Article pubs.acs.org/JPCB
New Scenario of Dynamical Heterogeneity in Supercooled Liquid and Glassy States of 2D Monatomic System Vo Van Hoang* Computational Physics Lab, Institute of Technology, Vietnam National University - HCM City, Vietnam
Victor Teboul Department of Physics, Angers University, Angers, France
Takashi Odagaki Division of Science, Tokyo Denki University, Tokyo, Japan ABSTRACT: Via analysis of spatiotemporal arrangements of atoms based on their dynamics in supercooled liquid and glassy states of a 2D monatomic system with a doublewell Lennard-Jones-Gauss (LJG) interaction potential, we find a new scenario of dynamical heterogeneity. Atoms with the same or very close mobility have a tendency to aggregate into clusters. The number of atoms with high mobility (and size of their clusters) increases with decreasing temperature passing over a maximum before decreasing down to zero. Position of the peak moves toward a lower temperature if mobility of atoms in clusters is lower together with an enhancement of height of the peak. In contrast, the number of atoms with very low mobility or solidlike atoms (and size of their clusters) has a tendency to increase with decreasing temperature and then it suddenly increases in the vicinity of the glass transition temperature leading to the formation of a glassy state. A sudden increase in the number of strongly correlated solidlike atoms in the vicinity of a glass transition temperature (Tg) may be an origin of a drastical increase in viscosity of the glass-forming systems approaching the glass transition. In fact, we find that the diffusion coefficient decays exponentially with a fraction of solidlike atoms exhibiting a sudden decrease in the vicinity of the glass transition region.
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INTRODUCTION Heterogeneous dynamics of atoms in supercooled liquids and increase of dynamic correlation length with decreasing temperature toward the glass transition have been studied intensively since it is a key problem of the liquid-glass transition.1−5 Dynamical heterogeneity (DH) has been observed in 2D simple atomic systems by computer simulation6−14 while heterogeneous dynamics in 3D atomic liquids has been first studied via MD simulation by Kob et al.15 and Donati et al.16−18 In particular, visualization of atomic configurations of 2D binary soft disk sytems obtained at two selected temperatures presented that the slowest 40% of the particles formed distinct domains which grew with decreasing temperature.8 Extended analysis of DH in the same system at the same two temperatures provided a clearer picture of the phenomenon.9 At the higher temperature, neither the fastest nor the slowest particles show any spatial correlation, while at lower temperature the development of clear spatial correlations is found, i.e., the fastest particles form islands of clusters connected by stringlike features, while the slowest ones form large compact domains.9 Moreover, a clear evidence of the correlation between DH and medium-range crystalline order in 2D glass-forming liquids has been found in studies.11,12 It is © XXXX American Chemical Society
found that slow regions having crystalline order emerge below the melting point, their characteristic size and lifetime steeply increase on cooling.11,12 These crystalline regions lead to DH in supercooled liquids and cause the slow dynamics which characterizes the glass transition.11,12 DH in 3D supercooled liquids shares similar trends found in 2D ones.15−18 In particular, it is found that mobile particles form clusters whose sizes grow with decreasing temperature.15 The extended studies in this direction can be found,19−23 and the results are roughly consistent with those found by Kob et al.15 However, no work related to DH in 2D system based on analysis of dynamics for the whole set of atoms in the system has been found in the literature. Moreover, it has been found recently that there are fundamental differences between glassy dynamics in 2D and 3D systems.24 It motivates us to carry out the study in this direction in order to highlight the situation in more details. Received: September 13, 2015 Revised: November 16, 2015
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DOI: 10.1021/acs.jpcb.5b08912 J. Phys. Chem. B XXXX, XXX, XXX−XXX
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The Journal of Physical Chemistry B
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CALCULATION There are two reasons that the 2D monatomic system containing 6400 atoms interacted via LJG interatomic potential25 is adopted for study in the present work: (i) LJG potential leads to the formation of a long-lived glass in both 2D and 3D monatomic systems;26,27 (ii) it is easier to analyze spatio-temperal arrangements of atoms approaching glass transition via using a simple monatomic 2D system since one can focus attention only on the topological order of atomic arrangement rather than on both topological and chemical ones if one employs a binary system. This simple system can mimic more conplicated systems found in practice,25−27 and we use the same parameters for LJG potential like those used in ref 26. The LJG potential form is given below:25 ⎡⎛ r ⎞12 ⎡ (r − r )2 ⎤ ⎛ r ⎞6 ⎤ G ⎥ V (r ) = ε0⎢⎜ 0 ⎟ − 2⎜ 0 ⎟ ⎥ − ε exp⎢ − ⎝r⎠ ⎦ 2σ 2 ⎦ ⎣ ⎣⎝ r ⎠
