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Thermal motions in complex liquids: the 2D Lennard-Jones liquid

Thermal motions in the 2D Lennard-Jones liquid near solidification are studied at equilibrium and under shear flow conditions. At the temperatures of the study, the liquid is significantly aggregated. On times of few to few tens of particles vibration periods, the dominant features are particles in-cage vibrations and the highest frequency longitudinal and transverse Hypersound. On time-scales of hundreds to thousands of vibration periods, the liquid appears spatially heterogeneous. On these times, slow non-oscillatory fluctuating currents persist for surprisingly long times; the hierarchical dynamics of the heterogeneous liquid results in changing temperature, density, and velocity profiles across the system. Heterogeneity fades, and a crossover to non-fluctuational Hydrodynamics is observed for smoothing times of many tens of thousands vibration periods. On these asymptotically-large times, the liquid is spatially homogeneous except for thin layers near the boundaries where the degree of crystallinity increases and the mobility decreases due to liquid-boundary interactions.


Introduction
In a body, molecules randomly move with average thermal velocity [1,2] v T =(Dk B T/m) 1/2 where D=1, 2, 3 is the number of spatial dimensions, T the temperature, m the molecular mass, and k B the Boltzmann constant. The direction of th5e actual velocity of a molecule changes due to interaction with other molecules; the characteristic time τ of this change serves as a natural time unit for thermal fluctuations in the system. Molecular interactions lead to correlations between molecules positions, and also their velocities. A known example of correlated, cooperative motion is sound. In sound motions, the center-of-mass velocity, density, and pressure and shear stress in a small volume oscillate with periods proportional to the wavelength λ of the soundwave. On larger times, especially in complex liquids ("soft-matter") [1,[3][4][5][6][7][8][9][10][11], one also observes non-oscillatory collective motions. Visco-elastic effects in liquids can be explained by assuming that molecules form transient [7][8][9][10] aggregates. Then, on time-scales defined by the lifetimes of these aggregates, the liquid is heterogeneous [9], with the spatial scale of this heterogeneity determined by the size of the aggregates. Direct observation and measurement of aggregation dynamics is a challenging experimental task, so the current picture of the fluctuating local structure in complex liquids lacks important details.
An alternative possibility of studying microscopic-scale motions in a liquid is provided by Molecular Dynamics (MD) simulations in combination with specially devised visualization, recognition, and measurement programs. Here, we use this approach to study fluctuations in a two-dimensional (2D) Lennard-Jones (LJ) liquid. The choice of the model takes into account that, in a certain sense, the 2D liquid near solidification represents the simplest, and easiest to study, complex liquid: previous studies [5,[7][8][9][10] found that at temperatures close to solidification, particles of this liquid aggregate into transient crystalline-ordered clusters ("crystallites") intertwined with lessordered amorphous clusters. This dynamic mosaic of crystallites and amorphous regions appears statistically-stable, and bears some general resemblance to much more complicated complex 3D liquids [6,11].
In recent decades, 2D systems have been intensely studied to test the nature of 2D melting [13][14][15][16]. The results of intensive computer simulations generally confirm the defect-unbinding nature of melting, although close to the triple point there are some subtle violations of this scenario [14]. Here, we study the liquid at temperatures above melting, in states where the correlation length for the local order is a fraction of the system size so the liquid is locally ordered but globally disordered. At these temperatures, a particle in the liquid is surrounded by several (4-6) close neighbors (referred to as cage particles), and vibrates in this environment with the average period τ. Unlike the case of a crystal where a particle vibrates in same cage for macroscopically large times, in a liquid the lifetime τ r of the near environment of a particle (the cage time [17]) is microscopic; this time varies substantially depending on thermodynamic parameters. In simple 2D and 3D liquids [2,9], this lifetime is of the order of τ. In the 2D Lennard-Jones liquid, τ r~τ at a temperature T that is ~50% above the melting temperature, increases one-two orders of magnitude on cooling to the aggregated state studied here, and becomes large beyond direct observation on further cooling/compression to crystalline state.
To perturb the thermodynamic equilibrium, the liquid in our study is placed between solid boundaries moving in opposite directions (see Fig. 1). For a macroscopic system, similar conditions generate the Couette flow [18,19]. The current particles configuration is visualized, and instant (at a current iteration step) or smoothed (timeaveraged) profiles of the local temperature T(r,) density ρ(r), and velocity v(r) across the system are studied. For sufficiently large smoothing times θ>θ H , these profiles become (see Fig. 2) the non-fluctuating, smooth profiles predicted by the Hydrodynamics [18,19], but for smaller smoothing times θ<θ H , the profiles fluctuate in space and time ( Fig. 3-6).
Factors that make the aggregated state of the 2D liquid stable are specific for 2D liquids: in 3D liquids, significant aggregation is observed or assumed only in systems with complicated and competing interactions. In quasi-2D liquid of hard and soft spheres confined between two parallel planes, aggregation diminishes with increasing distance between these planes [7,8]. We believe that general laws of complex behavior are common for complex liquids, so studies of the relatively simple 2D liquids can give an insight into the internal dynamics of structurally much more complicated 3D liquids. Particles in the temperature-monitored part are colored red, particles in some crystallites-blue, boundary particles -black; the measuring column is indicated by a gray background.

