Heterogeneous dynamics in polycrystal and glass in a binary mixture ...

PHYSICAL REVIEW E 75, 041503 共2007兲

Heterogeneous dynamics in polycrystal and glass in a binary mixture with changing size dispersity and composition Toshiyuki Hamanaka and Akira Onuki Department of Physics, Kyoto University, Kyoto 606-8502, Japan 共Received 26 December 2006; published 25 April 2007兲 Using molecular dynamics simulation we investigate the dynamics in a two-dimensional binary mixture at a low temperature and at high densities. We increase the size ratio of the diameters of the two components or the fraction of the larger particles. Then changeovers occur from polycrystal to glass with proliferation of defects. The relationship between the degree of disorder and the slow dynamics is studied by simultaneous visualization of a disorder variable D j introduced in our previous paper T. Hamanaka and A. Onuki关Phys. Rev. E 74, 011506 共2006兲兴 and the particle displacement ⌬r j in a long time interval. In polycrystal, the particles in the grain boundary regions have large D j and a relatively large mobility, producing significant dynamic heterogeneity on long time scales. In the crossover from polycrystal to glass, the crystalline regions become narrow, but the particles with relatively large D j still trigger the formation of chainlike particle motions, leading to the dynamic heterogeneity. DOI: 10.1103/PhysRevE.75.041503

PACS number共s兲: 61.43.⫺j, 61.72.⫺y, 64.70.Pf

I. INTRODUCTION

A large number of molecular dynamics simulations have been performed to study the liquid-glass transition in twodimensional 共2D兲 and three-dimensional 共3D兲 binary mixtures 关1–10兴, where the time scale of the particle motions becomes dramatically slowed down with lowering of the temperature T. The size ratio ␴2 / ␴1 of the diameters ␴1 and ␴2 of the two components has been chosen to maximize disruption of the crystal structure. As a salient feature, significant heterogeneity in the cooperative particle motions on long time scales has been reported, where successive stringlike motions form large-scale clusters growing at low temperatures. In particular, the heterogeneity correlation length ␰ was determined from the distribution of broken bonds in 2D and 3D 关3兴 and by a four-point dynamic susceptibility in 3D 关8兴. Recently attention has been paid to the connection between the structure and the slow dynamics. Vollmayr-Lee et al. 关11兴 found in 3D that mobile particles 共in their definition兲 are surrounded by fewer neighbors than the others. WidmerCooper and Harrowell 关12兴 detected a clear correlation between the short-time heterogeneity in a local Debye-Waller factor and the long-time dynamic heterogeneity in 2D. In their one-component model with fivefold pair interaction in 2D, Shintani and Tanaka 关13兴 introduced medium-range crystalline order strongly correlated to the slow dynamics in glass. Also varying the size ratio ␴2 / ␴1 as well as T 关14–16兴, we have recently investigated changeovers among crystal, polycrystal, and glass at fixed composition in 2D 关17兴. For weak size dispersity and at low T, the mixture is in a crystal state with a small number of defects. With increasing ␴2 / ␴1, polycrystal and glass states are subsequently realized. These three states can be distinguished using a disorder variable D j supported by each particle j, since snapshots of D j are highly heterogeneous in polycrystal and glass. The grain boundaries are pinned in the presence of size dispersity, while they emerge as rapidly varying thermal fluctuations near the 2D melting in one-component systems. 1539-3755/2007/75共4兲/041503共8兲

In this paper, we aim to investigate the dynamics in polycrystal and glass in a model binary mixture with size dispersity using the disorder variable D j, the particle displacement ⌬r j, and the bond breakage 关3兴. We may then study the relationship between the local disorder at a particular time and the dynamic heterogeneity obtained from the particle configurations at two times much separated. The origin of the dynamic heterogeneity will turn out to be rather obvious for polycrystal. We thus study the crossover from polycrystal to glass, where local crystalline order is appreciable. The organization of this paper is as follows. In Sec. II A, we will explain our method and introduce the disorder variable D j. We will then visualize D j and ⌬r j in Sec. II B and calculate the time correlation function and the mean square displacement in Sec. II C. In Sec. II D, we will examine the distribution of the displacements and that of the broken bonds as a function of a parameter S representing the degree of disorder. II. NUMERICAL RESULTS A. Method and disorder variable

As in our previous paper 关17兴, we performed molecular dynamics simulation of a 2D binary mixture interacting via a truncated Lenard-Jones potential of the form v␣␤共r兲 = 4⑀关共␴␣␤ / r兲12 − 共␴␣␤ / r兲6兴 − C␣␤ 共␣ , ␤ = 1 , 2兲, characterized by the energy ⑀ and the soft-core diameter ␴␣␤ = 共␴␣ + ␴␤兲 / 2. For r ⬎ rcut = 3.2␴1, we set v␣␤共r兲 = 0 and the constant C␣␤ ensures the continuity of v␣␤共r兲 at the cutoff r = rcut. The total particle number is fixed at N = N1 + N2 = 1000. The volume V is chosen such that the volume fraction of the soft-core regions is fixed at 0.9 or

␾ = 共N1␴21 + N2␴22兲/V = 0.9.

