Mean-field theory of glass transitions

Mean-field theory of glass transitions

ARTICLE IN PRESS Physica A 364 (2006) 23–62 www.elsevier.com/locate/physa Mean-field theory of glass transitions Michio Tokuyama Institute of Fluid ...

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ARTICLE IN PRESS

Physica A 364 (2006) 23–62 www.elsevier.com/locate/physa

Mean-field theory of glass transitions Michio Tokuyama Institute of Fluid Science, Tohoku University, Sendai 980-8577, Japan Received 27 June 2005; received in revised form 25 July 2005 Available online 15 September 2005

Abstract The experimental and the simulation results for the mean-square displacements in various systems are analyzed near glass transitions by employing the mean-field theory recently proposed. By comparing different glass transitions with each other from a unified point view, it is then shown that the mean-square displacements obey a logarithmic growth followed by a power-law growth of a super-diffusion type in b-relaxation stage. It is also shown that although the long-time selfdiffusion coefficients are approximately described by a singular function of a control parameter with the same critical exponent for different systems where the singular point depends on the system, no divergence of any characteristic times, such as a b-relaxation time, is found near the singular point nor the glass transition point. A crossover point over which the supercooled liquid phase appears is also predicted theoretically. It is thus suggested that the present theory provides an useful tool to understand phenomena near the glass transition within the framework of the mean-square displacements. r 2005 Elsevier B.V. All rights reserved. Keywords: Glass transition; Mean-square displacement; Logarithmic growth; Power-law growth; Supercooled liquids

1. Introduction In recent years, considerable attention has been drawn to the glass transitions [1]. This is mainly due to the reason that there exist remarkable similarities between apparently different glass transitions for their dynamical behavior. In fact, the dynamical behavior near the glass transition can be approximately described by a singular function of a control parameter p, which depends only on a separation parameter given by ð¼ 1  p=pc Þ near pc , where pc is a singular point. This was first predicted by the so-called mode-coupling theory (MCT), which was proposed by Bengtzelius et al. [2], and independently, by Leutheusser [3]. Since then, this was the origin of all later experimental, theoretical, and numerical studies of glass transitions. Although this theory provides a qualitatively correct view near the glass transition, it does not agree quantitatively with experimental observations and simulation results. In fact, for hard-sphere colloidal suspensions, MCT shows that the long-time self-diffusion coefficient DLS ðfÞ obeys a singular function DLS / j1  f=fc jg , where g is an exponent to be determined and f is a particle volume fraction. MCT then finds that fc ’ 0:516 and g ’ 2:46 [4]. Although the singular point 0.516 is much smaller than the melting volume fraction 0.545 for a monodisperse hard-sphere suspension, MCT can predict the dynamical behavior quite similar to the Tel./fax: +81 22 217 5327.

E-mail address: [email protected]. 0378-4371/$ - see front matter r 2005 Elsevier B.V. All rights reserved. doi:10.1016/j.physa.2005.08.041

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experimental results if the value of  is adjusted with each other. This is due to the fact that the long-time dynamics depends only on  near pc . However, we should mention here that the singular point is not identical to a glass transition point. As if there exists a singular point similar to that in critical phenomena, the glass transition is neither a critical phenomenon nor a phase transition because of the following two reasons. The first reason is that there is no correlation length diverging even at the singular point, while it diverges at the critical point. The second reason is that the relative magnitude of the density fluctuation to the mean density is small even near the singular point, while the relative magnitude of the fluctuation to the order parameter is larger than one near the critical point. Since the glass transition is not fully elucidated theoretically yet, it may be still useful to employ positively such a singularity concept to study the dynamical behavior near the glass transition. Then, we find that the long-time self-diffusion coefficient DLS ðpÞ are approximately described by a singular function of p with the same exponent g ¼ 2:0, where pc depends on the system. However, we show that the experimental data and the simulation data always seem to deviate from the singular function beyond pc , showing no divergence of the characteristic times. Thus, we suggest that a non-singular behavior is rather essential near the glass transition [5]. In this paper, we focus only on the dynamics of the mean-square displacement M 2 ðtÞ but ignore the very interesting topics on the dynamics of long-lived, spatial heterogeneities. The main subject is what can be learned within the framework of the mean-square displacement. We compare two types of systems, colloidal suspensions and molecular systems, with each other from an unified theoretical point of view recently obtained. Then, we show that both systems behave quite similarly in various aspects near the glass transition. By comparing different systems near pc with each other, we thus suggest how one can consistently predict a crossover point pb theoretically, over which the supercooled liquid phase appears. We also discuss the characteristic stages with the characteristic times such as a b-relaxation time. The mean-square displacement M 2 ðtÞ is given by M 2 ðtÞ ¼

N 1X h½X i ðtÞ  X i ð0Þ2 i , N i¼1

(1)

where X i ðtÞ denotes a position vector of the ith particle, N the total number of particles, and the brackets the equilibrium ensemble average. Near the glass transition point, the self-intermediate scattering function F S ðk; tÞ can be written as [6] F S ðk; tÞ ¼ hexp½ik  fX i ðtÞ  X i ð0Þgi " #   1 4 M 2 ðtÞ 2 ðdÞ 2 M 2 ðtÞ þ k a2 ðtÞ þ Oðk6 Þ ’ exp k 2d 2 2d

ð2Þ

with the d-dimensional non-Gaussian parameter aðdÞ 2 ðtÞ ¼

d M 4 ðtÞ 1, d þ 2 M 2 ðtÞ2

(3)

where M 4 ðtÞ is the mean-fourth displacement given by M 4 ðtÞ ¼ hð½X i ðtÞ  X i ð0Þ2 Þ2 i . Here d is the spatial dimensionality. Then, M 2 ðtÞ and d M 2 ðtÞ ¼ f 2 ðM 2 ðtÞ; tÞ , dt

(4) aðdÞ 2 ðtÞ

in general obey the following non-linear equations: (5)

d ðdÞ a ðtÞ ¼ f 4 ðM 2 ðtÞ; aðdÞ (6) 2 ; tÞ , dt 2 where f 2 and f 4 are unknown functions to be determined. There are two types of glass transitions. One is a glass transition [A] where aðdÞ 2 ðtÞ51. Typical example is a suspension of neutral hard spheres with a small polydispersity [7,8]. The other is a glass transition [B] where aðdÞ 2 ðtÞb1. Most examples belong to this type. For

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type [A], we have recently proposed the following mean-field equation for M 2 ðtÞ by studying the dynamics of the equilibrium density fluctuations in a colloidal suspension of neutral hard spheres with both hydrodynamic and direct interactions between colloids [9,10]: d M 2 ðtÞ ¼ 2dDLS ðpÞ þ 2d½DSS ðpÞ  DLS ðpÞelðpÞM 2 ðtÞ , dt

(7)

where DLS ðpÞ and DSS ðpÞ denote the long- and the short-time self-diffusion coefficients, respectively. Here lðpÞ is a free parameter to be determined and is originally related to the static structure factor SðkÞ. In order to study the molecular system for type [A], one can also generalize Eq. (7) as [11]  2  d v M 2 ðtÞ ¼ 2dDLS ðpÞ þ 2d 0 t  DLS ðpÞ elðpÞM 2 ðtÞ , (8) dt d p where v0 ðTÞð¼ dkB T=mÞ denotes the average velocity of a particle, T being the temperature. In this paper, we analyze experimental data and simulation data by using the mean-field equations given by Eqs. (7) and (8) and show that both equations describe not only the systems for type [A] but also the systems for type [B] well. Thus, we discuss what can be learned by comparing different glass transitions with each other from a unified point of view. We first review the mean-field equations for two kinds of systems, a colloidal suspension of hard spheres and a molecular system. Then, we analyze the experimental and the simulation results for the mean-square displacement in both systems by using the mean-field theory. Thus, we show that the long-time self-diffusion coefficient DLS ðpÞ is well described by the following singular function of p in a dimensionless form: DLS ðpÞ ¼



DSS ðpÞð1  npÞ S kDS ðpÞðp=pc Þð1  ðp=pc ÞÞg

,

(9)

where k and n are the positive constants to be determined and DSS ðpÞ a short-time self-diffusion coefficient due to short-time interactions. This coefficient shows a singular behavior near pc as DLS ðpÞjjg . Here we note that the time exponent g can always be given by 2.0 for any systems. For suspensions, the coefficient n results from the coupling between the collision interactions and the short-range hydrodynamic interactions. We note here that the original form of a singular function given by Eq. (9) was theoretically derived for the suspension of neutral hard spheres with both hydrodynamic interactions and direct interactions, where 9 k ¼ 1; n ¼ 32 ; g ¼ 2:0, and pc ¼ ð43Þ3 =ð7 ln 3  8 ln 2 þ 2Þ ’ 0:57184 . . . [12]. For molecular systems, we have S DS ¼ 1 and n ¼ 0 because of no short-time diffusion process. By comparing theory, simulation, and experiment, we thus suggest that in any systems the singular term is considered to result from the many-body correlation effects due to the relevant interactions and also that the exponent g ¼ 2:0 does not depend on the type of interactions but the singular point pc does. The analyses also show that although Eq. (9) can describe the data well up to pc , the experimental data seem to deviate from it beyond pc , suggesting a non-singular behavior [5]. In Section 2, the important theoretical results from the mean-field equations for the mean-square displacements—the logarithmic growth and the power-law growth of a super-diffusion type—are introduced. The time exponents of those growths and the characteristic relaxation times are then discussed. Thus, how to determine the crossover point pb is suggested from a new point of view. In Sections 3 and 4, the parameter p dependence of the adjustable parameter l is also discussed in different systems near pc together with the free volume of a particle given by V f ðpÞ ¼ ld=2 . Then, it is realized from unified analyses that there exist remarkable similarities among different glass transitions. The present approach is thus shown to provide a useful tool to describe quite marked similarities between apparently different glass transitions.

2. Mean-field equations In this section we briefly summarize the mean-field equations for the mean-square displacements in two cases, colloidal suspensions of hard spheres and hard-sphere fluids, separately.

