# Long-wavelength fluctuations and anomalous dynamics in 2-dimensional liquids

^{a}Division of Physics and Applied Physics, School of Physical and Mathematical Sciences, Nanyang Technological University, Singapore 637371, Singapore;^{b}Chemistry and Physics of Materials Unit, Jawaharlal Nehru Centre for Advanced Scientific Research, Jakkur, Bangalore 560064, India;^{c}Department of Physics and Astronomy, University of Pennsylvania, Philadelphia, PA 19104;^{d}State Key Laboratory of Polymer Physics and Chemistry, Changchun Institute of Applied Chemistry, Chinese Academy of Sciences, Changchun 130022, China;^{e}University of Science and Technology of China, Hefei 230026, China;^{f}Key Laboratory of Systems Bioengineering (Ministry of Education), School of Chemical Engineering and Technology, Tianjin University, Tianjin 300072, China;^{g}Department of Chemistry and Biochemistry, University of California, Los Angeles, CA 90095;^{h}Department of Physics and Astronomy, University of California, Los Angeles, CA 90095;^{i}International Centre for Materials Science, Jawaharlal Nehru Centre for Advanced Scientific Research, Jakkur, Bangalore 560064, India;^{j}Institute for Superconductors, Oxides and Other Innovative Materials and Devices, Consiglio Nazionale delle Ricerche, Dipartimento di Scienze Fisiche, Università di Napoli Federico II, I-80126 Napoli, Italy

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Edited by Eric R. Weeks, Emory University, Atlanta, GA, and accepted by Editorial Board Member Pablo G. Debenedetti October 1, 2019 (received for review May 30, 2019)

## Significance

Long-wavelength elastic modes, which have an infinitesimal energy cost, destroy the long-range translational order of 2D solids at finite temperatures. Here we demonstrate that these long-wavelength fluctuations also influence the dynamical properties of 2D systems in their normal liquid regimes. Hence, long-wavelength fluctuations make 2- and 3D molecular particulate systems behave differently from high- to very low-temperature regimes.

## Abstract

In 2-dimensional systems at finite temperature, long-wavelength Mermin–Wagner fluctuations prevent the existence of translational long-range order. Their dynamical signature, which is the divergence of the vibrational amplitude with the system size, also affects disordered solids, and it washes out the transient solid-like response generally exhibited by liquids cooled below their melting temperatures. Through a combined numerical and experimental investigation, here we show that long-wavelength fluctuations are also relevant at high temperature, where the liquid dynamics do not reveal a transient solid-like response. In this regime, these fluctuations induce an unusual but ubiquitous decoupling between long-time diffusion coefficient D and structural relaxation time τ, where

The dimensionality of a system strongly influences the equilibrium properties of its solid phase (1). According to the Mermin and Wagner (2) theorem, indeed, systems with continuous symmetry and short-range interactions lack true long-range translational order at finite temperature, in d ≤ 2 dimensions. This occurs as in small spatial dimensions the elastic response is dominated by the Goldstone modes, elastic excitations that in the limit of long wavelength (LW) have vanishing energy and a diverging amplitude. A signature of these LW fluctuations is system size-dependent dynamics, which arise as the system size provides a cutoff for the maximum wavelength. This dependence occurs in both ordered and disordered solids, as LW fluctuations are insensitive to the local order (3). This dynamical signature of the LW fluctuations also appears in supercooled liquids, which are liquids cooled below their melting temperature without crystallization occurring. In particular, LW fluctuations affect the transient solid-like response observed in the supercooled regime, which we recap in Fig. 1 via the investigation of the mean-square displacement (MSD), *Materials and Methods*). In the supercooled regime the ISF and the MSD develop a plateau revealing a solid-like response in which particles vibrate in cages formed by their neighbors (4). However, this is a transient response, as at longer times the ISF relaxes and the MSD enters a diffusive behavior. Flenner and Szamel (5) have demonstrated that such glassy relaxation dynamics depend on the system size; the signatures of a transient solid-like response disappear in the thermodynamic limit. This trend can be appreciated in Fig. 1, where we compare the dynamics for 3 different system sizes (see also ref. 5 for a full account). Subsequent works have then demonstrated that this size dependence results from the LW fluctuations (3, 6) by showing that the glassy features of the relaxation dynamics are recovered when the effect of LW fluctuations is filtered out (3, 6⇓–8).

