- Email: [email protected]
Slow dynamics of supercooled colloidal fluids: spatial heterogeneities and nonequilibrium density fluctuations
Physica A 270 (1999) 380–402
www.elsevier.com/locate/physa
Slow dynamics of supercooled colloidal uids: spatial heterogeneities and nonequilibrium density uctuations a Statistical
M. Tokuyamaa;∗ , Y. Enomotob , I. Oppenheimc
Physics Division, Tohwa Institute for Science, Tohwa University, Fukuoka 815, Japan School of Engineering, Nagoya Institute of Technology, Nagoya 466, Japan c Department of Chemistry, Massachusetts Institute of Technology, Cambridge, MA 02139, USA b Graduate
Received 23 February 1999
Abstract The coupled di usion equations recently proposed by Tokuyama for concentrated hard-sphere suspensions are numerically solved, starting from nonequilibrium initial con gurations. The most important feature of those equations is that the self-di usion coecient DS () becomes zero at the glass transition volume fraction g as DS () ∼ D0 |1 − (x; t)= g | with = 2 where (x; t) is the local volume fraction of colloids, D0 the single-particle di usion constant, and g = ( 43 )3 =(7 ln 3 − 8 ln 2 + 2). This dynamic anomaly results from the many-body correlations due to the long-range hydrodynamic interactions. Then, it is shown how small initial disturbances can be enhanced by this anomaly near g , leading to long-lived, spatial heterogeneities. Those heterogeneities are responsible for the slow relaxation of nonequilibrium density uctuations. In fact, the self-intermediate scattering function is shown to obey a two-step relaxation around the -relaxation time t ∼ |1 − = g |−1 , and also to be well approximated by the Kohlrausch– Williams–Watts function with an exponent around the -relaxation time t ∼ |1 − = g |− , where = = , and is the particle volume fraction. Thus, the nonexponential relaxation is c 1999 Elsevier shown to be explained by the existence of long-lived, spatial heterogeneities. Science B.V. All rights reserved. PACS: 82.70.Dd; 05.40.+j; 51.10.+y Keywords: Dynamic anomaly; Hard-sphere suspensions; Nonequilibrium density uctuations; Slow dynamics; Spatial heterogeneities
∗
Corresponding author. Fax: +81-92-542-0813. E-mail address: [email protected] (M. Tokuyama)
c 1999 Elsevier Science B.V. All rights reserved. 0378-4371/99/$ - see front matter PII: S 0 3 7 8 - 4 3 7 1 ( 9 9 ) 0 0 1 7 2 - 7
M. Tokuyama et al. / Physica A 270 (1999) 380–402
381
1. Introduction Supercooled colloidal uids are known to exhibit a transition from a uid phase to a glass phase, similar to that in glass-forming liquids [1–3]. One of striking features in supercooled liquids is the nonexponential behavior with the time scale diverging near the glass-transition temperature. The time evolution of the self-intermediate scattering function associated with relaxation can be described by the Kohlrausch–Williams– Watts formula (KWW), often also called stretched exponential. There has been several studies for an explanation of this empirical function [4,5]. This stretched behavior is also observed experimentally in the hard-sphere colloidal suspension [6]. In this paper, we show that the stretched behavior in such a suspension can be explained by the presence of long-lived, nite-sized, irregularly shaped glassy domains. Most of experiments in colloidal suspensions has been done in an equilibrium state which is attained for long times after the quench. In the equilibrium state, however, it might be dicult to observe the spatial heterogeneities experimentally. This is mainly because the heterogeneities may easily be destroyed by the equilibrium density uctuations although those heterogeneities are generated by the uctuations themselves. Here we should note that the equilibrium density uctuations obey a nonlinear stochastic process near the glass transition point, although the relative magnitude of the density uctuations n(x; t) to the mean density neq is small even near the transition point; |n=neq | 1 [7,8]. In order to see the heterogeneities, therefore, Tokuyama has recently proposed a linear response-like theory based on the coupled di usion equations for the average number density n(x; t) and the nonequilibrium density uctuations n(x; t) around n(x; t) [9]. This theory is applicable to a suspension in a nonequilibrium state before equilibration after the quench, where the initial state of the system is spatially nonuniform. Since |n=n| 1; n(x; t) can then describe the dynamics of heterogeneous structure, while n(x; t) describes a linear relaxation of the nonequilibrium density uctuations on such nonuniform structure. In fact, the nonequilibrium e ect has been observed as the waiting time dependence of the intermediate scattering function [10]. The most important feature of this theory is that the self-di usion coecient DS () contained in the coupled equations becomes dynamically anomalous at the glass transition volume fraction g as DS ∼ D0 |1 − (x; t)= g | with = 2, where (x; t) = 4a3 n(x; t)=3 is the average local volume fraction, g = ( 43 )3 =(7 ln 3 − 8 ln 2 + 2) ' 0:57184 : : : , and D0 the single-particle di usion coecient. We note here that this dynamic anomaly results from the many-body correlations between particles due to the long-range hydrodynamic interactions through the Oseen tensor [11]. When the system is initially in a nonequilibrium state, this theory can describe how even the small disturbances are enhanced by the dynamic anomaly near g , leading to long-lived, cluster-like glassy domains with (x; t)¿ g , and in uence the dynamics of the density uctuations, leading to a two-step relaxation and a nonexponential decay. In Section 2, we brie y summarize and discuss the coupled di usion equations for the average local volume fraction and the self-intermediate scattering function. In Section 3,
382
M. Tokuyama et al. / Physica A 270 (1999) 380–402
we show the numerical solutions of those equations, including the mean-square displacement and the self part of the dynamic susceptibility. Section 4 is devoted to conclusions.
2. Coupled di usion equations We consider a three-dimensional hard-sphere colloidal suspension with N identical spherical particles with mass m and radius a in an incompressible uid with shear viscosity at the suspension-hydrodynamic stage where the space–time cuto s (xcuto ; tcuto ), the minimum wavelength and time of the dynamic process of interest, are set as xcuto a and S ¿tcuto tB , where S ∼ a2 =D0 denotes the structural relaxation time, which is a time required for a particle to di use over a distance a, and tB the Brownian relaxation time of the particle. Here the whole system is enclosed in a volume V . Let us de ne the single-particle number density N (x; t) at position x and time t by N (x; t) =
N X
(x − Xi (t)) ;
(2.1)
i=1
where Xi (t) (i = 1; 2; : : : ; N ) denotes the position vector from the origin to the center of the particle i. As was shown in Refs. [9,12], one can then derive the nonlinear stochastic di usion equation for N (x; t), @ N (x; t) = ∇ · [DS (Q(x; t))∇N (x; t)] + R(x; t) @t
(2.2)
with the self-di usion coecient (1 − 9Q=32) ; (2.3) [1 + QDSS (Q)=( g D0 (1 − Q= g )2 )] R and the conservation law (1=V ) dxQ(x; t) = , where Q(x; t) = (4=3)a3 N (x; t) is the local volume fraction, D0 the single-particle di usion coecient, and = (4=3)a3 neq the particle volume fraction with the equilibrium number density neq =N=V . Here R(x; t) denotes a Gaussian, Markov random force with zero mean and satis es 4 3 0 0 0 0 0 a nx nx (x − x ) ; hR(x; t)R(x ; t ); ni = 2(t − t )∇ · ∇ DS 3 DS (Q) = DSS (Q)
where the angular brackets h· · · ; ni denote the conditional average over a canonical ensemble with the value of N (0) = {N (x; 0)} being xed so as to be n = {nx }. The short-time self-di usion coecient DSS (Q) is given by DSS (Q) = D0 =[1 + L(Q)]
(2.4)
M. Tokuyama et al. / Physica A 270 (1999) 380–402
383
with the many-body static e ects due to the hydrodynamic interactions 6bc c 2bc 2bc 2b2 1− − − + L(Q) = 1 − b 1 + 2c 1 − b + c 1 − b + c + 4bc 1 − b + c + 2bc 3bc2 bc2 1+ + (1 + c)(1 − b + c) (1 + c)(1 − b + c) − 2bc2 bc2 ; (2.5) − (1 + c)(1 − b + c) − bc2 where b(Q) = (9Q=8)1=2 and c(Q) = 11Q=16. The factor (9=32)Q in the numerator of Eq. (2.3) results from the coupling between the direct interactions and the short-range hydrodynamic interactions. The second singular term in the denominator of Eq. (2.3) results from the many-body correlations between particles due to the long-range hydrodynamic interactions. We note here that the short-time self-di usion coecient DSS (Q) still depends on space and time through Q(x; t). In the previous calculations in Refs. [13,14], it was simply replaced by the constant value DSS ( ). In this paper, however, its space-time dependence is shown to play an important role in smoothing the glassy phase with Q(x; t)¿ g . In order to solve Eq. (2.2) numerically, in the following we focus only on the case where the initial state of the system is out of equilibrium. The case where it is in equilibrium will be discussed elsewhere [8]. Similarly to van Kampen’s theory of