共2.1兲

With the mass ratio being m1 / m2 = 共␴1 / ␴2兲2, we integrated the Newton equations using the leapfrog algorithm under the periodic boundary condition. The temperature was controlled

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©2007 The American Physical Society

PHYSICAL REVIEW E 75, 041503 共2007兲

TOSHIYUKI HAMANAKA AND AKIRA ONUKI

with the Nose-Hoover thermostat 关18兴. The time step of integration is 0.002␶ with

兺

exp关6i␪ jk兴

k苸bonded

=兩␹ j兩e6i␣ j ,

共2.3兲

where ␪ jk is the angle of the relative vector rk − r j with respect to the x axis. The angle ␣ j defined in the second line represents the sixfold orientation order. The correlation function g6共r兲 of the space-dependent quantity 兺 j␹ j␦共r j − r兲 has been calculated for one-component systems 关20兴. See item 共viii兲 in the last section for results on g6共r兲 in our system. In our previous work 关17兴, we constructed another nonnegative-definite variable representing the degree of disorder for each particle j by Dj = 2

兺

关1 − cos 6共␣ j − ␣k兲兴.

共2.4兲

k苸bonded

For a perfect crystal at low T this quantity arises from the thermal vibrations and is nearly zero, but for particles around defects it assumes large values in the range 5–20. The aver¯ = 兺 D / N over the particles conveniently represents the age D j j degree of overall disorder, which is small in crystal and large in glass and liquid. To distinguish between glass and liquid, we furthermore need to examine the dynamics. We performed two series of simulations. In one series we increase the size ratio ␴2 / ␴1 at N1 = N2 = 500. In the other series we increase the fraction of large particles, c = N2/N,

c

0.2

共2.2兲

Hereafter we will measure time in units of ␶. We first equilibrated the system in a liquid state at T = 2⑀ / kB in a time interval of 103 and then quenched it to the final temperature T = 0.2⑀ / kB. The total simulation time was 2.2⫻ 104 for each run. After a relaxation time of order 5000, there was no appreciable time evolution in various quantities obtained as an average over the particles 共see Fig. 7 of Ref. 关17兴兲. There was also no tendency of phase separation in all the simulations in our previous and present papers 共see Fig. 1 of Ref. 关17兴, as an example兲. However, our simulation time became comparable to the structural relaxation time ␶␣ in glass at the largest ␴2 / ␴1 = 1.4. Much longer simulation times are thus needed for a more quantitative analysis of such slow structural relaxations. We consider a deviation from the hexagonal order for each particle j. In this work, the two particles j 苸 ␣ and k 苸 ␤ are bonded if their distance 兩r j − rk兩 is shorter than 1.5␴␣␤ 关3兴. For each j, we consider the sum 关19兴

␹j =

0.1

共2.5兲

¯ vs ␴ / ␴ at c = 0.5 at ␴2 / ␴1 = 1.4. In Fig. 1, the curve of D 2 1 ¯ vs c at ␴ / ␴ = 1.4 are presented 关21兴. They are and that of D 2 1 taken in steady states at T = 0.2⑀ / kB and ␾ = 0.9. The former curve was already presented in Fig. 8 in our previous paper 关17兴. As can be seen in Fig. 2, crystal, polycrystal, and glass are realized with increasing ␴2 / ␴1 or c. On the curve for the first

0.3

0.4

0.5

0.3 6

1.35

c

0.2 1.3

D

␶ = ␴1冑m1/⑀ .

