Spatial heterogeneities in structural temperature cause Kovacs’ expansion gapparadox in aging of glasses

Matteo Lulli,1, ∗ Chun-Shing Lee,1 Hai-Yao Deng,2 Cho-Tung Yip,3 and Chi-Hang Lam1

1Department of Applied Physics, Hong Kong Polytechnic University, Hong Kong, China2School of Physics and Astronomy, The University of Manchester, Manchester M13 9PL, United Kingdom

3School of Science, Harbin Institute of Technology,Shenzhen Graduate School, Shenzhen, Guangdong 518055, China

(Dated: September 10, 2019)

Volume and enthalpy relaxation of glasses after a sudden temperature change has been extensivelystudied since Kovacs seminal work. One observes an asymmetric approach to equilibrium uponcooling versus heating and, more counter-intuitively, the expansion gap paradox, i.e. a dependenceon the initial temperature of the effective relaxation time even close to equilibrium when heating.Here we show that a distinguishable-particles lattice model can capture both the asymmetry and theexpansion gap. We quantitatively characterize the energetic states of the particles configurationsusing a physical realization of the fictive temperature called the structural temperature, which,in the heating case, displays a strong spatial heterogeneity. The system relaxes by nucleation andexpansion of warmer mobile domains having attained the final temperature, against cooler immobiledomains maintained at the initial temperature. A small population of these cooler regions persistsclose to equilibrium, thus explaining the paradox.

Kovacs’ series of experiments [1] is fundamental to ourpresent understanding of aging and memory propertiesin glassy materials [2–4]. In [1], the volume relaxation ofpolymer glasses has been analyzed by performing rapidtemperature changes, or temperature jumps, focusing onexperimental protocols implementing one or two succes-sive temperature shifts. The renowned Kovacs effect,observed in experiments involving two successive rapidtemperature changes, or a double temperature jump,has been theoretically studied using empirical mean-fieldmodels including the Tool-Narayanaswamy-Moynihan(TNM) [5–7] and the Kovacs-Aklonis-Hutchinson-Ramos(KAHR) [8] models. A temperature jump more preciselyacts directly only on the phonon temperature. The Ko-vacs effect can be satisfactorily accounted for in thesetheories using a fictive temperature TF(t), which de-scribes some internal state of the material with a dy-namics generally lagging behind that of the phonons [5].In contrast, the expansion gap paradox, also called theτeff paradox, from single-jump experiments is much morepuzzling [9–16]. Specifically, the effective relaxation rateτeff of the polymeric system studied by Kovacs, after thetemperature jump, depends persistently on the initialtemperature, and apparently, even arbitrarily close toequilibrium. Such a strong material memory, however,cannot be reproduced by TNM or KAHR models, andhas only been accounted for by their stochastic coun-terparts, namely the stochastic version of a free-volumemodel [17] and, more recently, the stochastic constitutivemodel (SCM) [18]. The reasons for the failure of simplemean-field models and why stochastic fluctuations areextraordinarily important for this experimental protocolare not well understood.

∗ matteo.lulli@gmail.com

Here, we successfully reproduce Kovac’s expansiongap for the first time using a microscopic particlemodel, going beyond mean-field descriptions. Specifi-cally, we adopt the distinguishable-particle lattice model(DPLM) [19]. The phonon temperature, which is sub-jected to a single jump, is modeled by the bath temper-ature of the kinetic Monte Carlo simulation algorithm ofthe DPLM. We observe an expansion gap in the systemenergy relaxation, analogous to enthalpy relaxation inexperiments [20]. By studying spatial profiles of particledisplacements and interactions, we provide an intuitiveresolution of the paradox.