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2.1. Colloidal suspensions of hard spheres We first discuss a three-dimensional, equilibrium colloidal suspension, which consists of N identical hard spheres with radius a and an incompressible fluid with viscosity Z in a total volume V . Let p be a control parameter of the system. Then, there exist mainly two stages. The first is a kinetic stage, where the space–time cutoffs ðrcut ; tcut Þ, which are the minimum wavelength and time of interest, are set as aXrcut br0 and tB Xtcut bt0 . Here tB ð¼ m=6pZaÞ denotes the Brownian relaxation time of a particle, r0 the molecular length, and t0 the molecular time, where m is a mass of a particle. In this stage, the relevant variables are a set of the position vectors of particles given by fX 1 ; . . . ; X N g and a set of momenta of particles given by fP1 ; . . . ; PN g, where X i and Pi denote the position vector and the momentum of ith particle, respectively. Those are in general described by the Langevin equations, which are derived from Newtonian equations. The second is a suspension-hydrodynamic stage [12], where rcut ba and tD btcut btB . Here tD ð¼ a2 =D0 Þ is the structuralrelaxation time which is a time required for a particle to diffuse over a distance of order a, where D0 is a single particle diffusion constant. In this stage, the relevant variables are given by fX 1 ; . . . ; X N g. The mean-square displacement M 2 ðtÞ is a main physical quantity to calculate here. In the following, we focus only on the suspension-hydrodynamic stage. Recently, we have theoretically studied the slow dynamics of equilibrium density fluctuations in a colloidal suspension of neutral hard spheres with both hydrodynamic and direct interactions, where p is given by the particle volume fraction f [9,10]. Then, we have shown that the non-linear density fluctuations play an important role in forming long-lived, spatial heterogeneities and causing non-linear relaxation near fc , although the relative magnitude of the fluctuation to the mean density is always small even very near the singular point. We have thus derived the non-linear stochastic diffusion equation for the density fluctuation dfðr; tÞ [9,10] q dfðr; tÞ ¼ r2 ½DLS ðfÞdf þ DSS ðfÞfudf2 þ gdf3 g þ xðr; tÞ , (10) qt where the coefficients uðfÞ and gðfÞ are the known functions of f. The coefficient DLS ðfÞ is the long-time selfdiffusion coefficient given by [12] DLS ðfÞ ¼

DSS ðfÞð1  9f=32Þ , 1 þ DSS ðfÞðf=fc Þð1  ðf=fc ÞÞ2

(11)

9 where g ¼ 2:0; k ¼ 1:0; n ¼ 32 , and fc ¼ ð43Þ3 =ð7 ln 3  8 ln 2 þ 2Þ ’ 0:57184 . . . . Here the short-time selfS diffusion coefficient DS ðfÞ is given by [12]

DSS ðfÞ ¼

D0 , 1 þ 2q2 =ð1  qÞ  w=ð1 þ 2wÞ  qwð2 þ wÞ=ð1 þ wÞð1  q þ wÞ

(12)

where q ¼ ð9f=8Þ1=2 and w ¼ 11f=16. Here DSS ðfÞ results from the many-body effects due to the short-time hydrodynamic interactions. The singular term in Eq. (11) results from the many-body correlation effects due to the long-range hydrodynamic interactions between particles. The factor ð9=32Þf in the numerator of Eq. (11) results from the coupling between the direct interactions and the short-range hydrodynamic interactions. Here the random force xðr; tÞ obeys a Gaussian Markov process with zero mean and satisfies the fluctuationdissipation relation hxðr; tÞxðr0 ; t0 Þi ¼ 2DLS ðfÞdðt  t0 Þr2 hdfðrÞdfðr0 Þi .

(13)

The coefficient DLS ðfÞ shows the singular behavior as DLS ðfÞð1  f=fc Þ2 near fc . Hence it turns out that the linear term in Eq. (10) becomes very small near fc and thus balances with the square and cubic non-linear terms, even though those terms are quite small. Eq. (10) is a starting equation to study the slow dynamics of spatial heterogeneities generated by the non-linear fluctuations near fc . There are at least three ways to analyze the dynamics of density fluctuations dfðr; tÞ. The first is to solve Eq. (10) directly by a numerical simulation. We have done this recently and shown an important role of the long-lived, spatial heterogeneities generated by the non-linear density fluctuations near fc [13]. The second is to derive a closed non-linear equation for the intermediate-scattering function, which is a Fourier transform of the density–density correlation function. This can be done by applying the mode-coupling method used in critical phenomena to Eq. (10). The last is to derive

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a closed non-linear mean-field equation for the mean-square displacement from Eq. (10) by employing a meanfield approach [9,10]. Here we note that in any cases the static correlation function hdfðrÞdfðr0 Þi, whose Fourier transform gives a static structure factor SðkÞ, is in general unknown. In this paper, we discuss the last case, leading to Eq. (7). Thus, we show how the mean-field equation can describe the experimental results well by adjusting the parameter l which is related to SðkÞ. The mean-square displacement M 2 ðtÞ is described by the non-linear mean-field (7). Its formal solution is given by   S n o 1 DS 2dlDLS t M 2 ðtÞ ¼ ln 1 þ  1 . (14) e l DLS Solution (14) suggests the following asymptotic forms: ( S M 0 ðtÞ ¼ l1 lnð1 þ 2dt=tc Þ for tptc ; M 2 ðtÞ ’ 2dDLS t for tc 5tL pt ;

(15)

where tc ð¼ 1=lDSS Þ is a short time for a particle to diffuse over a distance of order l1=2 with the diffusion coefficient DSS and tL ð¼ a2 =DLS Þ a long-diffusion time for a particle to diffuse over a distance of order a with the diffusion coefficient DLS . In a late stage [L] for tL pt, M 2 ðtÞ thus grows linearly in time. In an early stage [E] for t5tc , it also grows linearly in time as M 2 ðtÞ2dDSS t. Thus, Eq. (7) describes the dynamic of a crossover from the short-time self-diffusion process characterized by DSS to the long-time self-diffusion process characterized by DLS . Near pc , there exists another time stage, the so-called b-relaxation stage ½b for tc 5t5tL . In fact, between two time scales, tc and tL , one can further define two more time scales, tg and tb , where tg 5tb . In order to find them, it is convenient to calculate the logarithmic derivatives given by jS1 ðt; pÞ ¼

q log jM 2 ðtÞ  M S0 ðtÞj , q log t

(16)

jS2 ðt; pÞ ¼

q jS . q log t 1

(17)

Then, jS2 ¼ 0 gives two time roots, tg and tb for p4ps , which reveal two fairly flat regions: 8 < bSb ðpÞ at t ¼ tb ; S j1 ¼ : bSg ðpÞ at t ¼ tg ;

(18)

and which satisfy the relations tb 1=ðlDLS Þ;

tg ðtc tb Þ1=2 ðl2 DSS DLS Þ1=2 , bSg

(19) bSb

¼ and tg ¼ tb at p ¼ ps . Thus, we find two time stages for where tc 5tg 5tb 5tL near pc . Here we have p4ps : a fast b-relaxation stage ½bf  for tc 5t5tb and a slow b-relaxation stage ½bs  for tg 5t5tL . Here tb is identical to the so-called b-relaxation time, while tg is a time scale to describe a plateau. By expanding M S0 ðtÞ in powers of lnðt=tg Þ, in stage ½bf , we obtain the asymptotic form S

M 2 ðtÞ ’ M S0 ðtÞ þ A1 ðpÞðt=tg Þbg (      bSg ) 1 tg t t ln 1 þ 2d ’ þ ln þ lA1 ðpÞ , l tg tg tc

ð20Þ

where A1 ðpÞ is a positive constant to be determined. Since lA1 51 near pc , M 2 ðtÞ is mostly dominated by the logarithmic growth given by lnðt=tg Þ around tg [14]. Similarly, in stage ½bs , we also obtain the asymptotic form S

M 2 ðtÞ ’ M S0 ðtÞ þ A2 ðpÞðt=tb Þbb (      bSb ) 1 tb t t ln 1 þ 2d ’ þ ln þ lA2 ðpÞ , l tb tb tc

ð21Þ

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where A2 ðpÞ is a positive constantsS to be determined. Since lA2 b1 near pc , M 2 ðtÞ is mostly dominated by the power-law growth given by ðt=tb Þbb around tb . It is shown that as p approaches to pc , both exponents bSg and bSb decrease and become constant in a glass phase beyond pc as bSb ¼ 1:1929 and bSg ¼ 1:0. Especially, the exponent bSg shows an existence of an inflection point at which the slope dbSg =dp becomes minimal and the exponent bSb nearly reduces to 1.1929. Hence this inflection point is considered to be a crossover point pb over which the supercooled liquid phase appears. This is confirmed later by analyzing the experimental results. For earlier times M 2 ðtÞ thus obeys a logarithmic growth in time, while for later times it obeys a power-law growth of a super-diffusion type with bSb 41:0. Here we note that since bSg 41:0, this power-law growth is different from the so-called von Schweidler type where the exponent is less than 1.0. The origin of the logarithmic growth is more clearly seen by introducing the time-dependent self-diffusion coefficient DS ðtÞ by DS ðt; pÞ ¼

1 d M 2 ðtÞ 2d dt L

¼

DSS ðpÞ

e2dlDS t L

1 þ ðDSS =DLS Þfe2dlDS t  1g

.

ð22Þ

Since tc 5tg 5tb 5tL near pc , by expanding Eq. (22) in powers of t=tg , one can easily find, on the time scale of tg , DS ðt; pÞ ’

DSS ð1 þ 2dlDLS tÞ 1 1 .  2dl t 1 þ 2dlDSS t

(23)

Near pc , DS ðtÞ thus obeys a t1 decay around tg , while DS ðtÞ ’ DSS in stage [E] for t5tc and DS ðtÞ ’ DLS in stage [L] for tbtL . This is an origin of a logarithmic growth of M 2 ðtÞ around tg . Hence this behavior is considered to be one of common features for equilibrium suspensions near pc . In Fig. 1, the time evolution of M 2 ðtÞ is shown for two typical three-dimensional systems, the system near pc and the system away from pc , where the control parameter p is given by the particle volume fraction fð¼ 4pa3 N=3V Þ. Here space is scaled with a, time is scaled with tD , and the diffusion coefficient DLS is scaled with D0 . The values of l, DSS , and DLS are chosen from Ref. [9,10], where fc ’ 0:57184 . . . . At fs ’ 0:364, we find bSg ¼ bSb and tg ¼ tb . For fpfs , the b-relaxation stage with two time scales tb and tg disappears. Hence a simple crossover from a short-time diffusion process to a long-time diffusion process is just seen. On the other hand, for f4fs , the b-relaxation stage appears. Especially, for fXfb , the supercooled region exists, where the asymptotic growths given by Eqs. (20) and (21) hold since tc 5tg 5tb 5tL . Thus, the dynamical behavior is quite different from that for small volume fractions. For comparison, the following power-law growth of von Schweidler type is also shown: M 2 ðtÞ ’ 100:81 þ 0:1ðt=tb Þ0:92 .