A transient solid-like response is observed only below the onset temperature, where the relaxation time exhibits a super-Arrhenius temperature dependence, in fragile systems (*SI Appendix*, Fig. S5), and dynamical heterogeneities (DHs) affect the relation between diffusion coefficient and relaxation time (4). In the normal liquid regime that occurs at higher temperatures in molecular liquids or at lower densities in colloidal systems, the MSD and the ISF do not exhibit plateaus possibly associated with a transient particle localization and are system size independent. This is apparent in Fig. 1, and it suggests that the normal liquid regime is not affected by LW fluctuations. Is this true? And more generally, how far must a system be from the solid phase for its LW fluctuations to have a negligible influence on its relaxation dynamics? Here we show that, surprisingly, LW fluctuations affect the structural relaxation dynamics of 2D systems even in their normal liquid regimes. Specifically, they induce a stretched-exponential relaxation, qualitatively distinct from that observed in the supercooled regime, and an unusual decoupling between the structural relaxation time τ and the long-time diffusion coefficient, D (*Materials and Methods*). This decoupling has been previously observed, both in experiments (9) of colloidal systems and in numerical simulations (10, 11), but its physical origin has remained mysterious. Our results are based on the numerical investigation of the relaxation dynamics of 2 model glass-forming liquids and on the experimental study of a quasi-2D suspension of ellipsoids (9). Numerically, we consider the 3-dimensional (3D) Kob–Andersen (KA) binary mixture (12) and its 2D variant (mKA) (13), as well as the harmonic model (14) (Harmonic) in both 2D and 3D. Numerical details are in *Materials and Methods* and *SI Appendix*. Details on the experimental systems are in refs. 9 and 15. Our results thus demonstrate that LW fluctuations are critical for understanding the properties of 2D systems not only in the crystalline phase (2, 16), the amorphous solid state (3), and the supercooled state (6, 7), but also, surprisingly, in the normal liquid regime.

## Results

### Dynamics in the 2D Normal Liquid Regime.

If the long-time relaxation dynamics of a liquid are not influenced by its energy landscape, as suggested by Fig. 1 and by analogous results for the 2D Harmonic model and 3D KA model reported in *SI Appendix*, Fig. S1, then at the relaxation timescale the displacements of the particles should be uncorrelated and their van-Hove distribution function a Gaussian. This is observed in Fig. 2*A*, where we plot the dependence of the non-Gaussian parameter *Materials and Methods*), on the relaxation time, for 2D mKA and the 3D KA models. In Fig. 2 and subsequent figures, the shaded area identifies the supercooled regime, as defined in *SI Appendix*, Figs. S5 and S6. Indeed, above the onset temperature *Materials and Methods*), behaves similarly in 2D and 3D, as in Fig. 2*B*. However, landscape-independent relaxation dynamics are also expected to lead to an exponential decay of the ISF, *C* illustrates that indeed *SI Appendix*, Fig. S4 to well describe the time evolution of the van-Hove distribution close to the relaxation time. We argue in a following section that this process signals the influence of the LW fluctuations on the relaxation dynamics.

### Decoupling between Relaxation and Diffusion.

In the supercooled regime, the behavior of the non-Gaussian parameter and the 4-point dynamical susceptibility points toward the coexistence of particles with small and large displacements, on a timescale of the relaxation time. These DHs, a hallmark of the supercooled liquid dynamics (18), induce a breakdown of the inverse relationship between the diffusion coefficient and the structural relaxation time, leading to *B*. The reader is warned to take with care the value of κ in the deeply supercooled regime. Indeed, as already shown in Fig. 1, the supercooled regime dynamics are influenced by finite-size effects. Conversely, we stress that results in the normal liquid regime, which is our main focus, are not influenced by finite-size effect, as we show in a subsequent section and in *SI Appendix*.

