Multi-time density correlation functions in glass-forming liquids: Probing dynamical heterogeneity and its lifetime Kang Kim1, a) and Shinji Saito1, b) Institute for Molecular Science, Okazaki 444-8585, Japan
arXiv:1004.2380v3 [cond-mat.soft] 24 Jun 2010
(Dated: 28 June 2010)
A multi-time extension of a density correlation function is introduced to reveal temporal information about dynamical heterogeneity in glass-forming liquids. We utilize a multi-time correlation function that is analogous to the higher-order response function analyzed in multidimensional nonlinear spectroscopy. Here, we provide comprehensive numerical results of the four-point, three-time density correlation function from longtime trajectories generated by molecular dynamics simulations of glass-forming binary soft-sphere mixtures. We confirm that the two-dimensional representations in both time and frequency domains are sensitive to the dynamical heterogeneity and that it reveals the couplings of correlated motions, which exist over a wide range of time scales. The correlated motions detected by the three-time correlation function is divided into mobile and immobile contributions that are determined from the particle displacement during the first time interval. We show that the peak positions of the correlations are in accord with the information on the non-Gaussian parameters of the van-Hove self correlation function. Furthermore, it is demonstrated that the progressive changes in the second time interval in the three-time correlation function enable us to analyze how correlations in dynamics evolve in time. From this analysis, we evaluated the lifetime of the dynamical heterogeneity and its temperature dependence systematically. Our results show that the lifetime of the dynamical heterogeneity becomes much slower than the α-relaxation time that is determined from the two-point density correlation function when the system is highly supercooled. I.
INTRODUCTION
When liquids are supercooled below their melting temperatures while avoiding crystallizations, they eventually undergo a glass transition to become amorphous solids. The glass transition is ubiquitous among a wide variety of materials. There are many known properties associated with the glass transitions.1–3 In particular, when the glass transition is approached, various time correlation functions decay with non-exponential relaxations. Moreover, dynamical properties such as the structural relaxation time and the viscosity of the system tend to diverge, whereas the static structures remain unchanged and thus similar to those of normal liquids. Despite a large number of theoretical, experimental, and numerical studies over the past decades, the understanding of the mechanisms behind this drastic slowing down remains one of the most challenging problems in condensed matter.4–6 To address this problem from the microscopic level, various experiments have been employed including nuclear magnetic resonance and optical spectroscopies.7–10 These studies have shown that the dynamics do not follow the “homogeneous” scenario, but instead follow the “heterogeneous” scenario in glass-forming liquids.11–13 In the heterogeneous scenario, the non-exponential relaxation is explained by the superposition of individual particle contributions with different relaxation rates. Recent molecular dynamics (MD) simulations of model
a) Electronic b) Electronic
mail:
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glass-forming liquids14–28 and experiments performed on colloidal dispersions using particle tracking techniques29–34 have provided direct evidence that the structural relaxation in glassy states occurs heterogeneously, i.e., there is a coexistence of mobile and immobile states moving within correlated regions. These studies have also shown that the sizes of the correlated regions gradually grow beyond the microscopic molecular length scale with decreasing temperature (increasing the volume fraction in the case of colloidal dispersions). Such recent efforts have established the concept of “dynamical heterogeneity” (DH), an idea that advocates for a key mechanistic role underlying the drastic slowing down of the glass transition. Thus, to understand the details of the relaxation processes involved in DH, we must systematically characterize and quantify its spatiotemporal structures. The questions we seek to answer, then, include “how large are the heterogeneities?” and “how log do they last?” as discussed in Ref. 12. Recently, the determination of the size and length scale of the DH has attracted much attention. The correlations in dynamics can be measured in terms of four-point correlation functions and their associated dynamical susceptibility. This approach has been successful in extracting and characterizing the growing length scale with approaching the glass transition.18,24,25,28,35–48 The fourpoint dynamical susceptibility has also been investigated by the mode-coupling theory49–51 and experiments.52–54 However, knowledge and measurements relating to the time scale and lifetime of the DH are still limited. Moreover, the temperature dependence of the lifetime remains controversial. It has been observed in some numerical simulations that the lifetime and characteristic time scale of the DH are comparable to the α-relaxation time, τα ,
2 as determined by the two-point density correlation function.20,23,55 On the contrary, other simulations show that the lifetime becomes much slower than the τα as temperature decreases, and indicate that there exist deviations between the two time scales in glass-forming models.19,28,56–59 The aim of the present paper is to investigate the DH by numerically calculating the multi-time density correlation function, which is an elaboration of our previous study.60 In this paper, we emphasize the essential consideration of the multi-time extension of the four-point correlation function that can aid in the elucidation of the time evolution of the correlated particle motions in the DH. We show numerical results of the multi-time correlation function via the two-dimensional (2D) representation analogous to the multidimensional spectroscopy techniques. The 2D representations in both the time and frequency domains enable us to explore the couplings of particle motions in the DH. Furthermore, the multi-time correlation function is divided into mobile and immobile contributions from the single-particle displacement. It is demonstrated that this decomposition provides additional information regarding detailed relaxation processes of both mobile and immobile correlated motions in the DH. From extensive numerical results of the multitime correlation function, we determine the lifetime of the DH and resolve all controversy regarding the temporal details of the DH. The paper is organized as follows. In Sec. II, we briefly review recent studies that have used the four-point correlation function and its associated dynamical susceptibility to characterize the correlation length of the DH in glassy systems. Furthermore, we highlight how the multi-time extension is a crucial element in the discovery of temporal information of the DH. In Sec. III, we briefly review our MD simulations and summarize some numerical results using conventional time correlation functions. In Sec. IV, we present numerical calculations of the multitime correlation function and the time evolution of the correlated motions of the DH. We also determine the lifetime of the DH and its temperature dependence. In Sec. V, we summarize our results and give our concluding remarks.