uctuations [15], let us decompose the number density N (x; t) into the average number density n(x; t) = hN (x; t); n0 i and a uctuating part n(x; t) as N (x; t) = n(x; t) + n(x; t) with the conservation laws Z (1=V ) dx n(x; t) = neq ;
(2.6) Z dx n(x; t) = 0 ;
(2.7)
where the initial values n0 = {nx;0 } are set in an appropriate nonequilibrium state. This decomposition is essential since the relative magnitude of the density uctuations n(x; t) to the causal part n(x; t) is usually small in a system away from a critical point; |n=n| 1. The dynamics of spatial heterogeneities of colloidal suspensions is described by the average local volume fraction (x; t)=hQ(x; t); n0 i. On the other hand, the dynamics of density uctuations can be measured by dynamic light scattering through the intermediate scattering function [16] which is given by F(k; t) = hnk (t)n∗k (0); n0 i=S(k) with the static√structure factor S(k) = h|nk (0)|2 ; n0 i, where nk (t) is a Fourier transform of n(x; t)= N . For scattering vectors larger than the maximum position km of S(k), the scattering function F(k; t) reduces to the self-intermediate scattering function FS (k; t), where S(k) = 1. In the suspension-hydrodynamic stage, therefore, we start with the
384
M. Tokuyama et al. / Physica A 270 (1999) 380–402
following coupled di usion equations already described elsewhere [9]: @ (x; t) = ∇ · [DS ((x; t))∇(x; t)] ; @t
(2.8)
X @ DS (k − q; t)FS (q; t) FS (k; t) = −k 2 @t q
(2.9)
with the Fourier transform, DS (k; t), of the self-di usion coecient DS ((x; t)). Eq. (2.8) describes the dynamics of heterogeneous structure from nonuniform initial con guration with (x; 0) to a nal con guration with (x; ∞) = . Eq. (2.9) describes a linear relaxation of the density uctuations around the nonuniform state determined by (x; t). The most important feature of those equations is that the diffusion coecient DS () becomes zero at g as DS () ∼ D0 (1 − = g )2 . For short times t t ; DS () is shown to reduce to DSS ( ) since the direct interactions and the correlations are negligible, where t ∼ a2 =DSS denotes the characteristic time of the short-time self-di usion process. In fact, the numerical calculation of Eqs. (2.8) and (2.9) shows that DS (k; t) ' DSS ( )k;0 within errors. For long times t tL , it reduces to the long-time self-di usion coecient DSL ( ) since (x; t) reaches , where tL ∼ a2 =DSL denotes the characteristic time of the long-time self-di uction process. In Fig. 1 we plot the normalized self-di usion coecient, DS =D0 , as a function of for short and long times. A good agreement is indeed seen between the theoretical results [11] and the experimental data [17–19]. Thus, there exists a crossover from the short-time self-di usion process to the long-time self-di usion process for intermediate times, where the dynamic anomaly plays an important role. 3. Numerical results In order to solve the coupled di usion equations (2.8) and (2.9) self-consistently, we rst x the values of the two parameters, and z0 = z(0), as the initial conditions, where the state parameter z(t) measures how the system is spatially nonuniform at time t and is given by [13] Z z(t) = 1 − (1=V ) dx|1 − (x; t)= | ; (3.1) where 0 ¡ z(t) ¡ 1 for nonuniform states and z(t) = 1 for uniform states. To integrate Eqs. (2.8) and (2.9), we employ the forward Euler di erence scheme with the time step 0:01A20 =D0 and the lattice spacing 0:5a in the volume (128a)3 of the three-dimensional simulation system with periodic boundary conditions. The initial value (x; 0) is chosen at each position x from a random number with a Gaussian distribution, which is characterized by a mean value 1 and a standard deviation s, where s is adjusted so as to nd a given value of z0 . Quantities evaluated here include the local volume fraction (x; t), the mean-square displacement M2 (t), the self-intermediate scattering function FS (k; t), and the self-part of the dynamic susceptibility S00 (k; !).
M. Tokuyama et al. / Physica A 270 (1999) 380–402
385
Fig. 1. A log plot of self-di usion coecient DS ( ) versus for short and long times. The solid line indicates the long-time self-di usion coecient DSL ( ), while the dotted line is the short-time self-di usion coecient DSS ( ). The symbols indicate the experimental data from Ref. [14] (+), Ref. [15] ( and ), and Ref. [16] ( and ).
•
3.1. Average local volume fraction (x; t) We rst discuss the numerical solutions of Eq. (2.8), starting in two kinds of completely random initial con gurations; (I) a slightly nonuniform con guration with z0 = 0:8 and (II) a nearly uniform con guration with z0 = 0:95. In Figs. 2(A)–2(C) we show the space–time dependence of the local volume fraction (x; t) in case (I) for di erent volume fractions; (A) =0:543 in the normal liquid region where 06 ¡ , (B) 0.571 in the supercooled liquid region where 6 ¡ g , and (C) 0.573 in the glass region where ¿ g . Here (z0 ; ka) denotes the crossover volume fraction over which the scattering function FS (k; t) has a shoulder for intermediate times and the susceptibility S00 (k; !) has one minimum and two peaks for intermediate frequencies [20]. Figs. 3(A)–3(C) show a sequence of snapshots projected onto a plane of a typical con guration of the glassy domains where the local volume fraction (x; t) is larger than g . In Figs. 4(A)–4(C), we also show the space–time dependence of (x; t) in case (II) for di erent volume fractions; (A) = 0:543, (B) 0.571, and (C) 0.573. Figs. 5(A)–5(C) show a sequence of snapshots. In order to see the long-lived behavior qualitatively, we also calculate the time evolution of the total volume fraction of the
386
M. Tokuyama et al. / Physica A 270 (1999) 380–402
Fig. 2. Spatial dependence of the local volume fraction (x; t) at z0 = 0:8 for (A) = 0:543, (B) 0.571, and (C) 0.573 for dimensionless times: (A) (a) 1, (b) 3.4 (t ), (c) 10, (d) 102 , (e) 184 (tL ), and (f) 103 ; (B) (a) 1, (b) 3.96 (t ), (c) 102 , (d) 853 (t ), (e) 104 , (f) 2:8 × 104 (t ), (g) 105 , (h) 3:4 × 105 (tL ), and (i) 106 ; (C) (a) 1, (b) 3.96 (t ), (c) 102 , (d) 1578 (t ), (e) 104 , (f) 4:4 × 104 (t ) (g) 105 , (h) 7:3 × 105 (tL ), and (i) 106 . The system size is (128a)3 .
M. Tokuyama et al. / Physica A 270 (1999) 380–402
387
Fig. 2. Continued.
glassy domains given by Z dx ((x; t) − g )(x; t) ; g (t) = V
(3.2)
where the step function (X ) satis es (X ) = 1 for X ¿0, and (X ) = 0 for X ¡ 0. In Fig. 6 we plot g (t) versus time at z0 = 0:95 for di erent volume fractions; (A) = 0:543, (B) 0.571, and (C) 0.573. In the early stage [E] on the time scale of order t , the spatial con gurations are random. The short-time di usion process causes a smoothing of (x; t) and leads to a power law decay of g (t), as t −p , for all values of , where the exponent p is numerically found to be approximately 1=3. After this stage, the nite-sized, glassy domains seem to be formed for volume fractions ¿ and hence the smoothing process of (x; t) and the decay of g (t) are slowing down for ¿ . This is the so-called -relaxation stage [ ] on the time scale of order t , where t is the -relaxation time given by t ∼ a2 =(DSS DSL )1=2 ∼ ||−1 [21]. Thus, the system exhibit the coexistence of
uid phase with (x; t) ¡ g and glassy phase with (x; t)¿ g . In the normal region for 06 ¡ , however, there are no nite-sized, long-lived glassy domains. After this stage, the spatial rearrangement of the glassy domains starts to occur and continues up to the time scale of order t , where t is the -relaxation time discussed later. This is the so-called -relaxation stage [ ]. Hence one can assume that the local volume
388
M. Tokuyama et al. / Physica A 270 (1999) 380–402
Fig. 3. A sequence of snapshots, projected onto a plane, of a typical con guration of glassy domains. Details are the same as in Fig. 2. The glassy phase is colored black.
M. Tokuyama et al. / Physica A 270 (1999) 380–402
389
Fig. 3. Continued.
fraction (x; t) is scaled for ¿ as (x; t) = [1 + (t x)] ;
R
(3.3)
where (z0 ) is a small exponent to be determined, and dx (x) = 0. Use of Eqs. (3.1) and (3.3) then leads to 1 − z(t) ∼ t −d . From the numerical calculations, therefore, we nd (0:95) ' 0:086 and (0:8) ' 0:15. After this stage, in the supercooled region the heterogeneous domains start to be dissolved, disappearing very slowly and g (t) decays again, while in the glass region those domains start to form clusters, covering a whole space very slowly and g (t) grows. In both regions the system gradually reaches the nal state for long times, where (x; ∞) = . This is the late stage [L]. The above behavior is also seen in the mean-square displacement M2 (t; z0 ; ), which is given by the time integral of the spatially averaged di usion coecient Z t Z dx DS ((x; s)) ; ds (3.4) M2 (t; ; z0 ) = 2d V 0
390
M. Tokuyama et al. / Physica A 270 (1999) 380–402
Fig. 4. Spatial dependence of the local volume fraction (x; t) at z0 = 0:95 for (A) = 0:543, (B) 0.571, and (C) 0.573 for dimensionless times: (A) (a) 1, (b) 3.96 (t ), (c) 10, (d) 102 , (e) 359 (tL ), and (f) 103 ; (B) (a) 1, (b) 8.54 (t ), (c) 102 , (d) 1578 (t ), (e) 104 , (f) 6:2 × 104 (t ), (g) 105 , (h) 5:4 × 105 (tL ), and (i) 106 ; (C) (a) 1, (b) 8:54 (t ), (c) 102 , (d) 5388 (t ), (e) 104 , (f) 1:02 × 105 (t ), (g) 5 × 105 , (h) 1:1 × 106 (tL ), and (i) 2 × 106 .
