Heterogeneous Structure, Heterogeneous Dynamics, and Complex

Aug 16, 2012 - Analysis of the metrical and topological features of the local structure in a freezing two-dimensional Lennard-Jones system found that ...
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Heterogeneous Structure, Heterogeneous Dynamics, and Complex Behavior in Two-Dimensional Liquids A. Z. Patashinski,*,† M. A. Ratner,† B. A. Grzybowski,*,† R. Orlik,‡ and A. C. Mitus*,§ †

Department of Chemistry, Northwestern University, 2145 Sheridan Road, Evanston, Illinois 60208, United States Orlik Software, ul. Lniana 22/12, 50-520 Wroclaw, Poland § Insitute of Physics, Wroclaw University of Technology, Wybrzeze Wyspianskiego 27, 50-370 Wroclaw, Poland ‡

ABSTRACT: Analysis of the metrical and topological features of the local structure in a freezing two-dimensional Lennard-Jones system found that in a narrow strip 3 of thermodynamic states close to the melting line, the liquid becomes a complex liquid characterized by a super-Arrhenius increase of relaxation times, stretched-exponential decay of correlations in time, and a power-law distribution of waiting times for changes in the local order. In 3 , the structure of the liquid and its dynamics are spatially heterogeneous; the sizes of ordered clusters are power-law distributed. Those features are governed by local structure evolution between solid-like and liquid-like (disordered) patterns. The liquid inside the strip 3 gives a unique opportunity to study how heterogeneous structure, dynamics and complexity are intertwined with each other on a microscopic level.

SECTION: Kinetics and Dynamics

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particles, the local structure can be quantified in terms of instantaneous solid-like and liquid-like patterns in the framework of probabilistic pattern recognition formalism.18−20 This approach shifts the main focus of studies from ensemble averages to single configurations (as advocated in ref 15 and partially started in ref 8), and makes possible to study the microscopic intertwine between static and dynamic heterogeneity and complexity under the verifiable assumption that the 2D liquid actually shows typical features of complex behavior. The verification of this assumption, and the study of microscopic details of the heterogeneous structure and dynamics are the aims of this paper. We show that in a narrow strip of thermodynamic states, a freezing 2D Lennard-Jones (LJ) liquid displays all the characteristic features of complex behavior. We have simulated a system of 2500 atoms interacting via LJ potential ULJ(r) = 4ε[(σ/r)12 − (σ/r)6] using Constant Temperature Molecular Dynamics (see ref 20 for details) in the range of temperatures and densities from liquid (left from line L) to crystal (right from line S) (see Figure 1a, which shows a part of phase diagram of 2D LJ system in variables T* − ρ* (T* ≡ kBT, ρ* ≡ ρσ2) calculated using local structure analysis).19 We use the vibration period τLJ for a particle in the harmonic part of LJ potential as the time unit. A typical configuration of the system in the center of the density interval between lines L and S is a fluctuating mosaic of

omputer simulations and experiments of past decades have given convincing evidence that the origin of the dynamical heterogeneity in equilibrium and supercooled liquids in two (2D) and three (3D) dimensions is the underlying structural heterogeneity.1−17 In 2D, this relation was quantified using short-time averaged trajectories4−6,8,11,12 and local as well global static order parameters,1 confirming that molecules perform oscillatory motions inside structurally ordered domains and fast string-like movement at the disordered boundaries of these domains.4−6,8,11,12 However, structural features underlying the causal link between structure and dynamic heterogeneity remain obscure,14 partly because of lacking clear definition of local order. This prevents establishing the causal link between a particle configuration and the spatial distribution of dynamics to which it gives rise.15 Structural and dynamic heterogeneity in glass-formers is strongly correlated with complex behavior of these materials. One suggests that this complexity is a manifestation of dynamic competition between the process creating ordered structures in small volumes and disordering mechanisms preventing the local order from becoming global. These processes are also responsible for the heterogeneity in time and space. The heterogeneity of both structure and dynamics and the complex behavior of the materials are then inextricably intertwined with each other. Understanding and quantification of these three factors and of their mutual relations remains one of the most challenging problems in condensed matter. A necessary first step in this direction requires a framework for treating heterogeneity in space and time as well as complexity on the microscopic level and base this framework on well-defined concept of liquid’s local structure. For 2D liquids of point-like © 2012 American Chemical Society

