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Effects of spatial heterogeneities on the slow dynamics of density fluctuations near the colloidal glass transition
Physica A 307 (2002) 27 – 40
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E ects of spatial heterogeneities on the slow dynamics of density "uctuations near the colloidal glass transition Michio Tokuyamaa;∗ , Yayoi Teradab , Irwin Oppenheimc a Institute
of Fluid Science, Tohoku University, Sendai 980-8577, Japan of Fluid Science, Tohoku University, Sendai 980-8577, Japan c Department of Chemistry, Massachusetts Institute of Technology, Cambridge, MA 02139, USA b Institute
Received 1 October 2001
Abstract The e ects of spatial heterogeneities on the slow dynamics of density "uctuations are discussed both in an equilibrium colloidal suspension and in a slightly nonequilibrium colloidal suspension. The long-lived, heterogeneous glassy domains appear near the glass transition and in"uence the dynamics of density "uctuations, leading to the - and -relaxation processes. The spatial heterogeneities related to the process are less noticeable in an equilibrium case than in a nonequilibrium case. Thus, a logarithmic decay is shown to be a new critical decay in an equilibrium case, instead of the power-law decay followed by the von Schweidler decay in a c 2002 Elsevier Science B.V. All rights reserved. nonequilibrium case. PACS: 82.70.Dd; 05.40.+j; 51.10.+y Keywords: Hard-sphere suspensions; Long-range hydrodynamic interactions; Nonlinear density "uctuations; Spatial heterogeneities; Slow dynamics
1. Introduction The important role of spatially heterogeneous structure in the slow dynamics of density "uctuations near the glass transition has been pointed out in glass-forming materials for the last decade [1–5]. Recently, Tokuyama et al. [6] have shown that in the nonequilibrium colloidal suspensions of hard spheres the spatial heterogeneities a ect the dynamics of density "uctuations near the colloidal glass transition, leading ∗
Corresponding author. Tel.: +81-22-217-5327; fax: +81-22-217-5327. E-mail address: [email protected] (M. Tokuyama).
c 2002 Elsevier Science B.V. All rights reserved. 0378-4371/02/$ - see front matter PII: S 0 3 7 8 - 4 3 7 1 ( 0 1 ) 0 0 5 7 9 - 9
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to the two distinct slow-relaxation processes around the -relaxation time t and the -relaxation time t . Near the glass transition, the small glassy domains are formed on the time scale of order t and lead to the power-law decay of the von Schweilder type for the intermediate scattering function. On the other hand, on the time scale of order t , the long-lived, large glassy domains are formed. The stretched exponential decay of the Kohlrausch–Williams–Watts (KWW) type is well explained by the existence of those large domains. Thus, the spatial heterogeneities were shown to be responsible for the slow dynamics of the nonequilibrium density "uctuations near the glass transition. Such nonequilibrium e ects are also observable as the waiting time e ect in experiments and simulations [7,8]. Most of the experiments in colloidal suspensions have been done in an equilibrium state, which is attained for long times after the quenched, and recovered long-known phenomena, such as the stretching of the process, similar to those in glass-forming materials near the glass transition [9 –11]. In contrast to a nonequilibrium case, however, it would be diGcult to observe the spatial heterogeneities experimentally in an equilibrium case. This is mainly because the equilibrium "uctuations can easily destroy those heterogeneities even though those heterogeneities are generated by nonlinear density "uctuations themselves [12–14]. In fact, the large glassy domains related to the -relaxation process can survive, while the small glassy domains are easily destroyed by the "uctuations. Hence the -relaxation process is expected to be less apparent in an equilibrium suspension than in a nonequilibrium suspension. In this paper, we compare the spatial heterogeneities obtained numerically in an equilibrium suspension with those obtained in a nonequilibrium suspension and then show that the in"uence of spatial heterogeneities on the dynamics of the density "uctuations are less noticeable in an equilibrium suspension than in a nonequilibrium suspension. In this paper, we discuss a concentrated suspension of neutral hard spheres. There exist two kinds of interactions between particles. One is the direct interaction between particles. The other is the hydrodynamic interaction between particles. For short times, the hydrodynamic interactions simply lead to corrections to the single-particle di usion coeGcient D0 , giving the volume fraction dependence of the short-time self-di usion coeGcient DSS (e ). For longer times, however, the long-range hydrodynamic interactions become important through the many-body correlations. Especially for larger volume fractions, they lead to the dynamic anomaly of the di usion coeGcient, in which the di usion coeGcient becomes zero at the volume fraction g , which we call the glass transition volume fraction. Near g , therefore, they in"uence the dynamics of density "uctuations signiJcantly [15]. Thus, the long-time processes are governed by the many-body correlations between particles, which also leads to the dynamic anomaly of the long-time self-di usion coeGcient DSL (e ). In order to see the importance of the many-body correlations, we plot the long-time self-di usion coeGcient DSL (e ) in Fig. 1 together with the experimental data. For comparison, the computer simulation results without the hydrodynamic interactions between particles are also shown. The large deviations of the simulation results from the experimental data are seen. This is due to the lack of hydrodynamic interactions. In order to check the e ect of the short-time self-di usion process on the long-time process, we also plot the results obtained by
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Fig. 1. The long-time self-di usion coeGcient DSL versus e . The solid line indicates the theoretical results from Ref. [15]. The symbol () indicates the experimental data from Ref. [11]. The symbols (4); ( ), and ( ) indicate the simulation results from Refs. [16 –18], respectively. The open symbols (); ( ), and ( ) indicate the results obtained by multiplying the simulation results (4); ( ), and ( ) by DSS . The dotted and the dashed lines indicate the glass transition volume fraction g and the melting volume fraction m , respectively.