M. Tokuyama et al. / Physica A 270 (1999) 380–402
391
Fig. 4. Continued.
where d is a spatial dimensionality. In Fig. 7 we show a log–log plot of the quantity M2 (t) versus dimensionless time =D0 t=a2 for di erent volume fractions at z0 =0:8 and 0.95. In stage [E], M2 (t) grows as M2 (t) = 6DSS t, obeying the short-time self-di usion process for all values of . In stage [ ], the growth of M2 (t) is slowing down for ¿ , and the shape of M2 (t) is very sensitive to the volume fraction, forming a shoulder. The shoulder becomes a plateau at g with the height mc0 (z0 ) = lim M2 (t; g ; z0 ) : t=t 1
(3.5)
In stage [ ]; M2 (t) grows again. In order to nd such a growth, it is convenient to introduce the separation parameter by = = g − 1. By expanding Eq. (2.3) in powers of , and using Eq. (3.3), on the time scale of order t , one can then write Eq. (3.4), to order 2 , as M2 (t; ; z0 ) = mc0 (z0 ) + m1 (z0 ) + 2 m2 (t) + O(3 ) with mc0 (z0 ) =
Z 0
t
Z ds
[ − 9=32 g (x; s)] dx Z(x; s)2 ; V [(x; s) + (x; s)2 D0 =DSS ]
(3.6)
(3.7)
392
M. Tokuyama et al. / Physica A 270 (1999) 380–402
Fig. 5. A sequence of snapshots, projected onto a plane, of a typical con guration of glassy domains. Details are the same as in Fig. 4. The glassy phase is colored black.
M. Tokuyama et al. / Physica A 270 (1999) 380–402
393
Fig. 5. Continued.
Z m1 (z0 ) =
0
Z m2 (t) =
0
t
t
Z ds Z
ds
dx (x; s)[ + (x; s) − 9=16( g = )(x; s)2 ] Z(x; s) ; V [(x; s) + (x; s)2 D0 =DSS ]2
dx (x; s)2 [ − 9=32( g = 2 )(x; s)3 ] ; V [(x; s) + (x; s)2 D0 =DSS ]3
(3.8) (3.9)
where we have used the fact that |(x; t)| 1 for t¿t . The function m2 (t) describes a long-time behavior when 6= 0, while the function m1 (z0 ) gives correction to the plateau value mc0 . Use of Eqs. (3.3) and (3.9) leads to m2 (t) ∼ c2 t with the exponent (z0 ) = 1 − d(z0 ), where c2 is a positive constant. From Eq. (3.6), we thus nd M2 (t) ' m(z0 ; ) + c2 2 t ;
(3.10)
where m(z0 ; ) = mc0 + m1 , and (0:8) = 3:65 and (0:95) = 2:69. This slow growth is due to the rearrangement of the long-lived glassy domains. Later, this growth leads to a stretched exponential decay in the self-intermediate scattering function FS (k; t). In fact, the heterogeneous domains do a ect the relaxation of the density uctuations
394
M. Tokuyama et al. / Physica A 270 (1999) 380–402
Fig. 6. A log–log plot of g (t) versus time at z0 = 0:95 for (A) = 0:559, (B) 0.571, and (C) 0.573. The symbols indicate the time scales: t (); t ( ); t ( ), and tL ( ).
•
since Eq. (2.9) contains the same di usion coecients as that in Eq. (2.8). We discuss this next. 3.2. Self-intermediate scattering function FS (k; t) We next discuss the numerical solutions of Eq. (2.9). In Fig. 8 we show a log–log plot of the quantity (−6=k 2 )ln[FS (k; t)] versus time at z0 = 0:95 and = 0:571 for di erent wavevectors; ka = 1; 3, and 5. Then the self-intermediate scattering function FS (k; t) can be written in the Gaussian form as 2 k (3.11) FS (k; t) = exp − M2 (t) : 2d Thus, the nonGaussian e ect is shown to be small for hard-sphere suspensions [6,22]. From Eqs. (3.10) and (3.11), at g FS (k; t) reaches the plateau with the height 1 2 c c (3.12) fk (z0 ) = FS (k; t → ∞; = 0) = exp − k m0 (z0 ) : 6 In Fig. 9, we show the Gaussian behavior of the plateau height fkc (z0 ) for wave vector at di erent values of z0 , where fkc = 0:879 (z0 = 0:8) and fkc = 0:921 (z0 = 0:95) at ka = 3. Here we note that as the initial state of the system is far from the equilibrium, the plateau height fkc (z0 ) becomes lower.
M. Tokuyama et al. / Physica A 270 (1999) 380–402
395
Fig. 7. A log–log plot of the mean-square displacement M2 (t) versus time for = 0:543; 0:559 ( ); 0:571, and 0.573 (dotted lines from left to right) at z0 = 0:8, and = 0:559, 0:565 ( ); 0:571, and 0:573 (solid lines from left to right) at z0 = 0:95. The details are the same as in Fig. 6. Table 1 Time exponents be and bl for di erent values z0 and at ka = 3 z0 = 0:8
z0 = 0:95
be
bl
be
bl
0.571 0.573
0.32 0.41
0.49 0.52
0.26 0.33
0.68 0.71
In the early stage [E], the spatial inhomogeneities are described by the solution of Eq. (2.8) as (x; t) = exp[ − tDSS ( ) 2 ](x; 0), and the density uctuations obey the short-time exponential decay, FS (k; t) = exp[ − k 2 DSS ( )t]. In the -relaxation state [ ] after this stage, the dynamical behavior becomes complicated because of the anomalous property of DS (). In order to see the crossover behavior around t , it is convenient to calculate the logarithmic derivatives given by ’ = @ log |fkc − FS (k; t)|=@ log t and ’1 = @’=@ log t [13]. Then, ’1 = 0 is shown to give two time roots, tbe ( ; z0 ; ka) and tbl ( ; z0 ; ka), which reveal two fairly at regions; ’ = be ( ; z0 ; ka) at t = tbe where t tbe t , and ’ = bl ( ; z0 ; ka) at t = tbl where t tbl t . The exponents be and bl are listed in Table 1. The crossover volume fraction (z0 ; ka) is thus determined by the equal root tbe ( ; z0 ; ka) = tbl ( ; z0 ; ka)
or be ( ; z0 ; ka) = bl ( ; z0 ; ka)
(3.13)
396
M. Tokuyama et al. / Physica A 270 (1999) 380–402
Fig. 8. A log–log plot of the quantity (−6=k 2 )ln[FS (k; t)] versus time for z0 = 0:95 and = 0:571 at ka = 1 (solid line), 3( ), and 5(+).
at xed z0 and k [20]. In Fig. 10 we plot the initial state parameter z0 versus for di erent wavevectors, ka = 3 and 5, where (0:95; 3) = 0:565 and (0:8; 3) = 0:559. With increasing volume fraction at a xed z0 , we thus observe a progression from normal colloidal liquid region [N] for 0 ¡ ¡ , to supercooled colloidal liquid region [S] for 6 ¡ g , and to glass region [G] for ¿ g (see Fig. 10). In the stage, the nite-sized, glassy domains start to be formed, and FS (k; t) thus obeys two kinds of power-law decays with exponents be and bl around t . In the early -relaxation stage [ E ] with t t6t ; FS (k; t) obeys FS (k; t) = fkc (z0 ) − Ak (z0 )(t=t )be ;
(3.14)
where Ak and A are positive constants. This power-law decay continues up to the crossover time t . In the late -relaxation stage [ L ] with t 6t t ; FS (k; t) obeys the so-called von Schweidler decay FS (k; t) = fkc (z0 ) − Bk (z0 )(t=t )bl ;
(3.15)
where Bk and B are positive constants. This power-law decay continues up to the -relaxation time t . In the -relaxation stage [ ], the spatial rearrangement of glassy domains occurs, and (x; t) obeys the scaling given by Eq. (3.3). From Eq. (3.10), one thus nd the KWW function FS (k; t) = fk (z0 ; )exp[ − (t=t ) (z0 ) ]
(3.16)
M. Tokuyama et al. / Physica A 270 (1999) 380–402
397
Fig. 9. The wavevector dependence of the plateau height fkc (z0 ) at z0 = 0:8 (dotted line) and 0.95 (solid line).
with the -relaxation time t ( ; z0 ; k) ˙ (k||)−(z0 ) ;
(3.17)
where fk (z0 ; ) = exp(−k 2 m=6). Here the time exponent (z0 ) satis es the relation (z0 ) = = (z0 ) ;
(3.18)
where the typical values of (z0 ) are listed in Table 2. Thus, the KWW formula turns out to be explained by the existence of long-lived, glassy domains. This stretched behavior continues up to the time scale of order tL . In the late stage [L] on the time scale of order tL , the di usion coecient DS () reduces to the long-time self-di usion coecient DSL ( ), and hence the uctuations obey the long-time exponential decay FS (k; t) = exp[ − k 2 DSL ( )t] :
(3.19)
In Fig. 11 we show the time evolution of FS (k; t) at z0 = 0:8 and 0.95 for di erent volume fractions, where ka = 3. As is seen from Figs. 11, below ; FS (k; t) decays quickly to zero, obeying a nearly simple exponential decay. On the other hand, above , the shape of FS (k; t) becomes very sensitive to the value of , forming a shoulder whose height increases as increases. We mention here that even above g ; FS (k; t) still decays to zero for much longer times. This tendency agrees with experiments [6,10]. We also note here that the shape of FS (k; t) is sensitive to the value of z0 for ¿ , and the plateau height increases as z0 increases, while it is independent of z0 for ¡ . This behavior can be observable experimentally as the waiting time e ects [10].