II. MULTI-POINT AND MULTI-TIME CORRELATION FUNCTION A. Four-point correlation function to measure the dynamical correlation length
As mentioned in the introduction, the concept of the DH indicates that the mobility of individual particles largely fluctuate in the slow dynamics. Furthermore, particles that have similar mobility form cooperative correlated regions. Conventional analysis that is based on the use of two-point density correlation functions, for example the intermediate scattering function F (k, t) =
hρ(k, t)ρ(−k, 0)i61 , cannot detect large fluctuations in local mobility because two-point correlation functions averP age over all particles. Here, ρ(k, t) ≡ N j=1 exp(ik·r j (t)) is the Fourier transform of the density field of the N particles in the system. r j (t) is the jth particle position at time t and k is a wave vector with k = |k|. To characterize and quantify the correlations of the local mobilities, we need to analyze the correlations of the fluctuations in the two-point density correlation function.18,24,25,28,35–48,62 This leads to the following fourpoint correlation function, χ(4) q (k, t) = N hδFq (k, t)δF−q (k, t)i,
(1)
with δFq (k, t) =
N 1 X iq ·r j (0) e (exp[ik · ∆r j (t)] − F (k, t)). N j=1
(2) Here, q is a wave vector with q = |q|. Integrating over the volume and setting q → 0, we obtain the so-called fourpoint dynamical susceptibility, χ4 (t). If the DH becomes dominant in the slow dynamics and if the fluctuations in the particle mobilities become large, then the χ4 (t) will be able to show the growth of its correlation length, ξ. Recently, the four-point dynamical susceptibility χ4 (t) has been intensely applied to study physical implementations of the DH in various systems that include sheared supercooled liquids,63 aging in structural glasses,64 supercooled water,65 slow dynamics confined in random media,66 colloidal gelations,67 and sheared granular materials.68 It is remarked that the value of χ4 (t) depends on the choice of the ensemble. This ensemble dependence influences the estimation of the correlation length ξ for q → 0.42,43,52 B.
Why use a multi-time correlation?
The four-point correlation function defined by Eq. (1) is a one-time correlation function, as is schematically illustrated in Fig. 1(a). In order to quantify the temporal details of the DH and its lifetime τhetero , it is essential to analyze how the correlated particle motions decay with time. This requires a multi-time extension of the four-point correlation function. In practice, by following Fig. 1(b), the four-point correlation for the density field ρ(k, t) can be generalized to the three-time correlation function with correlations at times 0, τ1 , τ2 , and τ3 given by F4 (k1 , k3 , t1 , t2 , t3 ) = hρ(k3 , τ3 )ρ(−k3 , τ2 )ρ(k1 , τ1 )ρ(−k1 , 0)i. (3) This equation takes into account three time intervals, t1 , t2 , and t3 . The definition of the time interval ti is ti = τi − τi−1 , where τ0 = 0.
3
t
(a)
time 0
t
t1
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t2
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t3
τ2
(b)
τ3
time
FIG. 1. Schematic illustrations of the time configuration of the four-point correlation function; (a) four-point correlation function denoting the correlation of fluctuations in the twopoint correlation function between two times 0 and t, and (b) the three-time correlation function with correlations at four times 0, τ1 , τ2 , and τ3 .