(24)

It can be seen that both Eqs. (21) and (24) describe the stage ½bs  well around tb . In Fig. 2, DS ðtÞ is also plotted versus time. Near fc , the t1 decay is seen, while there is no such a decay far away from fc . In the suspensions of neutral hard spheres, the non-Gaussian parameter aðdÞ 2 ðtÞ is known to be always negligibly small [7,8]. For such a special case, therefore, one can write F S ðk; tÞ in terms of M 2 ðtÞ only as 

L DS F S ðk; tÞ ¼ 1 þ LS fe2dlDS t  1g DS

ae ,

(25)

where ae ðk; pÞ ¼ k2 =ð2dlÞ. Near pc , there also exist three characteristic stage. In stage [E], we have F S ðk; tÞ ’ exp½k2 DSS t. In stage ½bf , from Eq. (20), the scattering function obeys a logarithmic decay "    bSg # 1 t t 1  ae ln F S ðk; tÞ ’  ae lA1 ðpÞ . (26) ð1 þ 2dtg =tc Þae tg tg

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2

1

log10(M2(t))

0

-1

-2

-3

-4

-2

0

2 log10(6t)

4

6

8

Fig. 1. A log–log plot of the mean-square displacement M 2 ðtÞ versus time at d ¼ 3. The solid line indicates the result for f ¼ 0:573, where l ¼ 70; DSS ¼ 0:1577, and DLS ¼ 1:436  106 and the dot-long-dashed line for f ¼ 0:466, where l ¼ 15, DSS ¼ 0:2183, and DLS ¼ 0:0135. The long-dashed line indicates M S0 ðtÞ, the dotted line the growth by Eq. (20) with bSg ð0:573Þ ’ 1:0; A1 ð0:573Þ ’ 1:4  104 , and tg ’ 101:235 , and the long-long-dashed line by Eq. (21) with bSb ð0:573Þ ’ 1:1929, A2 ð0:573Þ ’ 0:5, and tb ’ 103:997 . The symbols indicate the time scales: tc (filled square), tg (filled diamond), tb (filled circle), and tL (open square). The dashed line indicates the power-law growth of von Schweidler type given by Eq. (24).

0

-1

log10(DS(t))

-2

-3

-4 t-1 -5

-6

-7

-4

-2

0

2

4

6

8

log10(6t) Fig. 2. A log–log plot of the diffusion coefficient DS ðtÞ versus time. The details are the same as in Fig. 1.

In stage ½bs , from Eq. (21), F S ðk; tÞ obeys a power-law decay "    bSb # 1 t t F S ðk; tÞ ’ 1  ae ln  ae lA2 ðpÞ . ð1 þ 2dtb =tc Þae tb tb

(27)

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In stage [L], we have F S ðk; tÞ ’ exp½k2 DLS t. In Fig. 3, the scattering function F S ðk; tÞ is shown for the systems discussed in Fig. 1. For comparison, the asymptotic functions given by Eqs. (26) and (27) are also shown. Here we note that the exponent bSb depends only on f but not on k. We also mention here that the stretched exponential decay of the so-called Kohlrausch–Williams–Watts type does not exist for the equilibrium suspensions since aðdÞ 2 ðtÞ51 because of the existence of the long-range hydrodynamic interactions. This situation is different from that in non-equilibrium suspensions, where the stretched exponent b is much less than 1.0, even if aðdÞ 2 ðtÞ51. This difference is considered to result from the fact that the spatial heterogeneities in equilibrium suspensions, which are generated by the non-linear fluctuations, are unstable to fluctuations [9,10], while those in non-equilibrium suspensions are stable [15]. Once the self-intermediate scattering function F S ðk; tÞ is known analytically as Eq. (25), it is easy to calculate the self-part of the dynamic susceptibility given by Z 1 w00S ðk; oÞ ¼ o F S ðk; tÞ cosðotÞ dt . (28) 0

In Fig. 4, the susceptibility w00S ðk; oÞ is shown for different wave vectors. Near the singular point fc , the double peaks, a peak and b peak, are seen around oL and og at the peak position k ¼ 3:4 of the static structure factor SðkÞ, where oi ¼ 2p=ti . 2.2. Molecular systems We next discuss the molecular system, which contains N identical particles with mass m and radius a in the total volume V . The particles are described by Newtonian equations. In the following, we focus only on the hydrodynamic stage where rcut b‘f and tcut btf . Here ‘f and tf denote the mean-free path and the mean-free time, respectively. The relevant variables are a set of the position vectors of particles, fX 1 ; . . . ; X N g. The mean-square displacement M 2 ðtÞ is described by Eq. (8). The formal solution is given by 2 3 !2 L 1 4 v0 M 2 ðtÞ ¼ ln 1 þ 2 fe2dlDS t  1  2dlDLS tg5 . (29) l 2dl1=2 DLS 1.2

1

FS(k,t)

0.8

0.6

0.4

0.2

0

-2

0

2 log10(6t)

4

6

8

Fig. 3. A self-intermediate scattering function F S ðk; tÞ versus time for different volume fractions at k ¼ 2:0. The details are the same as in Fig. 1.

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0

log10(χ"S(k,ω))

0.5

1

-1.5

-2

-6

-4

-2

0

2

log10(ω) Fig. 4. A log–log plot of w00S ðk; oÞ versus frequency for different wave vectors k ¼ 2:0; 3:4, and 5.0 (from bottom to top). The symbols indicate the results at the characteristic frequencies foi g. The details are the same as in Fig. 3.

Similarly to the suspension, Solution (29) also suggests four different time scales; tA ð¼ 1=ðv0 l1=2 ÞÞ, which is a short time for a particle to move over a distance of order l1=2 , tg , tb , and tL , where tA 5tg 5tb 5tL . In fact, one can find the following asymptotic forms: ( M M 0 ðtÞ ¼ l1 ln½1 þ ðt=tA Þ2  for tptA ; M 2 ðtÞ ’ (30) 2dDLS t for tL pt : Similarly to the suspensions, therefore, there also exist three characteristic time stages near pc ; an early stage [E] for t5tA , a b-relaxation stage ½b for tA 5t5tL , and a late stage [L] for tL pt. In stage [E], we have M 2 ðtÞ ’ ðv0 tÞ2 , where the ballistic motion dominates the system. In stage [L], we have M 2 ðtÞ ’ 2dDLS t, where the long-time diffusion dominates the system. Between two time scales, tA and tL , one can also define a b-relaxation stage ½b with two time scales, tg and tb for p4ps by calculating the logarithmic derivatives jM 1 ðt; pÞ ¼

q log jM 2 ðtÞ  M M 0 ðtÞj , q log t

(31)

jM 2 ðt; pÞ ¼

q jM . q log t 1

(32)

Then, jM 2 ¼ 0 gives two time roots, tg and tb , which reveal two fairly flat regions for p4ps ; 8 < bM b ðpÞ at t ¼ tb ; jS1 ¼ : bM g ðpÞ at t ¼ tg ;

(33)

and which satisfy the relations tb 1=ðlDLS Þ;

tg ðtA tb Þ1=2 ðv0 l3=2 DLS Þ1=2 ,

(34)

M where tA 5tg 5tb 5tL near pc . Here we have bM g ¼ bb and tg ¼ tb at p ¼ ps . Similarly to the suspensions, it is M also shown that as p approaches to pc , both exponents bM g and bb decrease and become constant in a glass M M phase beyond pc as bb ¼ 1:3301 and bg ¼ 1:0. Especially, the exponent bM g shows an inflection point at which M the slope dbM g =dp becomes minimal and the exponent bb nearly reduces to 1.3301. Hence the inflection point

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32

must be a crossover point over which the supercooled liquid phase appears. This is confirmed later by analyzing the simulation results. Similarly to the suspensions, there exist two time stages in stage ½b; a fast b-relaxation stage ½bf  for tA 5t5tb and a slow b-relaxation stage ½bs  for tg 5t5tL . Similarly to Eq. (20), in stage ½bf , we obtain the asymptotic form M

bg M 2 ðtÞ ’ M M 0 ðtÞ þ B1 ðpÞðt=tg Þ (  2 !    bMg ) 1 tg t t ln 1 þ ’ þ 2 ln , þ lB1 ðpÞ l tg tg tA

ð35Þ

where B1 ðpÞ is a positive constant to be determined. Since lB1 51 near pc , M 2 ðtÞ is mostly dominated by the logarithmic growth given by lnðt=tg Þ around tg . In stage ½bs , we also obtain the asymptotic form M

bb M 2 ðtÞ ’ M M 0 ðtÞ þ B2 ðpÞðt=tb Þ ( !  2    bMb ) 1 tb t t ln 1 þ ’ þ lB2 ðpÞ þ 2 ln , l tb tb tA

ð36Þ

where B2 ðpÞ is a positive constant to be determined. Since lB2 41 near pc ; M 2 ðtÞ is mostly dominated by the M power-law growth given by ðt=tb Þbb around tb . Here we note that since bM g 41:0, this power-law growth is different from the so-called von Schweidler type with the exponent less than 1.0. Eq. (8) describes the dynamics of a crossover from the ballistic motion characterized by v0 to the long-time self-diffusion process characterized by DLS . The logarithmic growth of M 2 ðtÞ is then related to the t1 decay of the time-dependent diffusion coefficient DM ðtÞ given by 1 d M 2 ðtÞ 2d dt L ð2=lÞðv0 =2dDLS Þ2 fe2dlDS t  1g ¼ DLS . L 1 þ ð2=lÞðv0 =2dDLS Þ2 fe2dlDS t  1  2dlDLS tg

DM ðt; pÞ ¼

ð37Þ

Since tA 5tg 5tb 5tL near pc , by expanding Eq. (37) in powers of t=tg , one can easily find, on the time scale of tg , DS ðt; pÞ ’

v20 tð1 þ dlDLS tÞ 1 1 .  dð1 þ ðt=tA Þ2 Þ dl t

(38)

Near pc , DM ðtÞ thus obeys a t1 decay around tg , while DM ðtÞ ’ ðv20 =dÞt in stage [E] for t5tc and DS ðtÞ ’ DLS in stage [L] for tbtL . This is an origin of a logarithmic growth of M 2 ðtÞ around tg . Hence this behavior is also considered to be one of common features for equilibrium molecular systems near pc . In Fig. 5 a log–log plot of M 2 ðtÞ is shown for the hard-sphere fluid in which the control parameter is given by the volume fraction f. Here space is scaled with a, time is scaled with a=v0 , and the diffusion coefficient DLS is scaled with av0 . The value of l and DLS are chosen from Ref. [11], where fc ’ 0:5845. At fs ’ 0:458, we find tg ’ tb . For fpfs , therefore, the b-relaxation stage with two time scales tb and tg disappears. Hence a simple crossover from a short-time diffusion process to a long-time diffusion process is just seen. On the other hand, for f4fs , the b-relaxation stage appears and the asymptotic growths given by Eqs. (35) and (36) hold near fc . For comparison, the following power-law growth of von Schweidler type is also shown: M 2 ðtÞ ’ 100:59 þ 0:12ðt=tb Þ0:92 .