398
M. Tokuyama et al. / Physica A 270 (1999) 380–402
Fig. 10. Schematic phase diagram in z0 − plane; [N] a normal liquid state, [S] a supercooled state, and [G] a glass state. The solid and dotted lines indicate the crossover volume fraction at ka = 3, and 5, respectively. The dashed line indicates the glass transition volume fraction g . Table 2 Time exponents (z0 ); (z0 ) (= =) and = (1 − )=d for di erent values z0 z0
0.8 0.95
3.41 2.69
0.586 0.744
0.138 0.086
In order to show the existence of the above four characteristic stages numerically, in Fig. 12 we also plot FS (k; t) versus time for = 0:571 at z0 = 0:95 and ka = 3 together with Eqs. (3.14)–(3.16) and (3.19). 3.3. Self-parts of the dynamic susceptibility In order to see the crossover behavior in the intermediate-time region more clearly, we nally discuss the self-part of the generalized susceptibility S (k; !). Let us de ne the Fourier–Laplace transformation, GS (k; !), of the self-intermediate scattering function FS (k; t) by Z ∞ FS (k; t)e−i!t dt = GS0 (k; !) + iGS00 (k; !) : (3.20) GS (k; !) = 0
M. Tokuyama et al. / Physica A 270 (1999) 380–402
399
Fig. 11. Self-intermediate scattering function FS (k; t) versus time at ka = 3. The details are the same as in Fig. 7.
Fig. 12. Self-intermediate scattering function FS (k; t) versus time at ka = 3 and z0 = 0:95 for = 0:571. The dashed line indicates Eq. (3.15), the dot-dashed line Eq. (3.16), the dotted line Eq. (3.17), and the long-dashed line Eq. (3.20). The symbols indicate the time scales: t (); t ( ); t ( ), and tL ( ).
•
400
M. Tokuyama et al. / Physica A 270 (1999) 380–402
Fig. 13. A log–log plot of S00 (k; !) versus frequency for = 0:543; 0:559 ( ); 0:571, and 0.573 (dotted lines from right to left) at z0 = 0:8, and = 0:543; 0:565 ( ); 0:571, and 0.573 (solid lines from right to left) at z0 = 0:95, where ka = 3. The symbols indicate the characteristic frequencies: ! (); ! ( ); ! ( ), and !L ( ).
•
Then, the self-part of the generalized susceptibility, S (k; !) = S0 (k; !) + i S00 (k; !), is known to be related to GS (k; !) through the relation S0 (k; !) = 1 + !GS00 (k; !) = 1 − ! S00 (k; !)
=
!GS0 (k; !)
Z =!
0
∞
Z 0
∞
FS (k; t)sin(!t) dt ;
FS (k; t)cos(!t) dt :
(3.21) (3.22)
In Fig. 13 we plot the self-part of the dynamic susceptibility, S00 (k; !) at z0 =0:8 and 0.95 for di erent volume fractions, where ka = 3. For volume fractions larger than , it has two peaks and one minimum. The rst peak is the so-called peak at ! = !L = 2=tL ∼ || is in the lower-frequency region and describes the long-time relaxation process on the time scale of order tL . The second peak is the so-called peak at ! = ! = 2=t ∼ ||0 is in the higher-frequency region and describes and short-time relaxation process on the time scale of order t . The minimum is the so-called minimum at the frequency ! = ! = 2=t ∼ ||−1 , which corresponds to the crossover point in FS (k; t) at the time t , where !L ! ! . On the other hand, for volume fractions smaller than , the susceptibility S00 (k; !) has only one peak and has no minimum. Hence the crossover volume fraction (z0 ; k) is also de ned by the value of at which the minimum of S00 (k; !) appears. In each stage, S00 (k; !) obeys the following asymptotic power
M. Tokuyama et al. / Physica A 270 (1999) 380–402
401
laws: [E] S00 (k; !) ∼ !−1 ; [ E ] S00 (k; !) ∼ (!=! )be ; [ L ] S00 (k; !) ∼ (!=! )−bl ; [ ] S00 (k; !) ∼ !− , and [L] S00 (k; !) ∼ !. Finally, we should mention that if the initial state of the system is far from the equilibrium, the peak becomes lower and the peak becomes higher. This might be observed experimentally as the waiting time e ects. 4. Conclusion In conclusion, we have solved the coupled di usion equations, Eqs. (2.8) and (2.9), starting from a nonuniform initial con guration. Thus, we have shown that the dynamic anomaly of the self-di usion coecient plays as important role in a supercooled
uid of hard-sphere colloids, leading to the formation of long-lived, glassy domains and the slow relaxation of the nonequilibrium density uctuations. The main results reported here are as follows. (i) The KWW formula in the hard-sphere suspension can be explained by the existence of long-lived, irregularly shaped glassy domains. (ii) The self-intermediate scattering function can be written in the Gaussian form for whole times. (iii) There are four characteristic stages, obeying di erent types of relaxations of the density uctuations. (iv) If the initial state of the system is far from the equilibrium, the plateau height and the peak become lower. This nonequilibrium effects can be observable experimentally as the waiting time e ects. The above dynamic anomaly results from the many-body correlations due to the long-range, hydrodynamic interactions between particles. Hence we should nally mention that the above features are also seen for highly charged colloidal suspensions since the dynamic anomaly results from the pair correlations due to the long-range, Coulomb attractive interactions between macroions and counterions [23]. Acknowledgements This work was supported by the Tohwa Institute for Science, Tohwa University. References [1] P.N. Pusey, Colloidal suspensions, in: D. Levesque, J.P. Hansen, J. Zinn-Justin (Eds.), Liquids, Freezing and the Glass Transition, Elsevier, Amsterdam, 1991. [2] P.N. Pusey, W. van Megen, Nature 320 (1986) 340. [3] W. van Megen, S.M. Underwood, Phys. Rev. E 49 (1994) 4206. [4] K.L. Ngai, G.B. Wright (Eds.), Relaxation in complex systems, Proceedings of the International Discussion Meeting, Heraklion, 1990, parts I and II, North-Holland, Amsterdam, 1991. [5] E.W. Fischer, Physica A 201 (1993) 183. [6] W. van Megen, T.C. Mortensen, S.R. Williams, J. Muller, Phys. Rev. E 58 (1998) 6073. [7] W. Gotze, L. Sjogren, Phys. Rev. A 43 (1991) 5442. [8] M. Tokuyama, unpublished. [9] M. Tokuyama, Physica A 229 (1996) 36. [10] T.C. Mortensen, W. van Megen, in: M. Tokuyama, I. Oppenheim (Eds.), Slow Dynamics in Complex Systems, AIP, New York, 1999.