We provide an insight on the physical origin of the observed *Materials and Methods*), where the displacement of a particle is evaluated with respect to average displacement of its neighbors. In Fig. 3 we compare the standard and the CR measures by plotting both D vs. τ (black squares) and *SI Appendix*, Fig. S9 the presence of a plateau region when the product

CR measures have been suggested to filter out the effect of the LW fluctuations on the dynamics (6), in the supercooled regime, as they subtract correlated particle displacements. Our results suggest that the LW fluctuations are also responsible for the *C*. According to this picture, LW fluctuations affect the relaxation time and not the diffusivity; this interpretation is supported by the results of Fig. 3, from which we understand that the standard and the CR measures mainly differ in their estimates of the relaxation time, which is smaller for the standard measures (see *SI Appendix*, Fig. S1 for a direct comparison of the standard and CR MSD and ISF). The standard and the CR diffusion coefficients have indeed an expected small difference,

The above interpretation is also consistent with the investigation of the difference between the standard and the CR measures in 3D. Indeed, Fig. 4 clearly shows that in 3D, where LW fluctuations play a minor role, there are minor differences between the 2 measures, both in the normal liquid and in the supercooled regime.

### Long-Wavelength Fluctuations in the 2D Normal Liquid Regime.

The above results suggest that in 2D the dynamics of normal liquids are strongly influenced by the LW modes. This is a counterintuitive speculation since the relaxation dynamics are expected to be weakly dependent on the features of the underlying energy landscape (4, 24) above the onset temperature. In *SI Appendix*, Fig. S10, we show that the density of states of 2D normal liquids satisfies Debye scaling at LW, thus proving that LW fluctuations exist in this regime. We demonstrate that these LW fluctuations also influence the relaxation dynamics by performing 2 additional investigations.

First, we consider the effect of the microscopic dynamics on the value of κ, comparing the 2 limiting cases of underdamped and overdamped dynamics, corresponding to Langevin dynamics (*SI Appendix*) with Brownian time *A*, as expected. Conversely, the overdamped results strongly differ in that

As a second check we explicitly evaluate the relevance of the LW modes to the overall particle displacement, as a function of time. To this end, we project the normalized displacement *Materials and Methods*) of the

### Viscosity and Size Effects.

The study of the shear viscosity η, which is the Green–Kubo integral (26) of the shear stress autocorrelation function (*SI Appendix*, Fig. S3), offers an alternative approach to filter out the effect of the LW fluctuations. This is because the stress is insensitive to the LW fluctuations, as it depends on the interparticle forces and hence on the relative distances between the particles. Indeed, Fig. 7 shows that *A*). This result indicates that 2D and 3D systems behave analogously concerning the breakdown of the Stokes–Einstein (SE) relation

Fig. 7*B* demonstrates that size effects diminish as the system size increases, as we highlight also considering data from ref. 21 for systems with up to 4 million particles, and become negligible for large enough systems. This behavior results from the competition of 2 timescales. One timescale is the CR relaxation time *SI Appendix*, Fig. S11. Finite system size effects, which are present when the LW modes develop before the system relaxes, *B* also clarifies that for the 2D mKA model, above the onset temperature size effects are lost in our smallest system. Therefore, all our results obtained in the liquid regime do not suffer from finite system size effects.

## Discussion

In conclusion, our results indicate that LW fluctuations affect the structural relaxation of 2D liquids in the normal liquid regime, where the relaxation dynamics do not suggest a transient solid response, even when evaluated using CR measures (*SI Appendix*, Fig. S1). In this regime, the LW fluctuations induce stretched exponential relaxations qualitatively different from those observed in the supercooled regime, which is not associated with the coexistence of particles with markedly different displacements. This result allows us to rationalize an open issue in the literature (9⇓–11), namely the physical origin of the decoupling between relaxation and diffusion *SI Appendix*, Fig. S15). Thus, our findings appear extremely robust as they do not depend on whether the interaction potential is finite or diverging at the origin, attractive or purely repulsive, or isotropic or anisotropic.