•
•
◦
multiplying the simulation results by the theoretical values of DSS (e ) [15]. Even in this case, there still exist distinct deviations especially for volume fractions higher than the melting volume fraction m = 0:545, while for small volume fractions the correlation e ects are negligible. Hence those deviations are expected to be due to lack of many-body correlations through the long-range hydrodynamic interactions. We show that those correlations play an important role in forming long-lived, spatial heterogeneities and lead to two distinct slow relaxation processes, the - and -relaxation processes. In Section 2, we summarize and discuss the basic nonlinear di usion equations and concepts. In Section 3 we discuss the numerical results obtained in a slightly nonequilibrium suspension. In Section 4 we discuss the numerical results obtained in an equilibrium suspension. We thus explore the main di erence between the spatial heterogeneities in an equilibrium suspension and those in a nonequilibrium suspension
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and show how such a di erence a ect the dynamics of density "uctuations near the glass transition. Section 5 is devoted to conclusions. 2. A nonlinear stochastic diusion equation We consider a concentrated, neutral hard-sphere suspension with both the hydrodynamic and the direct interactions between particles. Let e (=4 a30 neq =3) denote the particle volume fraction, where neq (=N=V ) is the equilibrium particle number density, N and V being the total number of colloidal particles and the total volume of the system, respectively. In this paper, we focus only on a suspension-hydrodynamic stage [15], where the space–time cuto s (rc ; tc ), which are the minimum wavelength and time of the dynamic process of interest, are set as rc ¿ a0 and tc ¿ tD , where tD = a20 =D0 is the structural-relaxation time which is a time required for a particle to di use over a distance a0 . Here D0 is a di usion constant of a single particle. The relevant variable here is the slowly-varying, local volume fraction given by [15] N
(r; t) =
4 3 a (Xi (t) − r) ; 3 0
(2.1)
i=1
where Xi (t) is the position vector of particle i and (r) denotes the coarse-grained function given by (r) = (1=V ) k exp (−ik · r) with (|k| 6 1=a0 ). We should mention here that the other hydrodynamic variables, such as the velocity density and the energy density are rapidly varying variables and can be written in terms of (r; t) [15]. Hence, we start with the following nonlinear stochastic di usion equation already described elsewhere [14]: @ (r; t) = ∇ · [DS ((r; t))∇(r; t)] + (r; t) @t
(2.2)
with the self-di usion coeGcient DS () = DSS ()
1 − 9=32 ; 1 + (DSS =g D0 )(1 − =g )−2
(2.3)
where (r; t) is the Gaussian, Markov random force and satisJes (r; t)(r ; t ) = −2(t − t )∇ · [DS ((r; t))∇(r)(r ) −(1=2)(r)(r )∇DS ((r; t))] :
(2.4)
Here the bar denotes the average over an appropriate initial ensemble. The glass transition volume fraction g is given by g = (4=3)3 =(7 ln 3 − 8 ln 2 + 2) 0:57184 : : : :
(2.5)
Here we note that g is in a good agreement with the experimental values of 0.571– 0.580 [10,11]. The short-time self-di usion coeGcient DSS () is given by [15] DSS () = D0 =[1 + L()]
(2.6)
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with the nonmemory e ect
6xy 2x2 y 2xy 1− L() = − + − 1−x+y 1 − x + y + 4xy 1 − x 1 + 2y xy2 2xy + + 1 − x + y + 2xy (1 + y)(1 − x + y) xy2 3xy2 ; − × 1+ (1 + y)(1 − x + y) − xy2 (1 + y)(1 − x + y) − 2xy2 (2.7) 1=2
where x() = (9=8) and y() = 11=16. The nonmemory e ect L() results from the many-body static e ect due to the short- and the long-range hydrodynamic interactions. The factor 9=32 in the numerator of Eq. (2.3) comes from the many-body couplings between the two-body direct interactions and the short-range hydrodynamic interactions, where the factor reduces to 2 if the short-range hydrodynamic interactions are not taken into account. The singular term in the denominator of Eq. (2.3) results from the many-body correlation e ect due to the long-range hydrodynamic interactions. Here we should mention that the long-range interaction means an interaction which leads to a divergent integral. Eq. (2.2) is a starting equation to study the dynamics of hard-sphere suspensions from a liquid state to a glass state. The most important feature of that equation is that the self-di usion coeGcient becomes dynamically anomalous near the glass transition g as DS ((r; t)) ˙ D0 |1 − (r; t)=g | ;
(2.8)
where = 2 here. Hence the di usion coeGcient DS ((r; t)) becomes smaller and smaller as time goes on, showing a crossover from the short-time self-di usion process characterized by the short-time self-di usion coeGcient DSS (e ) to the long-time self-di usion process characterized by the long-time self-di usion coeGcient DSL (e ) = DS (e ). We should note here that the distinct slow relaxation processes, such as the and -relaxation processes, are always recovered for whatever original mechanisms if the di usion coeGcient DS ((r; t)) shows the dynamic anomaly discussed above [21]. In the following sections, we discuss the numerical solutions of Eq. (2.2) in two cases separately, an equilibrium case and a slightly nonequilibrium case. 3. Nonequilibrium case It is convenient to split the local volume fraction (r; t) into two parts, an average part (r; t) = (r; t) and a "uctuation (r; t); (r; t) = (r; t) + (r; t). This is because the glass transition is not a critical phenomenon since there is no correlation length diverging even at the transition point. As long as the system is away from a critical point, therefore, the relative magnitude of the density "uctuations to the mean density should be small even near the glass transition point, where |=| 1.
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Since (r; t = 0) = e in a nonequilibrium case, Eq. (2.2) can be decomposed into the following two types of equations: @ (r; t) = ∇ · [DS ((r; t))∇(r; t)] ; @t @ (r; t) = ∇ · [DS ((r; t))∇(r; t)] + (r; t) @t with the conservation laws 1 1 dr (r; t) = 0 ; dr (r; t) = e ; V V
(3.1) (3.2)
(3.3)
where the random force (r; t) satisJes the "uctuation–dissipation relation [14] (r; t)(r ; t ) = −2(t − t )∇ · [DS ((r; t))∇(r)(r ) −(1=2)(r)(r )∇DS ((r; t))] :
(3.4)
The dynamics of spatial heterogeneities is then described by the nonlinear deterministic di usion equation (3.1). The dynamics of the nonequilibrium density "uctuations obeys the linear stochastic di usion equation (3.2) and is in"uenced by the spatial heterogeneities through the di usion coeGcient DS ((r; t)). The intermediate scattering function F(k; t) is given by the Fourier transform of the autocorrelation function of the density "uctuations F(r; t)=(r; t)(0; 0)=[(4 a30 =3)2 N ]. For scattering vectors much larger than the maximum position km of the structure factor S(k) = F(k; 0), the scattering function F(k; t) reduces to the self-intermediate scattering function FS (k; t), where S(k)=1. Hence FS (k; t) can be easily calculated from Eq. (3.2) since (r; t)(r ; 0)=0. In order to solve Eq. (3.1) numerically, one must Jx the values of two parameters e and z0 , as the initial conditions. Here the state parameter z0 measures how the system is spatially nonuniform initially and is given by (3.5) z0 = 1 − (1=V ) dr|1 − (r; t)=e | ; where z0 = 1:0 in an equilibrium case. Suppose that we start from a completely random nonequilibrium conJguration. Then the smoothing process of (r; t) starts to occur, following Eq. (3.1), and (r; t) Jnally reaches the equilibrium volume fraction e for long times of order tL (=a20 =DSL ). As discussed in Ref. [6], near the glass transition there are four characteristic time stages. The Jrst is the early stage where t t =a20 =DSS (e ). The spatial conJgurations are random and are described by (r; t)=exp [tDSS (e )∇2 ](r; 0). The self-intermediate scattering function FS (k; t) obeys an exponential decay FS (k; t) = exp [ − k 2 DSS (e )t] ;
(3.6) DSS .
from which one can determine the short-time self-di usion coeGcient After this stage, the Jnite-sized, glassy domains with (r; t) ¿ g are formed for volume fractions larger than the crossover volume fraction [19]. The smoothing process of (r; t) is then slowing down due to those domains. On the time scale of order t , therefore, those glassy domains a ect the dynamics of the density "uctuations. This is the so-called -relaxation stage where t t 6 t . Depending on the time scale, the scattering function
M. Tokuyama et al. / Physica A 307 (2002) 27 – 40
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Fig. 2. Snapshots of typical conJgurations of glassy domains in an x–y plane for times t ∼ 4a20 =D0 , t ∼ 800a20 =D0 , and t ∼ 2 × 104 a20 =D0 at e = 0:5704 and z0 = 0:8. The glassy domains with (r; t) ¿ g are colored black.