We can define the following function as the difference between the four-point and three-time correlation function, F4 (k1 , k3 , t1 , t2 , t3 ), and the product of the twopoint correlation functions: ∆F (k1 , k3 , t1 , t2 , t3 ) = F4 (k1 , k3 , t1 , t2 , t3 ) − F (k3 , t3 )F (k1 , t1 ). (4) This can be regarded as a multi-time extension of Eq. (1). If the dynamics are homogeneous and if the motions between the two intervals t1 and t3 are uncorrelated and decoupled, then the three-time correlation function ∆F4 should become zero. On the other hand, if the dynamics become heterogeneous, the dichotomy between the mobile and immobile regions would lead to finite values of ∆F4 because of the correlated motions between the two intervals t1 and t3 . Furthermore, the progressive changes in the second time interval t2 = τ3 − τ2 of ∆F4 enable us to investigate how the correlated motions between two time intervals t1 and t3 decay with the waiting time t2 . This can provide the temporal information regarding the DH that is relevant in the quest to quantify its lifetime, τhetero .60 Some computational studies have already utilized multi-time correlations to examine the heterogeneous dynamics.19,20,23,55,56,69–72 However, in these calculations, only limited information has resulted (for example, results for t1 = t3 = τα has been successfully provided). This lack of results has caused the aforementioned controversy regarding the temporal information that is relevant to the DH. Throughout this paper, we present the comprehensive numerical results of a four-point, three-
time density correlation function without fixing any time intervals. This multi-time correlation function is used to quantify the lifetime of the DH, τhetero and to determine its temperature dependence. It is of interest to note that the multi-time correlation function can be regarded as an analogue of the nonlinear response functions of a molecular polarizability and dipoles as analyzed using the multidimensional spectroscopies such as 2D Raman spectroscopy and infrared (IR) spectroscopy.73–78 There exist promising theoretical treatments for the multi-time correlation function based on the mode-coupling theory.79,80 These techniques have now become powerful and standard tools to study condensed phase dynamics. For example, there are often used to study the ultrafast dynamics of liquid water.81–88 The utility of these techniques is enabled by the ability of the nonlinear response function to reveal details about the couplings between motions. This information is not available in the one-time linear response function. The present study analogously employs the underlying strategies and concepts of these multidimensional spectroscopy techniques to study the heterogeneous dynamics of the glass transition. It should be remarked that unique experiments have been recently proposed to examine heterogeneous dynamics in various chemical systems, which are referred to as 2D Fourier imaging correlation spectroscopy89 and multiple population period transition spectroscopy.90,91 These techniques provide information based on fourpoint correlation functions, which are basically the same as Eqs. 3 and 4. III. SIMULATION MODEL AND SOME DYNAMICAL CONSIDERATIONS A.
Model
We carried out MD simulations for a three-dimensional binary mixture. Our system consists of N1 = 500 particles of component 1 and N2 = 500 particles of component 2. They interact via a soft-core potential σ 12 ab vab (r) = ǫ , (5) r
where
σab =
σa + σb , 2
(6)
and a, b ∈ {1, 2}. The interaction was truncated at r = 3σ1 . The size and mass ratios were σ1 /σ2 = 1/1.2 and m1 /m2 = 1/2, respectively. The total number density was fixed at ρ = N1 + N2 /L3 = 0.8σ1−3 , where the system length was L = 10.77σ1 under periodic boundary conditions. In this paper, numerical results will p be presented in terms of reduced units σ1 , ǫ/kB , τ = m1 σ12 /ǫ for length, temperature, and time, respectively. The velocity Verlet algorithm was used with a time step of
3
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where ∆r j (0, t) ≡ r j (t) − r j (0) is the jth particle displacement vector during the two times 0 and t. The behavior of Fs (k, t) is often utilized to study the nonexponential decay of the structural relaxation as demonstrated in Fig. 2(a). Here the wave vector is chosen as k = 2π. This corresponds to the wave vector of the first peak of the static structure factor. Also well known is the fact that when the temperature decreases, this function plateaus during the β-relaxation regime. In this regime, a tagged particle is trapped by its surrounding caged particles. Eventually, the tagged particle escapes from the cage on a much longer time scale, which is referred to as the α-relaxation regime. In this paper, we define the α-relaxation time τα as Fs (k, τα ) = e−1 with k = 2π. In the inset of Fig. 2(a), the temperature dependence of τα is plotted as a function of the inverse of the temperature 1/T . We observe that the structural α-relaxation time is drastically increased and exhibits super-Arrhenius behavior as the temperature is decreased. Particle motions are also analyzed through the mean square displacement (MSD). We calculated the MSD for the particles of component 1, + * N1 1 X 2 2 (8) |∆r j (0, t)| , hδr (t)i = N1 j=1
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FIG. 2. Time correlation functions of glass-forming liquids at temperatures T = 0.772, 0.473, 0.352, 0.306, and 0.289 from left to right; (a) the self-part of the intermediate scattering function Fs (k, t) with k = 2π, (b) the mean square displacement hδr 2 (t)i, (c) the non-Gaussian parameter α2 (t), (d) the new non-Gaussian parameter γ(t), and (e) the fourpoint dynamical susceptibility χ4 (k, t) with k = 2π. Inset of (a): α-relaxation time τα as a function of the inverse temperature 1/T . The dotted line in (b) refers to the diffusion asymptote 6Dt at T = 0.289.
and displayed the results in Fig. 2(b) for various temperature. As shown in Fig. 2(b), at lower temperatures a plateau develops during the intermediate β-relaxation regime, when the cage effect is dominant. Diffusive behavior, hδr2 (t)i = 6Dt , eventually sets in over the time scale of t ≃ τα . C.