402 [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23]
M. Tokuyama et al. / Physica A 270 (1999) 380–402 M. Tokuyama, I. Oppenheim, Physica A 216 (1995) 85. M. Tokuyama, Physica A 265 (1999) 333. M. Tokuyama, Y. Enomoto, I. Oppenheim, Phys. Rev. E 55 (1997) R29. M. Tokuyama, Y. Enomoto, I. Oppenheim, Phys. Rev. E 56 (1997) 2302. N.G. van Kampen, Can. J. Phys. 39 (1961) 551. B.J. Berne, R. Pecora, Dynamic Light Scattering, Wiley, New York, 1976. P.N. Segre, O.P. Behrend, P.N. Pusey, Phys. Rev. E 52 (1995) 5070. W. van Megen, S.M. Underwood, J. Chem. Phys. 91 (1989) 552. W. van Megen, S.M. Underwood, J. Muller, T.C. Mortensen, S.I. Henderson, J.L. Harland, P. Francis, Prog. Theor. Phys. (Suppl.) 126 (1997) 171. M. Tokuyama, Prog. Theor. Phys. (Suppl.) 126 (1997) 43. M. Tokuyama, Phys. Rev. E 54 (1996) R1062. M. Fuchs, W. Gotze, M.R. Mayr, Phys. Rev. E 58 (1998) 3384. M. Tokuyama, Phys. Rev. E 58 (1998) R2729.
www.elsevier.com/locate/physa
Slow dynamics of supercooled colloidal uids: spatial heterogeneities and nonequilibrium density uctuations a Statistical
M. Tokuyamaa;∗ , Y. Enomotob , I. Oppenheimc
Physics Division, Tohwa Institute for Science, Tohwa University, Fukuoka 815, Japan School of Engineering, Nagoya Institute of Technology, Nagoya 466, Japan c Department of Chemistry, Massachusetts Institute of Technology, Cambridge, MA 02139, USA b Graduate
Received 23 February 1999
Abstract The coupled di usion equations recently proposed by Tokuyama for concentrated hard-sphere suspensions are numerically solved, starting from nonequilibrium initial con gurations. The most important feature of those equations is that the self-di usion coecient DS () becomes zero at the glass transition volume fraction g as DS () ∼ D0 |1 − (x; t)= g | with = 2 where (x; t) is the local volume fraction of colloids, D0 the single-particle di usion constant, and g = ( 43 )3 =(7 ln 3 − 8 ln 2 + 2). This dynamic anomaly results from the many-body correlations due to the long-range hydrodynamic interactions. Then, it is shown how small initial disturbances can be enhanced by this anomaly near g , leading to long-lived, spatial heterogeneities. Those heterogeneities are responsible for the slow relaxation of nonequilibrium density uctuations. In fact, the self-intermediate scattering function is shown to obey a two-step relaxation around the -relaxation time t ∼ |1 − = g |−1 , and also to be well approximated by the Kohlrausch– Williams–Watts function with an exponent around the -relaxation time t ∼ |1 − = g |− , where = = , and is the particle volume fraction. Thus, the nonexponential relaxation is c 1999 Elsevier shown to be explained by the existence of long-lived, spatial heterogeneities. Science B.V. All rights reserved. PACS: 82.70.Dd; 05.40.+j; 51.10.+y Keywords: Dynamic anomaly; Hard-sphere suspensions; Nonequilibrium density uctuations; Slow dynamics; Spatial heterogeneities
∗
Corresponding author. Fax: +81-92-542-0813. E-mail address: [email protected] (M. Tokuyama)
c 1999 Elsevier Science B.V. All rights reserved. 0378-4371/99/$ - see front matter PII: S 0 3 7 8 - 4 3 7 1 ( 9 9 ) 0 0 1 7 2 - 7
M. Tokuyama et al. / Physica A 270 (1999) 380–402
381
1. Introduction Supercooled colloidal uids are known to exhibit a transition from a uid phase to a glass phase, similar to that in glass-forming liquids [1–3]. One of striking features in supercooled liquids is the nonexponential behavior with the time scale diverging near the glass-transition temperature. The time evolution of the self-intermediate scattering function associated with relaxation can be described by the Kohlrausch–Williams– Watts formula (KWW), often also called stretched exponential. There has been several studies for an explanation of this empirical function [4,5]. This stretched behavior is also observed experimentally in the hard-sphere colloidal suspension [6]. In this paper, we show that the stretched behavior in such a suspension can be explained by the presence of long-lived, nite-sized, irregularly shaped glassy domains. Most of experiments in colloidal suspensions has been done in an equilibrium state which is attained for long times after the quench. In the equilibrium state, however, it might be dicult to observe the spatial heterogeneities experimentally. This is mainly because the heterogeneities may easily be destroyed by the equilibrium density uctuations although those heterogeneities are generated by the uctuations themselves. Here we should note that the equilibrium density uctuations obey a nonlinear stochastic process near the glass transition point, although the relative magnitude of the density uctuations n(x; t) to the mean density neq is small even near the transition point; |n=neq | 1 [7,8]. In order to see the heterogeneities, therefore, Tokuyama has recently proposed a linear response-like theory based on the coupled di usion equations for the average number density n(x; t) and the nonequilibrium density uctuations n(x; t) around n(x; t) [9]. This theory is applicable to a suspension in a nonequilibrium state before equilibration after the quench, where the initial state of the system is spatially nonuniform. Since |n=n| 1; n(x; t) can then describe the dynamics of heterogeneous structure, while n(x; t) describes a linear relaxation of the nonequilibrium density uctuations on such nonuniform structure. In fact, the nonequilibrium e ect has been observed as the waiting time dependence of the intermediate scattering function [10]. The most important feature of this theory is that the self-di usion coecient DS () contained in the coupled equations becomes dynamically anomalous at the glass transition volume fraction g as DS ∼ D0 |1 − (x; t)= g | with = 2, where (x; t) = 4a3 n(x; t)=3 is the average local volume fraction, g = ( 43 )3 =(7 ln 3 − 8 ln 2 + 2) ' 0:57184 : : : , and D0 the single-particle di usion coecient. We note here that this dynamic anomaly results from the many-body correlations between particles due to the long-range hydrodynamic interactions through the Oseen tensor [11]. When the system is initially in a nonequilibrium state, this theory can describe how even the small disturbances are enhanced by the dynamic anomaly near g , leading to long-lived, cluster-like glassy domains with (x; t)¿ g , and in uence the dynamics of the density uctuations, leading to a two-step relaxation and a nonexponential decay. In Section 2, we brie y summarize and discuss the coupled di usion equations for the average local volume fraction and the self-intermediate scattering function. In Section 3,
382
M. Tokuyama et al. / Physica A 270 (1999) 380–402
we show the numerical solutions of those equations, including the mean-square displacement and the self part of the dynamic susceptibility. Section 4 is devoted to conclusions.
2. Coupled di usion equations We consider a three-dimensional hard-sphere colloidal suspension with N identical spherical particles with mass m and radius a in an incompressible uid with shear viscosity at the suspension-hydrodynamic stage where the space–time cuto s (xcuto ; tcuto ), the minimum wavelength and time of the dynamic process of interest, are set as xcuto a and S ¿tcuto tB , where S ∼ a2 =D0 denotes the structural relaxation time, which is a time required for a particle to di use over a distance a, and tB the Brownian relaxation time of the particle. Here the whole system is enclosed in a volume V . Let us de ne the single-particle number density N (x; t) at position x and time t by N (x; t) =
N X
(x − Xi (t)) ;
(2.1)
i=1
where Xi (t) (i = 1; 2; : : : ; N ) denotes the position vector from the origin to the center of the particle i. As was shown in Refs. [9,12], one can then derive the nonlinear stochastic di usion equation for N (x; t), @ N (x; t) = ∇ · [DS (Q(x; t))∇N (x; t)] + R(x; t) @t
(2.2)
with the self-di usion coecient (1 − 9Q=32) ; (2.3) [1 + QDSS (Q)=( g D0 (1 − Q= g )2 )] R and the conservation law (1=V ) dxQ(x; t) = , where Q(x; t) = (4=3)a3 N (x; t) is the local volume fraction, D0 the single-particle di usion coecient, and = (4=3)a3 neq the particle volume fraction with the equilibrium number density neq =N=V . Here R(x; t) denotes a Gaussian, Markov random force with zero mean and satis es 4 3 0 0 0 0 0 a nx nx (x − x ) ; hR(x; t)R(x ; t ); ni = 2(t − t )∇ · ∇ DS 3 DS (Q) = DSS (Q)
where the angular brackets h· · · ; ni denote the conditional average over a canonical ensemble with the value of N (0) = {N (x; 0)} being xed so as to be n = {nx }. The short-time self-di usion coecient DSS (Q) is given by DSS (Q) = D0 =[1 + L(Q)]
(2.4)
M. Tokuyama et al. / Physica A 270 (1999) 380–402
383
with the many-body static e ects due to the hydrodynamic interactions 6bc c 2bc 2bc 2b2 1− − − + L(Q) = 1 − b 1 + 2c 1 − b + c 1 − b + c + 4bc 1 − b + c + 2bc 3bc2 bc2 1+ + (1 + c)(1 − b + c) (1 + c)(1 − b + c) − 2bc2 bc2 ; (2.5) − (1 + c)(1 − b + c) − bc2 where b(Q) = (9Q=8)1=2 and c(Q) = 11Q=16. The factor (9=32)Q in the numerator of Eq. (2.3) results from the coupling between the direct interactions and the short-range hydrodynamic interactions. The second singular term in the denominator of Eq. (2.3) results from the many-body correlations between particles due to the long-range hydrodynamic interactions. We note here that the short-time self-di usion coecient DSS (Q) still depends on space and time through Q(x; t). In the previous calculations in Refs. [13,14], it was simply replaced by the constant value DSS ( ). In this paper, however, its space-time dependence is shown to play an important role in smoothing the glassy phase with Q(x; t)¿ g . In order to solve Eq. (2.2) numerically, in the following we focus only on the case where the initial state of the system is out of equilibrium. The case where it is in equilibrium will be discussed elsewhere [8]. Similarly to van Kampen’s theory of