FS (k; t) obeys two types of power-law decays. In the fast -relaxation stage where t t 6 t , the small aggregates of glassy domains are formed and a ect the dynamics of the density "uctuations, leading to the critical decay with the exponent bf (k; e ; z0 ) FS (k; t) = fc (k; e ; z0 ) − f1 (k; e ; z0 )(t=t )bf ;
(3.7)
where fc and f1 are positive constants. This power-law decay continues up to the crossover time t . In the slow -relaxation stage where t 6 t 6 t , the aggregates grow to the larger clusters and lead to the power-law decay of von Schweilder type with the exponent bs (k; e ; z0 ) FS (k; t) = fc (k; e ; z0 ) − f2 (k; e ; z0 )(t=t )bs ;
(3.8)
where f2 is a positive constant. After this stage, the glassy domains further grow to larger clusters and continue up to the time scale of order t . Those domains in"uence the dynamics of the density "uctuations leading to the stretched exponential decay of KWW type FS (k; t) = f3 (k; e ; z0 ) exp [ − (t=t ) ] ;
(3.9)
where (e ; z0 ) is a stretched exponent to be determined, and f3 is a positive constant. This is the so-called -relaxation stage where t 6 t 6 tL =a20 =DSL . Here the -relaxation time t is deJned by the time which satisJes the relation FS (k; t ) = f3 e−1 and satisJes t ˙ |1 − e =g |−% with % = = [6]. After this stage, the glassy domains start to disappear very slowly and the system gradually reaches the equilibrium state on the time scale of order tL . Then, the spatial conJgurations become random. Hence the density "uctuations obey the exponential decay FS (k; t) = exp [ − k 2 DSL (e )t] ;
(3.10) DSL .
This is the from which one can determine the long-time self-di usion coeGcient late stage where t ¿ tL . In Fig. 2 the numerical solutions of Eq. (3.1) under a slightly nonequilibrium initial condition are shown at four di erent characteristic times, t , t , t , and tL . The self-intermediate scattering function FS (k; t) is also shown in Fig. 3. Thus, it turns out from Figs. 2 and 3 that on the time scale of order t , the small glassy domains are responsible for the power-law decay of von Schweidler type, while
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M. Tokuyama et al. / Physica A 307 (2002) 27 – 40
Fig. 3. Self-intermediate scattering function FS (k; t) versus time at e = 0:5704 and ka0 = 1:3. The dotted line indicates the results for a slightly nonequilibrium case at z0 = 0:8, the solid line the results for an equilibrium case at z0 = 1:0, and the open circles the experimental data from Ref. [11].
on the time scale of order t , the long-lived, large glassy domains are responsible for the nonexponential decay of KWW type. 4. Equilibrium case Next, we discuss the equilibrium case, where (r; t=0)=e (z0 =1:0). Since the total volume fraction (r; t) is given by (r; t) = e + (r; t), near the glass transition, one can write the di usion coeGcient DS () given by Eq. (2.3) as DS ((r; t)) ˙ D0 |& + (r; t)=g |2 , where & = e =g − 1 is the separation parameter and its magnitude is small near g . As long as the relative magnitude |=g | is the same order as |&|, therefore, Eq. (2.1) reduce to the cubic nonlinear stochastic di usion equation, up to order |=g |3 , [12–14] @ (r; t) = ∇2 [DSL (e )(r; t) + &A(e )(r; t)2 @t +B(e )(r; t)3 ] + (r; t) :
(4.1)
Here the random force (r; t) satisJes [14]
(r; t)(r ; t ) = −2(t − t )∇2 [DSL (e ) (r)(r ) +(1=2)&A(e ) (r)2 (r ) ] ;
(4.2)
M. Tokuyama et al. / Physica A 307 (2002) 27 – 40
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where the brackets denote the average over an equilibrium ensemble. Here the coeGcients A(e ) and B(e ) are given by A = DSS
(1 − 9e =32)ds ; g (ds + &2 )2
B=
(ds − 3&2 )A ; 3g (ds + &2 )
(4.3)
where ds = e DSS =(g D0 ). Eq. (4.1) is a nonlinear stochastic equation that describes the slow dynamics of the equilibrium density "uctuations in real space near the glass transition. Similarly to the nonequilibrium case, the numerical calculation of Eq. (4.1) shows that the aggregates of glassy domains with (r; t) ¿ g are formed. Fig. 4 shows the snapshots of the nonlinear density "uctuations on x–y plane for di erent volume fractions, e = 0:5704 and 0.502. The snapshot at e = 0:502 shows a random conJguration in space, while the snapshot at e = 0:5704 seems to show the clusters of the glassy domains. In order to see this point clearly, we also calculate Eq. (4.1) without the nonlinear terms. This is because the solutions of the linear stochastic equation for (r; t) show random conJgurations in space. In fact, Fig. 5 shows the snapshots of the linear density "uctuations on an x–y plane for di erent volume fractions, e = 0:5704 and 0.502. The di erence between two types of conJgurations is now obvious for higher volume fractions, where the nonlinear "uctuations play an important role in forming clusters. On the other hand, there is no clear di erence between them for lower volume fractions, where the nonlinear "uctuations are negligible. Although, it might be diGcult to observe such spatial heterogeneities experimentally, they are important because they are the origin of the slow relaxation of the equilibrium density "uctuations. Finally, we should mention here that the static correlation functions appearing in Eq. (4.2) are in general not known. In order to perform the above numerical calculations of Eq. (4.1), therefore, we have simply assumed as a Jrst approximation that (r)(r ) (4 a30 =3)e (r−r ) and (r)2 (r ) 0. Since the static paircorrelation function is not in general -correlated near the glass transition, however, the random force (r; t) would be more strongly correlated in space. Hence more appreciable aggregates than those shown in Fig. 4 are expected to be seen. This will be discussed elsewhere. In contrast to the spatial heterogeneities in the nonequilibrium case, in the equilibrium case only the glassy domains related to the -relaxation process seem to survive because the small domains related to the -relaxation process are easily destroyed by "uctuations, although the glassy domains are generated by the nonlinear density "uctuation themselves. In order to see this situation clearly, we
∞ next calculate the self-part of the dynamic susceptibility given by *S (k; t) = ! 0 FS (k; t) cos(!t) dt, where the self-intermediate scattering function FS (k; t) is now given by FS (k; t) =
(k; t)(−k; 0) =(4 a30 =3)2 . To Jnd an equation for the self-intermediate scattering function FS (k; t) from Eq. (4.1), we simply use a mean-Jeld approach discussed elsewhere [12–14]. Since FS (k; t) is related to the mean-square displacement M2 (t) through the relation FS (k; t) = exp [ − k 2 M2 (t)=6] [20], one can thus derive a nonlinear equation for M2 (t) d (4.4) M2 (t) = 6D SL (e ) + 6[DSS (e ) − D SL (e )]e−-(e )M2 (t) ; dt
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M. Tokuyama et al. / Physica A 307 (2002) 27 – 40
Fig. 4. The snapshots of the nonlinear density "uctuations, projected onto a x–y plane, at e = 0:5704 and 0.502. The glassy, the supercooled, and the liquid states are colored red, yellow, and blue, respectively.
Fig. 5. The snapshots of the linear density "uctuations, projected onto a x–y plane. The details are the same as in Fig. 4.
where -(e ) is a free parameter and is determined from the Jtting with experimental data. Eq. (4.4) is easily solved to give DS 1 M2 (t) = ln 1 + SL {exp (t=t ) − 1} ; (4.5) DS −a DS FS (k; t) = 1 + SL {exp (t=t ) − 1} ; (4.6) DS where the -relaxation time t is given by t (k; e ) = 1=(6-DSL ) and a(k; e ) = k 2 =(6-).