(39)

This equation has the same exponent as that of Eq. (24). It can be seen that both Eqs. (36) and (39) describe the stage ½bs  well around tb . In Fig. 6 a log–log plot of DM ðtÞ is also shown. Near fc , the t1 decay is seen around tg , while there is no such a decay far away from fc . The situation discussed above is the same as that in the suspension.

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2

1

log10(M2(t))

0

-1

-2

-3

-4

-2

0

2

4

6

8

log10(t) Fig. 5. A log–log plot of the mean-square displacement M 2 ðtÞ versus time. The solid line indicates the result for f ¼ 0:585, where l ¼ 70 and DLS ¼ 2:917  106 and the dot-dashed line for f ¼ 0:5, where l ¼ 18 and DLS ¼ 0:0243. The long-dashed line indicates M M 0 ðtÞ, the 4 1:00126 dotted line the growth by Eq. (35) with bM , and the long-long-dashed line by g ð0:585Þ ’ 1:0; B1 ð0:585Þ ’ 1:42  10 , and tg ’ 10 3:778 Eq. (36) with bM . The open diamond indicates the time tA . The dashed line indicates b ð0:585Þ ’ 1:33014; B2 ð0:585Þ ’ 0:07, and tb ’ 10 the power-law growth of von Schweidler type given by Eq. (39). The details are the same as in Fig. 1.

0 -1

log10(D(t))

-2 -3 -4

t-1

-5 -6 -7 -2

0

2 log10(t)

4

6

8

Fig. 6. A log–log plot of the diffusion coefficient DM ðtÞ versus time. The details are the same as in Fig. 5.

If the non-Gaussian parameter aðdÞ 2 ðtÞ is negligible, one can write the self-intermediate scattering function F S ðk; tÞ in terms of M 2 ðtÞ only. Then, one finds 2 3ae !2 L v 0 F S ðk; tÞ ¼ 41 þ 2 fe2dlDS t  1  2dlDLS tg5 . (40) 2dl1=2 DLS

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1.2

1

FS(k,t)

0.8

0.6

0.4

0.2

0

-2

0

2 log10(t)

4

6

8

Fig. 7. A self-intermediate scattering function F S ðk; tÞ versus time for different volume fractions at k ¼ 2:0. The details are the same as in Fig. 5.

0

log10(χ"S(k,ω))

-1

-2

-3

-6

-4

-2

0

2

log10(ω) Fig. 8. A log–log plot of w00S ðk; oÞ versus frequency for different wave vectors k ¼ 2:0; 3:4, and 5.0 (from left to right). The symbols indicate the results at the characteristic frequencies foi g. The details are the same as in Fig. 7.

In stage [E], we have F S ðk; tÞ ’ exp½k2 ðv0 tÞ2 =2d. In stage ½bf , from Eq. (35), the scattering function obeys a logarithmic decay "    bMg # 1 t t F S ðk; tÞ ’ 1  2ae ln  ae lB1 ðpÞ . 2 ae tg tg ½1 þ ðtg =tA Þ 

(41)

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In stage ½bs , from Eq. (36), F S ðk; tÞ obeys a power-law decay "    bMb # 1 t t F S ðk; tÞ ’ 1  2ae ln .  ae lB2 ðpÞ 2 ae tb tb ½1 þ ðtb =tA Þ 

35

(42)

In stage [L], we have F S ðk; tÞ ’ exp½k2 DLS t. Here we note that the exponent bSb depends only on f but not on k. We also mention here that the stretched exponential decay of the so-called Kohlrausch–Williams–Watts type does not exist if aðdÞ 2 ðtÞ51. In Fig. 7, the scattering function F S ðk; tÞ is shown for the systems discussed in Fig. 5. For comparison, the asymptotic forms given by Eqs. (41) and (42) are also shown. In Fig. 8, the selfpart of the dynamic susceptibility w00S ðk; oÞ is also shown for different wave vectors. The double peaks, a peak and b peak, are seen around oL and og , respectively, at the peak position k ¼ 3:4 of SðkÞ, where oi ¼ 2p=ti . The situation discussed above is quite similar to that in equilibrium suspensions. 3. Colloidal suspensions In this section, we analyze the experimental data and the simulation data for colloidal suspensions by using the mean-field equation (7). 3.1. Dense suspensions of neutral hard spheres—experiment We here analyze the experimental data obtained by van Megen et al. [8] for colloidal suspensions of neutral hard spheres with 6% polydispersity. The experimental suspension comprises mixtures of polymer particles (98% of total particle volume) and silica particles (2%) suspended in sis-decalin. The control parameter is given by the volume fraction f. Here space is scaled with a, time is scaled with tD , and the diffusion coefficients DLS and DSS are scaled with D0 , where a ¼ 200 nm, D0 ¼ 0:31  1012 m2 =s, and T ¼ 20:4  C. There exist three types of interactions acting on hard spheres. The first is a force exerted by the fluctuating fluid on spheres, leading to Brownian motion. The second is a direct force between particles. The last is a hydrodynamic interaction between particles [12]. In Fig. 9, we show the fitting results for M 2 ðtÞ for different volume fractions. The fitting values of DLS are then plotted in Fig. 10. For comparison, the theoretical result given by Eq. (11) is also plotted. Although the old experimental data are well described by that theoretical function, the recent data deviate slightly from the theoretical ones for higher volume fractions. In fact, the following singular function rather fits them well: DLS ðfÞ ¼

DSS ðfÞð1  1:2fÞ , 1 þ DSS ðfÞðf=fc Þð1  ðf=fc ÞÞ2

(43)

where g ¼ 2:0; k ¼ 1:0; n ¼ 1:2, and fc ¼ 0:564. Hence the recent experimental data suggests that by contrast with the theoretical singular function given by Eq. (11), the coupling effect between the direct interactions and 9 the hydrodynamic interactions given by n ¼ 1:2 is larger than that given by n ¼ 32 in Eq. (11) and the singular point fc ¼ 0:564 is smaller than 0:57184 . . . in Eq. (11). As discussed in Refs. [9,10], the experimental data can also be described by the law of Vo¨gel–Tamman–Fulcher (VTF) type (see Fig. 10)   0:471 L DS ’ 7:94 exp  , (44) 1  ðf=fc Þ where fc ’ 0:594. As mentioned by van Megen et al. [8], the experimental data at f ¼ 0:566 is the most concentrated for which M 2 ðtÞ grows linearly in time within the experimental time. For volume fractions higher than 0.566, the data fail to show a linear growth in time completely in the experimental time window although they show some upward curvature at longer times. Even for such higher volume fractions, however, the mean-field theory can predict the long-time self-diffusion coefficient approximately. The predicted values are also shown in Fig. 10. Thus, it can be seen that the experimental values deviate from any singular functions for volume fractions higher than fc . In fact, as discussed in Refs. [9,10], the experimental data are rather described well by the nonsingular diffusion coefficient predicted theoretically whose inflection point coincides with the singular point

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2

1

log10(M2(t))

0

-1

-2

-3

-4

-2

0

2 log10(6t)

4

6

Fig. 9. A log–log plot of M 2 ðtÞ versus time for different volume fractions f ¼ 0:466; 0:502; 0:529; 0:538; 0:553; 0:558; 0:566; 0:573; 0:578, and 0.583 (from left to right). The open circles indicate the experimental data from Ref. [8] and the solid lines the mean-field results given by Eq. (14).

0 -1 -2

log10(DSL)

-3 -4 -5 -6 -7 -8 0.3

0.35

0.4

0.45

0.5

0.55

0.6

0.65

φ Fig. 10. A log plot of DLS ðfÞ. The filled circles indicate the experimental data from Ref. [8], the filled diamonds the experimental data from Ref. [16], and the open circles the values predicted by the fitting. The solid line indicates the asymptotic fitting line given by Eq. (43), the dashed line Eq. (11), the long-dashed line VTF, and the dotted line the non-singular theoretical result from Refs. [9,10].

fc ¼ 0:564. This non-singular behavior must be a common feature around the glass transition. This situation could be more clearly seen if experiments and simulations are carefully done beyond the singular point. As shown in Ref. [8], the non-Gaussian parameter aðdÞ 2 ðtÞ does not play any role in the dynamics of this system for all volume fractions. Hence the self-intermediate scattering function F S ðk; tÞ is given by Eq. (25).

