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Viscoelastic shear stress relaxation in two-dimensionalglass-forming liquidsElijah Flennera,1 and Grzegorz Szamela

aDepartment of Chemistry, Colorado State University, Fort Collins, CO 80523

Edited by Giorgio Parisi, University of Rome, Rome, Italy, and approved December 19, 2018 (received for review August 31, 2018)

Translational dynamics of 2D glass-forming fluids is strongly influ-enced by soft, long-wavelength fluctuations first recognized byD. Mermin and H. Wagner. As a result of these fluctuations, char-acteristic features of glassy dynamics, such as plateaus in themean-squared displacement and the self-intermediate scatteringfunction, are absent in two dimensions. In contrast, Mermin–Wagner fluctuations do not influence orientational relaxation,and well-developed plateaus are observed in orientational cor-relation functions. It has been suggested that, by monitoringtranslational motion of particles relative to that of their neigh-bors, one can recover characteristic features of glassy dynamicsand thus disentangle the Mermin–Wagner fluctuations from the2D glass transition. Here we use molecular dynamics simulationsto study viscoelastic relaxation in two and three dimensions.We find different behavior of the dynamic modulus below theonset of slow dynamics (determined by the orientational or cage-relative correlation functions) in two and three dimensions. Thedynamic modulus for 2D supercooled fluids is more stretched thanfor 3D supercooled fluids and does not exhibit a plateau, whichimplies the absence of glassy viscoelastic relaxation. At lowertemperatures, the 2D dynamic modulus starts exhibiting an inter-mediate time plateau and decays similarly to the 2D dynamicmodulus. The differences in the glassy behavior of 2D and 3Dglass-forming fluids parallel differences in the ordering scenariosin two and three dimensions.

long-wavelength fluctuations | two-dimensional physics |phase transition | dimensionality

Upon approaching the glass transition, a supercooled fluidexhibits a pronounced viscoelastic response with interme-

diate time elasticity followed by viscous flow. This responseimplies a two-step decay of the viscoelastic response function,i.e., of the dynamic modulus. When a fluid’s viscosity reaches1013 poise, it appears solid on typical laboratory time scales,and this value of the viscosity is often used to define the glasstransition-state point. For 3D systems, together with pronouncedviscoelastic response and increasing of a fluid’s viscosity, onefinds that particles’ positions become localized for increasinglylonger times (1, 2). This localization results in intermediate timeplateaus in correlation functions monitoring particles’ transla-tional motion, such as the mean-squared displacement and theintermediate scattering function. Flenner and Szamel (3) usedmolecular dynamics simulations to show that there is no tran-sient localization of particles in 2D glass-forming liquids. Thelack of localization results in an absence of plateaus in themean-squared displacement and the self-intermediate scatter-ing function. This finding has been verified in experiments ofquasi-2D colloidal particles (4, 5). The absence of transientlocalization of translational motion originates from soft, long-wavelength fluctuations (4, 5), which were first mentioned byPeierls (6) and then analyzed by D. Mermin and H. Wagnerin the context of 2D magnetic systems (7) and crystallinesolids (8). Importantly, Mermin–Wagner fluctuations do notstrongly influence orientational relaxation. This is consistent withFlenner and Szamel’s finding of a decoupling between trans-lational and orientational relaxation and the presence of well-

developed plateaus in orientational correlation functions in twodimensions.

It was recently shown that the difference between the 2D and3D glass transition scenarios evident from Flenner and Szamel’swork could be suppressed by monitoring translational motion ofparticles with respect to their local environment, i.e., their “cage”(4, 5, 9). This is achieved by introducing “cage-relative” variantsof the mean-squared displacement and the intermediate scatter-ing function, which use displacements of the particles measuredrelative to their neighbors. These cage-relative functions exhibitwell-developed intermediate time plateaus at state points wherethe orientational correlation functions exhibit plateaus. Thisobservation suggests that, to study glassy dynamics of 2D fluids,one needs to remove the effects of Mermin–Wagner fluctua-tions on the translational motion by focusing on cage-relativedisplacements rather than absolute displacements. The picturethat emerged is that, when 2D glassy dynamics is viewed in termsof local properties, as in bond angle correlations (3, 4), bond-breaking correlations (10–12), or cage-relative displacements (4,5, 12), it is quite similar to the 3D glassy dynamics (13).