M. Tokuyama et al. / Physica A 307 (2002) 27 – 40
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For short times t t0 = 1=(6-DSS ), Eq. (4.5) reduces to M2 (t) 6DSS t, while for long times tt , it reduces to M2 (t) 6DSL t. Thus, Eq. (4.4) describes the dynamics of a crossover from a short-time di usion process to a long-time di usion process. In fact, as discussed in Ref. [13], there are four characteristic time stages near g . The Jrst is the early stage where t 6 t0 . The scattering function FS (k; t) obeys the exponential decay given by Eq. (3.6). The second is the -relaxation stage, where t0 t 6 t . Similarly to the nonequilibrium case, the scattering function FS (k; t) obeys two kinds of power-law decays near g . In the fast -relaxation stage where t0 6 t 6 t , it obeys a critical decay FS (k; t) = (1 + t=t0 )−a Q0 − Q1 ln(D0 t=a20 ) + O((ln(D0 t=a20 ))2 ) ;
(4.7)
where Q0 (k; e ) = (1 + a20 =D0 t0 )−a and Q1 (k; e ) = aQ0 =(1 + D0 t0 =a20 ). We note here that the scattering function FS (k; t) can be approximately described by the logarithmic decay. This decay continues up to the time scale of order t . In the slow -relaxation stage where t 6 t 6 t , FS (k; t) obeys the power law of the von Schweidler type FS (k; t) = (1 + t=t0 )−a − Qs (k; e )(t=t )b ;
(4.8)
where b(k; e ) is a time exponent to be determined, and Qs (k; e ) is a positive constant. This decay continues up to the time scale of order t . After this stage, the dynamics of the density "uctuations are governed by the large aggregates of the glassy domains. This is the so-called -relaxation stage where t 6 t 6 tL . The scattering function obeys the stretched exponential decay of KWW type FS (k; t) = (1 + t=t0 )−a exp [ − (t=t ) ] ;
(4.9)
where is an exponent to be determined. Here the -relaxation time t is deJned by the time which satisJes the relation FS (k; t ) = (1 + t =t0 )−a e−1 . We should mention here that the exponent is expected to be close to 1 since the size and magnitude of spatial heterogeneities are very small compared to those in a nonequilibrium case. After this stage, the glassy domains becomes random and the long-time self-di usion process dominates the system. This is the late stage where t ¿ tL . The scattering function obeys the exponential decay given by Eq. (3.9). One can now adjust the free parameter -(e ) so that the solution (4.5) Jts with experimental data in equilibrium systems. In Fig. 3 the numerical result with -(e = 0:5704) 52:62 [13] is thus shown together with a comparison with the experimental data [11]. In Fig. 6 the characteristic decays described by Eqs. (4.7) – (4.9) are also shown at e = 0:5704 and ka0 = 1:3 together with the theoretical result given by Eq. (4.6) and the experimental data. In an equilibrium case the logarithmic decay turns out to be one of characteristic decays. In Fig. 7 we show a log–log plot of the self-part of the dynamic susceptibility *S (k; !) versus frequency ! near the glass transition volume fraction both in an equilibrium case (z = 1:0) and in a slightly nonequilibrium case (z0 = 0:8). In a nonequilibrium case near the glass transition, there are two peaks, the so-called and peaks, and one minimum. This is also clearly seen in Fig. 3 as a two-step relaxation. On the other hand, in an equilibrium case the peak and the
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M. Tokuyama et al. / Physica A 307 (2002) 27 – 40
Fig. 6. Characteristic decays of FS (k; t) versus time at e = 0:5704 and ka0 = 1:3. The solid line indicates Eq. (4.6), the dot–dashed line the logarithmic decay given by Eq. (4.7), the dashed line the von Schweidler decay given by Eq. (4.8), and the dotted KWW type given by Eq. (4.9), where -=52:62; a=0:0054; b=0:972, and = 0:95. The open circles indicate the experimental data from Ref. [11].
Table 1 Time exponents for di erent values z0 at e = 0:5704 and ka0 = 1:3 z0
%(= =)
bf
bs
b
0. 8 1. 0
2.0 2.0
0.59 0.95
3.41 2.10
1.00 1.59
0.37 —
0.72 —
— 0.97
minimum seem to disappear, leading to the so-called excess wing. In Fig. 6 this is seen as the logarithmic decay followed by the power-law decay of the von Schweidler-type with the time exponent b close to one. Hence these situations are quite di erent from those in a nonequilibrium case (see Table 1). This is mainly because in an equilibrium case the small glassy domains are easily destroyed by the equilibrium "uctuations and only the large glassy domains can survive as aggregates, although those domains are both generated by the nonlinear "uctuations themselves.
M. Tokuyama et al. / Physica A 307 (2002) 27 – 40
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Fig. 7. A log–log plot of *S (k; !) versus frequency ! at ka0 = 1:3 for e = 0:5704: The dotted line indicates the results for a slightly nonequilibrium case at z0 = 0:8, and the solid line for an equilibrium case at z0 = 1:0.
5. Conclusions We have shown that in an equilibrium case the small glassy domains related to the -relaxation process are not stable compared to those in a nonequilibrium case because those patterns are easily destroyed by the "uctuations. Hence the -relaxation process is weakened in an equilibrium case, leading to a disappearance of a peak and a formation of an excess wing. Thus, the logarithmic decay followed by the power-law decay of von Schweidler type was shown to hold. On the other hand, the large glassy domains related to the -relaxation process are stable and lead to the stretched exponential decay of KWW type, although the stretched exponent is rather closer to one than that in a nonequilibrium case. Thus, the relaxation process of the density "uctuations is strongly related to the spatial pattern of the glassy domains. In an equilibrium suspension the nonlinear "uctuations cause the spatial heterogeneities near the glass transition and their existence is indispensable to discuss the slow relaxation of the density "uctuations. On the other hand, in a nonequilibrium suspension, the spatial heterogeneities are described by the nonlinear deterministic equation and in"uence the dynamics of the linear density "uctuations signiJcantly near the glass transition. More detailed analyses of those spatial patterns will be discussed elsewhere together with their correlation lengths. References [1] [2] [3] [4] [5]
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M. Tokuyama et al. / Physica A 307 (2002) 27 – 40 M. Tokuyama, Y. Enomoto, I. Oppenheim, Physica A 270 (1999) 380. W. Kob, H.C. Andersen, Phys. Rev. Lett. 73 (1994) 1376. P. Allegrini, J.F. Douglas, S.C. Glotzer, Phys. Rev. E 60 (1999) 5714. P.N. Pusey, W. van Megen, Nature (London) 320 1986 (340). W. van Megen, S.M. Underwood, Phys. Rev. E 49 (1994) 4206. W. van Megen, T.C. Mortensen, S.R. Williams, J. MPuler, Phys. Rev. E 58 (1998) 6073. M. Tokuyama, Phys. Rev. E 62 (2000) R 5915. M. Tokuyama, Physica A 289 (2001) 57. M. Tokuyama, Physica A 294 (2001) 23. M. Tokuyama, I. Oppenheim, Physica A 216 (1995) 85. B. Cichocki, K. Hinsen, Physica A 187 (1992) 133. D.R. Foss, J.F. Brady, J. Rheol. 44 (2000) 629. B. Doliwa, A. Heuer, Phys. Rev. E 61 (2000) 6898. M. Tokuyama, Prog. Theor. Phys. 126 (Suppl.) (1997) 43. P.N. SegrQe, P.N. Pusey, Phys. Rev. Lett. 77 (1996) 771. M. Tokuyama, in: M. Tokuyama, H.E. Stanley (Eds.), Statistical Physics, AIP, New York, 2000.
