Unveiling Dimensionality Dependence of Glassy Dynamics: 2D Infinite Fluctuation Eclipses Inherent Structural Relaxation

Abstract

By using large-scale molecular dynamics simulations, the dynamics of two-dimensional (2D) supercooled liquids turns out to be dependent on the system size, while the size dependence is not pronounced in three dimensional (3D) systems. It is demonstrated that the strong system-size effect in 2D amorphous systems originates from the enhanced fluctuations at long wavelengths, which are similar to those of 2D crystal phonons. This observation is further supported by the frequency dependence of the vibrational density of states, consisting of the Debye approximation in the low-wavenumber-limit. However, the system-size effect in the intermediate scattering function becomes negligible when the length scale is larger than the vibrational amplitude. This suggests that the finite-size effect in a 2D system is transient and also that the structural relaxation itself is not fundamentally different from that in a 3D system. In fact, the dynamic correlation lengths estimated from the bond-breakage function, which do not suffer from those enhanced fluctuations, are not size dependent in either 2D or 3D systems. PACS numbers: 62.60.+v, 61.20.Lc, Dimensionality plays a key role in the physics of solids and liquids -from high to low dimensions -and fluctuation shows up differently, as typically observed in phase transitions [1,2]. Indeed, two-dimensional (2D) systems often exhibit enhanced fluctuations, leading to various anomalies that are not experienced in three-dimensional (3D) systems. The melting of a 2D solid is a marked example [3][4][5][6][7][8][9], where the long-wavelength structural correlation is induced by thermal fluctuations sthat span an infinite length. For the glass transition from supercooled liquids to amorphous solids, the dimensionality dependence of the fluctuation has become an issue only recently. Gigantic fluctuation in 2D supercooled liquids has been observed that is far stronger than that in their 3D counterparts [10][11][12]. The aim of this Letter is to elucidate the similarity of this fluctuation to that in crystals [13], and also to investigate the heterogeneous dynamics in both 2D and 3D systems.For a crystalline solid of monodisperse particle assemblies, the mean-squared thermal displacement (MSTD) is given by using the vibrational density of state (VDOS) g(ω) as a function of angular frequency ω aswhere m the particle mass, d the spatial dimension, and (k B T ) −1 the inverse temperature. Under the Debye approximation for the VDOS of acoustic plane waves, g(ω) becomes proportional to ω d−1 [14]. It leads to divergence of the integral in 2D systems owing to the low-frequency acoustic waves, while it converges in 3D systems. As a result, the long-range translation order is prohibited in 2D systems [15,16]. Integration of Eq. (1) over ω ≥ 2πc/L provides us with its dependence on the linear system size L aswhere µ and K are shear and bulk moduli, σ 0 is the particle radius, and c is the velocity of sound. Such fluctuation is the source of the size-dependent behavior of 2D solids undergoing melting...

Local Density Fluctuation Governs the Divergence of Viscosity Underlying Elastic and Hydrodynamic Anomalies in a 2D Glass-Forming Liquid

“…This result implies that other mechanisms besides the emerging elasticity may underlie the stress relaxation in two dimensions, which motivates us to investigate long-wavelength hydrodynamic fluctuations that may influence the long-time relaxation of 2D glass-forming liquids. That is, while the transient elastic vibrations cause an anomalous divergence of thermal fluctuation amplitudes, as discovered recently [7], an anomalous logarithmic divergence of transport coefficients, including diffusivity and viscosity, may possibly be induced by the longtime diffusive relaxation of hydrodynamic fluctuations, as in normal 2D liquids [17][18][19][20][21][22]. Although this logarithmic divergence is weak compared to the viscosity enhancement associated with vitrifcation and remains difficult to be observed [23][24][25], if it exists, it would be easier to see by measuring the diffusivity of single particle trajectories, rather than by measuring the viscosity from the stress relaxations.In this Letter, large-scale MD simulations of a 2D glassforming liquid are used to examine how the growth of various relaxation times is related to the divergence of macroscopic viscosity in the presence of long-wavelength fluctuations.…”

“…Although retaining a random amorphous structure, elastic vibrations appear as the rigidity emerges with the decrease in temperature. The excess of low-frequency phonons in two dimensions [7,11] leads to enhanced thermal elastic vibrations at long wavelengths. Even in the presence of these long-wavelength fluctuations, the structural relaxation in 2D and 3D supercooled liquids appears to be similar, once the effect of these fluctuations has been eliminated by introducing quantities that characterize the local switching between neighbor particles [7,9,10].The glass transition is marked by a drastic increase in macroscopic viscosity, which is intimately related to the divergence of the microscopic structural relaxation time.…”

“…The excess of low-frequency phonons in two dimensions [7,11] leads to enhanced thermal elastic vibrations at long wavelengths. Even in the presence of these long-wavelength fluctuations, the structural relaxation in 2D and 3D supercooled liquids appears to be similar, once the effect of these fluctuations has been eliminated by introducing quantities that characterize the local switching between neighbor particles [7,9,10].The glass transition is marked by a drastic increase in macroscopic viscosity, which is intimately related to the divergence of the microscopic structural relaxation time. As such, theoretical and computational studies have focused on the dynamical mechanism of growth in the microscopic structural relaxation time, most typically the α-relaxation time [12][13][14][15], which is defined as the decay time of the autocorrelation function for the local density i.e., the intermediate scattering function.…”

“…The production runs are then performed as Newtonian (NVE) dynamics simulations to prevent the damping of longwavelength fluctuations. The data presented in the remainder of this Letter are averaged over four or eight independent simulations [27].First, we revisit three relaxation functions that have been considered in recent studies [7][8][9][10]16]. The first is the standard self-intermediate scattering function (SISF) F s (k, t) = (1/N) N j=1 exp{ik · [r j (t + t 0 ) − r j (t 0 )]} , with the wavevector set to |k| = 2π/σ 11 so that its decay represents particle…”

Lindemann melting criterion in two dimensions

“…Although MSD diverges with system size in 2D crystalline and amorphous solids [4,30], the squared deviation of the difference between the positions of two particles remains finite [31]. This can serve as a basis to construct modified Lindemann-like criteria of 2D melt-ing.…”