However, a recent study suggests that two dimensions mightplay a special role for the glass transition and, therefore, glassydynamics. Berthier et al. (14) found that, in two dimensions, theideal glass transition, defined as the state point at which the con-figurational entropy vanishes, occurs at zero temperature, whichis in contrast to the vanishing of the configurational entropy at anonzero temperature in three dimensions (15).

In this work, we address a spectacular manifestation of theincipient glass transition found in a laboratory: We study thetemperature dependence of the viscoelastic response and ofthe shear viscosity. To this end, we monitor the dynamic mod-ulus, which is proportional to the shear stress autocorrelationfunction. We find that, in two dimensions, in the temperatureregime where translational relaxation does not exhibit typicalfeatures of glassy dynamics but orientational and cage-relative

Significance

A phase transition from a liquid to an ordered solid statein two dimensions is different from that in three dimen-sions. In two dimensions, the appearance of quasi–long-rangeorientational correlations is decoupled from the that of quasi–long-range translational correlations, which is correlated withthe emergence of a nonzero shear modulus. Here, we showthat a similar decoupling also exists in 2D glassy phenomena,albeit with transitions replaced by crossovers. The onset ofslow dynamics is signaled by the appearance of a two-stepdecay in orientational correlations, but a glassy viscoelasticresponse develops at a lower temperature.

Author contributions: E.F. and G.S. designed research; E.F. performed research; E.F.analyzed data; and E.F. and G.S. wrote the paper.y

The authors declare no conflict of interest.y

This article is a PNAS Direct Submission.y

Published under the PNAS license.y1 To whom correspondence should be addressed. Email: flennere@gmail.com.y

Published online January 22, 2019.

www.pnas.org/cgi/doi/10.1073/pnas.1815097116 PNAS | February 5, 2019 | vol. 116 | no. 6 | 2015–2020

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A

B

Fig. 1. Dynamic modulus for (A) a model 2D glass former for T = 1.0, 0.8,0.7, 0.6, 0.5, 0.45, 0.4, 0.38, 0.36, and 0.34 listed from left to right and (B)a model 3D glass-former for T = 1.0, 0.7, 0.6, 0.55, 0.5, 0.47, 0.45, and 0.44listed from left to right. The solid orange lines are stretched exponential fits.

translational correlation functions exhibit well-developed pla-teaus, the dynamic modulus does not have a plateau. Thus, inthis temperature regime, there is no well-developed transientelastic response. At lower temperatures, the dynamic modulusdevelops a plateau, signaling emerging viscoelasticity. We alsofind that the shear viscosity grows more slowly with decreasingtemperature in two dimensions than in three dimensions, andthat its growth is decoupled from that of the orientational andcage-relative relaxation times.

Time-Dependent Viscoelastic ResponseThe time-dependent viscoelastic response is quantified by thedynamic modulus G(t), which is proportional to the shear stressautocorrelation function (16), G(t) = 〈Σxy(t)Σxy(0)〉 /(VkBT ),where Σxy(t) is the xy component of the stress tensor (seeMaterials and Methods). We note that the modulus depends oninterparticle distances rather than on absolute values of particles’coordinates, and thus, conceptually, it resembles orientationaland cage-relative correlation functions.

In Fig. 1, we compare the dynamic modulus of model 2D and3D glass-forming systems for temperatures below the onset tem-perature of slow dynamics (as determined by the appearance ofintermediate time plateaus in the orientational and cage-relativecorrelation functions). We checked, by simulating systems of dif-ferent sizes, that there are no finite size effects for the dynamicmodulus in two dimensions, in contrast to what was observedfor the mean-squared displacement and the self-intermediatescattering function (3, 12). Upon initial visual inspection, at thelowest temperatures accessible in our simulations, there appearsto be little difference in the time dependence of the dynamicmodulus for 2D and 3D glass-formers. They both exhibit an ini-tial decay, an intermediate time plateau, and a final decay fromthe plateau. However, a closer examination of Fig. 1 and a com-parison of the temperature dependence of G(t) with that ofcorrelation functions sensitive to translational, orientational, andcage-relative motions reveals important differences between twoand three dimensions.