Non-Gaussian parameter
We next employ the non-Gaussian parameter (NGP) α2 (t) defined as 0.005τ in the microcanonical ensemble. The states investigated here were T = 0.772, 0.473, 0.352, 0.306, and 0.289. Below, we present and summarize the key numerical results regarding the dynamic properties of the glassy dynamics. Other information regarding this model, in particular static properties such as static structure factors can be found in some previous works.18,21
B. Intermediate scattering function and mean square displacement
To begin, we examined the density fluctuation in terms of the self-part of the intermediate scattering function of component 1 particles that is defined as Fs (k, t) =
*
+ N1 1 X exp[ik · ∆rj (0, t)] , N1 j=1
(7)
α2 (t) =
3hδr4 (t)i − 1, 5hδr2 (t)i2
(9)
PN1 with hδr4 (t)i = h(1/N1 ) j=1 |∆r j (0, t)|4 i. The α2 (t) reveals how the distribution of the single-particle displacement, δr, at time t deviates away from the Gaussian distribution.61 As is well documented15 and shown in Fig. 2(c), α2 (t) begins to grow as the temperature is decreased. The growth of α2 (t) means that the distribution of the displacement will have two peaks. These peaks indicate the existence of both mobile and immobile particles, the main feature of DH. However, it is noted that α2 (t) mainly grows in the β-relaxation regime, when the Fs (k, t) plateaus (see Fig. 2(a)). In practice, in the time scales at the beginning of the β-relaxation regime, α2 (t) begins to grow, whereas on the time scale of τα , α2 (t) begins to decrease to zero. This is due to the fact that α2 (t) is strongly dominated by the mobile particles which move faster than particles with a Gaussian distribution.
5 2
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P(log10 (δr); t)
P(log10 (δr); t)
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time 0 and time t. Alternatively, χ4 (k, t) can be expressed by38
(c)
T=0.289
χ4 (k, t) = N1 [hFˆs (k, t)2 i − hFˆs (k, t)i2 ].
1
Here we adopt Fˆs (k, t) as 0
−1
0
N1 1 X Fˆs (k, t) = cos[k · ∆r j (0, t)], N1 j=1
1
log10 (δr)
FIG. 3. Distribution of the logarithm of the single-particle displacement P (log10 (δr); t) at T = 0.772 (a), 0.352 (b), and 0.289 (b). For each temperature, the times shown are t = τα (solid curve), 2τα (dashed curve), 4τα (short dashed curve), and 10τα (dash-dotted curve). The dotted curve in (c) refers to the Gaussian distribution Gs (δr, t) = [1/(4πDt)3/2 ] exp(−δr 2 /4Dt) at t = 10τα , where the diffusion constant D is evaluated by the asymptote 6Dt of the mean squared displacement (see Fig. 2(b)).
The time τNGP during α2 (t) has a peak thus becomes smaller than τα at lower temperatures. It is remarked that a similar behavior in α2 (t) has been demonstrated using the mode-coupling theory, which incorporates hopping motions.92 Instead of using the NGP α2 (t), Flenner and Szamel have recently proposed a new non-Gaussian parameter (NNGP), γ(t), defined as 1 1 γ(t) = hδr2 (t)i − 1, (10) 3 δr2 (t) PN1 |∆rj (0, t)|−2 i. The γ(t) with h1/δr2 (t)i = h(1/N1 ) j=1 strongly weights the immobile particles which have not move as far as the Gaussian distribution would predict.93 Figure 2(d) demonstrates γ(t) for various temperatures. It is observed that the time τNNGP at which γ(t) has a peak is of a longer time scale than τα . This is in contrast to the results of the conventional NGP analysis as shown in Fig. 2(c). D.
(13)
Four-point dynamical susceptibility
As mentioned in Sec. II A, the four-point correlation function that is defined as the correlation function of the fluctuations in the two-point correlation functions has become a powerful tool to determine the correlation length of the DH. Although there are several definitions for χ4 (k, t), one is given by,40 2 + * N1 X 1 χ4 (k, t) = N1 (11) δFj (k, 0, t) , N1 j=1
(14)
with Fs (k, t) = hFˆs (k, t)i. The χ4 (k, t) shows the correlation of the fluctuation in the two-point correlation function Fs (k, t). This reveals how the particle motions (or trajectories) between times 0 and t are correlated. In other words, the amplitude of χ4 (k, t) signals the total amount of spatial correlations in the particle displacements within the given time interval t. As seen in Fig. 2(e), the χ4 (k, t) typically presents non-monotonic time behavior. The peak of χ4 (k, t) appears on a time scale that is comparable to τα . Note that the ensemble dependence of the dynamical susceptibility is not taken into account since the microcanonical dynamics is employed in our simulations. E.