uctuations [15], let us decompose the number density N (x; t) into the average number density n(x; t) = hN (x; t); n0 i and a uctuating part n(x; t) as N (x; t) = n(x; t) + n(x; t) with the conservation laws Z (1=V ) dx n(x; t) = neq ;
(2.6) Z dx n(x; t) = 0 ;
(2.7)
where the initial values n0 = {nx;0 } are set in an appropriate nonequilibrium state. This decomposition is essential since the relative magnitude of the density uctuations n(x; t) to the causal part n(x; t) is usually small in a system away from a critical point; |n=n| 1. The dynamics of spatial heterogeneities of colloidal suspensions is described by the average local volume fraction (x; t)=hQ(x; t); n0 i. On the other hand, the dynamics of density uctuations can be measured by dynamic light scattering through the intermediate scattering function [16] which is given by F(k; t) = hnk (t)n∗k (0); n0 i=S(k) with the static√structure factor S(k) = h|nk (0)|2 ; n0 i, where nk (t) is a Fourier transform of n(x; t)= N . For scattering vectors larger than the maximum position km of S(k), the scattering function F(k; t) reduces to the self-intermediate scattering function FS (k; t), where S(k) = 1. In the suspension-hydrodynamic stage, therefore, we start with the
384
M. Tokuyama et al. / Physica A 270 (1999) 380–402
following coupled di usion equations already described elsewhere [9]: @ (x; t) = ∇ · [DS ((x; t))∇(x; t)] ; @t
(2.8)
X @ DS (k − q; t)FS (q; t) FS (k; t) = −k 2 @t q
(2.9)
with the Fourier transform, DS (k; t), of the self-di usion coecient DS ((x; t)). Eq. (2.8) describes the dynamics of heterogeneous structure from nonuniform initial con guration with (x; 0) to a nal con guration with (x; ∞) = . Eq. (2.9) describes a linear relaxation of the density uctuations around the nonuniform state determined by (x; t). The most important feature of those equations is that the diffusion coecient DS () becomes zero at g as DS () ∼ D0 (1 − = g )2 . For short times t t ; DS () is shown to reduce to DSS ( ) since the direct interactions and the correlations are negligible, where t ∼ a2 =DSS denotes the characteristic time of the short-time self-di usion process. In fact, the numerical calculation of Eqs. (2.8) and (2.9) shows that DS (k; t) ' DSS ( )k;0 within errors. For long times t tL , it reduces to the long-time self-di usion coecient DSL ( ) since (x; t) reaches , where tL ∼ a2 =DSL denotes the characteristic time of the long-time self-di uction process. In Fig. 1 we plot the normalized self-di usion coecient, DS =D0 , as a function of for short and long times. A good agreement is indeed seen between the theoretical results [11] and the experimental data [17–19]. Thus, there exists a crossover from the short-time self-di usion process to the long-time self-di usion process for intermediate times, where the dynamic anomaly plays an important role. 3. Numerical results In order to solve the coupled di usion equations (2.8) and (2.9) self-consistently, we rst x the values of the two parameters, and z0 = z(0), as the initial conditions, where the state parameter z(t) measures how the system is spatially nonuniform at time t and is given by [13] Z z(t) = 1 − (1=V ) dx|1 − (x; t)= | ; (3.1) where 0 ¡ z(t) ¡ 1 for nonuniform states and z(t) = 1 for uniform states. To integrate Eqs. (2.8) and (2.9), we employ the forward Euler di erence scheme with the time step 0:01A20 =D0 and the lattice spacing 0:5a in the volume (128a)3 of the three-dimensional simulation system with periodic boundary conditions. The initial value (x; 0) is chosen at each position x from a random number with a Gaussian distribution, which is characterized by a mean value 1 and a standard deviation s, where s is adjusted so as to nd a given value of z0 . Quantities evaluated here include the local volume fraction (x; t), the mean-square displacement M2 (t), the self-intermediate scattering function FS (k; t), and the self-part of the dynamic susceptibility S00 (k; !).
M. Tokuyama et al. / Physica A 270 (1999) 380–402
385
Fig. 1. A log plot of self-di usion coecient DS ( ) versus for short and long times. The solid line indicates the long-time self-di usion coecient DSL ( ), while the dotted line is the short-time self-di usion coecient DSS ( ). The symbols indicate the experimental data from Ref. [14] (+), Ref. [15] ( and ), and Ref. [16] ( and ).
•
3.1. Average local volume fraction (x; t) We rst discuss the numerical solutions of Eq. (2.8), starting in two kinds of completely random initial con gurations; (I) a slightly nonuniform con guration with z0 = 0:8 and (II) a nearly uniform con guration with z0 = 0:95. In Figs. 2(A)–2(C) we show the space–time dependence of the local volume fraction (x; t) in case (I) for di erent volume fractions; (A) =0:543 in the normal liquid region where 06 ¡ , (B) 0.571 in the supercooled liquid region where 6 ¡ g , and (C) 0.573 in the glass region where ¿ g . Here (z0 ; ka) denotes the crossover volume fraction over which the scattering function FS (k; t) has a shoulder for intermediate times and the susceptibility S00 (k; !) has one minimum and two peaks for intermediate frequencies [20]. Figs. 3(A)–3(C) show a sequence of snapshots projected onto a plane of a typical con guration of the glassy domains where the local volume fraction (x; t) is larger than g . In Figs. 4(A)–4(C), we also show the space–time dependence of (x; t) in case (II) for di erent volume fractions; (A) = 0:543, (B) 0.571, and (C) 0.573. Figs. 5(A)–5(C) show a sequence of snapshots. In order to see the long-lived behavior qualitatively, we also calculate the time evolution of the total volume fraction of the
386
M. Tokuyama et al. / Physica A 270 (1999) 380–402
Fig. 2. Spatial dependence of the local volume fraction (x; t) at z0 = 0:8 for (A) = 0:543, (B) 0.571, and (C) 0.573 for dimensionless times: (A) (a) 1, (b) 3.4 (t ), (c) 10, (d) 102 , (e) 184 (tL ), and (f) 103 ; (B) (a) 1, (b) 3.96 (t ), (c) 102 , (d) 853 (t ), (e) 104 , (f) 2:8 × 104 (t ), (g) 105 , (h) 3:4 × 105 (tL ), and (i) 106 ; (C) (a) 1, (b) 3.96 (t ), (c) 102 , (d) 1578 (t ), (e) 104 , (f) 4:4 × 104 (t ) (g) 105 , (h) 7:3 × 105 (tL ), and (i) 106 . The system size is (128a)3 .
M. Tokuyama et al. / Physica A 270 (1999) 380–402
387
Fig. 2. Continued.
glassy domains given by Z dx ((x; t) − g )(x; t) ; g (t) = V
(3.2)
where the step function (X ) satis es (X ) = 1 for X ¿0, and (X ) = 0 for X ¡ 0. In Fig. 6 we plot g (t) versus time at z0 = 0:95 for di erent volume fractions; (A) = 0:543, (B) 0.571, and (C) 0.573. In the early stage [E] on the time scale of order t , the spatial con gurations are random. The short-time di usion process causes a smoothing of (x; t) and leads to a power law decay of g (t), as t −p , for all values of , where the exponent p is numerically found to be approximately 1=3. After this stage, the nite-sized, glassy domains seem to be formed for volume fractions ¿ and hence the smoothing process of (x; t) and the decay of g (t) are slowing down for ¿ . This is the so-called -relaxation stage [ ] on the time scale of order t , where t is the -relaxation time given by t ∼ a2 =(DSS DSL )1=2 ∼ ||−1 [21]. Thus, the system exhibit the coexistence of
uid phase with (x; t) ¡ g and glassy phase with (x; t)¿ g . In the normal region for 06 ¡ , however, there are no nite-sized, long-lived glassy domains. After this stage, the spatial rearrangement of the glassy domains starts to occur and continues up to the time scale of order t , where t is the -relaxation time discussed later. This is the so-called -relaxation stage [ ]. Hence one can assume that the local volume
388
M. Tokuyama et al. / Physica A 270 (1999) 380–402
Fig. 3. A sequence of snapshots, projected onto a plane, of a typical con guration of glassy domains. Details are the same as in Fig. 2. The glassy phase is colored black.
M. Tokuyama et al. / Physica A 270 (1999) 380–402
389
Fig. 3. Continued.
fraction (x; t) is scaled for ¿ as (x; t) = [1 + (t x)] ;
R
(3.3)
where (z0 ) is a small exponent to be determined, and dx (x) = 0. Use of Eqs. (3.1) and (3.3) then leads to 1 − z(t) ∼ t −d . From the numerical calculations, therefore, we nd (0:95) ' 0:086 and (0:8) ' 0:15. After this stage, in the supercooled region the heterogeneous domains start to be dissolved, disappearing very slowly and g (t) decays again, while in the glass region those domains start to form clusters, covering a whole space very slowly and g (t) grows. In both regions the system gradually reaches the nal state for long times, where (x; ∞) = . This is the late stage [L]. The above behavior is also seen in the mean-square displacement M2 (t; z0 ; ), which is given by the time integral of the spatially averaged di usion coecient Z t Z dx DS ((x; s)) ; ds (3.4) M2 (t; ; z0 ) = 2d V 0
390
M. Tokuyama et al. / Physica A 270 (1999) 380–402
Fig. 4. Spatial dependence of the local volume fraction (x; t) at z0 = 0:95 for (A) = 0:543, (B) 0.571, and (C) 0.573 for dimensionless times: (A) (a) 1, (b) 3.96 (t ), (c) 10, (d) 102 , (e) 359 (tL ), and (f) 103 ; (B) (a) 1, (b) 8.54 (t ), (c) 102 , (d) 1578 (t ), (e) 104 , (f) 6:2 × 104 (t ), (g) 105 , (h) 5:4 × 105 (tL ), and (i) 106 ; (C) (a) 1, (b) 8:54 (t ), (c) 102 , (d) 5388 (t ), (e) 104 , (f) 1:02 × 105 (t ), (g) 5 × 105 , (h) 1:1 × 106 (tL ), and (i) 2 × 106 .