First, we examine the change of the time dependence of thedynamic modulus upon decreasing temperature by fitting its finaldecay to a stretched exponential Gpe

−(t/τs )β . We fit results from

four to eight independent simulations in the range between aninitial time ti and a final time tf . We choose tf = 2tc , where tc isthe time at which the dynamic modulus is equal to 0.1, to avoidfitting the large fluctuations due to having poor statistics in G(t)at later times. We vary ti to make sure the results are not sensi-tive to the fitting range. We observe that the fit parameters aremost sensitive to the range of the fit for T = 0.7, with Gp system-atically decreasing with an increasing ti . The stretching exponentβ gives deviations from exponential decay, which is expected for3D liquids well above the onset temperature of slow dynamics.The stretched exponential function fits the final decay well inevery case, and three fits are shown in Fig. 1 for both the 2Dand the 3D systems. We note that, for 2D systems at higher tem-peratures, Gp should only be considered a fit parameter; the fitdoes not does imply existence of an intermediate time plateauin G(t).

As shown in Fig. 2, we find very different behavior of theamplitude of the stretched exponential fit Gp and of the stretch-ing exponent β for 2D and 3D glass-forming systems. For 2Dglass-forming systems at higher temperatures, we find that Gp

decreases significantly with decreasing temperature, by a factorof 3 between the onset of supercooling and the low-temperaturelimit, and β increases by a factor of almost 2. This temperaturedependence of Gp is in stark contrast to what is observed in 3Dsystems, where Gp is approximately temperature-independentand β decreases with decreasing temperature. These observa-tions and a visual inspection of the dynamic modulus show thatstress fluctuations relax differently below the onset of supercool-ing in 2D and 3D systems, but become similarly deep within thesupercooled liquid. This leads to different viscoelastic relaxation;in 2D glass-forming fluids, intermediate-time elastic responseappears only upon deep supercooling, whereas, in 3D systems,it appears just below the onset of slow dynamics.

Shear ViscosityDramatic increase of the shear viscosity, η,

η=

∫ ∞0

dt G(t)≡ 1

VkBT

∫ ∞0

dt 〈Σxy(t)Σxy(0)〉, [1]

is a signature of the incipient glass transition. In simulations, therange of the increase is limited to four or five orders of magni-tude. In Fig. 3, we show the shear viscosity for our 2D and 3D

A

B

Fig. 2. The amplitude of (A) the stretched exponential fit Gp and (B) thestretching exponent β as functions of temperature. Just below the onsetof slow dynamics, there is a stark contrast between 2D and 3D systems,but similar behavior is observed in deeply supercooled fluids. The error barsrepresent the uncertainty in the fits when varying the range of the fits.

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Fig. 3. Shear viscosity η as a function of inverse temperature 1/T . Filledsymbols denote η calculated from the exact formula [1], and open symbolsdenote η as given by approximate relation [2]. Dashed lines indicate Vogel–Fulcher fits, η= A exp(−B/(T − T0)), and solid lines indicate modified Vogel–Fulcher fits, η= C exp(−D/(T − T1)2). Error bars are shown in the figure, butmost are smaller than the points.

glass-forming systems. We note that, technically, in two dimen-sions, η diverges, since there is a long time tail in 〈Σxy(t)Σxy(0)〉.However, these (hydrodynamic) long time tails are difficult toobserve in simulations of glassy systems. In two dimensions,η increases by almost six orders of magnitude over the fulltemperature range.

We observe that the shear viscosity of the 2D system increasesrather gradually with decreasing temperature below the onsetof slow dynamics, in contrast to that of the 3D system. Weused two different popular fitting functions, the Vogel–Fulcherfit η=A exp(−B/(T −T0)) and the modified Vogel–Fulcherfit η=C exp(−D/(T −T1)2), to quantify these temperaturedependencies. These fitting functions result in apparent glasstransition temperatures T0 and T1. Note that, while we do notassign any fundamental significance to these temperatures, theyare useful quantitative characteristics of the onset of solid-likeresponse on macroscopic time scales.

The apparent glass transition temperatures obtained forthe 2D glass-former, T 2d

0 = 0.203± 0.010 and T 2d1 = 0.068±

0.020, are significantly smaller than those obtained for the 3Dglass-former, T 3d

0 = 0.347± 0.010 and T 3d1 = 0.243± 0.010. This

observation, combined with the fact that the onset of slowdynamics is the same for both systems, agrees with the obser-vation made in Time-Dependent Viscoelastic Response. In twodimensions, normal features of glassy viscoelastic behavior areobserved only for deeply supercooled systems.