Distribution of single-particle displacements
We end this section with a discussion of the distribution of single-particle displacements, as alluded to above. Following Flenner and Szamel,93 we calculated the distribution P (log10 (δr); t) of the logarithm of the particle displacements, δr, at time t. These displacements are obtained from the self-part of the van-Hove correlation function Gs (δr, t) as P (log10 (δr); t) = ln(10)4πδr3 Gs (δr, t).
(15)
Figure 3 shows P (log10 (δr); t) at various times t for T = 0.772, 0.352, and 0.289. The dotted red curve in Fig. 3(c) refers to the distribution of the Gaussian process, Gs (δr, t) = [1/(4πDt)3/2 ] exp(−δr2 /4Dt) with the diffusion constant D at T = 0.289. It is noted here that the Gaussian distribution is independent of time t, and the peak height is given by P (log10 (δr); t) ≈ 2.13.93 At the high temperature T = 0.772, there is only one peak during time t. This peak has the smallest deviation from the Gaussian distribution. On the contrary, at the lowest temperature T = 0.289, we clearly see the two distinct mobile and immobile peaks. These peaks clearly have large deviations from the Gaussian even for longer time scales than τα , implying the long-lived DH.57 IV. NUMERICAL RESULTS OF THE MULTI-TIME CORRELATION FUNCTION
where
δFj (k, 0, t) = cos[k · ∆r j (0, t)] − Fs (k, t),
(12)
represents the individual fluctuations in the real-part and self-part of the intermediate scattering function between
A.
Three-time density correlation function
Following the time configuration illustrated in Fig. 1(b) and Eq. (4), we extend the dynamical susceptibility
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FIG. 5. Diagonal parts of the three-time correlation functions, ∆F4 , ∆F4mo , and ∆F4im at waiting time t2 = 0 for T = 0.772 (a) 0.352 (b), and 0.289 (c). The solid, dashed, and short dashed curves correspond to total, mobile, and immobile parts, respectively.
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FIG. 4. 2D representations of the three-time correlation functions; (a) total ∆F4 (k, t1 , t2 , t3 ), (b) the mobile part ∆F4mo (k, t1 , t2 , t3 ), and (c) the immobile part ∆F4im (k, t1 , t2 , t3 ) at waiting time t2 = 0 for various temperatures T = 0.772, 0.352, and 0.289 from left to right. The wave vector k is chosen as k = 2π.
χ4 (k, t) defined by Eq. (11) to the three-time density correlation function with times 0, τ1 , τ2 , and τ3 . This is defined as ∆F4 (k, t1 , t2 , t3 ) N1 1 X = δFj (k, τ2 , τ3 )δFj (k, 0, τ1 )) . (16) N1 j=1 We note that Eq. (16) is regarded as the self-part of Eq. (4). Moreover, the wave vector is chosen as k = k1 = k3 in our numerical calculations. As discussed in Sec. II B, ∆F4 (k, t1 , t2 , t3 ) denotes the correlations of fluctuations in the two-point correlation function Fs (k, t) between two time intervals, t1 = τ1 and t3 = τ3 − τ2 . Furthermore, the three-time correlation function given by Eq. (16) can be divided into two parts as ∆F4 (k, t1 , t2 , t3 ) = ∆F4mo (k, t1 , t2 , t3 ) + ∆F4im (k, t1 , t2 , t3 ),
(17)
where ∆F4mo (∆F4im ) represents the mobile (immobile) part, arising from the contribution of mobile (immobile) particles during the first time interval, t1 . Practically, we defined the mobile (immobile) particles as those particles
that move more (less) than p the mean value of the singleparticle displacement, hδr2 (t1 )i, for the first time interval t1 . During t1 , the function δFj (k, 0, τ1 ) selects the sub-ensemble of mobile (immobile) contributions in the DH, and the total function δFj (k, τ2 , τ3 )δFj (k, 0, τ1 ) contains information to determine how long the mobile (immobile) particles remain during the waiting time t2 . This information gained from the three-time correlation function ∆F4 is related to the joint probability P (δr(t3 )|δr(t1 )) of two successive particle displacements, δr(t3 ) and δr(t1 ). This joint probability describes the probability of the particle being mobile (immobile) at t1 and remaining mobile (immobile) at t3 after the waiting time t2 . More details will be given elsewhere by analyzing the multi-time correlation functions of the particle displacements.94
B.