Received: July 22, 2012 Accepted: August 16, 2012 Published: August 16, 2012 2431

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Figure 1. (a) Phase diagram of 2D LJ liquid in variables T*−ρ*. Dashed line - isothermic scan; shaded area - strip 3 . (b) Typical configuration (“mosaic state”) of 2D LJ liquid for T* = 0.7 and ρ* = 0.845 in strip 3 , corresponding to the black circle in panel a.

small crystallites separated by amorphous matter19,20 (Figure 1b). Black circles represent solid-like atoms (SLA), being the centers of seven-atom clusters classified as crystalline by the probabilistic classification rules for small fluctuating clusters in 2D;18,19 white circles are liquid-like atoms (LLA) representing noncrystalline seven-atom clusters. The system displays a heterogeneous spatial structure which substantially changes on a time-scale τ0 ∼ τLJ, the shortest scale in the hierarchy of structural relaxation times. To avoid misunderstanding, we point out that SLA clusters do not constitute a phase in a thermodynamic sense, but rather a dynamic structure with a finite lifetime. In a narrow strip 3 in the T*−ρ* plane, centered around ρ* = 0.845 at T* = 0.7 (see Figure 1a), the heterogeneity of spatial distribution of SLA and LLA atoms has a number of surprising physical and geometric consequences. We present the results for an isothermic scan T* = 0.7, the results for other temperatures are similar. In closest vicinity of line L (ρ* ≃ 0.82519), neighboring SLA atoms percolate,19 the distribution N(s) of clusters’ sizes s (number of SLA atoms in a cluster) has a scaling form N(s) ∼ s−α (Figure 2), with α close to its 2D

lattice counterpart 187/91, and the correlation length23 diverges to infinity. The scaling law breaks down away from line L. At densities well below line L, the fraction of SLA crystallites is small, and the system is a matrix of LLA material with small islands of order, while at higher densities, close to line S, SLA crystallites merge and form a crystalline matrix with small isolated islands of disorder hosting vacancies and dislocations. The onset of percolation of the SLA component is not immediately accompanied by the onset of long-range orientational order of seven-atom SLA clusters. The latter takes place in strip states 3 (ρ*1≃ 0.842 for T* = 0.720) and marks the onset of shear rigidity (solidity in terms of ref 13). Strip states 3 also mark the crossover between the ensembles of fluctuations for seven-atom SLA clusters, from independent Gaussian at lower densities to correlated (at higher densities) fluctuations.19,24 Finally, in strip 3 , a physical anomaly occurs: heat capacity has a well-defined maximum.24 We conclude that in a narrow strip 3 , the system is neither completely ordered nor completely disordered, and its properties are characteristic for complex systems. In what follows, we study microscopic temporal and spatial characteristics of the system to further reveal its complex-like behavior in strip 3 . For this task we use local topological, and not simply geometric, characteristics, as advocated in the context of disordered networks.13 However, unlike the typical approach, we shift the main focus to the basic structural unit, which is a cluster of atoms self, and not its function (like local order parameter). Any structural changes in the system involve changes in the nearest-neighbor (nn) relationships characterized by a dynamic list La(t) of six nearest-neighbors for each particle a.20 The dynamics of changes of particle’s local environment is characterized by the switches between initial list L0 = La(0) and other competing lists. The particle keeps list L0 during waiting time interval τ(0), then replaces it by list L1 in

Figure 2. Plot of a non-normalized distribution N(s) of SLA clusters’ sizes s close to line L (T* = 0.7, ρ* = 0.827) for 216 LJ atoms.

