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 D ∝ τ − κ , with κ > 1 . Long-wavelength fluctuations have a negligible influence on the relaxation dynamics only at extremely high temperatures in molecular liquids or at extremely low densities in colloidal systems.

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), ⟨ Δ r 2 ( t ) ⟩ , and of the self-intermediate scattering function (ISF), F s ( q , t ) , of the 2-dimensional (2D) modified Kob–Andersen (mKA) system, a prototypical model glass former (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).

Fig. 1. (A and B) Time dependence of the mean-square displacement (A) and of the self-intermediate scattering function (B) from molecular dynamics simulations of the 2D mKA model. Typical signatures of supercooled dynamics emerge as the temperature T decreases, including ISF oscillations associated with the Boson peak (17). The comparisons of systems with N = 2,000 , with N = 1 0 4 , and with N = 1 0 5 particles (see key defining symbols and lines in A) reveal that the dynamics in the supercooled regime are size dependent (5), while that is not the case in the normal liquid regime.

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.

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 D ∝ τ − κ with κ > 1 . In the main text, we have presented numerical data for the 2D mKA model and the 2D Harmonic model and their 3D counterpart for comparison, as well as experimental data for a 2D suspension of hard ellipses. We note, however, that we have also observed consistent results in a binary system with inverse power-law potential (28) and in monodisperse systems of Penrose kites (29, 30) (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 D ∝ τ − κ and κ = 2 , in both 2D and 3D, as we discuss and verify in SI Appendix, Fig. S2. We checked in SI Appendix, Figs. S7 and S8 that the value κ > 1 we have attributed to the LW fluctuations is not conversely the signature of a cross-over from the normal to the high-temperature liquid. In colloidal systems, particles perform independent Brownian motions in the low-density limit, where LW fluctuations are therefore negligible. We expect the cross-over density below which LW fluctuations are negligible to depend on the viscoelasticity of the solvent. 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 65 ( A ) : 35 ( B ) mixture, with interaction potential U α β ( r ) = 4 ϵ α β [ ( σ α β / r α β ) 12 − ( σ α β / r α β ) 6 + C α β ] , when r ≤ r α β = 2.5 σ α β , and U α β ( r ) = 0 otherwise. Here, α , β ∈ { A , B } . The interaction parameters are given by σ A B / σ A A = 0.8 , σ B B / σ A A = 0.88 , ϵ A B / ϵ A A = 1.5 , and ϵ B B / ϵ A A = 0.5 . C α β guarantees U α β ( r α β c ) = 0 . The number density is ρ = 1.2 . Length, energy, and time are recorded in units of σ A A , ϵ A A , and m σ A A 2 / ϵ A A , respectively. For this model, we consider N = 2,000 (if not otherwise stated), N = 1 0 4 , and N = 1 0 5 . The Harmonic model (14) is a 50 : 50 mixture of N = 3,000 particles with interaction potential U α β ( r ) = 0.5 ϵ ( 1 − r / σ α β ) 2 , for r < σ α β , and U α β ( r ) = 0 otherwise. The size ratios are σ A B / σ A A = 1.2 and σ B B / σ A A = 1.4 , and the number density is ρ = 0.699 . The units for energy, length, and time are ϵ, σ A A , and m σ A A 2 / ϵ , respectively. In 3D, we simulated the KA model (12), which consists of N = 3,074 with 80 % A and 20 % B particles, as well as the Harmonic model, with N = 3,000 . All of the results are averaged over at least 4 independent runs. We show the data for A particles if the system is a binary mixture. 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 ⟨ Δ r 2 ( t ) ⟩ = ⟨ 1 N ∑ i = 1 N Δ r i ( t ) 2 ⟩ , where Δ r i ( t ) = r i ( t ) − r i ( 0 ) is the displacement of particle i at time t. Its long-time behavior defines the diffusion coefficient D = lim t → ∞ ⟨ Δ r 2 ( t ) ⟩ 2 d t , with d the dimensionality. The ISF is F s ( q , t ) = ⟨ 1 N ∑ j = 1 N e i q ⋅ Δ r j ( t ) ⟩ with q = | q | the wavenumber of the first peak of the static structure factor. The relaxation time τ is such that F s ( q , τ ) = e − 1 . We have verified in SI Appendix, Figs. S13 and S14 that our results are robust with respect to the definition of τ. The non-Gaussian parameter is α 2 ( t ) = ⟨ Δ x ( t ) 4 ⟩ 3 ⟨ Δ x ( t ) 2 ⟩ 2 − 1 with Δ x ( t ) the displacement in the x coordinate (18). Finally, the 4-point dynamical susceptibility χ 4 ( t ) is defined as χ 4 ( t ) = N [ ⟨ F ^ s ( q , t ) 2 ⟩ ] − ⟨ F ^ s ( q , t ) ⟩ 2 ], with F ^ s ( q , t ) = 1 / N ∑ j = 1 N e i q ⋅ Δ r j ( t ) . CR quantities are defined by replacing the standard displacement Δ r i ( t ) with the CR one, Δ r i CR ( t ) = r i ( t ) − r i ( 0 ) − 1 / N i ∑ j = 1 N i [ r j ( t ) − r j ( 0 ) ] with N i the number of neighbors of particle i evaluated at time 0. Neighbors are identified via a Voronoi construction. Results of Fig. 6 are obtained by projecting the normalized particle displacement at time t on the modes of the inherent structures of the t = 0 configurations. We have obtained these modes by minimizing the energy of the t = 0 configurations using the conjugate gradient method and then diagonalizing their Hessian matrix.

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 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.