Zero waiting time t2 = 0
We first present the numerical results of the threetime density correlation function, ∆F4 (k, t1 , t2 , t3 ), at the waiting time t2 = 0. These are shown in Fig. 4(a) at various temperatures, T = 0.772, 0.352, and 0.289. It can be seen that the intensity of ∆F4 (k, t1 , t2 , t3 ) gradually grows with decreasing the temperature. This indicates that particles that are mobile (immobile) during the first time interval, t1 , tend to remain mobile (immobile) during the subsequent time interval, t3 . It can also be seen that the profile of ∆F4 (k, t1 , t2 , t3 ) is widely broadened, suggesting that the motions between various time scales, including α- and α-relaxation and α- and βrelaxation, are coupled. Furthermore, the time at which ∆F4 has its maximum value is approximately given by the α-relaxation time, τα . To describe the details of ∆F4 (k, t1 , t2 , t3 ) more fully, we show the mobile and immobile parts of the system, ∆F4mo (k, t1 , t2 , t3 ) and ∆F4im (k, t1 , t2 , t3 ) in Fig. 4(b) and (c), respectively. These diagonal parts at t1 = t3 are also drawn in Fig. 5 for T = 0.772 (a), 0.352 (b), and 0.289 (c). It can be seen that the peak of ∆F4 is composed of the two distinct mobile and immobile contributions particularly at the lower temperatures. The time scales of the two contributions are different, i.e., the peak of
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2
t
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10 10 10 10 10
0.04 0.03 0.02 0.01 0
1
10
0.02
FIG. 7. Diagonal parts of the three-time correlation functions; (a) total ∆F4 , (b) the mobile ∆F4mo , and (c) the immobile ∆F4im at T = 0.289. Waiting times are varied at t2 = 0, 0.5τα , τα , 2τα , 4τα , 6τα , and 10τα from top to bottom.
-1
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(k, t, t2, t)
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im
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T=0.289
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T=0.289
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(k, t, t2, t)
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T=0.289
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mo
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t2=3τα
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t3
t2=τα
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∆F4
t2=0
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∆F4 (k, t, t2, t)
(a)
10 -1
10
0
1
2
3
4
10 10 10 10 10
-1
10
0
1
2
3
4
10 10 10 10 10
t1
FIG. 6. 2D representations of the three-time correlation functions; (a) total ∆F4 (k, t1 , t2 , t3 ), (b) the mobile part ∆F4mo (k, t1 , t2 , t3 ), and (c) the immobile part ∆F4im (k, t1 , t2 , t3 ) at T = 0.289. Waiting times are varied for t2 = 0, τα , and 3τα from left to right. The wave vector k is chosen as k = 2π.
the mobile part, ∆F4mo , appears at t1 ≃ τNGP , while the peak of the immobile part, ∆F4mo , is pronounced on the time scale of t1 ≃ τNNGP . These findings are expected as per the discussion in Sec. III C. Specifically, the NGP focuses on the mobile particles and the NNGP weights the immobile contributions of the non-Gaussian distribution of the particle displacement (see Fig. 3(c)). C. Waiting time t2 dependence and the lifetime of the dynamical heterogeneity
As outlined in Sec. II B, the progressive changes in the waiting time t2 = τ3 − τ2 of the three-time correlation function ∆F4 (k, t1 , t2 , t3 ) make it possible to investigate how the correlated motions decay with time. Figure 6 shows the time evolutions of the three-time correlation functions, ∆F4 , ∆F4mo , and ∆F4im at the lowest temperature of T = 0.289. We also plot the diagonal parts of the evolution along t1 = t3 at various t2 in Fig. 7. It is demonstrated that the correlations gradually decay as the waiting time t2 increases. The values of the threetime correlation functions tend toward zero for t2 → ∞. We also find that for larger t2 , the peak of ∆F4 tends to shift to t ≃ 1000, which is close to τNNGP . This peak shift is attributed to the fact that immobile particles tend
FIG. 8. 2D representations of relaxation time distributions mo of three-time correlation functions; (a) τˆhetero , (b) τˆhetero , and im (c) τˆhetero . normalized by τα at T = 0.289. The dotted line refers to the iso-line of τhetero = τα in each panel.