The system
Molecular Dynamics (MD) with Verlet velocity algorithm is applied to simulate 2D systems of N=2500÷62500 Lennard-Jones particles in periodic boundary conditions; this rather standard algorithm is described in many publications -see, for example, [9, 10, 13-15, 20, 21]. We use a Euclidean coordinate system with the x-axis parallel and the y-axis orthogonal to the boundaries; due to periodicity along the x-direction, the system has the geometry of a cylindrical shell limited by the top and bottom boundaries (see Fig. 1).
To create solid boundaries, special Lennard-Jones particles (black circles in Fig.1) are placed at the bottom (y=0) and the top (y=L y ) of the simulation cell; the linear density of these boundary particles is chosen sufficiently large to prevent particles of the liquid from penetrating the boundaries. Boundaries move as solid bodies, parallel to the x-axis but in opposite directions, with velocities v for the top and -v for the bottom boundary.
The Lennard-Jones interaction potential is taken in the standard form 12 6 ( ) 4 U r r r Fig. 1). In the rest of the system, the temperature is determined by the balance between heat release due to velocity gradients and heat exchange with the T-maintained layer.
This way of temperature maintenance is chosen to mostly confine to a small layer the perturbations of particles "natural" motion by the (NVT) temperature maintenance procedure. For steady flows with average velocity gradients <dv y /dy>=2v/L<10 -4 , the smoothed temperature profile is linear within 2% accuracy. For larger gradients, we observed remarkable nonlinear regimes including phase separation and appearance of crystalline or gaseous sub-systems; although interesting, these regimes are beyond the scope of the current paper.
To facilitate data collection, the system is partitioned into measuring cells by n equidistant horizontal and n' equidistant vertical lines x=nL x /s, y=n'L y /s' (n =1, … , s-1; Here, L x and L y are the length and the width of the system (see Fig. 1). The give the data for a horizontal row of measuring cells. Smoothed profiles <X(y,t)> θ , <X(x,t)> θ are obtained by smoothing (time-averaging) instant characteristics over a time interval (t-θ/2, t+θ/2).
When the density increases along the T=2 isotherm, the fraction of particles in crystalline-ordered clusters grows from zero at ρ<0.65 to ~10% at ρ=0.87, ~20% at

Steady states, and crossover to Hydrodynamics
A sufficiently long simulation under steady (Tρv)-parameters (temperature T, density ρ, boundary velocity v) brings the system close to a (statistical) steady state; the time necessary to essentially "forget" the initial conditions and reach a "practical" steady state depends on system size and initial conditions, it is ~10 4 τ for a system of N=40000 particles at T=2, ρ=0.92 and crystalline initial configuration. As a practical definition, we