8

0

4 0.125 1.25

σ2/σ1

2 0.05 0.02 0

1

1.1 1.1

1.2 1.2

σ2/σ1

1.3

1.4

¯ = 兺 D / N at T = 0.2⑀ / k FIG. 1. Average disorder parameter D j j B and ␾ = 0.9 with changing the size ratio ␴2 / ␴1 at N1 = N2 = 500 共lower curve兲 关17兴 and with changing the fraction c = N2 / N of the larger particles at ␴2 / ␴1 = 1.4 共upper curve兲.

series, crystal states with only a few defects are realized for ␴2 / ␴1 ⱗ 1.15, but defects are abruptly proliferated for ␴2 / ␴1 ⲏ 1.2. On that for the second series, defects are always generated around each large particle 共see the right panel of ¯ ⬃ 30c for c Fig. 3 below兲 leading to a strong increase D ⱗ 0.2. In our calculations 共except those in Figs. 4–6兲, the particle positions used are the time averages taken over an interval with width 1. That is, the time average 兰t+1 t dt⬘r j共t⬘兲 was used in place of r j共t兲. This short-time smoothing of the particle positions was also used in our previous paper 关17兴. B. Changeover with increasing size dispersity and composition

In Fig. 2, we show snapshots of D j at t = 1.2⫻ 104 for 共a兲 ␴2 / ␴1 = 1.2, 共b兲 1.25, 共c兲 1.3, and 共d兲 1.35 in the first series and for 共a⬘兲 c = 0.02, 共b⬘兲 0.05, 共c⬘兲0.125, and 共d⬘兲 0.2 at ␴2 / ␴1 = 1.4 in the second series. The color of the particles varies in the order of rainbow, being violet for D j = 0 and red for the maximum of D j. The arrows are the displacement vectors from the particle position at this time to that at t = 2.2⫻ 104, where the width of the time interval is 104. Here the particles with displacement longer than ␴1 are called the mobile particles. In this visualization we clearly recognize polycrystal states intermediate between crystal and glass states. In the first series at c = 1 / 2, defects are accumulated around the grain boundaries and the crystalline regions become narrow and ill defined rather abruptly around ␴2 / ␴1 ⬃ 1.25. In the second series at ␴2 / ␴1 = 1.4, the large particles form grain boundaries and the crystalline regions composed of the small particles are well ordered for small c ⱗ 0.125. The crystalline regions are composed of the two components in the first series and of the smaller component only in the second series. We shall see a number of different dynamical effects in these two polycrystal states. In Fig. 3, the particle

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HETEROGENEOUS DYNAMICS IN POLYCRYSTAL AND…

(a) σ2/σ1 =1.2, c = 0.5

(a') c = 0.02, σ2/σ1 =1.4

(b) σ2/σ1 =1.25, c = 0.5

(b') c = 0.05, σ2/σ1 =1.4

(c) σ2/σ1 =1.3, c = 0.5

(c') c = 0.125, σ2/σ1 =1.4

(d) σ2/σ1 =1.35, c = 0.5 (d') c = 0.2, σ2/σ1 =1.4

find the grain boundaries for 共a兲 with Fig. 3 only, while the large particles are in the grain boundary regions for 共b⬘兲. We further discuss the dynamics revealed in Fig. 2. 共i兲 At any ␴2 / ␴1, mobile particles with large displacements form strings. With increasing ␴2 / ␴1 in 共a兲–共d兲 and c in 共a⬘兲–共d⬘兲, such strings tend to aggregate, resulting in significant dynamic heterogeneities. This feature has been observed numerically in supercooled liquids and glass 关3,4,7兴, but it can be seen here also in polycrystal with size dispersity. 共ii兲 The mobile particles have relatively large D j in many cases and, at least, particles with large D j trigger the formation of the strings. Small particles bonded to a large particle frequently become mobile even when the large one remains immobile, which is evident in the second series 共see the lower panel of Fig. 8 below also兲. 共iii兲 In polycrystal states such as 共a兲, 共b兲, and 共a⬘兲–共d⬘兲 the mobile particles are mostly in the grain boundary regions with large D j. The dynamic heterogeneity in polycrystal such as in 共b兲 and 共c⬘兲 is thus ascribed to a relatively high mobility of the particles in the grain boundary regions. 共iv兲 In the glass states 共c兲, 共d兲, and 共d⬘兲, the degree of disorder is enhanced and the grain boundaries become ill defined, but we can still see the aggregation of the strings extending longer than the heterogeneities of D j. For ␴2 / ␴1 = 1.4 and c = 1 / 2 共see Fig. 4 of Ref. 关17兴兲, the mobile particles are considerably fewer than in 共d兲, so a wider time interval is needed for visualization of the displacements. 共v兲 In addition, in polycrystal and glass, crystalline regions consisting of small D j can undergo collective motions induced by large displacements of the surrounding particles. Such motions will be clearly seen in expanded snapshots of Fig. 8. We recognize that they remain small in quiescent states, but we will show that they can be large in shear flow 关see item 共vii兲 in the last section兴. We may consider the particle-number density as a function of the disorder variable D j. Because the range 0 ⬍ D j ⱗ 20 realized is rather wide, we use S j = 冑D j

