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# Viscoelastic shear stress relaxation in two-dimensional glass-forming liquids

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

## Significance

A phase transition from a liquid to an ordered solid state in two dimensions is different from that in three dimensions. In two dimensions, the appearance of quasi–long-range orientational correlations is decoupled from the that of quasi–long-range translational correlations, which is correlated with the emergence of a nonzero shear modulus. Here, we show that a similar decoupling also exists in 2D glassy phenomena, albeit with transitions replaced by crossovers. The onset of slow dynamics is signaled by the appearance of a two-step decay in orientational correlations, but a glassy viscoelastic response develops at a lower temperature.

## Abstract

Translational dynamics of 2D glass-forming fluids is strongly influenced by soft, long-wavelength fluctuations first recognized by D. Mermin and H. Wagner. As a result of these fluctuations, characteristic features of glassy dynamics, such as plateaus in the mean-squared displacement and the self-intermediate scattering function, are absent in two dimensions. In contrast, Mermin–Wagner fluctuations do not influence orientational relaxation, and well-developed plateaus are observed in orientational correlation functions. It has been suggested that, by monitoring translational motion of particles relative to that of their neighbors, one can recover characteristic features of glassy dynamics and thus disentangle the Mermin–Wagner fluctuations from the 2D glass transition. Here we use molecular dynamics simulations to study viscoelastic relaxation in two and three dimensions. We find different behavior of the dynamic modulus below the onset of slow dynamics (determined by the orientational or cage-relative correlation functions) in two and three dimensions. The dynamic modulus for 2D supercooled fluids is more stretched than for 3D supercooled fluids and does not exhibit a plateau, which implies the absence of glassy viscoelastic relaxation. At lower temperatures, the 2D dynamic modulus starts exhibiting an intermediate time plateau and decays similarly to the 2D dynamic modulus. The differences in the glassy behavior of 2D and 3D glass-forming fluids parallel differences in the ordering scenarios in two and three dimensions.

Upon approaching the glass transition, a supercooled fluid exhibits a pronounced viscoelastic response with intermediate time elasticity followed by viscous flow. This response implies a two-step decay of the viscoelastic response function, i.e., of the dynamic modulus. When a fluid’s viscosity reaches

It was recently shown that the difference between the 2D and 3D glass transition scenarios evident from Flenner and Szamel’s work could be suppressed by monitoring translational motion of particles with respect to their local environment, i.e., their “cage” (4, 5, 9). This is achieved by introducing “cage-relative” variants of the mean-squared displacement and the intermediate scattering function, which use displacements of the particles measured relative to their neighbors. These cage-relative functions exhibit well-developed intermediate time plateaus at state points where the orientational correlation functions exhibit plateaus. This observation suggests that, to study glassy dynamics of 2D fluids, one needs to remove the effects of Mermin–Wagner fluctuations on the translational motion by focusing on cage-relative displacements rather than absolute displacements. The picture that emerged is that, when 2D glassy dynamics is viewed in terms of 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 might play a special role for the glass transition and, therefore, glassy dynamics. Berthier et al. (14) found that, in two dimensions, the ideal glass transition, defined as the state point at which the configurational entropy vanishes, occurs at zero temperature, which is in contrast to the vanishing of the configurational entropy at a nonzero temperature in three dimensions (15).

In this work, we address a spectacular manifestation of the incipient glass transition found in a laboratory: We study the temperature dependence of the viscoelastic response and of the shear viscosity. To this end, we monitor the dynamic modulus, which is proportional to the shear stress autocorrelation function. We find that, in two dimensions, in the temperature regime where translational relaxation does not exhibit typical features of glassy dynamics but orientational and cage-relative translational correlation functions exhibit well-developed plateaus, the dynamic modulus does not have a plateau. Thus, in this temperature regime, there is no well-developed transient elastic response. At lower temperatures, the dynamic modulus develops a plateau, signaling emerging viscoelasticity. We also find that the shear viscosity grows more slowly with decreasing temperature in two dimensions than in three dimensions, and that its growth is decoupled from that of the orientational and cage-relative relaxation times.

## Time-Dependent Viscoelastic Response

The time-dependent viscoelastic response is quantified by the dynamic modulus *Materials and Methods*). We note that the modulus depends on interparticle distances rather than on absolute values of particles’ coordinates, and thus, conceptually, it resembles orientational and cage-relative correlation functions.

In Fig. 1, we compare the dynamic modulus of model 2D and 3D glass-forming systems for temperatures below the onset temperature of slow dynamics (as determined by the appearance of intermediate time plateaus in the orientational and cage-relative correlation functions). We checked, by simulating systems of different sizes, that there are no finite size effects for the dynamic modulus in two dimensions, in contrast to what was observed for the mean-squared displacement and the self-intermediate scattering function (3, 12). Upon initial visual inspection, at the lowest temperatures accessible in our simulations, there appears to be little difference in the time dependence of the dynamic modulus for 2D and 3D glass-formers. They both exhibit an initial decay, an intermediate time plateau, and a final decay from the plateau. However, a closer examination of Fig. 1 and a comparison of the temperature dependence of

First, we examine the change of the time dependence of the dynamic modulus upon decreasing temperature by fitting its final decay to a stretched exponential

As shown in Fig. 2, we find very different behavior of the amplitude of the stretched exponential fit

## Shear Viscosity

Dramatic increase of the shear viscosity, η,

We observe that the shear viscosity of the 2D system increases rather gradually with decreasing temperature below the onset of slow dynamics, in contrast to that of the 3D system. We used two different popular fitting functions, the Vogel–Fulcher fit

The apparent glass transition temperatures obtained for the 2D glass-former, *Time-Dependent Viscoelastic Response*. In two dimensions, normal features of glassy viscoelastic behavior are observed only for deeply supercooled systems.