(1)
In general, the LJG potential is a sum of the Lennard-Jones potential and a Gaussian contribution. It forms a double-well potential for most values of the parameters with the second well located at r = rG, of the depth ε and width σ. Note that general form of pair potentials in metals consists of a strongly repulsive core plus a decaying oscillatory term.28 Therefore, the LJG potential can be considered as such an oscillatory one, cut off after the second minimum. Initially, this potential was proposed for the self-assembly of 2D monatomic complex crystals and quasicrystals.25 However, stable glassy states have been found at some parameter regions, in particular, for rG = 1.47r0, ε = 1.5ε0, and σ2 = 0.02r02 (see more details in ref 26). We take these values for LJG potential in the present work. It was found that the potential used in our work favors the formation of a pentagonal local order because of the existence of the second well and pentagons are essential for stability of a glassy state.26 We use “NVT” ensemble simulation and periodic boundary conditions (PBCs). We employ LJ reduced units as follows: energy in units of ε0, length in units of r0, temperature T in units of ε0/kB, and time in units of τ0 = r0(m/ε0)1/2 where kB is the Boltzmann constant, m is an atomic mass, r0 is atomic diameter, and ε0 is a depth of LJ part of LJG potential. The Verlet algorithm is employed and MD time step is dt = 0.001τ0. Initially, atomic configurations of 2D simple squared lattice structure of the size of S = 80.0r0 × 80.0r0 and at the fixed density ρ = (N/S) = 1.0 have been relaxed at a temperature as high as T = 2.5 for 2 × 106 MD steps in order to get an equilibrium liquid state. Then the system is cooled and the temperature is decreased linearly with time as T = T0 − γ × n via the simple atomic velocity rescaling until reaching T = 0.1. Here, γ = 10−6 per MD step is a cooling rate and n is the number of MD steps, so we employ the same cooling rate and the same density which lead to the formation of a glassy state like those used in ref 26. Note that lower cooling rate leads to the formation of 2D crystals.26 On the other hand, a melting point for the system (e.g., at fixed density ρ = 1.0 and under PBCs) is Tm = 0.43 (see ref 26). In order to improve the statistics, we average results over two independent runs. We use VMD software for 2D visualization of atomic configurations.29
Figure 1. Temperature dependence of potential energy per atom (a), heat capacity per atom (b), and static diffraction image of atomic configuration obatined at T = 0.1 (c). (d) Correlation between diffusion coefficient (D) and fraction of solidlike atoms (x = Nsolid/N) for the whole temperature range studied (empty circles are calculated data while the solid line is a an exponential decay law fitting).
dependence of potential energy per atom is rather continuous indicated a glass formation in the system (Figure 1a) and glass transition temperature is determined via the peak of the temperature dependence of the heat capacity, i.e. Tg = 0.31 (Figure 1b). Heat capacity is approximately calculated via simple relation: CV = ΔE/ΔT. Here, ΔE = E2 − E1 is the discrepancy of total energy for temperature range of ΔT = T2 − T1 = 0.01. On the other hand, static diffraction image of the final atomic configuration obtained at T = 0.1 exhibits clearly a glassy nature of atomic configuration confirming that indeed glass transition occurs in the system (Figure 1c). We also calculate diffusion constant of atoms in the system for the whole temperature range studied via the Einstein relation: D = lim
t →∞
r 2(t ) 4t
(2)
Here, ⟨r (t)⟩ is a mean squared displacement (MSD) of atoms and t is a diffusion time. In order to calculate diffusion constant, atomic configuration has been relaxed at a given temperature for 2 × 105 MD time steps. As shown in Figure 1d, diffusion constant strongly correlates with fraction of solidlike atoms in the system; that is, it decreases exponentially with the latter via fitting equation: 2
D = 0.1867exp( −x /0.0436) + 0.6898exp( −x /0.0078) (3)
Diffusion constant D is in LJ reduced unit and x = Nsolid/N, N is total number of atoms in the system. Atom is considered as solidlike if its atomic displacement (ad) is in the range ad = [0.0−0.2) ] after 5000 MD steps of relaxation at a given temperature. We have checked via using the critical Lindemann ratio and found that these atoms are indeed solidlike; that is, their Lindemann ratio is much less than the critical one.30 One can see in Figure 1d that the fitting is excellent, and a sudden decrease of diffusion constant in the vicinity of glass transition temperature is related to the sudden increase of fraction of solidlike atoms in the system. We will return to this problem later.