In 3D systems, where the intermediate time plateau appearsjust below the onset of slow dynamics, viscosity is usually inter-preted as a product of the plateau and the characteristic relax-ation time of the final decay. Here we make it more quantita-tive and approximate the viscosity by the integral of the finalstretched exponential decay,

η≈∫ ∞

0

dt Gpe−(t/τs )β =GpτsΓ[1 +β−1], [2]

where Γ denotes the Gamma function. As shown in Fig. 3, for alltemperatures below the onset of slow dynamics, approximation[2] agrees very well with the 3D shear viscosity.

More interestingly, as also shown in Fig. 3, if we apply approxi-mation [2] in two dimensions, the result still agrees very well withthe shear viscosity, for all temperatures below the onset of slowdynamics. This happens despite the fact that, for our 2D glass-

former, between the onset of slow dynamics and deep supercool-ing, the dynamic viscosity does not exhibit a clear plateau, andthe fit parameter Gp is strongly temperature-dependent.

Translational, Translational Cage-Relative, and OrientationalDynamic Correlation FunctionsWe now compare and contrast temperature dependence of the2D viscoelastic response with that of the 2D translational, cage-relative, and orientational time-dependent correlation functions.

We start by looking at the self-intermediate scattering function

Fs(k ; t) =1

N

⟨∑n

e ik·(rn (t)−rn (0))

⟩. [3]

In Eq. 3, rn(t) is the position of particle n at a time t , andk = |k|= 6.28 is the peak position of the static structure factor.The self-intermediate scattering function decay time is a mea-sure of the time it takes for a particle to move over a distanceof around 2π/k . For 2D glass-formers, due to Mermin–Wagnerfluctuations, one would expect Fs(k ; t) to decay, albeit possi-bly very slowly. In contrast, for 3D glass-formers, intermediatetime localization of the particle positions leads to a pronouncedplateau in Fs(k ; t).

The self-intermediate scattering function is shown in Fig. 4Afor T = 1.0, 0.8, 0.7, 0.6, 0.5, and 0.45. We note that, for tem-peratures below T = 0.45, we could not simulate systems largeenough to remove pronounced finite size effects. There is noplateau in the self-intermediate scattering function for the tem-perature range shown in Fig. 4A (the time dependence of thecollective intermediate scattering functions is similar to that ofthe self-intermediate scattering functions).

Recently, simulations and experiments analyzing 2D glass-forming systems have examined cage-relative translationaldynamics to remove the effects of Mermin–Wagner fluctuations(4, 5, 9). Examining cage-relative motion is motivated by thestudy of melting of 2D ordered solids. It was argued that, byexamining cage-relative motion, one can remove the size effectsshown in the dynamics of 2D simulations, restoring the plateauin the mean-squared displacement. Thus, 2D glassy dynamics

A

B

Fig. 4. (A) The self-intermediate scattering function shown for the tem-perature range where we can remove the finite size effects. (B) Thecage-relative self-intermediate scattering function for the same range oftemperatures.

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would then have similar characteristics as 3D glassy dynamics.A cage-relative self-intermediate scattering function,

FCRs (k ; t) =

1

N

⟨∑n

e ik·δrCRn (t)

⟩, [4]

involves the cage-relative displacements

δrCRi (t) = [ri(t)− ri(0)]− 1

Nnn(i)

Nnn (i)∑n=1

[rn(t)− rn(0)], [5]

where the summation is over Nnn(i) nearest neighbors of par-ticle i . For k around the peak of the static structure factor,FCR

s (k ; t) probes the escape of a particle from its cage. Shownin Fig. 4B is FCR

s (k ; t) for temperatures where we could simu-late large enough systems to remove finite size effects, T = 1.0,0.8, 0.7, 0.6, 0.5, and 0.45. We find that the shape of FCR

s (k ; t)is similar to what is observed for Fs(k ; t) for 3D systems. Thereis an initial decay to a plateau and then a final decay when theparticles begin to lose their neighbors. Additionally, we find thattime–temperature superposition holds, i.e., the final decays ofFCR

s (k ; t) overlap if scaled by their relaxation time. This is instark contrast to the lack of time–temperature superposition forFs(k ; t).

However, we recall that there is no plateau in the dynamicmodulus for temperatures between T = 1.0 and T = 0.45. Thus,by introducing cage-relative function FCR

s (k ; t), we overempha-size the glassy character of the dynamics.