In 3D systems, where the intermediate time plateau appears just below the onset of slow dynamics, viscosity is usually interpreted as a product of the plateau and the characteristic relaxation time of the final decay. Here we make it more quantitative and approximate the viscosity by the integral of the final stretched exponential decay,**2**] agrees very well with the 3D shear viscosity.

More interestingly, as also shown in Fig. 3, if we apply approximation [**2**] in two dimensions, the result still agrees very well with the shear viscosity, for all temperatures below the onset of slow dynamics. This happens despite the fact that, for our 2D glass-former, between the onset of slow dynamics and deep supercooling, the dynamic viscosity does not exhibit a clear plateau, and the fit parameter

## Translational, Translational Cage-Relative, and Orientational Dynamic Correlation Functions

We now compare and contrast temperature dependence of the 2D 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**3**,

The self-intermediate scattering function is shown in Fig. 4*A* for *A* (the time dependence of the collective intermediate scattering functions is similar to that of the self-intermediate scattering functions).

Recently, simulations and experiments analyzing 2D glass-forming systems have examined cage-relative translational dynamics to remove the effects of Mermin–Wagner fluctuations (4, 5, 9). Examining cage-relative motion is motivated by the study of melting of 2D ordered solids. It was argued that, by examining cage-relative motion, one can remove the size effects shown in the dynamics of 2D simulations, restoring the plateau in the mean-squared displacement. Thus, 2D glassy dynamics would then have similar characteristics as 3D glassy dynamics. A cage-relative self-intermediate scattering function,*B* is

However, we recall that there is no plateau in the dynamic modulus for temperatures between

A possible alternative to using cage-relative translational dynamic correlations is provided by an orientational correlation function,**6**,*Inset*. While we can see the emerging plateau in

Again, we emphasize that the orientational correlation function exhibits a plateau even when none exists in the dynamic modulus. This is despite the fact that both functions depend on interparticle distances and, thus, are only sensitive to local dynamics.

Finally, we comment on the expectation that pronounced finite size effects are removed when one studies cage-relative dynamics. For an ordered 2D solid, particles stay within a cage formed by their neighbors. While the absolute position of the particles drifts due to Mermin–Wagner fluctuations, one expects that the position relative to a particle’s neighbors remains fixed.

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

Shown in Fig. 6 is

The cage-relative mean-squared displacement is also system size-dependent in the liquid state. Shown in Fig. 6, *Inset* are

## Time Scales

We briefly compare the growth of the characteristic time scales related to particle motion to that of the shear viscosity. We note that the orientational and cage-relative correlation functions follow time–temperature superposition, and thus, for these functions, we use the usual definition of the relaxation time,

In Fig. 7, we compare the temperature dependence of the relaxation times and the shear viscosity. There is only about two decades in increase of

To examine the correlations between the relaxation times and the viscosity, we calculate the ratios of the relaxation times to the viscosity. The relaxation time *Inset*. We find that the cage-relative relaxation time and the orientational relaxation time increase at statistically the same rate, and, importantly, they increase faster than the shear viscosity. The former result agrees with the observations of refs. 4 and 5. Thus, available evidence suggests that cage-relative and orientational relaxations are correlated. However, in the temperature range just below the onset of slow dynamics, these dynamics do not seem to be correlated with viscoelastic relaxation.

The integrated relaxation time

## Discussion

In two dimensions, low-temperature phase behavior and properties of ordered solids are different from those in three dimensions. In particular, the freezing transition can be a two-step process (17⇓⇓–20), with semi–long-range orientational correlations appearing first and semi–long-range translational correlations and long-range (nondecaying) orientational correlations second. Elasticity appears discontinuously at the second transition (17).

We showed here that the 2D glass transition scenario reflects the above-described scenario. At the onset of slow dynamics, orientational correlations start exhibiting typical features of glassy dynamics: two-step decay with intermediate-time plateau whose duration increases rapidly with decreasing temperature. In contrast, the two-step decay is not seen in the self-intermediate scattering function (which is sensitive to translational relaxation) and in the dynamics modulus (which quantifies viscoelastic relaxation). At a somewhat lower temperature, the latter function (the modulus) starts exhibiting typical features of glassy dynamics. Unfortunately, simulating the self-intermediate scattering function in that temperature range would require using much bigger systems, which is not possible with our present computational resources. We note that, on the basis of general arguments invoking Mermin–Wagner fluctuations, we expect that translational motion is not localized even in a very deeply supercooled fluid. If the shear viscosity continued to grow in relation to the translational relaxation time, we would expect the glass transition to occur at

Finally, we elaborate on a remark we made in passing in *Shear Viscosity*. In two dimensions, the very existence of a consistent hydrodynamic description is in question. Specifically, in two dimensions, the so-called hydrodynamic long-time tails decay as

## Materials and Methods

We simulated the Kob–Andersen binary Lennard-Jones mixture in two dimensions and three dimensions (23⇓–25). The interaction potential is

The dynamic modulus **10**, since it is small at the temperatures we simulate.

To calculate the viscosity η, we examine

## Acknowledgments

We gratefully acknowledge partial support from National Science Foundation (NSF) Grant DMR-1608086. This research utilized the Colorado State University Information Science and Technology Center Cray high-performance computing system supported by NSF Grant CNS-0923386.

## Footnotes

- ↵
^{1}To whom correspondence should be addressed. Email: flennere{at}gmail.com.

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.

The authors declare no conflict of interest.

This article is a PNAS Direct Submission.

Published under the PNAS license.

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