共2.6兲

and define the number density N

n共S兲 = 兺 具␪共S − S j兲典/N,

共2.7兲

j=1

min

max

FIG. 2. 共Color online兲 Disorder variable D j in Eq. 共2.4兲 at T = 0.2⑀ / kB and ␾ = 0.9. Left: 共a兲 ␴2 / ␴1 = 1.2, 共b兲 1.25, 共c兲 1.3, and 共d兲 1.35 at c = 0.5. Right: 共a⬘兲 c = 0.02, 共b⬘兲 0.05, 共c⬘兲0.125, and 共d兲⬘ 0.2 at ␴2 / ␴1 = 1.4. The arrows represent the particle displacement ⌬r j in the subsequent time interval of width 104. The color is given to each picture independently, according to its minimum and maximum of D j. The average of D j in the crystalline regions is larger in 共a兲 than in 共a⬘兲and 共b⬘兲 共see Figs. 3 and 4兲, leading to the difference in color.

configurations around grain boundaries are shown for the polycrystal states 共a兲 and 共b⬘兲 in Fig. 2, where the circles have diameters ␴1 共in black兲 and ␴2 共in gray兲. We can hardly

where ␪共x兲 is the step function, being 1 for x ⬎ 0 and 0 for x ⱕ 0. The average 具¯典 in Eq. 共2.7兲 is over the time and over five independent runs. In Fig. 4, we show the derivative dn共S兲 / dS, which is the distribution of the density as a function of S. The area below each curve is unity or 兰⬁0 dSdn共S兲 / dS = 1. The peaks of the curves 共a兲 and 共a⬘兲 at small S arise from the particles forming crystalline regions. Notice that these peaks are separated by a small amount because the crystalline regions in 共a⬘兲are more frustrated 共see Fig. 3兲. The distributions for the glass states 共d兲 and 共d⬘兲 are broad in the range S ⱗ 4, but a mild peak representing crystalline order still remains at S ⬃ 0.5 for 共d⬘兲.

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(a) σ2/σ1 =1.2, c = 0.5

0.1

-0.5 -0.5

0.2

(b') c = 0.05,

σ2/σ1 =1.4 FIG. 3. Particles around grain boundaries, which are 36% of 共a兲 and 共b⬘兲 in Fig. 2. Left: the left and bottom parts of 共a兲, where strings of the small particles are conspicuous. Right: the right and bottom parts of 共b⬘兲, where the large particles form grain boundaries. The numbers at the corners are the coordinates. The total system is in the region −0.5ⱕ x ⱕ 0.5 and −0.5ⱕ y ⱕ 0.5 in the horizontal x and vertical y axes.

-0.4 0.1 -0.1

0.5

C. Time correlation function and mean-square displacement

We also examine the quantities constructed from the particle displacement, ⌬r j共t兲 = r j共t + t0兲 − r j共t0兲,

共2.8兲

In the first series, the long-time behaviors of these two quantities change abruptly from ␴2 / ␴1 = 1.2 to 1.25, where D j changes abruptly. In the second series, these two quantities change gradually with increasing c. We may determine the structural relaxation time ␶␣ by Fs共q , ␶␣兲 = e−1 at this

of the small particles for various ␴2 / ␴1 and c. Here we do not take the time average of the particle positions on a short time interval of width 1 to see the short-time ballistic motion 共see the last sentence of Sec. II A兲. In Fig. 5, we display the self-time-correlation function Fs共q,t兲 = 兺 具exp关iq · ⌬r j共t兲兴典/N1 ,

共2.9兲

j=1

at q = 2␲ / ␴1. In Fig. 6, we show the mean square displacement

1.1

0.8

Fs (q,t )

N1

1

1.2

0.6

0.4

σ2/σ1 = 1.1

0.2

N1

具关⌬r共t兲兴2典 = 兺 具关⌬r j共t兲兴2典/N1 .

共2.10兲 0 0.1

j=1

The average 具¯典 in Eqs. 共2.9兲 and 共2.10兲 关and in Eqs. 共2.11兲 and 共2.13兲 below兴 is taken over the initial time t0 and over five independent runs.