The recently introduced DPLM displays many featuresof particle dynamics characteristic of glasses and pos-sesses exactly solvable equilibrium statistics [19]. More-over, a wide range of values of the fragility index can beobtained by varying the interaction energy distributionof the model [21]. In the simulations we have performed,the DPLM is defined on a 2-dimensional square lattice oflinear size L = 100. The sites are occupied by N distin-guishable particles, each of them associated to a uniquelabel ranging from 1 to N . Then, si = 1, . . . , N denotesthe particle at site i. The key feature of the model isthat each particle is coupled to its nearest neighbors bymeans of site- and particle-dependent random interac-tions: a four-indices interaction energy Vijsisj is associ-ated to the particles si and sj sitting at sites i and j.In order to simulate the hopping dynamics of the par-ticles we allow for the presence of empty sites or voids.Considering a void density φv = 0.005, we allow the pres-ence of Nv = 50 voids with default label si = 0 so thatL2 = N +Nv. One can write the system energy as

E =∑

〈ij〉′Vijsisj , (1)

where the sum∑〈ij〉′ is restricted to occupied nearest

arX

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0368

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9 S

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neighboring sites. The interactions are symmetric un-der concurrent exchange of spatial and particle indices,i.e. Vijkl = Vjilk. The entire set of possible interactions{Vijkl} is drawn according to an a priori probability dis-tribution g(V ) and it is quenched. For each particle con-figuration, there is a set {Vijsisj} of interactions whichare referred to as realized. It has been proved in [19]that at equilibrium the probability distribution of the in-teractions factorizes over all occupied neighboring sites.The equilibrium probability distribution of any realizedinteraction at temperature T follows the form

peq(V, T ) =1

N (T )g(V ) e−V/kBT (2)

where N (T ) is a normalization constant. For the rest ofthe paper, we will work in natural units with kB = 1. Asin [19], we take g as a uniform distribution defined on theinterval V ∈ [V0, V1], with V1 = −V0 = 0.5. In this case,the equilibrium distribution in Eq. (2) yields a simpleexponential dependence on the interaction energy. Thesimulations are performed by means of a kinetic MonteCarlo algorithm for activated hopping dynamics in whicheach particle can hop to the position of a neighboring voidwith a rate

w = w0 exp

[− 1

T

(E0 +

∆E

2

)], (3)

where ∆E is the energy change of the system inducedby the hop. We set w0 = 106 and E0 = 1.5 so thatE0 + ∆E/2 ≥ 0.

In our simulations, we implement the single-jump pro-tocol analyzed in [1] and displayed in Fig. 1(a) startingfrom equilibrium configurations [19] at some initial tem-peratures Ti. Then, the bath temperature T in Eq. (3),representing the phonon temperature, is set instanta-neously at the final temperature Tf . It is a commonpractice to refer to the cases Ti > Tf (cooling protocol)as down-jumps, and to the cases Ti < Tf (heating proto-col) as up-jumps.

Analogously to [1], we study the fractional deviationδE(t) of the system energy E(t) from its equilibrium valueE∞ at Tf given by

δE(t) =E(t)− E∞|E∞|

, (4)

where E(t) and E∞ < 0 are computed using Eq. (1)and are averaged over 217 ' 1.3× 105 independent runswith different random number seeds. The analytical ex-pression for E∞ is reported in Eq.(2) of the Supplemen-tal Material and shows agreement with the numericallycomputed values. Fig. 1(b) reports the time evolution ofδE(t) for a set of symmetric temperature jumps, whichare chosen to be of similar relative magnitudes as in [1].As shown in Fig 1, δE from DPLM simulations closelyresembles experimental results in [1] and, in particular,correctly reproduces the up-down asymmetry of the ap-proach to equilibrium. The initial asymmetry at time 0

down-jumpup-jump

Tf TiTi

E∞

Eup(0)

Edown(0)(a)

−0.2

−0.15

−0.1

−0.05

0

0.05

0.1

0.15

10−1 100 101 102 103

down-jump

up-jump

(b)Tf = 0.25

−2.5

−2

−1.5

−1

−0.5

−0.2 −0.15 −0.1 −0.05 0 0.05 0.1 0.15

{Expansion Gap

down-jumpup-jump

(c)

−2.3

−2.1

−0.003 0 0.003

δ Et

Ti = 0.3125

0.303

0.292

0.280

0.266

0.234

0.220

0.208

0.197

0.1875

−log10τ eff

δE

FIG. 1. Panel (a): Schematic diagram for single-temperaturejump protocol for equilibrium samples at initial temperatureTi which are then cooled (for Ti > Tf , i.e. down-jump) orheated (for Ti < Tf , i.e. up-jump) to the final temperatureTf . Panel (b): data from DPLM simulations for Tf = 0.25and different values of Ti. The asymmetry of the approachbetween up- and down-jumps is observed. Panel (c): resultson τeff measured using data in middle panel. Data close toequilibrium with |δE(t)| ≤ 0.003 are shown in the inset.

is simply due to an equilibrium heat capacity decreasingwith T , of which we report the analytical expression inEq.(3) of the Supplemental Material. However, the muchslower relaxation for the up-jumps compared with thedown-jumps, successfully reproduced here, is non-trivialand has been the focus of many studies [9–16].