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RESULTS AND DISCUSSION Thermodynamic Properties and Evolution of Structure upon Cooling from the Melt. Temperature dependence of some thermodynamic quantities of the system upon cooling from the melt can be seen in Figure 1. Temperature B
DOI: 10.1021/acs.jpcb.5b08912 J. Phys. Chem. B XXXX, XXX, XXX−XXX
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The Journal of Physical Chemistry B Glass formation in the system is also confirmed via evolution of radial distribution function (RDF) upon cooling from the melt (Figure 2); that is, at high temperature RDF is rather
Figure 2. Evolution of radial distribution function upon cooling from the melt (for unrelaxed models).
smooth exhibiting a normal liquid state while additional peaks of a glassy state of 2D LJG system occur at low temperature. RDF at T = 0.1 is similar to that found in ref 26. For the model obtained at T = 0.1, the position of the first peak in RDF is located at 0.93 and second peak is at 1.50. The position of the first peak is slightly inner than the position of the first minimum, while the position of the second peak is at around the second minimum of the LJG potential. In order to get more detailed information on the structure of the model, we show ring distribution in Figure 3. We employ ISAACS software for
Figure 4. 2D visualization of atoms with the same (or close) atomic displacement (ad, in LJ reduced unit) after relaxation for 5000 MD steps at a given temperature, atoms are colored as follows: blue for ad = [0.0−0.2), red for ad = [0.2−0.4), gray for ad = [0.4−0.6), orange for ad = [0.6−0.8), yellow for ad = [0.8−1.0), tan for ad = [1.0−1.2), silver for ad = [1.2−1.4), green for ad = [1.4−1.6), white for ad = [1.6−1.8), pink for ad = [1.8−2.0).
region (not shown). Therefore, it is large enough for atom to overcome a plateau regime to diffuse if it is a liquid-like one. Moreover, it has been proposed that τC should not be larger than some atomic vibrations of picoseconds (see ref 27 and references therein) and adopted τC appropriates this criterion. Atoms with different atomic displacements (ad) are colored differently, and we find that atoms with the same or very close mobility are strongly correlated (Figure 4). At very high temperature, dynamics of atoms is rather homogeneous and heterogeneous dynamics occurs/enhances with lowering temperature (Figure 4a). We show atomic configurations obtained at temperature above and below Tg = 0.31 in Figure 4. Some important points can be drawn: (i) Atoms with the same or very close mobilities have a tendency to aggregate into clusters of different forms; (ii) Primarily, the population of atoms with high mobility has a tendency to decrease while population of atoms with low mobility has a tendency to increase with decreasing temperature; (iii) Evidently that atom is being slow if it is initially surrounded by other slow atoms, or being fast if it is initially in the midst of fast neighbors like that found in ref 9; (iv) Dynamics of atoms is heterogeneous not only in the supercooled liquid state (for temperature range between Tm = 0.43 and Tg = 0.31, see Figure 4b) but it is also heterogeneous in the glassy state (for T ≤ Tg, see Figure 4c,d); (v) Atoms with high mobility have a tendency to aggregate into stringlike form clusters (Figure 4a,b), while atoms with very slow mobility (the “blue” ones) have a tendency to aggregate into more compact clusters (Figure 4b,c), the latter grow into the largest one which spans almost throughout the model at a temperature much below Tg (Figure 4d). The blue atoms (i.e., atoms with ad = [0.0−0.2)) are considered as solidlike ones like that discussed above.