www.elsevier.com/locate/physa
E ects of spatial heterogeneities on the slow dynamics of density "uctuations near the colloidal glass transition Michio Tokuyamaa;∗ , Yayoi Teradab , Irwin Oppenheimc a Institute
of Fluid Science, Tohoku University, Sendai 980-8577, Japan of Fluid Science, Tohoku University, Sendai 980-8577, Japan c Department of Chemistry, Massachusetts Institute of Technology, Cambridge, MA 02139, USA b Institute
Received 1 October 2001
Abstract The e ects of spatial heterogeneities on the slow dynamics of density "uctuations are discussed both in an equilibrium colloidal suspension and in a slightly nonequilibrium colloidal suspension. The long-lived, heterogeneous glassy domains appear near the glass transition and in"uence the dynamics of density "uctuations, leading to the - and -relaxation processes. The spatial heterogeneities related to the process are less noticeable in an equilibrium case than in a nonequilibrium case. Thus, a logarithmic decay is shown to be a new critical decay in an equilibrium case, instead of the power-law decay followed by the von Schweidler decay in a c 2002 Elsevier Science B.V. All rights reserved. nonequilibrium case. PACS: 82.70.Dd; 05.40.+j; 51.10.+y Keywords: Hard-sphere suspensions; Long-range hydrodynamic interactions; Nonlinear density "uctuations; Spatial heterogeneities; Slow dynamics
1. Introduction The important role of spatially heterogeneous structure in the slow dynamics of density "uctuations near the glass transition has been pointed out in glass-forming materials for the last decade [1–5]. Recently, Tokuyama et al. [6] have shown that in the nonequilibrium colloidal suspensions of hard spheres the spatial heterogeneities a ect the dynamics of density "uctuations near the colloidal glass transition, leading ∗
Corresponding author. Tel.: +81-22-217-5327; fax: +81-22-217-5327. E-mail address: [email protected] (M. Tokuyama).
c 2002 Elsevier Science B.V. All rights reserved. 0378-4371/02/$ - see front matter PII: S 0 3 7 8 - 4 3 7 1 ( 0 1 ) 0 0 5 7 9 - 9
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M. Tokuyama et al. / Physica A 307 (2002) 27 – 40
to the two distinct slow-relaxation processes around the -relaxation time t and the -relaxation time t . Near the glass transition, the small glassy domains are formed on the time scale of order t and lead to the power-law decay of the von Schweilder type for the intermediate scattering function. On the other hand, on the time scale of order t , the long-lived, large glassy domains are formed. The stretched exponential decay of the Kohlrausch–Williams–Watts (KWW) type is well explained by the existence of those large domains. Thus, the spatial heterogeneities were shown to be responsible for the slow dynamics of the nonequilibrium density "uctuations near the glass transition. Such nonequilibrium e ects are also observable as the waiting time e ect in experiments and simulations [7,8]. Most of the experiments in colloidal suspensions have been done in an equilibrium state, which is attained for long times after the quenched, and recovered long-known phenomena, such as the stretching of the process, similar to those in glass-forming materials near the glass transition [9 –11]. In contrast to a nonequilibrium case, however, it would be diGcult to observe the spatial heterogeneities experimentally in an equilibrium case. This is mainly because the equilibrium "uctuations can easily destroy those heterogeneities even though those heterogeneities are generated by nonlinear density "uctuations themselves [12–14]. In fact, the large glassy domains related to the -relaxation process can survive, while the small glassy domains are easily destroyed by the "uctuations. Hence the -relaxation process is expected to be less apparent in an equilibrium suspension than in a nonequilibrium suspension. In this paper, we compare the spatial heterogeneities obtained numerically in an equilibrium suspension with those obtained in a nonequilibrium suspension and then show that the in"uence of spatial heterogeneities on the dynamics of the density "uctuations are less noticeable in an equilibrium suspension than in a nonequilibrium suspension. In this paper, we discuss a concentrated suspension of neutral hard spheres. There exist two kinds of interactions between particles. One is the direct interaction between particles. The other is the hydrodynamic interaction between particles. For short times, the hydrodynamic interactions simply lead to corrections to the single-particle di usion coeGcient D0 , giving the volume fraction dependence of the short-time self-di usion coeGcient DSS (e ). For longer times, however, the long-range hydrodynamic interactions become important through the many-body correlations. Especially for larger volume fractions, they lead to the dynamic anomaly of the di usion coeGcient, in which the di usion coeGcient becomes zero at the volume fraction g , which we call the glass transition volume fraction. Near g , therefore, they in"uence the dynamics of density "uctuations signiJcantly [15]. Thus, the long-time processes are governed by the many-body correlations between particles, which also leads to the dynamic anomaly of the long-time self-di usion coeGcient DSL (e ). In order to see the importance of the many-body correlations, we plot the long-time self-di usion coeGcient DSL (e ) in Fig. 1 together with the experimental data. For comparison, the computer simulation results without the hydrodynamic interactions between particles are also shown. The large deviations of the simulation results from the experimental data are seen. This is due to the lack of hydrodynamic interactions. In order to check the e ect of the short-time self-di usion process on the long-time process, we also plot the results obtained by
M. Tokuyama et al. / Physica A 307 (2002) 27 – 40
29
Fig. 1. The long-time self-di usion coeGcient DSL versus e . The solid line indicates the theoretical results from Ref. [15]. The symbol () indicates the experimental data from Ref. [11]. The symbols (4); ( ), and ( ) indicate the simulation results from Refs. [16 –18], respectively. The open symbols (); ( ), and ( ) indicate the results obtained by multiplying the simulation results (4); ( ), and ( ) by DSS . The dotted and the dashed lines indicate the glass transition volume fraction g and the melting volume fraction m , respectively.