1.4

1.2 1.25 1.3 1.35 1.4

1.35

1

10

1

t

100

1000

0.05

1.2

Fs (q,t )

dn(S)/ dS

a' a ) σ2/σ1 = 1.2, c=0.5 a') σ2/σ1 = 1.4, c=0.05 d ) σ2/σ1 = 1.35,c=0.5 d') σ2/σ1 = 1.4, c=0.2

a

0.6

0.125

0.4

c = 0.02 0.05 0.125 0.2 0.3 0.5

0.2

0.6

d

0

d'

1

2

3

4

S FIG. 4. Number-density distribution dn共S兲 / dS as a function of the disorder variable S in Eq. 共2.6兲 for the polycrystal states 共a兲 and 共a 兲 and for the glass states 共d兲 and 共d⬘兲.

⬘

10000

0.02

0.8

1.8

0 0

1.3

1.25

0.1

1

10

0.2 0.3 0.5

t

100

1000

10000

FIG. 5. Time correlation function Fs共q , t兲 of the small particles at q = 2␲ / ␴1 in Eq. 共2.9兲 for various ␴2 / ␴1 共upper plate兲 and for various c 共lower plate兲 on a semilogarithmic scale. The decay is fastest for ␴2 / ␴1 = 1.25 in the upper plate, while it becomes faster with increasing c in the lower plate.

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HETEROGENEOUS DYNAMICS IN POLYCRYSTAL AND…

t

1.2 1.25 1.3 1.35 1.4

1.3

500

1.25 1.35

1.4

0.1

1.2

σ2/σ1 = 1.1

1.25 1.3 1.4

300 200

1.3

1.4

100

0.01 0.1

σ2/σ1 = 1.2

1.25

400

dM(S)/ dS

1

σ2/σ1 = 1.1

2

< ∆r ( t ) >

600

1.2

1

10

100

t

1000

0 0

10000

1

2

S

300

c = 0.02 0.05 0.125 0.2 0.3 0.5

0.3 0.2

0.5

0.125 0.1

dM(S)/ dS

t

250

2

< ∆r ( t ) >

1

t

0.2

50

0.01 10

150

100

c = 0.05 0.125 0.2 0.5

0.125

0.02

1

0.5

200

4

100

0.05

0.1

3

1000

0

10000

FIG. 6. Mean-square displacement 具关⌬r共t兲兴2典 of the small particles in Eq. 共2.10兲 for various ␴2 / ␴1 共upper plate兲 and for various c 共lower plate兲 on a logarithmic scale. It is nonmonotonic with increasing ␴2 / ␴1 in the upper plate.

wave number 关3兴. In Fig. 6, we can see that the particle motions in short times 共ⱗ1兲 are ballistic or 具关⌬r共t兲兴2典 ⬀ t2. The diffusive behavior 具关⌬r共t兲兴2典 ⬵ 4Dt appears at large t for ␴2 / ␴1 ⱖ 1.3 in the upper panel and for c ⱖ 0.125 in the lower panel in our simulation time, where D is the diffusion constant of a small tagged particle. For ␴2 / ␴1 = 1.25, 1.3, 1.35, and 1.4 in the first series 共upper panels兲, we have ␶␣ = 1.9, 1.5, 4.3, and 6.1 in units of 103␶ and D = 2.3, 2.7, 1.6, and 1.1 in units of 10−5␴21 / ␶, respectively. Remarkably, the relaxation of Fs共q , t兲 is shortest and 具关⌬r共t兲兴2典 is largest for a polycrystal state with ␴2 / ␴1 ⬵ 1.25. For c = 0.125, 0.2, and 0.4 in the second series 共lower panels兲, we have ␶␣ = 2.2, 1.1, and 0.64 in units of 104␶ and D = 0.66, 0.97, and 1.3 in units of 10−5␴21 / ␶, respectively, where ␶␣ decreases and D increases monotonically with increasing c. See explanations of Figs. 7–9 below to interpret these results. D. Distributions of displacements and broken bonds parametrized by disorder

The particles with relatively large D j should have a higher probability to undergo large displacements 共ⲏ␴1兲 on the

0

0.05

1

2

S

3

4

FIG. 7. Distribution of the mean-square displacement dM共S兲 / dS as a function of the disorder variable S in Eq. 共2.6兲 for various ␴2 / ␴1 共upper plate兲 and for various c 共lower plate兲. M共s兲 is defined in Eq. 共2.10兲. The curve of ␴2 / ␴1 = 1.25 in the upper plate exhibits a large peak at small S exceptionally 共see the text兲.

time scale of ␶␣. Using S j = S j共t0兲 in Eq. 共2.6兲 at the initial time t0, we pick up the particles with S j ⬍ S in the meansquare displacement to define N