It is natural to ask whether, at high enough temperature or low enough density, the effect of LW fluctuations becomes negligible. The answer to this question is affirmative. Indeed, we do see in Fig. 3 that the difference between the standard and the CR measures, which is a proxy for the relevance of LW fluctuations, decreases as the relaxation time decreases. In the numerical model we have explicitly verified that the 2 measures coincide in this very high-temperature limit (*SI Appendix*, Fig. S1). Interestingly, in the Harmonic model, where the potential is bounded, we have found that in this limit the system relaxes before the ballistic regime of the MSD ends. This leads to *SI Appendix*, Fig. S2. We checked in *SI Appendix*, Figs. S7 and S8 that the value

We conclude with 2 more remarks. First, it is established that CR measures remove the effect of LW fluctuations (3, 6). Here we note that CR measures filter out all correlated displacements between close particles, regardless of their physical origin. In particular, in the supercooled regime, they suppress the effect of correlated particle displacements arising from DHs (see *SI Appendix*, Fig. S12 for the comparison of the 4-point dynamical susceptibility between standard and CR measures). This has to be taken into account when using 2D systems to investigate the glass transition. We note that it appears difficult to selectively suppress only the correlations arising from 1 of these 2 physical processes, as DHs are associated with the low-frequency vibrational modes (31). In this respect, perhaps one may consider that DHs in the supercooled regime are associated with localized modes, while LW fluctuations are signatures of extended modes.

Finally, we highlight that LW fluctuations are found in quasi-2D colloidal experiments of both spherical (3, 6) and ellipsoidal (32) particles, as we have shown. However, we have found no clear evidence of LW fluctuations in our overdamped numerical simulations. Hence, the overdamped simulations do not fully describe the behavior of colloidal suspensions. This is not a surprise, as it is indeed well known (33⇓⇓⇓–37) that, due to the presence of hydrodynamic interactions, the velocity autocorrelation function of colloidal systems does not decay exponentially as in the numerical simulations of the overdamped dynamics. The upshot of this consideration is that collective vibrations observed in colloidal systems, including the LW fluctuations, may stem from the hydrodynamic interparticle interaction. It would be of interest to better characterize these collective hydrodynamic induced modes.

## Materials and Methods

### Model Systems.

In 2D, we investigated the mKA model (13) and the harmonic model (14). The mKA model is a

We have performed Newtonian dynamics in different thermodynamic ensembles, as well as using a Langevin dynamics, as detailed in *SI Appendix*. All simulations are performed with the GPU-accelerated GALAMOST package (38).

### Calculation Details.

The MSD is *SI Appendix*, Figs. S13 and S14 that our results are robust with respect to the definition of τ. The non-Gaussian parameter is

CR quantities are defined by replacing the standard displacement

Results of Fig. 6 are obtained by projecting the normalized particle displacement at time t on the modes of the inherent structures of the

## Acknowledgments

M.P.C. and Y.-W.L. acknowledge support from the Singapore Ministry of Education through the Academic Research Fund (Tier 2) MOE2017-T2-1-066 (S) and from the National Research Foundation Singapore and are grateful to the National Supercomputing Center of Singapore for providing computational resources. K.Z. acknowledges support from the National Natural Science Foundation of China (NSFC) (21573159 and 21621004). Z.-Y.S. acknowledges support from the NSFC (21833008 and 21790344). T.G.M. acknowledges financial support from University of California, Los Angeles.

## Footnotes

- ↵
^{1}To whom correspondence may be addressed. Email: massimo{at}ntu.edu.sg.

Author contributions: K.Z., T.G.M., and M.P.C. designed research; Y.-W.L., C.K.M., Z.-Y.S., and R.G. performed research; and Y.-W.L., K.Z., T.G.M., and M.P.C. wrote the paper.

The authors declare no competing interest.

This article is a PNAS Direct Submission. E.R.W. is a guest editor invited by the Editorial Board.

This article contains supporting information online at www.pnas.org/lookup/suppl/doi:10.1073/pnas.1909319116/-/DCSupplemental.

Published under the PNAS license.

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