A possible alternative to using cage-relative translationaldynamic correlations is provided by an orientational correlationfunction,

Cψ =1

N

⟨∑n

Ψn6 (t)[Ψn

6 (0)]∗⟩

⟨∑n

|Ψn6 |

2

⟩ . [6]

In Eq. 6,

Ψn6 (t) =

1

Nb

∑m

e i6θnm (t), [7]

where Nb is the number of neighbors identified through Voronoitessellation, and θnm(t) is the angle that the bond between par-ticle n and particle m makes with an arbitrary fixed axis at atime t . Shown in Fig. 5 is Cψ(t) for our full range of tempera-tures [we verified that there are no finite size effects for Cψ(t)].We clearly see the onset of slow dynamics; at around T = 1,a plateau emerges. Its height increases slightly with decreasingtemperature, but the extent of the plateau and the final relax-ation time increase dramatically. To highlight the difference indynamic modulus and CΨ(t) for temperatures between 1.0 and0.45, we plot 20×CΨ(t) for T = 0.5 and G(t) in Fig. 5, Inset.While we can see the emerging plateau in CΨ(t), there is noindication of any plateau in the dynamic modulus.

Again, we emphasize that the orientational correlation func-tion exhibits a plateau even when none exists in the dynamicmodulus. This is despite the fact that both functions dependon interparticle distances and, thus, are only sensitive to localdynamics.

Finally, we comment on the expectation that pronounced finitesize effects are removed when one studies cage-relative dynam-ics. For an ordered 2D solid, particles stay within a cage formedby their neighbors. While the absolute position of the particlesdrifts due to Mermin–Wagner fluctuations, one expects that theposition relative to a particle’s neighbors remains fixed.

Fig. 5. The bond angle correlation function for the full range of temper-atures. The bond angle correlation function does not exhibit finite sizeeffects. Inset compares 20× CΨ(t) with 〈Σxy (t)Σxy (0)〉/(VkBT) for T = 0.5.While a plateau is emerging for CΨ(t), there is no evidence for a plateau for〈Σxy (t)Σxy (0)〉/(VkBT).

For a glass in two dimensions, we expect that the cage-relativemean-squared displacement,

⟨δr2(t)

⟩CR=

1

N

N∑i

⟨[δrCR

i (t)]2⟩

, [8]

will reach a constant value, but⟨δr2(t)

⟩=N−1∑

i〈[ri(t)−ri(0)]2〉 will drift due to Mermin–Wagner fluctuations. In aliquid, for long times, the displacements are uncorrelated,〈[ri(t)− ri(0)][rj (t)− rj (0)]〉=

⟨[ri(t)− ri(0)]2

⟩δij , and then⟨

δr2(t)⟩CR

=⟨δr2(t)

⟩+ 〈Nnn〉−1 ⟨δr2(t)

⟩. Thus, in the long-

time limit, a liquid’s cage-relative mean-squared displacement islarger than the mean–squared displacement.

Shown in Fig. 6 is⟨δr2(t)

⟩CR and⟨δr2(t)

⟩for T = 1.0, 0.6,

and 0.45. There is no plateau in⟨δr2(t)

⟩, but

⟨δr2(t)

⟩CR is sim-ilar to what is observed for 3D systems, and there is evidenceof an emerging plateau. While the particles displacements arecorrelated,

⟨δr2(t)

⟩CR<⟨δr2(t)

⟩. Once the particles start to

lose their neighbors,⟨δr2(t)

⟩CR grows faster than linearly witht until

⟨δr2(t)

⟩CR is approximately equal to⟨δr2(t)

⟩, and then⟨

δr2(t)⟩CR grows linearly with t . As shown by Shiba et al. (9),

we also find that [⟨δr2(t)

⟩−⟨δr2(t)

⟩CR]≈ tα with α< 1.0 for a

region of time, but this difference cannot continue to grow in theliquid, and, eventually, it goes through zero.

The cage-relative mean-squared displacement is also systemsize-dependent in the liquid state. Shown in Fig. 6, Inset are⟨δr2(t)

⟩CR and⟨δr2(t)

⟩for N = 900 and N = 4,000,000 cal-

culated for T = 0.45. We can see that⟨δr2(t)

⟩CR and⟨δr2(t)

⟩differ at long times. Importantly, there is no finite size effect inthe emerging plateau region of

⟨δr2(t)

⟩CR. This is in agreementwith the observation of Shiba et al. (9). These two findings leadus to speculate that, in the glass, the cage-relative mean-squareddisplacement is not system size-dependent.