3

FIG. 2. Snapshots of structural temperature TS(~x, t) (2nd and 3rd rows) and particle displacement d(~x, t) (1st and 4th rows)for down-jump from Ti = 0.3125 (1st and 2nd rows) and up-jump from Ti = 0.1 (3rd and 4th rows) to final temperatureTf = 0.25 in a system of linear size L = 200 with coarse graining scale ` = 5. Different columns refer to average overlapq = 1, 0.8, 0.6, 0.4, 0.2 and 0.05 (from left to right) corresponding to increasing time t. Both TS(~x, t) and d(~x, t) show relativelyhomogeneous evolution for the down-jump, but strongly heterogeneous evolution with large domains for the up-jump. Finally,one can notice for the up-jump case a clear coincidence between immobile domains and low structural temperature domains.

We next define the effective relaxation time τeff as in [1]

τ−1eff (t) = − 1

δE(t)

dδE(t)

dt, (5)

which would reduce to a constant for an exponentiallydecaying δE(t). Averages and errors for τeff have beencomputed with the jackknife resampling method. Resultsfor τeff(t) against δE(t) are reported in Fig. 1(c) and theyshow very similar features to those reported in [1]. Mostimportantly, we observe as in [1] that the data for τeff

have not converged to a single limiting value independentof Ti, even very close to equilibrium at |δE | ' 0, creatingthe expansion gap paradox. The inset reports the dataclose to equilibrium: while the down-jump data showa clear convergence among themselves and with respectto the up-jump data at small jumps, convergence is notobserved in the large up-jump cases even for the small-est δE studied. To the best of our knowledge, amongthe constitutive models [5–8, 17, 18] only the stochas-tic free-volume model [17] and the SCM [18], accountingfor dynamic heterogeneities, have been able to reproducethe gap. Being able to qualitatively recover the mostimportant experimental features by means of a micro-

scopic particle model is clearly important for a deeperunderstanding of the aging dynamics. Nevertheless, τ−1

effin Fig 1(c) does not exhibit a gentle rise at intermedi-ate values of δE for the up-jumps as observed in exper-iments [1]. This may happen because we have adopted,for simplicity, a constant void density φv in our simula-tions, which should instead increase upon heating. Suchan increase of φv would hence yield a faster dynamics.

Compared to the constitutive models, an advantage ofour approach is that it allows us to analyze the differencesbetween up- and down-jump dynamics from the real-space perspective, going beyond mean-field descriptions.We define a local particle displacement d(~x, t) = |~x− ~x0|as the distance of a particle located at ~x at time t rela-tive to its position ~x0 at time 0 at which the tempera-ture jump is imposed. If ~x is vacant at time t, we putd(~x, t) = 0 for simplicity. It is also useful to define a localparticle persistence, i.e. an overlap field, q(~x, t) as

q(~x, t) =

{1 if d(~x, t) = 00 if d(~x, t) > 0

(6)

The average overlap q(t), which equals q(~x, t) averagedover sites occupied at t, gives the fraction of particles still

4

10−3

10−2

10−1

100

101

−0.5 −0.25 0 0.25 0.5

up-jump(b)

10−1

100

−0.5 −0.25 0 0.25 0.5

down-jump(a)

p(V)

V

q = 1.0

0.8

0.6

0.4

0.2

p(V)

V

q = 1.0

0.8

0.6

0.4

0.2

FIG. 3. Panels (a) and (b): semi-log plot of the p(V ), in col-ored dots, for five values of the overlap q = 1, 0.8, 0.6, 0.4, 0.2,for the down-jump protocol with Ti = 1.0 and the up-jumpone with Ti = 0.1 to the common final temperature Tf = 0.25.The final equilibrium distribution is drawn in green solid lineswhile the initial distributions are drawn in dashed red andblue lines. Panel (a): Down-jump data superposed to single-temperature fits in black lines, showing a good agreement.Panel (b): Up-jump data superposed to Eq. (7).

located at their original positions at time t. In Fig.1 inthe Supplemental Material we show q(t) against t.