Figure 3. Parameters of stretched exponential fit to K(t): relaxation time θ for ρ* = 0.84 (a) and β for T* = 0.7 (b). Inset: function K(t) and stretched exponential fit for ρ* = 0.84. Vertical lines mark the lines L and S. 2432

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time interval τ(1), and so on. We quantify nn-changes on a “macroscopic” scale using the approach of ref 20: the nn-lists of all initially SLA particles were memorized at a reference time tm and compared with nn-lists at a time tm + t to calculate the average number Nnn(t) of particles with the same nn-lists at both times. The normalized function K(t) ≡ Nnn(t)/N was successfully fitted with stretched-exponential function K(t) ∝ exp(−(t/θ)β). Upon approaching line L from the liquid side, the function K(t) starts to display features characteristic of complex systems: it becomes stretched-exponential (inset in Figure 3a), and the stretching exponent β deviates from unity and becomes temperature/density-dependent (see Figure 3b). Inside the strip 3 , exponent β is less than 1 and attains the minimum βm ≃ 0.6 at density ρ* = 0.85 close to the end of strip states 3 . The relaxation time θ rapidly increased, from only few τLJ close to line L to (103−104) τLJ close to line S. Figure 3a shows the Arrhenius plot for θ calculated for density ρ* = 0.84. In the liquid phase, the plot is linear and corresponds to Arrhenius-like behavior. Close to line L, a nonlinear dependence sets in and becomes more distinctive upon crossing this line. This behavior is typical of a modestly fragile glass-former. Close to line S it gives way again to linear dependence. The linear fits intersect at temperature T* ≃ 0.7, close to strip 3 . Complex behavior in liquids is often accompanied by power laws for waiting times of specific microscopic events which, in our case, are waiting times τ for the lists La. Figure 4 shows

Figure 5. Trajectory of a single particle separated into SLA (red) and LLA (black) parts. T* = 0.7, ρ* = 0.84. Unit of length: σ. Large symbols: conserved initial list. Inset: time series (in units of τLJ) for lists: 1 - list L(0); 0 - other lists.

symbols) and explores an area with the size of an average interparticle distance. In this period, the particle changes its structural state from SLA (red) to LLA (black) and vice versa. Finally, the particle is released and moves to another long-lived state of mostly SLA character, in the upper-left part, through one - two short-lived states. The trace of the particle’s locations in SLA state consists of separate compact parts while its LLA counterpart is rather continuous. Large displacements occur mostly in the LLA state; this feature is well illustrated by the central part of the trajectory, which displays a ballistic motion of the particle with velocity much exceeding that in the SLA state, reminiscent of observed, but not interpreted in terms of structure, string-like motion.4−6,8,11,12 The characteristic intermittent mobility along a random walk trajectory of a diffuser is a known experimental fact (see, for example, observation of diffusion in a colloid system in ref 16, where these observations were interpreted in terms of density changes). The densities of SLA and LLA clusters differ, and this difference is larger than density fluctuations in these clusters, so our interpretation is consistent with that referring to density differences, and explains the difference in mobility between more and less dense clusters. The spatiotemporal features of unjamming processes are closely related to the mosaic state (spatial characterization) and function K(t) (temporal characterization), shown in Figure 6 for a state in the strip 3 (T* = 0.70, ρ* = 0.84) for t = 60. Black color denotes particles a, which initially (t = 0) were in

Figure 4. PDFs ρ(τ) of waiting times τ, calculated for ρ* = 0.85 and ρ* = 0.81, 0.86 (inset) at T* = 0.7.

log−log plots of probability density functions (PDFs) ρ(τ) at temperature T* = 0.7. In the liquid state at ρ* = 0.81 (close to line L) and at ρ* = 0.86 (close to line S), the plots have different convexities. In the vicinity of ρ* = 0.85, the plot becomes linear, indicating a scaling behavior ρ(τ)∝τ−γ with γ = 1.63 ± 0.01 and dynamic percolation.25 Away from a narrow interval of densities, which partly intersect with the strip 3 at T* = 0.7, the power laws break down. Dynamic heterogeneity implies the absence of a typical pattern of dynamic behavior of single particles. One of its manifestations is configuration jamming (the particle is trapped) and subsequent release. The structural aspects of this phenomenon are obscure; qualitative arguments stating that the release is triggered by a change of the particle’s surroundings were put forward in refs 8 and 16. We study this effect in the context of local structure dynamics and show that it is triggered by local structural changes from SLA to LLA. In Figure 5, the points show the successive positions of a single particle, initially located at point (0,0), separated by time intervals ≃τLJ/5. In the first phase of approximately 30 τLJ the jammed particle remembers its initial neighbors (inset, large