Time ScalesWe briefly compare the growth of the characteristic time scalesrelated to particle motion to that of the shear viscosity. Wenote that the orientational and cage-relative correlation func-tions follow time–temperature superposition, and thus, for thesefunctions, we use the usual definition of the relaxation time,

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Fig. 6. The cage-relative mean-squared displacement 〈δr2(t)〉CR (dashedlines) and the mean-squared displacement 〈δr2(t)〉 (solid lines) for T = 1.0,0.60, and 0.45. Inset shows 〈δr2(t)〉CR and 〈δr2(t)〉 for T = 0.45 and twosystem sizes. There is a pronounced finite size effect at this temperature.

Cψ(τψ) = e−1 and FCRs (k ; τc) = e−1. Since Fs(k ; t) does not

obey time–temperature superposition, we define two differentrelaxation times, Fs(k ; τα) = e−1 and τ int

α =∫Fs(k ; t)dt .

In Fig. 7, we compare the temperature dependence of therelaxation times and the shear viscosity. There is only about twodecades in increase of τα (black circles), τ int

α (green diamonds),and τc (orange triangles) over the temperature range where wecan calculate these relaxation times. However, we do find thatτ intα increases faster with decreasing temperature than τα. This

increase is due to the increasingly stretched behavior of Fs(k ; t).The orientational relaxation time τψ is initially slightly smallerthan τα and τ int

α , but it increases faster with decreasing temper-ature. We can calculate Cψ(t) over the whole temperature, andwe can see almost six orders of magnitude of the orientationalrelaxation slowdown and the relaxation time τs found throughfitting the dynamic modulus.

To examine the correlations between the relaxation times andthe viscosity, we calculate the ratios of the relaxation times tothe viscosity. The relaxation time τs does not track the viscos-ity and increases faster than the viscosity until a clear plateaudevelops in the dynamic modulus. Additionally, τs does not trackthe orientational relaxation time τΨ. These ratios, normalized totheir values at T = 1.0, are shown in Fig. 7, Inset. We find thatthe cage-relative relaxation time and the orientational relaxationtime increase at statistically the same rate, and, importantly, theyincrease faster than the shear viscosity. The former result agreeswith the observations of refs. 4 and 5. Thus, available evidencesuggests that cage-relative and orientational relaxations are cor-related. However, in the temperature range just below the onsetof slow dynamics, these dynamics do not seem to be correlatedwith viscoelastic relaxation.

The integrated relaxation time τ intα grows at the same rate

as the viscosity, while τα, defined using the standard defini-tion, increases slower than viscosity with decreasing temperature.Therefore, at least in the temperature range in which we cansimulate large enough systems to remove finite size effects inthe self-intermediate scattering function, surprisingly, viscoelas-tic relaxation seems to be correlated with translational dynamicsrather than with orientational relaxation.

DiscussionIn two dimensions, low-temperature phase behavior and proper-ties of ordered solids are different from those in three dimen-sions. In particular, the freezing transition can be a two-stepprocess (17–20), with semi–long-range orientational correlations

appearing first and semi–long-range translational correlationsand long-range (nondecaying) orientational correlations second.Elasticity appears discontinuously at the second transition (17).

We showed here that the 2D glass transition scenario reflectsthe above-described scenario. At the onset of slow dynamics, ori-entational correlations start exhibiting typical features of glassydynamics: two-step decay with intermediate-time plateau whoseduration increases rapidly with decreasing temperature. In con-trast, the two-step decay is not seen in the self-intermediatescattering function (which is sensitive to translational relaxation)and in the dynamics modulus (which quantifies viscoelastic relax-ation). At a somewhat lower temperature, the latter function (themodulus) starts exhibiting typical features of glassy dynamics.Unfortunately, simulating the self-intermediate scattering func-tion in that temperature range would require using much biggersystems, which is not possible with our present computationalresources. We note that, on the basis of general arguments invok-ing Mermin–Wagner fluctuations, we expect that translationalmotion is not localized even in a very deeply supercooled fluid.If the shear viscosity continued to grow in relation to the trans-lational relaxation time, we would expect the glass transition tooccur at T = 0. We note that this conclusion, and our modifiedVogel–Fulcher fit, is compatible with the result of Berthier et al.(14) of a zero-temperature transition.