to remain immobile on larger time scales as indicated in Fig. 6(c). Moreover, the presence of correlations on larger time scales than the α-relaxation time scale is observed, even for t2 = 10τα . It is also clearly seen that the offdiagonal parts of ∆F4 , ∆F4mo , and ∆F4im become noticeable with increasing t2 . These observations imply that the relaxation rates of ∆F4 , ∆F4mo , and ∆F4im largely depend on which time scale is examined. To explore the details of the time scale of the correlated motions, we define the relaxation time τˆhetero (t1 , t3 ) of the ∆F4 (k, t1 , t2 , t3 ) as ∆F4 (k, t1 , τˆhetero , t3 )/∆F4 (k, t1 , 0, t3 ) = e−1 ,
(18)
for various values of t1 and t3 . Similarly, the relaxation mo im times τˆhetero (t1 , t3 ) and τˆhetero (t1 , t3 ) are determined from ∆F4mo and ∆F4im , respectively. Figure 8 shows the 2D mo representations of the relaxation times τˆhetero , τˆhetero , and im τˆhetero at T = 0.289. We confirm that the relaxation time τˆhetero of the total function ∆F4 is described by mo the summation of the mobile and immobile parts, τˆhetero im and τˆhetero . Furthermore, it is of interest to note that the distribution of the τˆhetero has a multiple structure; the relaxation time τˆhetero becomes much larger than the time scale, τα , if the time interval t1 or the time interval t3 is examined for larger time scale than τα . On the other hand, τˆhetero becomes smaller than τα if t1 or t3 is examined for smaller time scale than τα . In order to obtain the average lifetime of the DH, we define the volume of the heterogeneities as Z ∞ Z ∞ dt3 ∆hetero (k, t2 ) = dt1 ∆F4 (k, t1 , t2 , t3 ). (19) 0
0
8
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5
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ω3
T=0.772 T=0.473 T=0.352 T=0.306 T=0.289
τhetero
10
(a)
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∆hetero(k, t2) / ∆hetero(k, 0)
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10-2 -3
10 10
Through the use of the analogy to the 2D IR spectroscopy, it is of interest to examine the three-time correlation function ∆F4 (k, t1 , t2 , t3 ) in the frequency domain. The Fourier transformed 2D spectrum is obtained by ∆F4 (k, ω1 , t2 , ω3 ) Z ∞ Z ∞ = dt3 dt1 ∆F4 (k, t1 , t2 , t3 )eiω1 t1 +iω3 t3 . (20)
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2D spectra of three-time correlation functions
-1
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-4
We examined the t2 dependence of ∆hetero (k, t2 ). Figure 9(a) shows ∆hetero (k, t2 )/∆hetero (k, 0) as a function of the waiting time t2 normalized by τα at each temperature. From Fig. 9(a), we see that ∆hetero rapidly decays to zero at higher temperatures and that the time scale is comparable to τα . In contrast, at lower temperatures, the relaxation of ∆hetero occurs on a time scale larger than τα . ∆hetero (k, t2 )/∆hetero (k, 0) can be fitted by the stretched-exponential function exp[−(t2 /τhetero )c ], where τhetero can be regarded as the average lifetime of the relation as τhetero ≃ R R DH. We obtain the approximate RR τˆhetero (t1 , t3 )dt1 dt3 / dt1 dt3 . We plot τhetero at each temperature T in Fig. 9(b). It is found in Fig. 9(b) that τhetero becomes much larger than τα as the temperature T decreases. In practice, the lifetime τhetero is approximately 6τα ≃ 2400 with c ≃ 0.5 at the lowest temperature of T = 0.289. Furthermore, as seen in the inset of Fig. 9(b), we observe the strong deviation between the two time scales τhetero and τα that follows the power low, τhetero ∼ τα 1.5 . Similarly, we determined the mo im average lifetimes τhetero and τhetero for the mobile and immobile parts. We confirm that t2 dependences of the mobile and immobile parts are close to that of the total im mo are comparable to function and that τhetero and τhetero τhetero for each temperature (data not shown).
t2=τα
10-2
-4
ω3
FIG. 9. (a) Waiting time t2 dependence of the integrated three-time correlation function ∆hetero (k, t2 )/∆hetero (k, 0) with k = 2π for various temperatures. The waiting times are normalized by τα for each temperature. The solid curve is determined by a fitting with the stretched-exponential form for each temperature. (b) Average lifetime of DH τhetero normalized by the α-relaxation τα versus temperature T . Inset: Relation between two time scales, τhetero and τα . The straight line with the slope 1.5 is presented as a viewing guide.
0
1 0.8 0.6 0.4 0.2 0
-2
0.8
T
-1
0
t2=3τα
10
10
10
D.
10-1
-2
10-5 10-4 10-3 10-2 10-1 0
t2=τα
10 -5
10
-4
10
-3
ω1
-2
10
10
-1
10
-5
10
-4
10
-3
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10
-1
FIG. 10. Imaginary parts of 2D spectra of the three-time correlation functions; (a) total ℑ[∆F4 (k, ω1 , t2 , ω3 )], (b) the mobile part ℑ[∆F4mo (k, ω1 , t2 , ω3 )], and (c) the immobile part ℑ[∆F4im (k, ω1 , t2 , ω3 )] at T = 0.289. Waiting times are varied at t2 = 0, τα , and 3τα from left to right. The wave vector k is chosen as k = 2π. The profile is normalized by the peak value at t2 = 0 for each panel.