M共S兲 = 兺 具␪共S − S j兲关⌬r j共t兲兴2典/N,

共2.11兲

j=1

where the small and large particles both contribute to the sum. In Fig. 7, we plot the derivative M ⬘ = dM共S兲 / dS for various ␴2 / ␴1 and c, which is the distribution of the displacements as a function of S. The area below each curve is equal to the mean-square displacement 具关⌬r共t兲兴2典 = 兺Nj=1具关⌬r j共t兲兴2典 / N. Characteristic features are as follows. 共i兲 In the crystalline states, M ⬘ mainly arises from the particle motions around defects and is very small. 共ii兲 In the first series, the area is largest for ␴2 / ␴1 = 1.25 due to enhancement of M ⬘ around S ⬃ 1. This indicates a high mobility of the particles with S ⬃ 1, giving rise to the behavior of Fs共t兲 and 具关⌬r共t兲兴2典 in the upper panels of Figs. 5 and 6. 共iii兲 In the first series, M ⬘decreases with increasing ␴2 / ␴1 for not large S 共ⱗ2兲, while it becomes insensitive to ␴2 / ␴1 for large S.

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(a) σ2/σ1 =1.2, c = 0.5

300

σ2/σ1 =1.2

0.1

1.25 1.3 1.4

dB(S)/dS

1.25

-0.5 -0.5 0.2

200 1.4 1.3

100

0

(b') c = 0.05, σ2/σ1 =1.2

0.1

1.2

0

1

2

3

180

c = 0.05 0.125 0.2 0.5

0.2

dB(S)/dS

120

-0.4 -0.1

0.5

0.125

60

0.05

0 0

0.5

FIG. 8. 共Color online兲 D j, displacements, and broken bonds in 共a兲 and 共b⬘兲 of Fig. 2 共the same parts as in Fig. 3兲. Arrows 共in black兲 denote the displacement for ⌬t = 104, while line segments 共in red兲 are broken bonds. Here most particles in the crystalline region are undergoing small collective motion with their bonds unbroken.

共iii兲 In the second series, M ⬘ increases with increasing c for S ⲏ 2.5, giving rise to the behavior of Fs共t兲 and 具关⌬r共t兲兴2典 in the lower panels of Figs. 5 and 6. Another method to visualize the configuration changes at high densities is to introduce “bond breakage” 关3兴. For each particle configuration given at time t0, a pair of particles i and j is considered to be bonded if rij共t0兲 = 兩ri共t0兲 − r j共t0兲兩 ⱕ A1␴␣␤ ,

4

S

共2.12兲

where i and j belong to the species ␣ and ␤, respectively, ␴␣␤ = 共␴␣ + ␴␤兲 / 2, and we set A1 = 1.5 here. After a time ⌬t, the bond is treated to be broken if the distance rij共t0 + ⌬t兲 exceeds the threshold A2␴␣␤ with A2 = 1.75. In Fig. 8, we display the parts of 共a兲 and 共b⬘兲 in Fig. 2 共the same parts as in Fig. 3兲, where the broken bonds written as segments are located around a grain boundary. We recognize that the bonds of the particles undergoing large displacements are mostly broken. However, the bonds between the particles undergoing small collective motion are not broken. In the same manner as in Eq. 共2.11兲 for M共S兲, we define B共S兲 such

1

2

3

4

S FIG. 9. Distribution of the broken bonds dB共S兲 / dS as a function of the disorder variable S in Eq. 共2.6兲 for various ␴2 / ␴1 共upper plate兲 and for various c 共lower plate兲. B共S兲 is defined in Eq. 共2.13兲.

that it is the broken bond number in the range Sij共t0兲 ⬅ 关Si共t0兲 + S j共t0兲兴 / 2 ⬍ S. That is, B共S兲 = 兺 具␪„S − Sij共t0兲…␪„A1␴␣␤ − rij共t0兲… ⫻ ␪„rij共t0 + ⌬t兲 ij

− A2␴␣␤…典.

共2.13兲

In Fig. 9, we plot the distribution of the broken bonds B⬘ = dB共S兲 / dS for various ␴2 / ␴1 and c with ⌬t = 104. In this quantity, the contribution from the collective motion without bond breakage in the crystalline regions should have been removed, as suggested by Fig. 8. We make comments. 共i兲 It is of interest to compare the curves of ␴2 / ␴1 = 1.25 in Figs. 7 and 9. The peak of M ⬘ at S ⬃ 1 in Fig. 7 is much reduced in B⬘ in Fig. 9, but B⬘ there is still not small, suggesting considerable bond breakage at S ⬃ 1 in this polycrystal state. 共ii兲 In the glassy configurations, the contributions around S ⬃ 3 are most dominant. 共iii兲 In the second series, M ⬘ in Fig. 7 and B⬘in Fig. 9 exhibit very similar behavior. III. SUMMARY AND REMARKS