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Then, one can easily calculate w00S ðk; oÞ at the peak position km ¼ 3:4 of the static structure factor SðkÞ [8] as in Fig. 11. The double peaks, a peak and b peak, are seen for the volume fractions higher than f ’ 0:544. This suggests that fb must be around 0.544. We here note that the b peak seems to approach the so-called excess wing for higher volume fractions, where bSb ¼ 1:1929 and bSg ¼ 1:0. As shown in Ref. [8], the glass transition occurs around fg ¼ 0:58 and the supercooled liquid phase is observed for fb pfofg , where fb ’ 0:544. As is discussed in the next subsection, the simulations for the suspension of hard spheres show that without the hydrodynamic interactions, the glass transition does not occurs even for 6% polydisperse case, which is the same condition as that in the experiment, but the first-order phase transition occurs at the melting volume fraction fm . Hence we conclude that when aðdÞ 2 is negligible, the many-body correlation effects due to the longrange hydrodynamic interactions between colloids are indispensable to explain the colloidal glass transition. This will be discussed later. The fitting values for the adjustable parameter lðfÞ are shown in Fig. 12. Here the fitting value of l is determined in such a way that the mean-field result of M 2 ðtÞ coincides with the experimental data or the simulation data at t ¼ tg . For fX0:544, l seems to increase smoothly but drastically beyond the singular points, where no change in l is seen at both singular points, 0.564 and 0.57184. . . . In Figs. 13 and 14, we also show the time exponents bSg and bSb calculated by using Eqs. (16) and (17) and the slope dbSg =df, respectively. It can be seen that as f increases, the exponent bSg decreases, showing a inflection point around f ¼ 0:544, and nearly reduces to 1.0 in a glass phase around f ¼ 0:583, while the exponent bSb nearly reduces to a constant value 1.1929 above 0.544. Hence the crossover volume fraction fb over which the supercooled liquid phase appears is also suggested to be 0.544. This is also consistent with the value obtained by using one of turning points for the derivative dlðfÞ=df [10], while the other point corresponds to the glass transition point fg ’ 0:58 (see Fig. 15). Here the inflection point of the derivative dlðfÞ=df coincides with fc ¼ 0:564, which is also a turning point of l. In Fig. 16 we also show the characteristic times fti g ¼ ðtc ; tg ; tb ; tL Þ. Here we find fs ’ 0:364. The slopes of ti become steep above fb and then become gradual beyond fc . Since tL 1=DLS , tb ðlDLS Þ1 , and tg ðl2 DSS DLS Þ1=2 , one may expect that those time scales show a singular behavior around fc from Eqs. (11) or (43). However, they do not show any singular behavior around fc . This is due to the fact that the experimental results for the diffusion coefficient do not show any singular behavior for higher volume fractions (see Fig. 10). 0

log10(χ"S(k,ω))

-0.5

-1

-1.5

-2

-2.5

-6

-4

-2

0

2

log10(ω) Fig. 11. A log–log plot of w00S ðk; oÞ versus frequency at k ¼ 3:4 for different volume fractions f ¼ 0:466; 0:502; 0:529; 0:538; 0:553; 0:558; 0:566; 0:573; 0:578, and 0.583 (from right to left). The dotted line indicates the theoretical result at fb ’ 0:544 and the dashed line the mean-field result at f ¼ 0:6.

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120

100

λ

80

60

40

20

0 0.46

0.48

0.5

0.52

0.54

0.56

0.58

0.6

φ Fig. 12. An adjustable parameter lðfÞ. The filled circles indicate the fitting results, the dotted line the crossover volume fraction fb ¼ 0:544, the long-dashed line fc ’ 0:57184, the dot-dashed line fc ¼ 0:564, and the solid line a guide to the eyes.

1.2

bγS, bβS

1.15

1.1

1.05

1 0.45

0.5

0.55

0.6

φ Fig. 13. A plot of the time exponents bSb ðfÞ and bSg ðfÞ. The filled circles stand for bSg and the open circles for bSb . The dashed lines indicate bSb ¼ 1:1929 and bSg ¼ 1:0. The solid lines indicate the mean-field results obtained by using the fitting functions for DLS and l discussed in Refs. [9,10]. The details are the same as in Fig. 12.

From Eq. (14), the parameter l is inversely proportional to M 2 ðtÞ at time t ¼ 1=ð2dlDLS Þ. As discussed in the previous paper [17], therefore, the function V f ðfÞð¼ ld=2 Þ can be considered to be proportional to a free volume of a particle. The function V f ðfÞ is then shown in Fig. 17. On the analogy of a compressibility in the

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0 -0.2 -0.4

dbγS/ dφ

-0.6 -0.8 -1 -1.2 -1.4 -1.6 0.5

0.52

0.54

0.56

0.58

0.6

φ Fig. 14. A plot of the slope dbSg ðfÞ=df. The details are the same as in Fig. 13.

φβ φc φg 1000

λ, 4-1dλ / dφ

800

600

400

200

0 0.45

0.5

0.55

0.6

0.65

0.7

φ Fig. 15. l and dl=df versus f. The solid line indicates the derivative 41 dl=df and the dotted line the fitting function for l given by Eq. (5.2) of Ref. [10]. The open circles indicate the fitting values from experiment [8]. Here fb ¼ 0:544; fc ¼ 0:564, and fg ¼ 0:58.

glass-forming materials, we also introduce a thermodynamic-like quantity KðfÞ by [5] KðfÞ ¼ 

1 qV f . V f qf

(45)

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40

8

log10(t i)

6

4

2

0

-2 0.45

0.5

0.55

0.6

φ Fig. 16. A log plot of the characteristic times fti ðfÞg. The symbols indicate the time scales: tc (filled square), tg (filled diamond), tb (filled circle), and tL (open square). The dotted line indicates the b-relaxation time given by tb ¼ 1=ðlDLS Þ. The details are the same as in Fig. 13.

0.025 Glass phase

Supercooled liquid phase

Liquid phase

0.02

Vf

0.015

0.01

0.05

0

0.6

0.56

0.52

0.48

0.44

φ Fig. 17. A free volume V f ðfÞ. The filled circles indicate the fitting values. The vertical dot-dashed lines indicate the crossover volume fraction fb ’ 0:544 and the glass transition volume fraction fg ’ 0:58 [8–10]. The solid line up to f ¼ 0:519 and the dotted line are guides to the eye. The solid line for fX0:519 results from the fitting function for l given by Eq. (5.2) of Ref. [10].

In Fig. 18 a plot of KðfÞ is shown versus f, where the solid line was calculated from the fitting function for l given by Eq. (5.2) of Ref. [10]. Here K is a smooth function of f and has a peak around f ¼ 0:5666. We note here that the volume fraction dependence of V f and K is very similar to the temperature dependence of the corresponding thermodynamic quantities in the glass-forming materials.

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80 70 60

K

50 40 30 20 10 Glass phase

0 0.65

Supercooled liquid phase

0.6

Liquid phase

0.55

0.5

0.45

0.4

φ Fig. 18. A compressibility KðfÞ. The details are the same as in Fig. 17.

3.2. Dense suspensions of neutral hard spheres—simulation We here analyze the simulation data for the suspension of hard spheres [11]. The system consists of 10 976 hard spheres with radius ai and mass mi in an equilibrium fluid with the total volume V at a constant temperature T, where the distribution of radii obeys a Gaussian distribution with standard deviation s divided by a, and mass mi is proportional to a3i . Here a is an average radius. The main forces acting on hard spheres are a direct interaction between spheres and a force exerted by the fluctuating fluid on a sphere, leading to Brownian motion. Here the hydrodynamic interactions between spheres are neglected. The control parameter p is given by the volume fraction f ¼ ð4pa3 N=3V Þð1 þ 3s2 Þ, where s ¼ 0 for monodisperse case and s ¼ 0:06 for polydisperse case. As in Fig. 5, space is scaled with a, time is scaled with tD , and the diffusion coefficients DLS and DSS are scaled with D0 . In Fig. 19, we show the fitting results for M 2 ðtÞ for different volume fractions. At the highest volume fraction f ¼ 0:56, the simulation result does not fit well with Eq. (14) in stage ½b, where it shows an overshoot from the mean-field result. This is because the simulation result does not reach its equilibrium value yet (see Fig. 51). The mean-field result at the artificial point A (f ¼ 0:585; l ¼ 41) is also shown for comparison. Although such a point A does not exist physically because the liquid–solid phase transition is expected to occur at the melting volume fraction fm ð0:06Þ ’ 0:563, one can show the existence of a supercooled state near fc mathematically. Here we note that our recent extensive simulations for the hard-sphere fluid [18] suggest fm ð0:06Þ ’ 0:563, which is smaller than the value 0.570 predicted in the previous paper [11]. The long-time self-diffusion coefficient DLS ðfÞ is plotted in Fig. 20. The fitting values are shown to be well described by the asymptotic equation DLS ðfÞ ¼

1 , 1 þ DSS ðfÞðf=fc Þð1  ðf=fc ÞÞ2

(46)

where fc ðsÞ ¼ 0:5845; g ¼ 2; n ¼ 0, and k ¼ 1. Here the numerator is given by 1 because there exists no shorttime self-diffusion coefficient since the hydrodynamic interactions are completely neglected here. The singular term of Eq. (46) is considered to result from the many-body correlation effects due to direct interactions between particles. The diffusion coefficient DLS ðfÞ does not depend on s for sp0:06. However, the present system shows the first-order liquid–solid phase transition at the melting volume fraction fm ðsÞ, which is very

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42

2

log10(M2(t))

1

0

-1

-2 -2

-1

0

1 2 log10(6t)

3

4

5

Fig. 19. A log–log plot of M 2 ðtÞ versus time at s ¼ 0:06 for different volume fractions f ¼ 0:45; 0:50; 0:51; 0:52; 0:53; 0:54; 0:55, and 0.56 (from left to right). The solid lines indicate the mean-field results given by Eq. (14) and the open circles the simulation results from Ref. [11]. The dotted line indicates the mean-field result at the artificial point A, where f ¼ 0:585 and l ¼ 41.

0

-1

log10(DSL)

-2

-3

-4

-5

-6 0.3

0.35

0.4

0.45

5

0.55

6

φ Fig. 20. A log plot of DLS ðfÞ. The open squares indicate the simulation results for monodisperse case and the filled squares for polydisperse case. Both simulation results are taken from Ref. [11]. The dashed line indicates the asymptotic line given by Eq. (46) and the vertical dotdashed line the melting volume fraction fm ð0:06Þ ¼ 0:563. The cross indicates the value at the artificial point A.

sensitive to s. In fact, we have fm ðs ¼ 0Þ ’ 0:54 and fm ð0:06Þ ’ 0:563. Here we note that the detailed analyses of the simulation results for higher volume fractions [18] lead to fc ð0:06Þ ¼ 0:5845 which is different from 0.586 suggested by the data up to f ¼ 0:560 in the previous paper [11]. Up to 6%, therefore, the present system does not show any glass transition. This is also seen in the self-part of the dynamic susceptibility 00 w00S ðk; oÞ. In fact, the non-Gaussian parameter aðdÞ 2 is small for all volume fractions. Hence wS ðk; oÞ is easily calculated by using Eqs. (25) and (28). As is shown in Fig. 21, there are no double peaks for fpfm .

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0

log10(χ"S(k,ω))

-0.5

-1

-1.5

-2

-6

-4

-2

0

2

log10(ω) Fig. 21. A log–log plot of w00S ðk; oÞ versus frequency at s ¼ 0:06 and k ¼ 3:4 for different volume fractions f ¼ 0:50; 0:51; 0:52; 0:53; 0:54; 0:55; 0:56, and 0.585 (from left to right). The solid lines indicate the mean-field results given by Eq. (28). The dotted line indicates the mean-field result at the artificial point A.