•
•
◦
multiplying the simulation results by the theoretical values of DSS (e ) [15]. Even in this case, there still exist distinct deviations especially for volume fractions higher than the melting volume fraction m = 0:545, while for small volume fractions the correlation e ects are negligible. Hence those deviations are expected to be due to lack of many-body correlations through the long-range hydrodynamic interactions. We show that those correlations play an important role in forming long-lived, spatial heterogeneities and lead to two distinct slow relaxation processes, the - and -relaxation processes. In Section 2, we summarize and discuss the basic nonlinear di usion equations and concepts. In Section 3 we discuss the numerical results obtained in a slightly nonequilibrium suspension. In Section 4 we discuss the numerical results obtained in an equilibrium suspension. We thus explore the main di erence between the spatial heterogeneities in an equilibrium suspension and those in a nonequilibrium suspension
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M. Tokuyama et al. / Physica A 307 (2002) 27 – 40
and show how such a di erence a ect the dynamics of density "uctuations near the glass transition. Section 5 is devoted to conclusions. 2. A nonlinear stochastic diusion equation We consider a concentrated, neutral hard-sphere suspension with both the hydrodynamic and the direct interactions between particles. Let e (=4 a30 neq =3) denote the particle volume fraction, where neq (=N=V ) is the equilibrium particle number density, N and V being the total number of colloidal particles and the total volume of the system, respectively. In this paper, we focus only on a suspension-hydrodynamic stage [15], where the space–time cuto s (rc ; tc ), which are the minimum wavelength and time of the dynamic process of interest, are set as rc ¿ a0 and tc ¿ tD , where tD = a20 =D0 is the structural-relaxation time which is a time required for a particle to di use over a distance a0 . Here D0 is a di usion constant of a single particle. The relevant variable here is the slowly-varying, local volume fraction given by [15] N
(r; t) =
4 3 a (Xi (t) − r) ; 3 0
(2.1)
i=1
where Xi (t) is the position vector of particle i and (r) denotes the coarse-grained function given by (r) = (1=V ) k exp (−ik · r) with (|k| 6 1=a0 ). We should mention here that the other hydrodynamic variables, such as the velocity density and the energy density are rapidly varying variables and can be written in terms of (r; t) [15]. Hence, we start with the following nonlinear stochastic di usion equation already described elsewhere [14]: @ (r; t) = ∇ · [DS ((r; t))∇(r; t)] + (r; t) @t
(2.2)
with the self-di usion coeGcient DS () = DSS ()
1 − 9=32 ; 1 + (DSS =g D0 )(1 − =g )−2
(2.3)
where (r; t) is the Gaussian, Markov random force and satisJes (r; t)(r ; t ) = −2(t − t )∇ · [DS ((r; t))∇(r)(r ) −(1=2)(r)(r )∇DS ((r; t))] :
(2.4)
Here the bar denotes the average over an appropriate initial ensemble. The glass transition volume fraction g is given by g = (4=3)3 =(7 ln 3 − 8 ln 2 + 2) 0:57184 : : : :
(2.5)
Here we note that g is in a good agreement with the experimental values of 0.571– 0.580 [10,11]. The short-time self-di usion coeGcient DSS () is given by [15] DSS () = D0 =[1 + L()]
(2.6)
M. Tokuyama et al. / Physica A 307 (2002) 27 – 40
31
with the nonmemory e ect
6xy 2x2 y 2xy 1− L() = − + − 1−x+y 1 − x + y + 4xy 1 − x 1 + 2y xy2 2xy + + 1 − x + y + 2xy (1 + y)(1 − x + y) xy2 3xy2 ; − × 1+ (1 + y)(1 − x + y) − xy2 (1 + y)(1 − x + y) − 2xy2 (2.7) 1=2
where x() = (9=8) and y() = 11=16. The nonmemory e ect L() results from the many-body static e ect due to the short- and the long-range hydrodynamic interactions. The factor 9=32 in the numerator of Eq. (2.3) comes from the many-body couplings between the two-body direct interactions and the short-range hydrodynamic interactions, where the factor reduces to 2 if the short-range hydrodynamic interactions are not taken into account. The singular term in the denominator of Eq. (2.3) results from the many-body correlation e ect due to the long-range hydrodynamic interactions. Here we should mention that the long-range interaction means an interaction which leads to a divergent integral. Eq. (2.2) is a starting equation to study the dynamics of hard-sphere suspensions from a liquid state to a glass state. The most important feature of that equation is that the self-di usion coeGcient becomes dynamically anomalous near the glass transition g as DS ((r; t)) ˙ D0 |1 − (r; t)=g | ;
(2.8)
where = 2 here. Hence the di usion coeGcient DS ((r; t)) becomes smaller and smaller as time goes on, showing a crossover from the short-time self-di usion process characterized by the short-time self-di usion coeGcient DSS (e ) to the long-time self-di usion process characterized by the long-time self-di usion coeGcient DSL (e ) = DS (e ). We should note here that the distinct slow relaxation processes, such as the and -relaxation processes, are always recovered for whatever original mechanisms if the di usion coeGcient DS ((r; t)) shows the dynamic anomaly discussed above [21]. In the following sections, we discuss the numerical solutions of Eq. (2.2) in two cases separately, an equilibrium case and a slightly nonequilibrium case. 3. Nonequilibrium case It is convenient to split the local volume fraction (r; t) into two parts, an average part (r; t) = (r; t) and a "uctuation (r; t); (r; t) = (r; t) + (r; t). This is because the glass transition is not a critical phenomenon since there is no correlation length diverging even at the transition point. As long as the system is away from a critical point, therefore, the relative magnitude of the density "uctuations to the mean density should be small even near the glass transition point, where |=| 1.
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M. Tokuyama et al. / Physica A 307 (2002) 27 – 40
Since (r; t = 0) = e in a nonequilibrium case, Eq. (2.2) can be decomposed into the following two types of equations: @ (r; t) = ∇ · [DS ((r; t))∇(r; t)] ; @t @ (r; t) = ∇ · [DS ((r; t))∇(r; t)] + (r; t) @t with the conservation laws 1 1 dr (r; t) = 0 ; dr (r; t) = e ; V V
(3.1) (3.2)
(3.3)
where the random force (r; t) satisJes the "uctuation–dissipation relation [14] (r; t)(r ; t ) = −2(t − t )∇ · [DS ((r; t))∇(r)(r ) −(1=2)(r)(r )∇DS ((r; t))] :
(3.4)
The dynamics of spatial heterogeneities is then described by the nonlinear deterministic di usion equation (3.1). The dynamics of the nonequilibrium density "uctuations obeys the linear stochastic di usion equation (3.2) and is in"uenced by the spatial heterogeneities through the di usion coeGcient DS ((r; t)). The intermediate scattering function F(k; t) is given by the Fourier transform of the autocorrelation function of the density "uctuations F(r; t)=(r; t)(0; 0)=[(4 a30 =3)2 N ]. For scattering vectors much larger than the maximum position km of the structure factor S(k) = F(k; 0), the scattering function F(k; t) reduces to the self-intermediate scattering function FS (k; t), where S(k)=1. Hence FS (k; t) can be easily calculated from Eq. (3.2) since (r; t)(r ; 0)=0. In order to solve Eq. (3.1) numerically, one must Jx the values of two parameters e and z0 , as the initial conditions. Here the state parameter z0 measures how the system is spatially nonuniform initially and is given by (3.5) z0 = 1 − (1=V ) dr|1 − (r; t)=e | ; where z0 = 1:0 in an equilibrium case. Suppose that we start from a completely random nonequilibrium conJguration. Then the smoothing process of (r; t) starts to occur, following Eq. (3.1), and (r; t) Jnally reaches the equilibrium volume fraction e for long times of order tL (=a20 =DSL ). As discussed in Ref. [6], near the glass transition there are four characteristic time stages. The Jrst is the early stage where t t =a20 =DSS (e ). The spatial conJgurations are random and are described by (r; t)=exp [tDSS (e )∇2 ](r; 0). The self-intermediate scattering function FS (k; t) obeys an exponential decay FS (k; t) = exp [ − k 2 DSS (e )t] ;
(3.6) DSS .
from which one can determine the short-time self-di usion coeGcient After this stage, the Jnite-sized, glassy domains with (r; t) ¿ g are formed for volume fractions larger than the crossover volume fraction [19]. The smoothing process of (r; t) is then slowing down due to those domains. On the time scale of order t , therefore, those glassy domains a ect the dynamics of the density "uctuations. This is the so-called -relaxation stage where t t 6 t . Depending on the time scale, the scattering function
M. Tokuyama et al. / Physica A 307 (2002) 27 – 40
33
Fig. 2. Snapshots of typical conJgurations of glassy domains in an x–y plane for times t ∼ 4a20 =D0 , t ∼ 800a20 =D0 , and t ∼ 2 × 104 a20 =D0 at e = 0:5704 and z0 = 0:8. The glassy domains with (r; t) ¿ g are colored black.