Using molecular dynamics simulation we have simultaneously visualized the local disorder and the slow dynamics

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HETEROGENEOUS DYNAMICS IN POLYCRYSTAL AND…

in polycrystal and glass in a 2D Lenard-Jones binary mixture. We have changed ␴2 / ␴1 at c = 1 / 2 in the first series and c at ␴2 / ␴1 = 1.4 in the second series, with T = 0.2⑀ / kB and ␾ = 0.9 held fixed. We summarize our main results and give comments. 共i兲 Figure 2 demonstrates strong correlations between the disorder variable D j and the long-time particle displacement. See Refs. 关11–13兴 for similar attempts. The structural heterogeneity represented by D j is a characteristic feature at high densities and can be seen even in liquid at higher temperatures and even in liquid of one-component systems 共see Fig. 10 of Ref. 关17兴兲, while the dynamic heterogeneity disappears in liquid. 共ii兲 For polycrystal-like configurations, the dynamic heterogeneity unambiguously stems from the relatively large mobility of the particles in the grain boundary regions. The dynamics in polycrystal is markedly different in the two series, as in Figs. 5–9. The crystalline regions in the first series are composed of the two components with size dispersity, as in Fig. 3, and are more frustrated than those in the second series. 共iii兲 In highly frustrated glass, such as for ␴2 / ␴1 = 1.4 and c = 1 / 2 in our model, the dynamic heterogeneity emerges on prolonged time scales. The configuration changes in glass are rare thermally activated events and should very sensitively depend on the degree of local disorder which is itself heterogeneous. In 共c兲, 共d兲, and 共d⬘兲 in Fig. 2, the aggregates of the chainlike motions extend longer than the spatial scale of D j. We note that no tendency of saturation has been observed in the correlation length of the dynamic heterogeneity with lowering of T in highly frustrated glass 关3,7–10兴. In our small system, however, we cannot obtain reliable results of the small-q behavior of the structure factor of D j and that of the broken bonds, while they both decay as q−2 at relatively large wave number q 关3兴 共not shown in this paper兲. In future work, we should thus examine the correlation length of D j and that of the dynamic heterogeneity at low T in a larger system. 共iv兲 We may furthermore envisage a crucial role of the local free volume in the slow dynamics. In the literature 关22,23兴, its slow diffusion was supposed to give rise to the aging effects. However, it is difficult to detect such behavior in simulations. In Fig. 3, if the local free volume is identified with the spacing outside the disks, it is nearly randomly distributed in 共a兲 共left兲 and is mostly around the large particles in 共b⬘兲 共right兲. We even notice that a particle densely surrounded by other particles is often pushed out into a new

cage. It is worth noting that Widmer-Cooper and Harrowell 关12兴 found no correlation between a local free volume 共in their definition using the inherent structure兲 and the slow dynamics. 共v兲 We have calculated the functions n共S兲 in Eq. 共2.7兲, M共S兲 in Eq. 共2.11兲, and B共S兲 in Eq. 共2.13兲 and have shown their derivatives dn / dS, dM / S, and dB / dS in Figs. 4, 7, and 9, respectively. These derivatives represent the distributions of the particle number, the displacements, and the broken bonds, respectively, as functions of the parameter S 关see Eq. 共2.6兲 for its definition兴. The dM / S and dB / dS exhibit similar behavior except the case of ␴2 / ␴1 = 1.25 and c = 1 / 2 in the upper panels of Figs. 7 and 9. 共vi兲 In future work we should investigate the relation between the size ␰ and the lifetime 共⬃␶␣兲 of the dynamic heterogeneity for general ␴2 / ␴1 and c, which has been done only at fixed ␴2 / ␴1 关3,5–11兴. 共vii兲 We will shortly report on the rheology of polycrystal states under shear flow. We shall see marked sliding motions of the particles in the grain boundary regions and large collective motions of the small crystalline regions, both being induced by shear. 共viii兲 In the literature of 2D melting 关19,20兴, the correlation function g6共r兲 of the sixfold orientation order has been calculated numerically and experimentally for onecomponent systems. In our binary systems, its spatial decay becomes shorter with increasing ␴2 / ␴1 in the first series and with increasing c in the second series 共not shown in this paper兲. In particular, the algebraic decay g6共r兲 ⬃ r−1/4 follows for ␴2 / ␴1 ⬃ 1.2 and c ⬃ 0.02 in the two series, respectively. This decay characterizes a transition from a “hexatic” to liquid state in one-component systems. We could discuss vitrification and melting in the same context in 2D binary mixtures 关17,24兴, where defects are proliferated and D j can be a convenient variable to describe these two transitions.