Collective thermal motions in the liquid
In the liquid at T=2, ρ=0.92, the number of particles in a large crystallite is δN~20-70. Connecting each internal (not at the border) crystallite particle with its cage particles creates a near-neighbors network that can be mapped onto a finite-size hexagonal crystal [9]. Borders of a crystallite visibly fluctuate in time due to small (3-5 particles) random increases or decreases of the crystallite at the border [10]. As a result of these micro-melting/micro-crystallization events, the crystallite (as a finite-size network) changes its shape and position without substantial changes in the positions of internal particles. At T=2, ρ=0.92 the average time for a particle to continuously remain in a crystallite is ~15τ. Direct observation proves that density differences between crystallineordered clusters are substantially smaller than the difference (~15%) between a crystalline and amorphous cluster. Direct observation reveals that the volume fraction of crystalline clusters in the small measuring cell fluctuates between close to 100% (when the entire cell is occupied by a crystallite) and 0% (when the cell is occupied by an amorphous cluster). The density then fluctuates between that of a crystal and of an amorphous cluster; the expected (and observed) amplitude of these fluctuations is of the order of density difference (~15%) between a crystallite and an amorphous cluster. In a small cell, this particles rearrangement mechanism gives the largest contribution to density fluctuations. A possible contribution of fluctuating currents (see next sub-Section) bringing differentlyordered clusters into the cell appears small due to the very small thermal displacements of particles caused by these currents.  The FFT (Fast Fourier Transform) power spectrum of the difference δρ=ρ(t)-<ρ> is shown in Fig. 5. The frequency range ν>0.05 of this spectrum is discussed in the next Section. In the range ν<0.01, density fluctuations are, roughly, frequencyindependent, indicating that structure changes at corresponding long times are uncorrelated. Spectra for even longer sampling times (∆t=10000) and/or larger (10X10) measuring cells support this assumption.  Fig. 4.

b. Thermal mixing and fluctuating currents
In a complex liquid, the trajectory of a single particle reveals a complex hierarchy of thermal motions [9], with the particle intermittently trapped in crystallites for  On a larger time-scale θ=1000 (the red line in Fig. 6), the average velocities both of the ∆R~40 blocks and the ∆R~10 regions are 2-4 times smaller than for θ=100, quantitatively consistent with the prediction of the random walk theory. However, for θ=5000 (the green line), all displacements decreased faster than this theory predicts. We suggest that for times of hundreds and few thousands of vibration periods, the viscoelastic liquid retains substantial elasticity that prevents large thermal displacements.
An independent proof of this short-times elasticity give the transverse Hypersound oscillations with periods of at least few hundreds of particles vibration periods -see the next Section. In addition to <v x (y,t)> θ , we also analyzed the profiles <v y (y,t)> θ , and also the profiles <v x (x,t)> θ and <v y (x,t)> θ obtained using a row of measuring cells stretched along the x-axis. All these profiles give the same picture of slow thermal currents and decaying correlations between particles velocities on times θ>1000.

c. Hypersound
At the high-frequency end of the sound spectrum, the periods of soundwaves approach the particles vibration period; the corresponding wavelengths are close to intermolecular distances. As discussed above, on these short times, the liquid is acoustically heterogeneous. Correspondingly presents the velocity profiles <v y (y,t)> θ  The decrease of fluctuation amplitudes with increasing smoothing time can be used to roughly estimate the wavelength λ(ν) to frequency ν relations for the sound modes. Fig.9 presents FFT transforms of the smoothed velocity <v y (y,t)> θ (as function of y) for θ=1,10, and 20. As the plot shows, the correspondence is rather loose, but the peaks indicating modes are distinguishable, especially for lower frequencies (and wave-vectors).

Discussion and conclusions
We presented a rather detailed picture of internal motions in an aggregated 2D liquid. On time-scales of up to few thousand of particles vibration, the liquid is heterogeneous and represent a mosaic of ordered and less-ordered clusters. There is increasing evidence [11,12] that this type of heterogeneity is a common feature of condensed but not long-range ordered systems. In some well-studied liquids, a local order resembling that in a crystal has been demonstrated long-ago. A known example here is liquid Benzene where some elements of a rather complex crystalline local arrangement have been found in X-ray studies [25]. We suggest that liquid-liquid phase transitions [26][27][28][29] are rather dramatic manifestations of this kind of the local order, and assume that a mosaic of ordered and disordered clusters is a common feature of locally-ordered but globally-disordered systems, and that molecules redistribution between ordered and disordered (or less-ordered) regions determines the unusual properties of these systems.
The simplest and easiest to study complex, aggregated liquids are the 2D-liquid near its crystallization temperature; these systems, in particular the 2D Lennard-Jones liquid, can be suggested as a test-bed for studies of the dynamics of particles aggregation, aggregates clustering, and the interaction of local heterogeneities with externally-imposed flows.