FS (k; t) obeys two types of power-law decays. In the fast -relaxation stage where t t 6 t , the small aggregates of glassy domains are formed and a ect the dynamics of the density "uctuations, leading to the critical decay with the exponent bf (k; e ; z0 ) FS (k; t) = fc (k; e ; z0 ) − f1 (k; e ; z0 )(t=t )bf ;
(3.7)
where fc and f1 are positive constants. This power-law decay continues up to the crossover time t . In the slow -relaxation stage where t 6 t 6 t , the aggregates grow to the larger clusters and lead to the power-law decay of von Schweilder type with the exponent bs (k; e ; z0 ) FS (k; t) = fc (k; e ; z0 ) − f2 (k; e ; z0 )(t=t )bs ;
(3.8)
where f2 is a positive constant. After this stage, the glassy domains further grow to larger clusters and continue up to the time scale of order t . Those domains in"uence the dynamics of the density "uctuations leading to the stretched exponential decay of KWW type FS (k; t) = f3 (k; e ; z0 ) exp [ − (t=t ) ] ;
(3.9)
where (e ; z0 ) is a stretched exponent to be determined, and f3 is a positive constant. This is the so-called -relaxation stage where t 6 t 6 tL =a20 =DSL . Here the -relaxation time t is deJned by the time which satisJes the relation FS (k; t ) = f3 e−1 and satisJes t ˙ |1 − e =g |−% with % = = [6]. After this stage, the glassy domains start to disappear very slowly and the system gradually reaches the equilibrium state on the time scale of order tL . Then, the spatial conJgurations become random. Hence the density "uctuations obey the exponential decay FS (k; t) = exp [ − k 2 DSL (e )t] ;
(3.10) DSL .
This is the from which one can determine the long-time self-di usion coeGcient late stage where t ¿ tL . In Fig. 2 the numerical solutions of Eq. (3.1) under a slightly nonequilibrium initial condition are shown at four di erent characteristic times, t , t , t , and tL . The self-intermediate scattering function FS (k; t) is also shown in Fig. 3. Thus, it turns out from Figs. 2 and 3 that on the time scale of order t , the small glassy domains are responsible for the power-law decay of von Schweidler type, while
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M. Tokuyama et al. / Physica A 307 (2002) 27 – 40
Fig. 3. Self-intermediate scattering function FS (k; t) versus time at e = 0:5704 and ka0 = 1:3. The dotted line indicates the results for a slightly nonequilibrium case at z0 = 0:8, the solid line the results for an equilibrium case at z0 = 1:0, and the open circles the experimental data from Ref. [11].
on the time scale of order t , the long-lived, large glassy domains are responsible for the nonexponential decay of KWW type. 4. Equilibrium case Next, we discuss the equilibrium case, where (r; t=0)=e (z0 =1:0). Since the total volume fraction (r; t) is given by (r; t) = e + (r; t), near the glass transition, one can write the di usion coeGcient DS () given by Eq. (2.3) as DS ((r; t)) ˙ D0 |& + (r; t)=g |2 , where & = e =g − 1 is the separation parameter and its magnitude is small near g . As long as the relative magnitude |=g | is the same order as |&|, therefore, Eq. (2.1) reduce to the cubic nonlinear stochastic di usion equation, up to order |=g |3 , [12–14] @ (r; t) = ∇2 [DSL (e )(r; t) + &A(e )(r; t)2 @t +B(e )(r; t)3 ] + (r; t) :
(4.1)
Here the random force (r; t) satisJes [14]
(r; t)(r ; t ) = −2(t − t )∇2 [DSL (e ) (r)(r ) +(1=2)&A(e ) (r)2 (r ) ] ;
(4.2)
M. Tokuyama et al. / Physica A 307 (2002) 27 – 40
35
where the brackets denote the average over an equilibrium ensemble. Here the coeGcients A(e ) and B(e ) are given by A = DSS
(1 − 9e =32)ds ; g (ds + &2 )2
B=
(ds − 3&2 )A ; 3g (ds + &2 )
(4.3)
where ds = e DSS =(g D0 ). Eq. (4.1) is a nonlinear stochastic equation that describes the slow dynamics of the equilibrium density "uctuations in real space near the glass transition. Similarly to the nonequilibrium case, the numerical calculation of Eq. (4.1) shows that the aggregates of glassy domains with (r; t) ¿ g are formed. Fig. 4 shows the snapshots of the nonlinear density "uctuations on x–y plane for di erent volume fractions, e = 0:5704 and 0.502. The snapshot at e = 0:502 shows a random conJguration in space, while the snapshot at e = 0:5704 seems to show the clusters of the glassy domains. In order to see this point clearly, we also calculate Eq. (4.1) without the nonlinear terms. This is because the solutions of the linear stochastic equation for (r; t) show random conJgurations in space. In fact, Fig. 5 shows the snapshots of the linear density "uctuations on an x–y plane for di erent volume fractions, e = 0:5704 and 0.502. The di erence between two types of conJgurations is now obvious for higher volume fractions, where the nonlinear "uctuations play an important role in forming clusters. On the other hand, there is no clear di erence between them for lower volume fractions, where the nonlinear "uctuations are negligible. Although, it might be diGcult to observe such spatial heterogeneities experimentally, they are important because they are the origin of the slow relaxation of the equilibrium density "uctuations. Finally, we should mention here that the static correlation functions appearing in Eq. (4.2) are in general not known. In order to perform the above numerical calculations of Eq. (4.1), therefore, we have simply assumed as a Jrst approximation that (r)(r ) (4 a30 =3)e (r−r ) and (r)2 (r ) 0. Since the static paircorrelation function is not in general -correlated near the glass transition, however, the random force (r; t) would be more strongly correlated in space. Hence more appreciable aggregates than those shown in Fig. 4 are expected to be seen. This will be discussed elsewhere. In contrast to the spatial heterogeneities in the nonequilibrium case, in the equilibrium case only the glassy domains related to the -relaxation process seem to survive because the small domains related to the -relaxation process are easily destroyed by "uctuations, although the glassy domains are generated by the nonlinear density "uctuation themselves. In order to see this situation clearly, we
∞ next calculate the self-part of the dynamic susceptibility given by *S (k; t) = ! 0 FS (k; t) cos(!t) dt, where the self-intermediate scattering function FS (k; t) is now given by FS (k; t) =
(k; t)(−k; 0) =(4 a30 =3)2 . To Jnd an equation for the self-intermediate scattering function FS (k; t) from Eq. (4.1), we simply use a mean-Jeld approach discussed elsewhere [12–14]. Since FS (k; t) is related to the mean-square displacement M2 (t) through the relation FS (k; t) = exp [ − k 2 M2 (t)=6] [20], one can thus derive a nonlinear equation for M2 (t) d (4.4) M2 (t) = 6D SL (e ) + 6[DSS (e ) − D SL (e )]e−-(e )M2 (t) ; dt
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M. Tokuyama et al. / Physica A 307 (2002) 27 – 40
Fig. 4. The snapshots of the nonlinear density "uctuations, projected onto a x–y plane, at e = 0:5704 and 0.502. The glassy, the supercooled, and the liquid states are colored red, yellow, and blue, respectively.