In Fig. 2 (and in the supplementary videossupvideo up.mp4 and supvideo down.mp4 together withthe voids positions), we report the evolution of the lo-cal displacement d(~x, t) for selected values of the averageoverlap q which provides a useful measure of the progressof the relaxation. Large jump magnitudes are used sothat qualitative features of the dynamics can be moreclearly observable. Fig. 2 shows that the growth of d(~x, t)is much more heterogeneous in the up-jump case. Well-separated domains with highly mobile particles nucleateand invade other immobile domains. Hence, a strongspatial heterogeneity dominates the up-jump relaxation.

In order to understand the emergence of the signifi-cantly more heterogeneous up-jump dynamics, we studythe energy states of the particle configurations by ana-lyzing the probability distribution p(V ) of the realizedinteractions Vijsisj . Results are reported in Fig. 3. Inthe panels (a) and (b), the computed p(V ), for up- and

down-jump dynamics respectively, are reported for differ-ent values of the average overlap q. The final equilibriumdistribution peq(V, Tf ) given in Eq.(2) is also reported,while the initial ones, i.e. peq(V, Ti) are reported in redand blue dashed lines for down- and up-jump respectively.Fig. 3(a) shows that in the down-jump case the evolutionof the probability distribution occurs simultaneously forthe whole range of V . Furthermore, p(V ) can be well ap-proximated by a single equilibrium distribution peq(V, T ),with T decreasing monotonically with 1−q and thus alsowith t, yielding reasonable fits. In contrast, for the up-jump data reported in Fig. 3(b), there is a remarkabledifference between the evolution of high- and low-energyinteractions: For example for q = 0.8, the distributionp(V ) at V & 0.25 has already attained the same slopein the semi-log plot as the equilibrium one at the finaltemperature Tf , while values of p(V ) for low-energy in-teractions V . −0.25 are still very close to those of theinitial temperature Ti. Indeed, p(V ) for a wide rangeof q can be very well fitted by a superposition of twoequilibrium distributions

p(V ) = q · peq(V, Ti) + (1− q) · peq(V, Tf ). (7)

This result suggests that the particle configurations inthe mobile regions, whose relative extent is 1 − q, havereached equilibrium at the final temperature Tf while theimmobile regions, whose extent is q, have an interactionpopulation distributed according to peq at the initial tem-perature Ti.

The previous results suggest the existence of a strongspatial heterogeneity in the distribution of the realizedparticle pair-interactions only for the up-jump case. Asa further step, we compute from the interactions a tem-perature TS we call the structural temperature. In phys-ical terms, TS measures how well particles are locallypacked, and hence a low temperature, for example, cor-responds to a better bonded and more stable configura-tion. Such a definition can be applied to a wide rangeof materials and it should not be regarded as specific tothe present case. For the DPLM, we define it as a localtemperature TS(~x, t) at position ~x based on the inter-action V `(~x, t) averaged over a square domain of linearsize ` centered at ~x. Requiring that TS should coincidewith the bath temperature within statistical fluctuationsat equilibrium, we define TS and solve for it numericallyfrom V `(~x, t) =

∫dV V peq(V, TS). Note that TS is anal-

ogous to Tool’s fictive temperature [5], local values ofwhich have also been studied before [22, 23].

Figure 2 also shows (along with the supplementaryvideos supvideo up.mp4 and supvideo down.mp4 ) theevolution of TS(~x, t) for ` = 5. By a direct comparisonwith the local displacement d(~x, t), we see that the evo-lution of TS(~x, t) is spatially homogeneous for the down-jump. For the up-jump, high-TS domains with TS ' Tfdevelop in good spatial correspondence with the highlymobility regions. The immobile regions in contrast main-tain the initial temperature, i.e. TS ' Ti. These re-sults are fully consistent with the good fits to p(V ) in

5

Fig. 3(b) using Eq.(7). Moreover, some low-TS immobiledomains remain even at the very late stage of relaxationat q = 0.05, thus constituting a remnant of the initialtemperature acting on the dynamics, although equilib-rium is already reached almost everywhere else.