40 35 30

λ

25 20 15 10 5 0 0.44

0.46

0.48

0.5

0.52

0.54

0.56

0.58

0.6

φ Fig. 22. A plot of lðfÞ. The open circles indicate the simulation results for monodisperse case in an equilibrium liquid phase and the filled circles for polydisperse case. The vertical long-dashed line indicates the singular point fc ¼ 0:5845 and the dot-dashed line fm ¼ 0:563. The cross indicates the artificial point A. The solid line is a guide to the eye.

The adjustable parameter l and the free volume V f are shown in Figs. 22 and 23, respectively. In both figures, the first-order phase transition from a liquid phase to a crystal phase occurs at the different melting volume fractions, fm ð0Þ ¼ 0:54 for monodisperse case and fð0:06Þ ¼ 0:563 for polydisperse case. In Figs. 24 and 25, the time exponents bSg and bSb and the slope dbSg =df are shown, respectively. Here we find fs ’ 0:44.

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0.03

0.025

Vf(φ)

0.02

0.015

0.01

0.005

0 0.6

0.58

0.56

0.54

0.52

0.5

0.48

0.46

0.44

φ Fig. 23. A plot of V f ðfÞ. The details are the same as in Fig. 22.

1.2

bγS(φ), bβS(φ)

1.15

1.1

1.05

1 0.45

0.47

0.49

0.51

0.53

0.55

0.57

0.59

φ Fig. 24. A plot of time exponents bSb ðfÞ and bSg ðfÞ. The filled circles stand for bSg and the open circles for bSb . The crosses indicate the results at the artificial point A. The horizontal dashed line indicates bSb ¼ 1:1929. The vertical dot-dashed line indicates the melting volume fraction fm ¼ 0:563 and the long-dashed line the singular point f ¼ 0:5845. The solid lines are guides to the eye.

Hence for f40:44 the b stage appears but there exists no supercooled region because up to fm ð0:06Þ ¼ 0:563, both exponents do not reach the final values yet and there is no inflection point for dbSg =df. In Fig. 26 the characteristic times fti g are also shown. They just increase monotonically up to f ¼ 0:56 as f increases. Finally, we mention here that M 2 ðtÞ and w00S ðk; oÞ suggest the existence of supercooled region near fc . In fact, the power-law growth and the double peaks are found at the artificial point A (see Figs. 19 and 21), where

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0

dbγS(φ)/dφ

-1

-2

-3 0.46

0.48

0.5

0.52

0.54

0.56

0.58

0.6

φ Fig. 25. A plot of the slope dbSg ðfÞ=df. The details are the same as in Fig. 24.

3

log10(t i(φ))

2

1

0

-1

-2 0.45

0.5

0.55

0.6

φ Fig. 26. A log plot of the characteristic times fti ðfÞg. The details are the same as in Figs. 16 and 24.

the parameter l increases drastically, the time exponents bSg and bSb become nearly constant, and the characteristic times fti g increase steeply. These situations are very similar to those discussed in the previous subsection. Hence we conclude that the supercooled state can be realized mathematically near fc . 3.3. Dilute suspensions of magnetic hard spheres—experiment We here analyze the experimental data by Hwang et al. [19] for the suspension of magnetic colloids. The system is an aqueous suspension of magnetic particles of 0:8 mm in diameter and 2% in concentration.

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The particles consist of ferromagnetic grains ðFe2 O3 Þ in the cores and are coated by polystyrene with a surface charge group (–COOH). The control parameter is given by the strength of the magnetic field 2



~ 4pm0 w2 a3 H s3=2 , 9kB T

(47)

~ the where w is the magnetic susceptibility of the colloid, m0 the magnetic permeability of the vacuum, H ~ external magnetic field, and s the area fraction. As Hð¼ jHjÞ increases, the magnetic particles are assembled and aligned in a magnetic field, forming a linear chain with n particles. Depending on the strength of H, the structure of the array changes remarkably. The chains interact via a repulsive dipole–dipole interaction and undergo Brownian motion due to the force from liquid particles. Since the area fraction s is small, the hydrodynamic interactions between chains are considered to be negligible. Here space is scaled with the average spacing ‘c between the chains, time is scaled with tl ð¼ ‘2c =Dc0 Þ which is a time for a chain to diffuse over a distance of order ‘c , and the diffusion coefficient DLS is scaled with the diffusion constant of a single chain Dc0 ð¼ D0 =nÞ. Here we have ‘c ¼ 7 mm, a ¼ 0:4 mm, D0 ¼ 0:548  1012 m2 =s, w ’ 0:07, s ¼ 0:03, 2 T ¼ 300 K, and n ¼ 15, leading to G ¼ 4:37336  106 H^ , where H^ is a dimensionless magnetic field defined by HOe ¼ ð103 =4pÞH^ A=m. The mean-square displacement M 2 ðtÞ of a tracer chain measures the displacement on the plain ~ In Fig. 27, we plot the fitting results for M 2 ðtÞ for different magnetic fields, where perpendicular to H. d ¼ 2 here. Below H^ ¼ 18, the experimental data show a linear growth in time for long times. Hence the long^ time self-diffusion coefficient DLS ðGÞ can be found by fitting. For H418, however, they fail to obey a linear growth in the experimental time window. Even in this case, one can predict the fitting values by using the mean-field equation (14). In Fig. 28 the fitting values for DLS are thus plotted versus G. The diffusion coefficient is thus shown to be well described by the singular function DLS ðGÞ ¼

DSS ðGÞ , 1 þ 8DSS ðGÞðG=Gc Þð1  ðG=Gc ÞÞ2

(48)

where k ¼ 8; g ¼ 2:0, and Gc ¼ 0:00193ðH^ c ¼ 21:0). Here n ¼ 0 because the hydrodynamic interactions do not play any role. Here DSS ðGÞ is a short-time self-diffusion coefficient due to the short-time dipole interactions whose asymptotic form is determined by using the fitting data for DSS . As H^ decreases, DSS reduces to 1, which 2

1

log10(M2(t))

0

-1

-2

-3

-4

-5

-6

-5

-4

-3

-2 log10(t)

-1

0

1

2

Fig. 27. A log–log plot of M 2 ðtÞ versus time for different values of magnetic field H^ ¼ 8; 10; 12; 14; 16; 18; 20; 25, and 30 (from left to right). The open circles indicate the experimental results from Ref. [19] and the solid lines the mean-field results given by Eq. (14).

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0

log10(DSL)

-1

-2

-3

-4

-5 0

0.0005 0.001 0.0015 0.002 0.0025 0.003 0.0035 0.004 Γ

DLS ðGÞ.

DLS

Fig. 28. A log plot of The filled circles indicate the values for obtained by fitting and the filled diamonds for DSS . The solid line L indicates Eq. (48) and the open circles the values for DS predicted by fitting.

1200

1000

λ

800

600

400

200

0 0

0.0005 0.001 0.0015 0.002 0.0025 0.003 0.0035 0.004 Γ

Fig. 29. A plot of lðGÞ. The long-dashed line indicates the singular point Gc ¼ 0:00193ðH^ c ¼ 21:0Þ and the dotted line the crossover point Gb ¼ 0:00138ðH^ b ’ 17:76Þ. The solid line is a guide to the eye.

is a diffusion constant of a single chain in a dimensionless unit. The singular term of Eq. (48) is related to the many-body correlation effects due to the dipole–dipole interactions between chains. Similarly to the experiment for the suspension of neutral colloids, the experimental data also deviate from Eq. (48) for G4Gc . The parameter l is determined by fitting within the experimental time. The fitting values for lðGÞ and ^ V f ð¼ ld=2 Þ are then shown in Figs. 29 and 30, respectively, where d ¼ 2 here. For GX0:00138ðHX17:76Þ, the

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0.0015 Supercooled liquid phase

0.00125

Liquid phase

Vf

0.001

0.00075

0.0005

0.00025

0 0.004 0.0035 0.003 0.0025 0.002 0.0015 0.001 0.0005

0

Γ Fig. 30. A plot of V f ðGÞ. The details are the same as in Fig. 29.

1.2

bγS, bβS

1.15

1.1

1.05

0 0

0.0005 0.001 0.0015 0.002 0.0025 0.003 0.0035 0.004 Γ

Fig. 31. A plot of time exponents

bSg ðGÞ

and

bSb ðGÞ.

The dashed line indicates bSb ¼ 1:1929. The details are the same as in Fig. 29.

slope of l seems to become slightly steep and that of V f becomes gradual. In Fig. 31, the time exponents bSg and bSb are shown. Both exponents decrease to constant values as G increases. In Fig. 32, the slope dbSg =dG is also shown. There exists a minimum point around G ’ 0:00138ðH^ ’ 17:76Þ, leading ð2Þ to Gb ’ 0:00138ðH^ b ’ 17:76Þ, where the peak height of að2Þ 2 is given by a2 ðlog t ¼ 0:032Þ ’ 0:843 [19]. In Fig. 33 we also show the characteristic times fti g ¼ ðtc ; tg ; tb ; tL Þ.The peak positions of að2Þ 2 ðtÞ obtained by Hwang et al. [19] are also plotted. Those correspond to the so-called a-relaxation time ta , which we do not

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0

dbγS / dΓ

-50

-100

-150

-200 0

0.0005 0.001 0.0015 0.002 0.0025 0.003 0.0035 0.004 Γ

Fig. 32. A plot of the slope dbSg ðGÞ=dG. The solid line indicates the mean-field result obtained by using the asymptotic fitting functions for l and DLS .

4

log10(t i)

2

0

-2

-4 0

0.0005 0.001 0.0015 0.002 0.0025 0.003 0.0035 0.004 Γ

Fig. 33. A log plot of the characteristic times fti ðGÞg. The dotted line indicates the b-relaxation time given by tb ¼ 1:26=ðlDLS Þ. The open circles indicate the peak positions of að2Þ 2 ðtÞ from Ref. [19]. The long-dashed line indicates Gc ¼ 0:00193. The details are the same as in Fig. 16.

discuss here. The slopes of ti seems to become slightly steep above Gb and then to become gradual beyond Gc . Here we also find Gs ’ 0:00052ðH s ’ 10:9Þ. Finally, we note that all typical features obtained here are qualitatively the same as those for the suspensions of neutral colloids, except that the non-Gaussian parameter að2Þ 2 ðtÞ becomes larger than 1 for GXGb .