Fig. 5. The snapshots of the linear density "uctuations, projected onto a x–y plane. The details are the same as in Fig. 4.
where -(e ) is a free parameter and is determined from the Jtting with experimental data. Eq. (4.4) is easily solved to give DS 1 M2 (t) = ln 1 + SL {exp (t=t ) − 1} ; (4.5) DS −a DS FS (k; t) = 1 + SL {exp (t=t ) − 1} ; (4.6) DS where the -relaxation time t is given by t (k; e ) = 1=(6-DSL ) and a(k; e ) = k 2 =(6-).
M. Tokuyama et al. / Physica A 307 (2002) 27 – 40
37
For short times t t0 = 1=(6-DSS ), Eq. (4.5) reduces to M2 (t) 6DSS t, while for long times tt , it reduces to M2 (t) 6DSL t. Thus, Eq. (4.4) describes the dynamics of a crossover from a short-time di usion process to a long-time di usion process. In fact, as discussed in Ref. [13], there are four characteristic time stages near g . The Jrst is the early stage where t 6 t0 . The scattering function FS (k; t) obeys the exponential decay given by Eq. (3.6). The second is the -relaxation stage, where t0 t 6 t . Similarly to the nonequilibrium case, the scattering function FS (k; t) obeys two kinds of power-law decays near g . In the fast -relaxation stage where t0 6 t 6 t , it obeys a critical decay FS (k; t) = (1 + t=t0 )−a Q0 − Q1 ln(D0 t=a20 ) + O((ln(D0 t=a20 ))2 ) ;
(4.7)
where Q0 (k; e ) = (1 + a20 =D0 t0 )−a and Q1 (k; e ) = aQ0 =(1 + D0 t0 =a20 ). We note here that the scattering function FS (k; t) can be approximately described by the logarithmic decay. This decay continues up to the time scale of order t . In the slow -relaxation stage where t 6 t 6 t , FS (k; t) obeys the power law of the von Schweidler type FS (k; t) = (1 + t=t0 )−a − Qs (k; e )(t=t )b ;
(4.8)
where b(k; e ) is a time exponent to be determined, and Qs (k; e ) is a positive constant. This decay continues up to the time scale of order t . After this stage, the dynamics of the density "uctuations are governed by the large aggregates of the glassy domains. This is the so-called -relaxation stage where t 6 t 6 tL . The scattering function obeys the stretched exponential decay of KWW type FS (k; t) = (1 + t=t0 )−a exp [ − (t=t ) ] ;
(4.9)
where is an exponent to be determined. Here the -relaxation time t is deJned by the time which satisJes the relation FS (k; t ) = (1 + t =t0 )−a e−1 . We should mention here that the exponent is expected to be close to 1 since the size and magnitude of spatial heterogeneities are very small compared to those in a nonequilibrium case. After this stage, the glassy domains becomes random and the long-time self-di usion process dominates the system. This is the late stage where t ¿ tL . The scattering function obeys the exponential decay given by Eq. (3.9). One can now adjust the free parameter -(e ) so that the solution (4.5) Jts with experimental data in equilibrium systems. In Fig. 3 the numerical result with -(e = 0:5704) 52:62 [13] is thus shown together with a comparison with the experimental data [11]. In Fig. 6 the characteristic decays described by Eqs. (4.7) – (4.9) are also shown at e = 0:5704 and ka0 = 1:3 together with the theoretical result given by Eq. (4.6) and the experimental data. In an equilibrium case the logarithmic decay turns out to be one of characteristic decays. In Fig. 7 we show a log–log plot of the self-part of the dynamic susceptibility *S (k; !) versus frequency ! near the glass transition volume fraction both in an equilibrium case (z = 1:0) and in a slightly nonequilibrium case (z0 = 0:8). In a nonequilibrium case near the glass transition, there are two peaks, the so-called and peaks, and one minimum. This is also clearly seen in Fig. 3 as a two-step relaxation. On the other hand, in an equilibrium case the peak and the
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M. Tokuyama et al. / Physica A 307 (2002) 27 – 40
Fig. 6. Characteristic decays of FS (k; t) versus time at e = 0:5704 and ka0 = 1:3. The solid line indicates Eq. (4.6), the dot–dashed line the logarithmic decay given by Eq. (4.7), the dashed line the von Schweidler decay given by Eq. (4.8), and the dotted KWW type given by Eq. (4.9), where -=52:62; a=0:0054; b=0:972, and = 0:95. The open circles indicate the experimental data from Ref. [11].
Table 1 Time exponents for di erent values z0 at e = 0:5704 and ka0 = 1:3 z0
%(= =)
bf
bs
b
0. 8 1. 0
2.0 2.0
0.59 0.95
3.41 2.10
1.00 1.59
0.37 —
0.72 —
— 0.97
minimum seem to disappear, leading to the so-called excess wing. In Fig. 6 this is seen as the logarithmic decay followed by the power-law decay of the von Schweidler-type with the time exponent b close to one. Hence these situations are quite di erent from those in a nonequilibrium case (see Table 1). This is mainly because in an equilibrium case the small glassy domains are easily destroyed by the equilibrium "uctuations and only the large glassy domains can survive as aggregates, although those domains are both generated by the nonlinear "uctuations themselves.
M. Tokuyama et al. / Physica A 307 (2002) 27 – 40
39
Fig. 7. A log–log plot of *S (k; !) versus frequency ! at ka0 = 1:3 for e = 0:5704: The dotted line indicates the results for a slightly nonequilibrium case at z0 = 0:8, and the solid line for an equilibrium case at z0 = 1:0.
5. Conclusions We have shown that in an equilibrium case the small glassy domains related to the -relaxation process are not stable compared to those in a nonequilibrium case because those patterns are easily destroyed by the "uctuations. Hence the -relaxation process is weakened in an equilibrium case, leading to a disappearance of a peak and a formation of an excess wing. Thus, the logarithmic decay followed by the power-law decay of von Schweidler type was shown to hold. On the other hand, the large glassy domains related to the -relaxation process are stable and lead to the stretched exponential decay of KWW type, although the stretched exponent is rather closer to one than that in a nonequilibrium case. Thus, the relaxation process of the density "uctuations is strongly related to the spatial pattern of the glassy domains. In an equilibrium suspension the nonlinear "uctuations cause the spatial heterogeneities near the glass transition and their existence is indispensable to discuss the slow relaxation of the density "uctuations. On the other hand, in a nonequilibrium suspension, the spatial heterogeneities are described by the nonlinear deterministic equation and in"uence the dynamics of the linear density "uctuations signiJcantly near the glass transition. More detailed analyses of those spatial patterns will be discussed elsewhere together with their correlation lengths. References [1] [2] [3] [4] [5]
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