M. Tokuyama et al. / Physica A 270 (1999) 380–402
391
Fig. 4. Continued.
where d is a spatial dimensionality. In Fig. 7 we show a log–log plot of the quantity M2 (t) versus dimensionless time =D0 t=a2 for di erent volume fractions at z0 =0:8 and 0.95. In stage [E], M2 (t) grows as M2 (t) = 6DSS t, obeying the short-time self-di usion process for all values of . In stage [ ], the growth of M2 (t) is slowing down for ¿ , and the shape of M2 (t) is very sensitive to the volume fraction, forming a shoulder. The shoulder becomes a plateau at g with the height mc0 (z0 ) = lim M2 (t; g ; z0 ) : t=t 1
(3.5)
In stage [ ]; M2 (t) grows again. In order to nd such a growth, it is convenient to introduce the separation parameter by = = g − 1. By expanding Eq. (2.3) in powers of , and using Eq. (3.3), on the time scale of order t , one can then write Eq. (3.4), to order 2 , as M2 (t; ; z0 ) = mc0 (z0 ) + m1 (z0 ) + 2 m2 (t) + O(3 ) with mc0 (z0 ) =
Z 0
t
Z ds
[ − 9=32 g (x; s)] dx Z(x; s)2 ; V [(x; s) + (x; s)2 D0 =DSS ]
(3.6)
(3.7)
392
M. Tokuyama et al. / Physica A 270 (1999) 380–402
Fig. 5. A sequence of snapshots, projected onto a plane, of a typical con guration of glassy domains. Details are the same as in Fig. 4. The glassy phase is colored black.
M. Tokuyama et al. / Physica A 270 (1999) 380–402
393
Fig. 5. Continued.
Z m1 (z0 ) =
0
Z m2 (t) =
0
t
t
Z ds Z
ds
dx (x; s)[ + (x; s) − 9=16( g = )(x; s)2 ] Z(x; s) ; V [(x; s) + (x; s)2 D0 =DSS ]2
dx (x; s)2 [ − 9=32( g = 2 )(x; s)3 ] ; V [(x; s) + (x; s)2 D0 =DSS ]3
(3.8) (3.9)
where we have used the fact that |(x; t)| 1 for t¿t . The function m2 (t) describes a long-time behavior when 6= 0, while the function m1 (z0 ) gives correction to the plateau value mc0 . Use of Eqs. (3.3) and (3.9) leads to m2 (t) ∼ c2 t with the exponent (z0 ) = 1 − d(z0 ), where c2 is a positive constant. From Eq. (3.6), we thus nd M2 (t) ' m(z0 ; ) + c2 2 t ;
(3.10)
where m(z0 ; ) = mc0 + m1 , and (0:8) = 3:65 and (0:95) = 2:69. This slow growth is due to the rearrangement of the long-lived glassy domains. Later, this growth leads to a stretched exponential decay in the self-intermediate scattering function FS (k; t). In fact, the heterogeneous domains do a ect the relaxation of the density uctuations
394
M. Tokuyama et al. / Physica A 270 (1999) 380–402
Fig. 6. A log–log plot of g (t) versus time at z0 = 0:95 for (A) = 0:559, (B) 0.571, and (C) 0.573. The symbols indicate the time scales: t (); t ( ); t ( ), and tL ( ).
•
since Eq. (2.9) contains the same di usion coecients as that in Eq. (2.8). We discuss this next. 3.2. Self-intermediate scattering function FS (k; t) We next discuss the numerical solutions of Eq. (2.9). In Fig. 8 we show a log–log plot of the quantity (−6=k 2 )ln[FS (k; t)] versus time at z0 = 0:95 and = 0:571 for di erent wavevectors; ka = 1; 3, and 5. Then the self-intermediate scattering function FS (k; t) can be written in the Gaussian form as 2 k (3.11) FS (k; t) = exp − M2 (t) : 2d Thus, the nonGaussian e ect is shown to be small for hard-sphere suspensions [6,22]. From Eqs. (3.10) and (3.11), at g FS (k; t) reaches the plateau with the height 1 2 c c (3.12) fk (z0 ) = FS (k; t → ∞; = 0) = exp − k m0 (z0 ) : 6 In Fig. 9, we show the Gaussian behavior of the plateau height fkc (z0 ) for wave vector at di erent values of z0 , where fkc = 0:879 (z0 = 0:8) and fkc = 0:921 (z0 = 0:95) at ka = 3. Here we note that as the initial state of the system is far from the equilibrium, the plateau height fkc (z0 ) becomes lower.
M. Tokuyama et al. / Physica A 270 (1999) 380–402
395
Fig. 7. A log–log plot of the mean-square displacement M2 (t) versus time for = 0:543; 0:559 ( ); 0:571, and 0.573 (dotted lines from left to right) at z0 = 0:8, and = 0:559, 0:565 ( ); 0:571, and 0:573 (solid lines from left to right) at z0 = 0:95. The details are the same as in Fig. 6. Table 1 Time exponents be and bl for di erent values z0 and at ka = 3 z0 = 0:8
z0 = 0:95
be
bl
be
bl
0.571 0.573
0.32 0.41
0.49 0.52
0.26 0.33
0.68 0.71
In the early stage [E], the spatial inhomogeneities are described by the solution of Eq. (2.8) as (x; t) = exp[ − tDSS ( ) 2 ](x; 0), and the density uctuations obey the short-time exponential decay, FS (k; t) = exp[ − k 2 DSS ( )t]. In the -relaxation state [ ] after this stage, the dynamical behavior becomes complicated because of the anomalous property of DS (). In order to see the crossover behavior around t , it is convenient to calculate the logarithmic derivatives given by ’ = @ log |fkc − FS (k; t)|=@ log t and ’1 = @’=@ log t [13]. Then, ’1 = 0 is shown to give two time roots, tbe ( ; z0 ; ka) and tbl ( ; z0 ; ka), which reveal two fairly at regions; ’ = be ( ; z0 ; ka) at t = tbe where t tbe t , and ’ = bl ( ; z0 ; ka) at t = tbl where t tbl t . The exponents be and bl are listed in Table 1. The crossover volume fraction (z0 ; ka) is thus determined by the equal root tbe ( ; z0 ; ka) = tbl ( ; z0 ; ka)
or be ( ; z0 ; ka) = bl ( ; z0 ; ka)
(3.13)
396
M. Tokuyama et al. / Physica A 270 (1999) 380–402
Fig. 8. A log–log plot of the quantity (−6=k 2 )ln[FS (k; t)] versus time for z0 = 0:95 and = 0:571 at ka = 1 (solid line), 3( ), and 5(+).
at xed z0 and k [20]. In Fig. 10 we plot the initial state parameter z0 versus for di erent wavevectors, ka = 3 and 5, where (0:95; 3) = 0:565 and (0:8; 3) = 0:559. With increasing volume fraction at a xed z0 , we thus observe a progression from normal colloidal liquid region [N] for 0 ¡ ¡ , to supercooled colloidal liquid region [S] for 6 ¡ g , and to glass region [G] for ¿ g (see Fig. 10). In the stage, the nite-sized, glassy domains start to be formed, and FS (k; t) thus obeys two kinds of power-law decays with exponents be and bl around t . In the early -relaxation stage [ E ] with t t6t ; FS (k; t) obeys FS (k; t) = fkc (z0 ) − Ak (z0 )(t=t )be ;
(3.14)
where Ak and A are positive constants. This power-law decay continues up to the crossover time t . In the late -relaxation stage [ L ] with t 6t t ; FS (k; t) obeys the so-called von Schweidler decay FS (k; t) = fkc (z0 ) − Bk (z0 )(t=t )bl ;
(3.15)
where Bk and B are positive constants. This power-law decay continues up to the -relaxation time t . In the -relaxation stage [ ], the spatial rearrangement of glassy domains occurs, and (x; t) obeys the scaling given by Eq. (3.3). From Eq. (3.10), one thus nd the KWW function FS (k; t) = fk (z0 ; )exp[ − (t=t ) (z0 ) ]
(3.16)
M. Tokuyama et al. / Physica A 270 (1999) 380–402
397
Fig. 9. The wavevector dependence of the plateau height fkc (z0 ) at z0 = 0:8 (dotted line) and 0.95 (solid line).
with the -relaxation time t ( ; z0 ; k) ˙ (k||)−(z0 ) ;
(3.17)
where fk (z0 ; ) = exp(−k 2 m=6). Here the time exponent (z0 ) satis es the relation (z0 ) = = (z0 ) ;
(3.18)
where the typical values of (z0 ) are listed in Table 2. Thus, the KWW formula turns out to be explained by the existence of long-lived, glassy domains. This stretched behavior continues up to the time scale of order tL . In the late stage [L] on the time scale of order tL , the di usion coecient DS () reduces to the long-time self-di usion coecient DSL ( ), and hence the uctuations obey the long-time exponential decay FS (k; t) = exp[ − k 2 DSL ( )t] :
(3.19)
In Fig. 11 we show the time evolution of FS (k; t) at z0 = 0:8 and 0.95 for di erent volume fractions, where ka = 3. As is seen from Figs. 11, below ; FS (k; t) decays quickly to zero, obeying a nearly simple exponential decay. On the other hand, above , the shape of FS (k; t) becomes very sensitive to the value of , forming a shoulder whose height increases as increases. We mention here that even above g ; FS (k; t) still decays to zero for much longer times. This tendency agrees with experiments [6,10]. We also note here that the shape of FS (k; t) is sensitive to the value of z0 for ¿ , and the plateau height increases as z0 increases, while it is independent of z0 for ¡ . This behavior can be observable experimentally as the waiting time e ects [10].