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4. Molecular systems In this section, we analyze the simulation data for molecular systems by using the mean-field equation (8). 4.1. Glass-forming Lennard– Jones liquids—simulation We here analyze the simulation data performed by Gallo et al. [20] for a Lennard–Jones binary mixture (LJBM) embedded in an off-lattice matrix of soft spheres. The system consists of 800 particles of type A, 200 particles of type B and 16 soft spheres of type M. The interaction pair potentials are given by   s 6  smn 12 mn U mn ðrÞ ¼ 4emn  Zmn , (49) r r where indices m,n run on the particle type A; B; M. The parameters of the LJBM have been chosen as in Ref. [21] (see Fig. 34). The control parameter is given by the inverse of temperature 1=T. In the following, energy is scaled with eAA , length with sAA , time with sAA =ð48eAA =mÞ1=2 , temperature with eAA =kB , and diffusion coefficient with sAA ð48eAA =mÞ1=2 . Here we only analyze the dynamics of particles of type A. In the dimensionless form, Eq. (29) can then be written as " # 1 T 2dlDLS t L M 2 ðtÞ ¼ ln 1 þ fe  1  2dlDS tg . (50) l 96dlðDLS Þ2 Fig. 35 shows the fitting results for M 2 ðtÞ for different temperatures ranging from T ¼ 0:37 to T ¼ 2:0. At the lowest temperature T ¼ 0:37, the simulation result does not fit well with Eq. (50) in stage ½b, where it shows an overshoot from the mean-field result. Similarly to the suspensions, this is also considered to be due to the fact that the simulation results do not reach the equilibrium values yet. This is discussed in the next subsection (see Fig. 51). The fitting values for the long-time self-diffusion coefficient DLS ðTÞ are then shown in Fig. 36. Those values are well described by the singular function DLS ðTÞ ¼

1 , 1 þ 330ðT c =TÞð1  ðT c =TÞÞ2

(51)

4 AA BB AB MA MB

3

Uµν(r)

2

1

0

-1

-2 0

1

2

3

4

5

6

r Fig. 34. An interaction pair potential U mn ðrÞ versus r. The parameter values are expressed in Lennard–Jones units and are chosen from Ref. [21].

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4

2

log10(M2(t))

0

-2

-4

-6 -2

-1

0

1

2 log10(t)

3

4

5

6

Fig. 35. A log–log plot of M 2 ðtÞ for particles of type A versus time for different temperatures T ¼ 0:37; 0:39; 0:41; 0:43; 0:465; 0:48; 0:538; 0:58; 0:8, and 2.0 (from right to left). The open circles indicate the simulation results from Ref. [20] and the solid lines the meanfield results given by Eq. (50).

0

-1

log10(DSL(T))

-2

-3

-4

-5

-6

-7

0

0.5

1

1.5

2

2.5

3

3.5

1/ T Fig. 36. A log plot of DLS ðTÞ. The closed circles indicate the fitting results and the solid line indicates Eq. (51).

where T c ¼ 0:345; g ¼ 2; k ¼ 330; n ¼ 0, and DSS ¼ 1. This singular term is considered to result from the many-body correlation effects due to attractive interactions between particles. The small deviation from the singular function is also seen for lowest temperature. In Figs. 37 and 38, the fitting values for l and V f ðTÞ are plotted versus temperature, respectively. Here the fitting value of l is determined in such a way that the mean-field result of M 2 ðtÞ coincides with the experimental data or the simulation data at t ¼ tg . For Tp0:58ð1=TX1:724Þ, the slope of l seems to become

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120

100

λ

80

60

40

20

0

0

0.5

1

1.5

2

2.5

3

3.5

1/ T Fig. 37. An adjustable parameter lðTÞ. The closed circles indicate the fitting results, the long-dashed line the singular point T c ¼ 0:345ð1=T c ¼ 2:899Þ, and the dotted line the crossover point T b ¼ 0:58ð1=T b ¼ 1:724Þ. The solid line is a guide to the eye.

0.01 Liquid phase

Supercooled liquid phase

0.008

Vf

0.006

0.004

0.002

0 3.5

3

2.5

2

1.5

1

0.5

0

1/ T Fig. 38. A plot of a free volume V f ðTÞ. The details are the same as in Fig. 37.

M M steep and that of V f becomes gradual. The time exponents bM g and bb and the slope dbg =dð1=TÞ are also M shown in Figs. 39 and 40, respectively. It can be seen that as T decreases, the exponent bg decreases, showing a inflection point around T ¼ 0:58, and is expected to nearly reduce to 1.0 in a glass phase beyond T c ¼ 0:345. The exponent bM b nearly reduces to 1.3301 below T ’ 0:58. Hence the crossover temperature T b over which ð3Þ the supercooled liquid phase appears must be around 0.58, where the peak height of að3Þ 2 is given by a2 ðlog t ¼ 1:47Þ ’ 0:6825 [20]. In Fig. 41, the characteristic times fti g are also plotted. The peak positions of að3Þ 2 ðtÞ

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1.35

1.3

bγM, bβM

1.25

1.2

1.15

1.1

1.05

1

0

0.5

1

1.5

2

2.5

3

3.5

1/ T M M M Fig. 39. A plot of the time exponents bM b ðTÞ and bg ðTÞ. The open circles stand for bb and the filled circles for bg . The horizontal dashed line indicates bM b ¼ 1:3301. The details are the same as in Fig. 37.

-0.06

dbγM/d(1/ T)

-0.08

-0.1

-0.12

-0.14

1

1.5

2

2.5

3

1/ T dbM g ðTÞ=dð1=TÞ.

Fig. 40. A plot of the slope The solid line indicates the result obtained by using the asymptotic fitting functions for l and DLS . The details are the same as in Fig. 37.

obtained by Gallo et al. [20] are also plotted. Those correspond to the so-called a-relaxation time ta , which we do not discuss here. Below T b , the slopes of the times tb and tL also increase drastically but they do not become gradual yet up to T c . Here we also find T s ’ 2:3ð1=T s ¼ 0:435Þ, above which the b stage does not exist. By comparing the present results with the experimental results discussed in the previous section, we M conclude that the glass region is far beyond T c since bM g ¼ 1 and dbg =dð1=TÞ ¼ 0 at T ¼ T g . Similarly to the

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5

4

log10(t i)

3

2

1

0

-1

-2

0

0.5

1

1.5

2

2.5

3

3.5

1/ T Fig. 41. A log plot of the characteristic times fti ðTÞg. The dotted line indicates the b-relaxation time given by tb ¼ 1:12=ðlDLS Þ. The open circles indicate the peak positions of a2ð3Þ ðtÞ from Ref. [20]. The details are the same as in Figs. 5 and 37.

suspensions, below T c the slopes of ti are then expected to become gradual and the diffusion coefficients are also expected to deviate from the singular function given by Eq. (51). Finally, we note that in the present system, the non-Gaussian parameter aðdÞ 2 becomes larger than 1.0 around ta for TpT b ’ 0:58 [20]. 4.2. Hard-sphere fluids—simulation We here analyze the simulation data for a hard-sphere fluid [11,18]. The system consists of 10 976 hard spheres with radius ai and mass mi in a cubic box of volume V at a constant temperature T, where the distribution of radii obeys a Gaussian distribution with standard deviation s divided by the average radius a, and mass mi is proportional to a3i . The main mechanism of this system is a direct interaction between particles. The control parameter p is then given by the volume fraction f ¼ ð4pa3 N=3V Þð1 þ 3s2 Þ, where s ¼ 0 for a monodisperse case and s ¼ 0:06 for a polydisperse case. As in Fig. 5, space is scaled with a, time is scaled with a=v0 , and the diffusion coefficients DLS and DSS are scaled with av0 . In Fig. 42, the fitting results for M 2 ðtÞ are shown for different volume fractions. Up to f ¼ 0:5625, the simulation results are well described by the mean-field equation given by Eq. (29). This means that they are all measured in an equilibrium state. This situation is quite different from that in the suspension of hard spheres where the simulation result at f ¼ 0:56 does not reach an equilibrium state yet. Hence it turns out that for higher volume fractions the suspension takes more time to be equilibrated than the hard-sphere fluid does. Similarly to Fig. 19, the mean-field result at the artificial point B (f ¼ 0:585 and l ¼ 70), whose asymptotic behavior were discussed in Fig. 5, is also plotted for comparison. The fitting results for the long-time self-diffusion coefficient are then shown in Fig. 43. Since we have DSS ¼ 1 and n ¼ 0 in the molecular systems, the symptotic singular form of DLS is expected to be DLS ðfÞ ¼

1 , 1 þ ðf=fc Þð1  ðf=fc ÞÞ2

(52)

where fc ðsÞ ¼ 0:5845; g ¼ 2, and k ¼ 1. In fact, this can describe the simulation results well for the volume fractions higher than 0.5. As discussed in the previous paper [11], the long-time behavior is governed by the many-body correlation effects due to the direct interactions both in the hard-sphere fluids and in the

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3

2

log10(M2(t))

1

0

-1

-2

-3

-4

-1

-2

0

1

2 log10(t)

3

4

5

6

Fig. 42. A log–log plot of M 2 ðtÞ versus time at s ¼ 0:06 for different volume fractions f ¼ 0:50; 0:51; 0:52; 0:53; 0:54; 0:55; 0:56, and 0.5625 (from left to right). The solid lines indicate the mean-field results given by Eq. (29) and the open circles the simulation results from Ref. [11]. The dotted line indicates the mean-field result at the artificial point B shown in Fig. 5.