Figure 3. Ring distribution in the model obtained at T = 0.1 after relaxation for 5000 MD steps at a given temperature.
calculating ring statistics following the “shortest path” rule with a cut off radius RC = 1.2r0 (see ref 31). This cutoff radius is equal to the position of the first minimum after the first peak in RDF of a glassy state obtained at T = 0.1. On the other hand, we find that pentagons (i.e., 5-fold rings) dominate in the model obtained at T = 0.1 and triangles, squares plus pentagons are the basic structural units in the glassy state (i.e., 3-fold, 4fold and 5-fold rings, see Figure 3 and Figure 4). It is in good accordance with that found previously in ref 26. Moreover, we find the existence of larger rings such as 6-fold, 7-fold, and 8fold rings (Figure 3) which have not been reported yet. Dynamical Heterogeneity. In order to study DH, models obtained by cooling from the melt have been relaxed for 5000 MD steps or 5τ0 at a given temperature before further spatiotemporal analysis of configurations based on dynamics of atoms. The value τC = 5τ0 is located at the end of a plateau regime of MSD at temperature in the deeply supercooled liquid C
DOI: 10.1021/acs.jpcb.5b08912 J. Phys. Chem. B XXXX, XXX, XXX−XXX
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temperature dependence of fraction of atoms with different mobilties moves toward a lower temperature accompanied by increasing of height of the peak if mobility of atoms is lower (Figure 5). It ensures two points: (i) There is a competition between two processes, i.e., an increase of fraction of atoms with the same or close mobility with starting of further cooling and a decrease of their number due to lowering temperature leading to the formation of a peak in temperature dependence of fraction of atoms with different ad; (ii) The higher mobility the lower lifetime of clusters of atoms is, and finally, atoms transform into population of the next lower mobility. However, quite different situation is found for atoms with very slow dynamics or solidlike ones (e.g., atoms with ad = [0.0−0.2]), i.e., their fraction is small in the high temperature region exhibiting slight inrease with lowering temperature. However, it strongly increases in the vicinity of Tg up to about 100% at temperature far below Tg leading to the formation of a glassy state. Temperature dependence of fraction of solidlike atoms in the system can be approximately described as an exponential decay law: Nsolid/N = 1.6588 exp(−T/0.1757) for temperature (T) ranged from 0.1 to 2.5. Such a full temperature dependence of fraction of atoms with different mobilities upon cooling from the melt has not been previuously reported yet. Clusering of atoms with the same or close mobility can be seen in Figures 6 and 7. We consider that if the distance between two atoms is not larger than the cutoff radius RC = 1.2r0 they are belonging to the same cluster. We find that atoms with the same or close mobility have a tendency to form clusters. Temperature dependence of size of the largest cluster (Smax, Figure 6) and mean cluster size (S̅, Figure 6) exhibits the same behavior like that found for the fraction of atoms with different mobilities (see Figure 5); that is, it also passes over a maximum with an exception for clusters of atoms with the slowest dynamics (solidlike ones). In the high temperature region, the higher mobility the larger Smax and S̅ are, and these quantities have a tendency to increase with decreasing temperature (see the inset of Figures 6 and 7). In contrast, in the low temperature region the situation is more complicated. In addition, we find that Smax and S̅ of solidlike atoms is small in the high temperature region then these quantities increase strongly in the vicinity of a glass transition (Figures 6 and 7). Below Tg, almost all atoms in the system become solidlike to form a percolation cluster which spans throughout the system to form a glassy state since the largest cluster contains almost all atoms in the system (see Figures 4d and 6). We pause here for a more detailed discussion about our results. Aggregation of the fastest or slowest atoms in supercooled liquids has been found in the past by both computer simulations and experiments.7−9,12,14,32 In particular, via video microscopy experiments and computer simulations it is found clearly that the fastest and slowest colloidal ellipsoids aggregate into clusters.32 On approaching the glass transition, the dynamics of atoms becomes not only drastically slower but also more spatially heterogeneous.2,5,32 The nature of such DH is still unclear and two types of structural signatures responsible for this dynamics arrest have been proposed, i.e., amorphous order33−36 and crystalline one.11,12,37 Recently, it has been found that kinetic slowing down is caused by a decrease in the structural entropy and an increase in the size of glassy clusters.32 As found and discussed above (see Figure 1d and related discussion), drastic slowing down of the diffusion constant in the vicinity of Tg associates with a sudden growth of
However, more details about DH in the system can be seen via analysis of temperature dependence of fraction of atoms with diffferent mobilities and their clustering (Figures 5, 6, and
Figure 5. Temperature dependence of fraction of atoms with various atomic displacements after relaxation at a given temperature for 5000 MD steps.