Similarly, the mobile part ∆F4mo and immobile part ∆F4im can be represented in the frequency domain with respect to t1 and t3 . In Fig. 10, we present the imaginary parts of 2D spectra, the total ℑ[∆F4 (k, ω1 , t2 , ω3 )] (a), the mobile part ℑ[∆F4mo (k, ω1 , t2 , ω3 )] (b), and the immobile part ℑ[∆F4im (k, ω1 , t2 , ω3 )] (c) at T = 0.289 for several waiting times. It is seen in Fig. 10(a) that the peak of ℑ[∆F4 ] appears near the detected slower time −1 −1 scale, (ω1 , ω3 ) ≃ (τhetero , τhetero ), which is longer-lived for larger waiting times. The peak is diagonally elongated at t2 = 0 because of the strong correlations between two the frequencies ω1 and ω3 . The elongation that is directed toward the high frequency side tends to be lost because of the loss of the frequency correlations at larger t2 . Moreover, the off-diagonal cross peaks −1 −1 become pronounced at (ω1 , ω3 ) ≃ (τNNGP , τhetero ) and −1 −1 (τhetero , τNNGP ). The time scale τNNGP corresponds to the peak position of ∆F4 (k, t1 , t2 , t3 ) at larger t2 , as seen in Fig. 6(a). Similar behaviors are observed in the 2D spectra of the mobile and immobile parts, ℑ[∆F4mo ] and ℑ[∆F4im ]. The mobile part ℑ[∆F4mo ] is rather horizontally elon−1 gated until ω1 ≃ τNGP because of the coupling between −1 ω3 ≃ τhetero and the higher frequency ω1 , which is correlated to the 2D representation in the time domain that is seen in Fig. 6(b). Furthermore, the immobile part
9 ℑ[∆F4im ] tends to be almost symmetric for the diagonal line ω1 = ω3 , although the 2D profile in time domain is largely asymmetric as seen in Fig. 6(c).
V.
CONCLUSIONS AND FINAL REMARKS
We have investigated the four-point, three-time density correlation function to quantitatively characterize the temporal structures of the DH. The correlations detected by the three-time correlation function can be divided into two parts, mobile and immobile contributions determined from the single-particle displacement during the first time interval. These 2D representations in both the time and frequency domains that are presented over a wide range of time scales enable us to explore the couplings of particle motions. It is shown that the peak positions of the mobile and immobile parts are correlated to the dominant time scales of the non-Gaussian parameters. These extracted results are not obtainable from a one-time correlation function. Furthermore, the progressive changes in the waiting time allow us to obtain detailed information regarding the correlations of motions decay with the time. The waiting time dependence of the multi-time correlation function shows the existence of the correlations on larger time scales between immobile particles. The multi-time correlations allow for the quantification of the average lifetime of the DH, τhetero , in glass-forming liquids. Our analysis can be regarded as an analogue of the multidimensional nonlinear spectroscopic analysis applied to liquids and biological systems to understand ultrafast dynamics, e.g., the transition from inhomogeneous to homogeneous broadening and the couplings between molecular motions. We have found that the τhetero becomes much slower than the α-relaxation time τα when the system is highly supercooled. This is due to the long-lived DH at lower temperatures. Our findings show that the presence of the new time scale τhetero exceeds that of the α-relaxation time, τα . These findings are correlated with recent numerical studies.19,28,56–59,95 Such strong deviations and decouplings between τhetero and τα when approaching the glass transition temperature have been observed in some experiments,96,97 in which the lifetime of a subensemble is measured after selective excitation. Recent single-molecule experiments that detect the local mobility of probe molecules dispersed in glassy materials have other relevance to our simulations98–101 . In these experiments, the lifetime of the dynamical heterogeneity is evaluated from the exchange time between mobile and immobile regions, which is found to be much slower than the structural relaxation time τα near the glass transition temperature. It is of great importance to examine the relation between the length and time scales of the DH in order to characterize the relevant spatiotemporal structures. So far, various relations such as τ ∼ ξ z as seen in critical
phenomena18,20,24,37,45 or τ ∼ exp((ξ/kB T )ζ ) based on the random first order transition46 have been proposed. In these studies, the four-point correlation function is used to extract the length scale ξ of the DH, where the fluctuation in the two-point correlation function δF (k, t) is considered as an order parameter as seen in Eq. (1). On the contrary, the time scale is typically chosen as the relaxation time of the two-point density correlation function, τα . Here, we show that the time scale τ associated with ξ should not be τα but should instead be the average lifetime of the order parameter, i.e., τhetero . This τhetero is hidden in the two-point correlation function and unveiled when we apply the the four-point, three-time correlation function. It is worth mentioning that several recent attempts have utilized the multi-time correlation function to detect heterogeneous dynamics. A third-order nonlinear susceptibility is theoretically applied to the glassy systems.102,103 . Recently this concept has been tested experimentally to measure the DH.104 Furthermore, as previously mentioned, recent 2D optical techniques89–91 can in principle provide information on heterogeneous dynamics via 2D representations of multi-time correlation functions. We hope that our numerical results will be directly compared with the experimental ones in the future.
ACKNOWLEDGMENTS
The authors thank Ryoichi Yamamoto, Kunimasa Miyazaki, and Takuma Yagasaki for helpful discussions and comments. This work was partially supported by KAKENHI; Young Scientists (B) No. 21740317, Scientific Research (B) No. 22350013, and Priority Area “Molecular Theory for Real Systems”. This work was also supported by the Molecular-Based New Computational Science Program, NINS and the Next Generation Super Computing Project, Nanoscience program. The computations were performed at Research Center of Computational Science, Okazaki, Japan. 1 P.
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