398
M. Tokuyama et al. / Physica A 270 (1999) 380–402
Fig. 10. Schematic phase diagram in z0 − plane; [N] a normal liquid state, [S] a supercooled state, and [G] a glass state. The solid and dotted lines indicate the crossover volume fraction at ka = 3, and 5, respectively. The dashed line indicates the glass transition volume fraction g . Table 2 Time exponents (z0 ); (z0 ) (= =) and = (1 − )=d for di erent values z0 z0
0.8 0.95
3.41 2.69
0.586 0.744
0.138 0.086
In order to show the existence of the above four characteristic stages numerically, in Fig. 12 we also plot FS (k; t) versus time for = 0:571 at z0 = 0:95 and ka = 3 together with Eqs. (3.14)–(3.16) and (3.19). 3.3. Self-parts of the dynamic susceptibility In order to see the crossover behavior in the intermediate-time region more clearly, we nally discuss the self-part of the generalized susceptibility S (k; !). Let us de ne the Fourier–Laplace transformation, GS (k; !), of the self-intermediate scattering function FS (k; t) by Z ∞ FS (k; t)e−i!t dt = GS0 (k; !) + iGS00 (k; !) : (3.20) GS (k; !) = 0
M. Tokuyama et al. / Physica A 270 (1999) 380–402
399
Fig. 11. Self-intermediate scattering function FS (k; t) versus time at ka = 3. The details are the same as in Fig. 7.
Fig. 12. Self-intermediate scattering function FS (k; t) versus time at ka = 3 and z0 = 0:95 for = 0:571. The dashed line indicates Eq. (3.15), the dot-dashed line Eq. (3.16), the dotted line Eq. (3.17), and the long-dashed line Eq. (3.20). The symbols indicate the time scales: t (); t ( ); t ( ), and tL ( ).
•
400
M. Tokuyama et al. / Physica A 270 (1999) 380–402
Fig. 13. A log–log plot of S00 (k; !) versus frequency for = 0:543; 0:559 ( ); 0:571, and 0.573 (dotted lines from right to left) at z0 = 0:8, and = 0:543; 0:565 ( ); 0:571, and 0.573 (solid lines from right to left) at z0 = 0:95, where ka = 3. The symbols indicate the characteristic frequencies: ! (); ! ( ); ! ( ), and !L ( ).
•
Then, the self-part of the generalized susceptibility, S (k; !) = S0 (k; !) + i S00 (k; !), is known to be related to GS (k; !) through the relation S0 (k; !) = 1 + !GS00 (k; !) = 1 − ! S00 (k; !)
=
!GS0 (k; !)
Z =!
0
∞
Z 0
∞
FS (k; t)sin(!t) dt ;
FS (k; t)cos(!t) dt :
(3.21) (3.22)
In Fig. 13 we plot the self-part of the dynamic susceptibility, S00 (k; !) at z0 =0:8 and 0.95 for di erent volume fractions, where ka = 3. For volume fractions larger than , it has two peaks and one minimum. The rst peak is the so-called peak at ! = !L = 2=tL ∼ || is in the lower-frequency region and describes the long-time relaxation process on the time scale of order tL . The second peak is the so-called peak at ! = ! = 2=t ∼ ||0 is in the higher-frequency region and describes and short-time relaxation process on the time scale of order t . The minimum is the so-called minimum at the frequency ! = ! = 2=t ∼ ||−1 , which corresponds to the crossover point in FS (k; t) at the time t , where !L ! ! . On the other hand, for volume fractions smaller than , the susceptibility S00 (k; !) has only one peak and has no minimum. Hence the crossover volume fraction (z0 ; k) is also de ned by the value of at which the minimum of S00 (k; !) appears. In each stage, S00 (k; !) obeys the following asymptotic power
M. Tokuyama et al. / Physica A 270 (1999) 380–402
401
laws: [E] S00 (k; !) ∼ !−1 ; [ E ] S00 (k; !) ∼ (!=! )be ; [ L ] S00 (k; !) ∼ (!=! )−bl ; [ ] S00 (k; !) ∼ !− , and [L] S00 (k; !) ∼ !. Finally, we should mention that if the initial state of the system is far from the equilibrium, the peak becomes lower and the peak becomes higher. This might be observed experimentally as the waiting time e ects. 4. Conclusion In conclusion, we have solved the coupled di usion equations, Eqs. (2.8) and (2.9), starting from a nonuniform initial con guration. Thus, we have shown that the dynamic anomaly of the self-di usion coecient plays as important role in a supercooled
uid of hard-sphere colloids, leading to the formation of long-lived, glassy domains and the slow relaxation of the nonequilibrium density uctuations. The main results reported here are as follows. (i) The KWW formula in the hard-sphere suspension can be explained by the existence of long-lived, irregularly shaped glassy domains. (ii) The self-intermediate scattering function can be written in the Gaussian form for whole times. (iii) There are four characteristic stages, obeying di erent types of relaxations of the density uctuations. (iv) If the initial state of the system is far from the equilibrium, the plateau height and the peak become lower. This nonequilibrium effects can be observable experimentally as the waiting time e ects. The above dynamic anomaly results from the many-body correlations due to the long-range, hydrodynamic interactions between particles. Hence we should nally mention that the above features are also seen for highly charged colloidal suspensions since the dynamic anomaly results from the pair correlations due to the long-range, Coulomb attractive interactions between macroions and counterions [23]. Acknowledgements This work was supported by the Tohwa Institute for Science, Tohwa University. References [1] P.N. Pusey, Colloidal suspensions, in: D. Levesque, J.P. Hansen, J. Zinn-Justin (Eds.), Liquids, Freezing and the Glass Transition, Elsevier, Amsterdam, 1991. [2] P.N. Pusey, W. van Megen, Nature 320 (1986) 340. [3] W. van Megen, S.M. Underwood, Phys. Rev. E 49 (1994) 4206. [4] K.L. Ngai, G.B. Wright (Eds.), Relaxation in complex systems, Proceedings of the International Discussion Meeting, Heraklion, 1990, parts I and II, North-Holland, Amsterdam, 1991. [5] E.W. Fischer, Physica A 201 (1993) 183. [6] W. van Megen, T.C. Mortensen, S.R. Williams, J. Muller, Phys. Rev. E 58 (1998) 6073. [7] W. Gotze, L. Sjogren, Phys. Rev. A 43 (1991) 5442. [8] M. Tokuyama, unpublished. [9] M. Tokuyama, Physica A 229 (1996) 36. [10] T.C. Mortensen, W. van Megen, in: M. Tokuyama, I. Oppenheim (Eds.), Slow Dynamics in Complex Systems, AIP, New York, 1999.
402 [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23]
M. Tokuyama et al. / Physica A 270 (1999) 380–402 M. Tokuyama, I. Oppenheim, Physica A 216 (1995) 85. M. Tokuyama, Physica A 265 (1999) 333. M. Tokuyama, Y. Enomoto, I. Oppenheim, Phys. Rev. E 55 (1997) R29. M. Tokuyama, Y. Enomoto, I. Oppenheim, Phys. Rev. E 56 (1997) 2302. N.G. van Kampen, Can. J. Phys. 39 (1961) 551. B.J. Berne, R. Pecora, Dynamic Light Scattering, Wiley, New York, 1976. P.N. Segre, O.P. Behrend, P.N. Pusey, Phys. Rev. E 52 (1995) 5070. W. van Megen, S.M. Underwood, J. Chem. Phys. 91 (1989) 552. W. van Megen, S.M. Underwood, J. Muller, T.C. Mortensen, S.I. Henderson, J.L. Harland, P. Francis, Prog. Theor. Phys. (Suppl.) 126 (1997) 171. M. Tokuyama, Prog. Theor. Phys. (Suppl.) 126 (1997) 43. M. Tokuyama, Phys. Rev. E 54 (1996) R1062. M. Fuchs, W. Gotze, M.R. Mayr, Phys. Rev. E 58 (1998) 3384. M. Tokuyama, Phys. Rev. E 58 (1998) R2729.