The structural temperature heterogeneity observed forthe up-jumps can be understood in terms of a stabil-ity argument of propagating fronts as follows. First, theheating up of a glass is an auto-catalytic process, sincethe excitation of particle arrangements to higher-energyconfigurations speeds up the particle dynamics and henceprovides a positive feedback to the further warming of thesystem. In d dimensions, TS(~x, t) can be seen as a suc-cession of equal-time d-dimensional surfaces in a (d+ 1)-dimensional space, representing a front propagating up-wards from Ti to Tf . The propagation is driven by theenergy influx from the bath and is stochastic because ofthe intrinsic noise of the particles dynamics. The evo-lution of the surfaces is unstable against small pertur-bations, meaning that a locally out-stretched (warmer)region will further advance much faster towards the finalvalue Tf as the auto-catalytic nature of the dynamicsamplifies the perturbations. For very low Ti, implyingan extreme sensitivity of the dynamics on temperature,TS can comparatively quickly reach Tf in localized do-mains, while being practically stuck at the initial valueTi elsewhere. This explains the nucleation of Tf domainsin a background of Ti regions. The fast dynamics in Tfdomains enhances the heating-up of neighboring regions,inducing domain-wall motions. Due to the very stableconfigurations of the Ti regions, the domain invasion canbe a slow process compared with the relaxation dynamicsin the Tf domains. Therefore, the particle displacementd(~x, t) can become very large in the mobile Tf domainseven close to their domain boundaries as observable inFig. 2. By contrast, cooling for the down-jump protocolis instead an auto-retarding process so that the down-ward propagating front TS(~x, t) is stable against pertur-bations. The dynamics is thus overall homogeneous withrelatively uniform TS(~x, t) as shown in Fig. 2.

In summary, Kovacs’ expansion gap paradox in energyrelaxation is reproduced based on kinetic Monte Carlosimulations of a particle model in two-dimensions. Astructural temperature is introduced to characterize theenergy states of the particle configurations. After an up-jump of the bath temperature from Ti to Tf , a largespatial heterogeneity is observed in both local particledisplacement and local structural temperature. The evo-lution of the latter is characterized by the nucleation andcoarsening of Tf domains invading the original Ti do-

mains. Relaxation dynamics persistently depend on Tibecause isolated Ti domains survive even close to the endof the relaxation. This leads to strong memory effectsand explains the paradox. We argue that the strong spa-tial fluctuations are caused by a spatial instability due tothe auto-catalytic nature of the heating of glasses. In con-trast, for a temperature down jump, the auto-retardingnature of cooling leads to a stable and thus homogeneousevolution of the structural temperature, resulting in weakmemory effects and converging relaxation rates.

The structural temperature introduced in this workis, to the best of our knowledge, the first example ofa physical realization of Tool’s fictive temperature [5].As an advancement, it is measurable from particle sim-ulations based on well-defined microscopic dynamics, incontrast to the fictive temperature which follows sepa-rate empirical evolution rules [4]. Our results stress theimportance of its spatial heterogeneity in understandingthe expansion gap paradox. This naturally explains whymean-field models with a global fictive temperature [5–8] in general have difficulty reproducing the paradox. Italso justifies the results of the stochastic models [17, 18]in which different stochastic realizations empirically rep-resent local regions at different stages of evolution. Webelieve that the structural temperature will be of generalimportance in the study of non-equilibrium behaviors ofglasses. It should be of interest to measure it experi-mentally by means of, for example, electron correlationmicroscopy [24].

Kovacs’ experiments are important because excep-tional material properties often provide the deepest in-sights. Overcoming the long-standing challenge of repro-ducing the expansion gap using a microscopic particlemodel, our results do not only provide a possible intu-itive understanding of the paradox, but also support thevalidity of the DPLM as a reliable computational tool forstudying glassy dynamics. Finally, our findings shouldalso be useful to scrutinize and to further develop theo-retical approaches on glasses [25, 26].

I. ACKNOWLEDGEMENTS

We gratefully acknowledge Chor-Hoi Chan, HaihuiRuan and Giorgio Parisi for interesting discussion andcomments. We thank the support of Hong Kong GRF(Grant 15330516) and PolyU (Grants 1-ZVGH and G-UAF7).