Figure 6. Visualization of mosaic state and function K(t) for T* = 0.70, ρ* = 0.84 and t = 60 τLJ. Color coding is explained in the text. 2433

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the SLA state and have the same lists at t = 0 and t = 60: La(0) = La(60). Blue particles, initially SLA, have changed the lists: La(0) ≠ La(60). For small gray particles, initially LLA, La(0)= L a(60), while for green particles, initially LLA, La(0) ≠ La(60). Finally, red particles have changed their lists shortly (1 τLJ) before t = 60. We observe a strong relation between heterogeneous structure and heterogeneous dynamics: most of the black and blue particles remain in the SLA state at t = 60, while most of gray and green particles remain in the LLA state. This implies that locally solid-like ordered clusters have a strong tendency to remain ordered, while disordered component remains mostly disordered. The dynamics of structural changes is strongly heterogeneous: nn-changes were only detected outside or at the borders of crystallites and frequently resulted in a small (1−3 particles) local increase or decrease of the crystallite. This local process of microcrystallization/micromelting substantially changed positions, shapes, and orientations of crystallites at times on the order of or larger than θ. The long-lived clusters of black particles correspond to the next scale τ1 ≫ τ0 in the hierarchy of temporal scales. The study of larger scales requires analysis of correlations between those clusters and goes beyond the scope of this Letter. To summarize, the quantification of the concept of local structure in a 2D Lennard-Jones system using metrical and topological characteristics and shifting the focus of studies from ensemble averages to single configurations makes it possible to set the clear connection between signatures of complexity and percolation of crystalline-ordered clusters that takes place in a narrow strip 3 of thermodynamic states: complexity is a manifestation of the adaptive hierarchical system that represents the percolation cluster. This percolation nature of complexity is demonstrated in 2D liquids, but may be assumed to determine the generic properties of all complex liquids. The function K(t) as defined in the Letter describes the changes in the SLA part of the system (crystallites), and it is an internal characteristic of structure relaxation underlying any macroscopic relaxation. The similarly defined function for all atoms (including LLA) gives a description of the part of a configuration that keeps the local structure for a given time. To relate these internal microscopic characteristics to macroscopically observable quantities, one needs a micromechanical model of the material (an example of a micromechanical model for a stressed dynamically heterogeneous material can be found in ref 26). Construction and applications of appropriate micromechanical models to describe the response of the system to perturbations is an important but separate task that is beyond the scope of this Letter. The mechanism of defect unbinding determining the properties of a 2D liquid very close to the melting transition, and in particular the appearance of a hexatic phase, assumes a very large radius of the positional order, and an even larger radius of orientation order. In most states studied here (for states left of the middle line of 3 ) both radii are smaller than the size of the system, so the system is a normal (not hexatic21,22) liquid. Near this middle line, the correlation radius for orientations reaches the size of the system,20 but still about half of the particles are SLA and the other half LLA, and the sizes of crystallites are smaller than the size of the system. At the low temperature/high density boundary of 3 , there is a crossover from the mosaic state to a multiconnected crystalline matrix with small islands of LLA atoms hosting defects, and the

defect-unbinding phenomena may play a role in system behavior.

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AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected](A.Z.P.); grzybor@ northwestern.edu (B.A.G.); [email protected] (A.C.M.). Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS This work was supported by the Nonequilibrium Energy Research Center (NERC), which is an Energy Frontier Research Center funded by the U.S. Department of Energy, Office of Science, Office of Basic Energy Sciences, under Award Number DE-SC0000989.

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