110022 110033 tt 110044 110055 FIG.4: Energydifference∆E(t)=E(t)−E0vstatT =TMC
with constrained dynamics, Qˆ = 0.25. E0 is a parameter of
FIG. 3: Left: A/B = ξ−2(t) (see Eq. 2) at T = TMC in the fit E(t)=E0+γt−1/3. The line is 1/t1/3, corresponding
the constrained (circles) and unconstrained (triangle) cases. to thesurface tension exponentθ=2.
Right: thesame at T =1.55T . L=16.
MC
parameter is conserved and constrained to take a value
unconstrained case ξ−2 keeps well clear of L−2, while in the non-convex interval, phase separation occurs. We
in the constrained case ξ−2(t) unmistakably goes below have clearly observed phase-separation. Can we define a
L−2. This is exactly what we expect in a system with thermodynamicpotentialdisplayingMaxwell’sconstruc-
nonzerosurfacetensionundergoingphaseseparation. We tion?
studied two other sizes, L = 8 and L = 25, and in both Let us proceedminimalistically. Our phase-separating
cases ξ−2(t) drops below L−2, indicating phase separa- order parameter is the overlap Q, so it is a potential
tion. At higher temperatures, however, though the cor- W(Q) we are after. Besides, the potential must deter-
relation is enhanced by the constraint, the latter is in- mine the observed probability distribution of Q through
effective to make ξ−2(t) drop below L−2 (Fig. 3, right). the relation, P(Q) = exp[−NW(Q)] θ(Q − Qˆ), where
These results are consistent with the idea that the sur- the θ-function enforces the constraint [18]. If we com-
facetensiondecaysathightemperature,thus preventing pute the average linear fluctuation of Q and expand the
phase separation [16]. exponential, we obtain,
In systems with conserved order parameter undergo- W′(Qˆ)∼N−1 hQ−Qˆi−1. (3)
ing phase separation the domains size ξ(t) grows as t1/3
andthedynamicsproceedsbyreducingthetotalamount This quantity is easy to compute: we let the system
of interfaces, and therefore of energy, in the system [15]. evolve until the constraint is hit, and then we measure
The interface energy per domain scales like ξθ, where θ the (very small) average fluctuation of the overlap Q
is the surface tensionexponent. The totalnumber ofdo- over Qˆ (see [19] for a different definition of the poten-
mainsisLd/ξd,sothatthe totalinterfaceenergydensity tial). We report W′(Qˆ) in Fig.(5). The second deriva-
is ∆E(t)∼1/ξ(t)d−θ ∼1/t(d−θ)/3. In the standard case tiveofthepotentialisclearlynonzeroathighT,whereas
θ = d − 1, so that ∆E(t) ∼ 1/t1/3 [15]. Fig.4 shows around the Mode Coupling temperature a finite region
that something remarkably similar happens in our case. with W′′(Qˆ)∼0 develops. This is evidence ofMaxwell’s
After the constraint kicks in, ∆E(t) decays compatibly constructionandit supportsthe link betweenphase sep-
with an exponent 1/3. Hence, even though fitting coars- aration and metastability in our system.
ening exponents is notoriouslydifficult, andone must be W(Q)isafinite-dimensionalvariantofthetwo-replica
careful in drawing any conclusion, our data seems to be potential originally introduced in mean-field spin-glasses
compatible with the ‘naive’ exponent θ =2 [16, 17]. [20],andlatergeneralizedtostructuralglasses[21]. This
In general, phase-separation is the landmark of first potential is the free energy cost to keep a configuration
order phase transitions and metastability. At the mean- (the running one in the present work) at fixed overlap
field level one can normally define a thermodynamic po- Q with a generic equilibrium configuration (the initial
tential as a function of the order parameter that, below referenceone). Belowa dynamictransition(roughly,the
some spinodal point, exhibits a stable and a metastable Mode Coupling temperature), the mean-field potential
minimum, corresponding to the two phases. In finite di- develops a metastable minimum at a finite value of Q.
mensionMaxwell’sconstructionmakesthepotentialcon- In this framework relaxation at low temperatures can
vex, so that the derivative of the potential is constant be interpreted as a barriercrossingprocess, bringing the
(zero second derivative) in a finite interval (Fig. 5, in- systemfromthemetastableminimum(shorttimes,finite
set). Maxwell’sconstructionimpliesthatwhentheorder Q)tothe stableminimum(longtimes,zeroQ)[22]. The
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relaxation and therefore force phase separation.
[19] L.Fernandez,V.Martin-Mayor,andD.Yllanes,Nuclear
We have studied dynamic heterogeneities in a glass- Physics B 807, 424 (2009).
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[24] T. S. Grigera and G. Parisi, Phys. Rev. E 63, 045102
occurs, supporting the view of a surface tension that de-
(2001).
creases at high T. The co-existence of regions belonging
[25] J.-N. Roux, J.-L. Barrat, and J.-P. Hansen, J. Phys.:
todifferentamorphous‘states’(herethehigh/lowoverlap Condens. Matt. 1, 7171 (1989).
patches) is reminiscent of the random first-order theory [26] We simulate the 3-d soft-sphere binary mixture [23]
of thermodynamic relaxation [9]. In the dynamical case with parameters as in ref. 7. Simulations were done
theevolutionoftheseregionsisdrivenbyaclassiccoars- with Metropolis Monte Carlo with particle swaps [24].
eningmechanism,whichisstablewiththeconstraint,but The mode-coupling temperature for this system is
T =0.226 [25]. Our largest system has N = 16384
ephemeral in absence of the constraint. In the thermo- MC
particles in a box of length L=25.4.
dynamic case, on the other hand, the evolution of these
[27] The side ℓ of the cells is such that the probability of
regions is presumably driven by an entropic mechanism
findingmorethanoneparticleinasingleboxisnegligible.
[9]. Our results show that surface tension and metasta- [28] The so-called second-moment correlation length is ob-
bility stand as key links between the two frameworks. tained by computing A and B using only the first two
We thank G. Biroli, J.-P. Bouchaud, L. Cugliandolo, points, ξ2 =[S−1(k1)/S−1(0)−1]/k12, where k1 =2π/L
S. Franz, W. Kob and F. Zamponi for several important [12]. However, we find that a linear fit to a few small-k
points gives equivalent results for ξ2 yet lowering statis-
remarks, and ECT* and CINECA for computer time.
tical errors. Of course, the free-field form (2) of S(k,t)
The work of TSG was supported in part by grants from
does not hold at generic values of k.
ANPCyT, CONICET, and UNLP (Argentina). [29] Actually, normalization is such that Q = ℓ3 = 0.062876
for completely uncorrelated configurations, while Q = 1
for two identical configurations.
[30] To enforce the constraint we modify the Metropolis
algorithm: the probability to accept a move is p =
[1] M. D. Ediger, Annu.Rev.Phys.Chem. 51, 99 (2000). min{1,exp−∆E/T}forQ(t)≥Qˆ,andp=0forQ(t)<Qˆ.
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[3] C.Donati,S.Franz,G.Parisi,andS.C.Glotzer,J.Non- [R drG(r,t)]−1 mustbeexcludedfromtheanalysis.This
Crys.Sol. 307, 215 (2002). also implies that the dynamical susceptibility, χ(t) =
[4] J.-P.BouchaudandG.Biroli, J.Chem.Phys.121, 7347 S(0,t)=V[hQ2(t)i−hQ(t)i2],astandardmarkerofhet-
(2004). erogeneous dynamics, is trivially zero and therefore use-
[5] A. Montanari and G. Semerjian, J. Stat. Phys. 125, 23 less with theconstraint.