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[19] Ling-Han Zhang and Chi-Hang Lam, “Emergent facilita-tion behavior in a distinguishable-particle lattice modelof glass,” Phys. Rev. B 95, 184202 (2017).

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Supplemental: Spatial heterogeneities in structural temperature cause Kovacs’expansion gap paradox in aging of glasses

Matteo Lulli,1 Chun-Shing Lee,1 Hai-Yao Deng,2 Cho-Tung Yip,3 and Chi-Hang Lam1

1Department of Applied Physics, Hong Kong Polytechnic University, Hong Kong, China2School of Physics, University of Exeter, Exeter EX4 4QL, United Kingdom

3School of Science, Harbin Institute of Technology,Shenzhen Graduate School, Shenzhen, Guangdong 518055, China

(Dated: September 9, 2019)

I. THERMAL PROPERTIES OF DPLM

We report here some further details about the DPLM.In the main text, we discuss about the final equilibriumvalue of the system energy E∞ and the specific heat as afunction of the temperature. Let us now derive in detailthese two quantities. In order to obtain the equilibriumenergy E∞ as a function of the temperature T one hasto compute the average of the system energy (see Eq.(1)in the main text)

E∞(T ) =∑〈ij〉′〈Vijsisj 〉T = Nb

∫ V1

V0

dV V peq(V, T ), (1)

where [V0, V1] is the range of variation of the couplings,Nb is the average number of bonds and peq is the equi-librium distribution of the interaction energies given inEq.(2) of the main text. A good estimate of Nb, ata low void density, can be obtained in the approxima-tion of isolated voids, i.e. a void does not have anothervoid as a nearest neighbor, yielding in two dimensionsNb = 2L2− 4Nv ' 2N(1−φv), where 2L2 is the numberof interactions for the fully occupied lattice while eachisolated void decreases this number by the 4 missing cou-plings with its nearest neighboring particles. Hence, theproblem reduces to computing the average interactionenergy as a function of the temperature, i.e. 〈V 〉T . Inthis paper we choose g(V ) as a uniform distribution, i.e.g(V ) = 1/∆V where ∆V = V1 − V0 is the range of vari-ation of the interactions. Hence, choosing the boundaryvalues as V1 = −V0 = 1/2 and keeping on using naturalunits as in the main text, we write

E∞(T ) = 2N(1− φv)

[T − 1

2coth

(1

2T

)]. (2)

It is easy to check that E∞(T ) ≤ 0 where the equalityholds in the limit T →∞. By taking the derivative withrespect to the temperature, we write the heat capacityas

CV (T ) = 2N(1− φv)

{1− 1

4T 2

[coth2

(1

2T

)− 1

]},

(3)which is a decreasing function of T .

II. OVERLAP AND VOIDS DYNAMICS

In the main text, we report the evolution of differentquantities, in Figures 2 and 3, as a function of the averageoverlap rather than as a function of time. This choice ismotivated by the fact that the differences in the dynam-ics between down- and up-jump are most clearly visiblewhen measuring the evolution in terms of the fraction ofparticles that at a given time t still retain the position ~x0at the moment when the temperature jump is performed,i.e. the average overlap q as discussed in the main text.Hence, we report in Fig. 1 the evolution of the averageoverlap as a function of time, where the much slower up-jump relaxation is clearly visible.

Finally, we report the positions of the voids for threeof the configurations shown in Fig.2 of the main text. Wenotice that for the down-jump dynamics all the voids be-long to the mobile region already at an early stage, whilefor the up-jump some voids stay trapped in the immobileregions. The voids and fields dynamics can be examinedalso from the supplementary videos supvideo up.mp4 andsupvideo down.mp4.

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FIG. 1. Average overlap evolution, q(t), for up-jump withTi = 0.1 and down-jump with Ti = 0.3125, with a commonfinal temperature Tf = 0.25. These values are the same asthose used in Fig.2 of the main text.

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FIG. 2. Three of the four snapshots reported in Fig.1 of the main text with the voids positions reported by black dots. Theinitial temperatures are Ti = 0.3125 and Ti = 0.1 for the down- and up-jump respectively, with the final temperature Tf = 0.25.