Finally, we elaborate on a remark we made in passing inShear Viscosity. In two dimensions, the very existence of a consis-tent hydrodynamic description is in question. Specifically, in twodimensions, the so-called hydrodynamic long-time tails decay as1/t (21). This suggests that the transport coefficients are diver-gent. Somewhat faster decay is obtained if one does a moreadvanced self-consistent calculation, but the divergence is noteliminated (21, 22). Here we followed many earlier studies of 2Dglassy phenomena and ignored all these effects. From the practi-cal standpoint, the hydrodynamic long-time tails are very difficultto observe in glassy fluids.

Materials and MethodsWe simulated the Kob–Andersen binary Lennard-Jones mixture in twodimensions and three dimensions (23–25). The interaction potentialis Vαβ (r) = 4εαβ [(σαβ/r)12− (σαβ/)6], where εBB = 0.5εAA, εAB = 1.5εAA,σBB = 0.8σAA, and σAB = 0.88σAA. The potential is truncated and shifted at2.5σαβ . The results are presented in reduced units, where σAA is the unitof length, εAA is the unit of energy, and

√σ2m/εAA is the unit of time. The

mass m is the same for both species. For the 3D system, the larger speciescomposed 80% of the particles, while the larger species composed of 65% ofthe particles in the 2D system. We used both Large-Scale Atomic/Molecular

Fig. 7. The temperature dependence of the relaxation times τs, τα, τ intα ,

τψ , and τc and the viscosity η. (Inset) The ratio of the relaxation times tothe viscosity normalized to this ratio at T = 1.0.

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Massively Parallel Simulator (LAMMPS) and Highly Optimized Object-Oriented Many-Particle Dynamics (HOOMD) blue for the 2D simulations,and LAMMPS for the 3D simulations. We simulated temperatures T = 0.34,0.36, 0.38, and 0.4 (in two dimensions), and 0.45, 0.5, 0.6, 0.7, 0.8, 0.9, and1.0 (in two and three dimensions). We simulated the system in a constantdensity and energy (NVE) ensemble for T ≥ 0.5, but used an NVT Nose–Hoover thermostat with a coupling constant of τ = 10 for T < 0.5. All resultsare averages over four or more production runs. In two dimensions, forT ≥ 0.9, we simulated 10,000 particles. We simulated 250,000 particles for0.5≤ T ≤ 0.8 and 4 million particles at T = 0.45. For T ≤ 0.4, we again sim-ulated 10,000 particles. We checked to see if there were finite size effectsin the shear stress autocorrelation function and the bond angle correlationfunction by simulating 250,000 particles systems at T = 0.4 and comparingthem with the 10,000 particle system. The results agreed to within error. Wesimulated 27,000 particles in three dimensions.

The dynamic modulus G(t) is proportional to the shear stress autocorrela-tion function (16),

G(t) = 〈Σab(t)Σab(0)〉/(VkBT), [9]

where a 6= b, V is the volume of the simulation cell, and kB is Boltzmann’sconstant. The a, b∈ (x, y, z) component Σab of the stress tensor is given by

Σab =N∑n

mnvanvb

n −1

2

N∑n

N∑m 6=n

ranmrb

nm

rnm

dV(rnm)

drnm, [10]

where V(rnm) is the interaction potential between particle n and m, andva

n and ranm refer to the component of the particle n’s velocity and the

projection of the vector rnm, respectively, along the Cartesian axis a. Wedrop the first term in Eq. 10, since it is small at the temperatures wesimulate.

To calculate the viscosity η, we examine η(t) =∫ t

0 G(t′)dt′. When G(t) hasdecayed close to zero, η(t) plateaus and fluctuates around an average value.For times t near the largest times in our calculation, there are large fluctu-ations in G(t), which results in large fluctuations in η(t). For four to eightproduction runs, we obtained η for each run by averaging over the plateauregion of η(t). The error bars shown in Fig. 3 are the SD of η(t) for theproduction runs.

ACKNOWLEDGMENTS. We gratefully acknowledge partial support fromNational Science Foundation (NSF) Grant DMR-1608086. This research uti-lized the Colorado State University Information Science and TechnologyCenter Cray high-performance computing system supported by NSF GrantCNS-0923386.

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2020 | www.pnas.org/cgi/doi/10.1073/pnas.1815097116 Flenner and Szamel

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