Figure 6. Temperature dependence of size of the largest cluster (Smax/ N) of atoms with 03 selected atomic displacement ranges after relaxation at a given temperature for 5000 MD steps. Inset shows the high temperature region.
Figure 7. Temperature dependence of the mean cluster size (S̅) of atoms with 03 selected atomic displacements after relaxation at a given temperature for 5000 MD steps. Inset shows the high temperature region.
7). Figure 5 shows that the fraction of atoms with different mobilities has a similar temperature dependence with an exception for those with the slowest dynamics; that is, it has a tendency to increase first and then it decreases downto zero passing over a maximum. In the high temperature region, it is logically that the higher atomic mobility the higher atomic fraction is. However, it is more complicated in the low temperature region although atoms with low mobility dominate in the region (Figure 5). The position of a maximum of D
DOI: 10.1021/acs.jpcb.5b08912 J. Phys. Chem. B XXXX, XXX, XXX−XXX
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sudden decrease in the vicinity of Tg. It is quite new since in the past we thought that DH in 2D supercooled liquids is caused only by correlation of the most mobile and/or the most immobile atoms. On the other hand, we also thought that correlation length of dynamical heterogeneity only increases with decreasing temperature (not passing over a maximum). It is true only for the correlation of atoms with the slowest mobility or solidike ones. We also find that glass formation in 2D supercooled monatomic liquids is homogeneous and it proceeds via several intermediate phases which have different structural orders including 5-fold symmetry, the latter enhances with decreasing temperature and then dominates in the final produced one. Note that a double-well LJG interaction potential is more relevant for metallic glasses and some other specific systems such as water.40 Therefore, the results of the present work should be found for these glass-forming systems. Further extended applicability of the results for other glassforming systems should be checked and it is going on.
fraction of solidlike atoms in the system. In order to highlight the situation, we show 2D visualization of solidlike atoms occurred in the system obtained at temperatures above and below Tg (Figure 8). We find that solidlike clusters (i.e., glassy
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Figure 8. 2D visualization of configuration of solidlike atoms in model obtained at T = 0.3, i.e., at around Tg = 0.31.
AUTHOR INFORMATION
Corresponding Author
*E-mail:
[email protected]. Notes
clusters) exhibit a glassy structure and have different shapes including stringlike, ringlike, and more compact ones (Figure 8). Note that 5-fold symmetry dominates in a stable glassy state of 2D system with LJG interatomic potential (see Figures 3 and 4d and ref 26). However, solidlike clusters found in atomic configurations obtained at temperature even around Tg exhibit different structural orders (without well-ordered or crystalline one): while 5-fold symmetry dominates in compact clusters, other structural orders of stringlike and ringlike clusters can be found (Figure 8). Compact clusters grow fast with decreasing temperature including transformation/growth of stringlike and ringlike clusters into the compact ones (Figure 4). Finally, glassy state with the domination of 5-fold symmetry is formed at a temperature much lower than Tg, and no periodic order is observed in the glassy state like that found in ref 26 (Figure 4d). This means that glass formation in our 2D supercooled liquid proceeds via several intermediate phases. It is found that when a system undergoes a transition from a liquid to a solid phase, it passes through multiple intermediate structures before reaching the final state.38
The authors declare no competing financial interest.
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ACKNOWLEDGMENTS This research is funded by Vietnam National Foundation for Science and Technology Development (NAFOSTED) under Grant 103.01-2014.86.
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CONCLUSION In summary, via intensive MD simualtion of DH in supercooled liquid and glassy states of the 2D monatomic system with a double-well LJG interaction potential, we find that atoms with the same or close mobility are strongly correlated and aggregate into clusters/domains like cooperatively rearranging regions (CRRs) proposed in the past.2,39 Fraction of atoms with high mobility increases with decreasing temperature passing over a maximum and then it decreases downto zero. In contrast, fraction of atoms with the slowest mobility (solidlike ones) has a tendency to increase passing over a sudden increase in the vicinity of Tg reaching almost 100% at low temperature to form a glassy state. The size of the largest cluster/domain of atoms with different mobilities also has the same tendency. Sudden increase of fraction of strongly correlated solidlike atoms may be an origin of a drastical increase of viscosity of glass-forming supercooled liquids in the vicinity of Tg found in practice. Indeed, we find that diffusion constant of atoms in the system decays exponentially with fraction of solidlike atoms including a E
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