关1兴 T. Muranaka and Y. Hiwatari, Phys. Rev. E 51, R2735 共1995兲. 关2兴 M. M. Hurley and P. Harrowell, Phys. Rev. E 52, 1694 共1995兲. 关3兴 R. Yamamoto and A. Onuki, J. Phys. Soc. Jpn. 66, 2545 共1997兲; Phys. Rev. E 58, 3515 共1998兲. 关4兴 W. Kob, C. Donati, S. J. Plimpton, P. H. Poole, and S. C. Glotzer, Phys. Rev. Lett. 79, 2827 共1997兲; C. Donati, S. C.

Glotzer, P. H. Poole, W. Kob, and S. J. Plimpton, Phys. Rev. E 60, 3107 共1999兲. 关5兴 Donna N. Perrera, J. Phys.: Condens. Matter 10, 10115 共1998兲. 关6兴 B. Doliwa and A. Heuer, Phys. Rev. E 61, 6898 共2000兲. 关7兴 R. Yamamoto and A. Onuki, Phys. Rev. Lett. 81, 4915 共1998兲; J. Phys. Condens. Matter 12, 6323 共2000兲.

ACKNOWLEDGMENTS

We would like to thank Hajime Tanaka and Takeshi Kawasaki for showing their work on the relation between the local crystalline order and the dynamic heterogeneity in colloidal glass 关24兴. The calculations of this work were performed at the Human Genome Center, Institute of Medical Science, University of Tokyo. This work was supported by Grants in Aid for Scientific Research and for the 21st Century COE project 共Center for Diversity and Universality in Physics兲 from the Ministry of Education, Culture, Sports, Science and Technology of Japan.

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TOSHIYUKI HAMANAKA AND AKIRA ONUKI 关8兴 N. Lačević, F. W. Starr, T. B. Schroder, and S. C. Glotzer, J. Chem. Phys. 119, 7372 共2003兲. 关9兴 L. Berthier, Phys. Rev. E 69, 020201共R兲 共2004兲; S. Whitelam, L. Berthier, and J. P. Garrahan, Phys. Rev. Lett. 92, 185705 共2004兲. 关10兴 A. C. Pan, J. P. Garrahan, and D. Chandler, Phys. Rev. E 72, 041106 共2005兲. 关11兴 K. Vollmayr-Lee, W. Kob, K. Binder, and A. Zippelius, J. Chem. Phys. 116, 5158 共2002兲. 关12兴 A. Widmer-Cooper and P. Harrowell, Phys. Rev. Lett. 96, 185701 共2006兲; J. Non-Cryst. Solids 352, 5098 共2006兲. 关13兴 H. Shintani and H. Tanaka, Nature 2, 200 共2006兲. 关14兴 E. Dickinson and R. Parker, Chem. Phys. Lett. 79, 578 共1981兲. 关15兴 L. Bocqueht, J. P. Hansen, T. Biben, and P. Madden, J. Phys.: Condens. Matter 4, 2375 共1992兲. 关16兴 W. Vermöhlen and N. Ito, Phys. Rev. E 51, 4325 共1995兲. 关17兴 T. Hamanaka and A. Onuki, Phys. Rev. E 74, 011506 共2006兲.

关18兴 关19兴

关20兴 关21兴

关22兴 关23兴 关24兴

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Here the size ratio ␴2 / ␴1 is incorrectly written as ␴1 / ␴2 in many places. S. Nóse, Mol. Phys. 52, 255 共1983兲. B. I. Halperin and D. R. Nelson, Phys. Rev. Lett. 41, 121 共1978兲; D. R. Nelson, Defects and Geometry in Condensed Matter Physics 共Cambridge University Press, Cambridge, England, 2002兲, p. 68. K. Zahn, R. Lenke, and G. Maret, Phys. Rev. Lett. 82, 2721 共1999兲. ¯ depends on the time t after quenching the To be precise, D system 关17兴. Hence we took its average over t in the period 关t f − td , t f 兴 with t f = 2.2⫻ 104 and td = 2 ⫻ 103. L. C. E. Struik, Physical Aging in Polymers and Other Amorphous Materials 共Elsevier, Amsterdam, 1978兲. A. Onuki, A. Furukawa, and A. Minami, Pramana, J. Phys. 64, 661 共2005兲. T. Kawasaki, T. Araki, and H. Tanaka 共unpublished兲.