0 -1 -2

log10(DSL)

-3 -4 -5 -6 -7 -8 0.4

0.45

0.5

0.55

0.6

0.65

φ Fig. 43. A log plot of DLS ðfÞ. The open circles indicate the simulation results for the monodisperse case and the filled circles for the polydisperse case. The dot-dashed line indicates the singular function given by Eq. (52) and the solid line Eq. (53). The filled diamonds indicate the experimental results from Ref. [8] and the open diamonds the values predicted by the mean-field theory. The long-dashed line indicates the singular function given by Eq. (43) and the dotted line the non-singular function from Refs. [9,10]. The details are the same as in Fig. 20.

suspensions of hard spheres. However, the long-time self-diffusion coefficient in the suspension differs from that in the hard-sphere fluid due to a lack of the short-time hydrodynamic interactions between particles which leads to the short-time self-diffusion coefficient DSS . In this respect, from Eq. (46) one can also find the

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asymptotic singular function of DLS as [11,18] DLS ðfÞ ¼

DSS ðfÞ , 1 þ DSS ðfÞðf=fc Þð1  ðf=fc ÞÞ2

(53)

where fc ðsÞ ¼ 0:5845, g ¼ 2, n ¼ 0, and k ¼ 1. As seen in Fig. 20, this singular function can describe the simulation results for a wider range of volume fractions than Eq. (52) does. Thus, it turns out from Eqs. (46) and (53) that the long-time behavior for the suspensions of hard spheres coincide with that for the hard-sphere fluids if the short-time hydrodynamic interactions are taken into account in the simulations from the beginning. We should note here that even if those interactions are included in the simulations in addition to the direct interactions, the simulation results for the suspension do not agree with the experimental results because the singular point fc ¼ 0:5845 is still much larger than the experimental one given by fc ¼ 0:564 (see Fig. 43). Hence one may expect that the long-time hydrodynamic interactions must play an important role in reducing the singular point from 0.5845 to 0.564. Thus, we conclude that in order to explain the experimental results for the suspension of neutral hard spheres qualitatively and quantitatively near the glass transition, the hydrodynamic interactions between particles are indispensable. The fitting values of the adjustable parameter lðfÞ and the free volume V f ðfÞ are shown in Figs. 44 and 45, M S respectively. In Figs. 46–48, the time exponents bM b ðfÞ and bg ðfÞ, the slope dbg ðfÞ=df, and the characteristic ð3Þ times fti ðfÞg are also shown, respectively. The peak positions of a2 ðtÞ obtained by the simulations are also plotted. Those correspond to the so-called a-relaxation time ta , which we do not discuss here. Here we also find fs ’ 0:458, below which the b stage does not exists. The features for this system are mostly the same as those for the suspension of hard spheres. However, the big difference is that in the present system the inflection point exists for bM g at f ¼ 0:5524 below fm ¼ 0:563 (see Fig. 47). Hence this suggests an existence of a supercooled phase for fb ð’ 0:5524Þpfofm ð’ 0:563Þ, although this system shows a first-order liquid–solid transition at the melting point fm . The existence of a supercooled state is also confirmed by calculating the self-part of the dynamic susceptibility w00S ðk; oÞ at the peak position k ¼ 3:4 of SðkÞ. In fact, as is seen in Fig. 49, only the shoulder appears clearly above fb ¼ 0:5524 since aðdÞ 2 ðtÞ is neglected. The double peaks should appear if one calculates the susceptibility, including aðdÞ 2 ðtÞ. We should mention here that this is a first evidence by simulations to show the existence of a phase transition from a supercooled liquid to a crystal for a hardsphere fluid with polydispersity s ¼ 0:06, although this kind of transition is already known experimentally for water. The fitting procedure also shows that even in the crystal region near fc the double peaks are seen in the 70 60 50

λ

40 30 20 10 0 0.44

0.46

0.48

0.5

0.52 φ

0.54

0.56

0.58

0.6

Fig. 44. A plot of lðfÞ. The dotted line indicates fb ¼ 0:5524, the dot-dashed line fm ¼ 0:563, and the long-dashed line fc ¼ 0:5845. The cross indicates the result at the artificial point B.

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0.03

0.025

Vf (φ)

0.02

0.015

0.01

0.005

0 0.6

0.58

0.56

0.54

0.52

0.5

0.48

0.46

0.44

φ Fig. 45. A plot of V f ðfÞ. The details are the same as in Fig. 44.

1.35

1.3

bγM, bβM

1.25

1.2

1.15

1.1

1.05

1 0.44

0.46

0.48

0.5

0.52

0.54

0.56

0.58

0.6

φ M M M Fig. 46. A plot of time exponents bM b ðfÞ and bg ðfÞ. The filled circles stand for bg and the open circles for bb . The dashed line indicates ¼ 1:3301. The vertical dotted line indicates the crossover volume fraction f ’ 0:5524, the vertical dot-dashed line the melting volume bM b b fraction fm ¼ 0:563, and the long-dashed line the singular point fc ¼ 0:5845. The solid lines are guides to the eye. The crosses indicate the results at the artificial point B.

susceptibility w00S ðk; oÞ (see Fig. 49). Hence this suggests a possibility of the glass transition beyond fc ¼ 0:5845. In fact, this is also confirmed by showing that the long-time self-diffusion coefficient for the simulation data obey the same type of non-singular function as that proposed for the experiment of the suspension. By comparing the singular functions given by Eqs. (43) and (53), one can shift the non-singular

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0

dbγ (φ) /dφ

-1

-2

-3

-4

-5 0.44

0.46

0.48

0.5

0.52

0.54

0.56

0.58

0.6

φ Fig. 47. A plot of the slope dbSg ðfÞ=df. The filled circles indicate the results obtained by fitting. The dotted line indicates fb ’ 0:5524, the dot-dashed line fm ¼ 0:563, and the dashed line fc ¼ 0:5845. The solid line indicates the mean-field result obtained by using the fitting functions for DLS and l.

4

3

log10(ti(φ))

2

1

0

-1

-2 0.44

0.46

0.48

0.5

0.52

0.54

0.56

0.58

0.6

φ Fig. 48. A log plot of the characteristic times fti ðfÞg. The dotted line indicates the b-relaxation time given by tb ¼ 1:17=ðlDLS Þ. The open circles indicate the peak positions of a2ð3Þ ðtÞ from the simulations. The details are the same as in Figs. 26 and 46.

function for the experiment to the simulation results. First we divide the non-singular function by the factor ð1  1:2  fÞ and then transform f into f þ fc  fSc , where fSc is the singular point for the suspension. The result is shown in Fig. 50 together with the experimental data and its fitting non-singular function. The

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0

log10(χ"S(k,ω))

-0.5

-1

-1.5

-5

-4

-3

-2

-1

0

1

log10(ω) Fig. 49. A log–log plot of w00S ðk; oÞ versus frequency at s ¼ 0:06 and k ¼ 3:4 for different volume fractions f ¼ 0:50; 0:51; 0:52; 0:53; 0:54; 0:549; 0:55; 0:555; 0:56; 0:5625, and 0.585 (from left to right). The solid lines indicate the mean-field results given by Eq. (28) and the dashed line for fb ¼ 0:5524. The dotted line indicates the mean-field result at the artificial point B.

0 -1 -2

log10(DSL)

-3 -4 -5 -6 -7 -8 0.4

0.45

0.5

0.55

0.6

0.65

0.7

φ Fig. 50. Non-singular behavior of the long-time self-diffusion coefficient DLS ðfÞ. The solid line indicates the resultant non-singular function for the hard-sphere fluid and the dotted line the non-singular function for the suspension from Refs. [9,10]. The filled circles indicate the simulation results and the open circles the experimental results for the suspension. The long-dashed line indicates the singular function given by Eq. (43) and the dashed line Eq. (53). The vertical dot-dashed line indicates the melting volume fraction fm ¼ 0:563.

simulation results are well described by the non-singular function. This may predict the glass transition point as fg ’ 0:60 since fg ¼ fSg þ fc  fSc where fSc ð¼ 0:58Þ is the glass transition point for the suspension. The dynamics and the f dependences of physical quantities in the hard-sphere fluid are thus very similar to those in

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2

log10(M2(t))

0

-2

-4

-6 -3

-2

-1

0

1 log10(t)

2

3

4

5

Fig. 51. A comparison of the mean-square displacement in an equilibrium state with that in a non-equilibrium state at f ¼ 0:56. The filled circles indicate the equilibrium results and the open circles the non-equilibrium results. The solid line indicates the mean-field result given by Eq. (29).

the other systems which show the glass transition. Hence the re-entrant melting (transition from crystal to supercooled liquid) suggested by free energy calculations [22] is also expected to occur at higher volume fractions at s ¼ 0:06. This will be discussed in detail elsewhere. Finally, we discuss the non-equilibrium effect on the dynamics of the mean-square displacement M 2 ðtÞ for the hard-sphere fluids. In Fig. 51, the time evolution of the mean-square displacement in an non-equilibrium state is shown. The simulation result in a non-equilibrium state deviates from the equilibrium one in stage ½b, although the short- and the long-time behavior coincide with each other. The deviation always overshoots the equilibrium results. As seen in the previous sections, this kind of deviation appears near pc since the systems take long times to be equilibrated near pc . 5. Summary and conclusions In this paper, we have analyzed the experimental data and the simulation data by employing the mean-field theory for the mean-square displacements. We have first shown that in the supercooled region for pb ppppg the mean-square displacement obeys a logarithmic growth in stage ½bf  and a power-law growth of a superdiffusion type in stage ½bs . Here this power-law growth is different from that of von Schweidler type because the power-law exponent bb ðpÞ is larger than 1.0, where bSb ðpÞX1:1929 and bM b ðpÞX1:3301. There are three important parameter points. One is a crossover point pb over which the supercooled state appears. We have proposed that this point can be theoretically determined by an inflection point of the power-law exponent bg ðpÞ. This was in fact confirmed by the existence of double peaks for the self-part of the susceptibility when the non-Gaussian parameter is negligible and also by the existence of the distinct shoulder for the self-part of the susceptibility when the non-Gaussian parameter is not negligible. The second is a point ps over which the b stage exists. This is theoretically determined by a point at which the two characteristic times tb and tg coincide with each other. The last is a glass transition point pg . As discussed in the previous paper [10], this can be theoretically predicted by one of the turning points for the derivative dlðpÞ=dp, while the other determines pb (see Fig. 15). However, the available present data are not enough to find pg by calculating such a derivative, except the experimental data for the suspension of neutral hard spheres. Those values are listed in Table 1. We have then shown that up to pc the fitting data of the long-time self-diffusion coefficient can be well described

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Table 1 pi ; bb , and bg for different systems System

pb

bg ðpb Þ

bb ðpb Þ

pg

bg ðpg Þ

bb ðpg Þ

ps

bg;b ðps Þ

HSS-E HSS-S MCS-E LJL-S HSF-S

0.544 – 17.76 0.58 0.5524

1.026 – 1.095 1.143 1.112

1.1929 – 1.1949 1.3301 1.3304

0.58 – ? ? 0.60

1.0 – 1.0 1.0 –

1.1929 – 1.1929 1.3301 –

0.364 0.44 10.9 2.3 0.458

1.223 1.223 1.223 1.351 1.351

HSS-E indicates the hard-sphere suspension (experiment), HSS-S the hard-sphere suspension (simulation), MCS-E the magnetic-colloid suspension (experiment), LJL-S the Lennard–Jones liquid (simulation), and HSF-S the hard-sphere fluid (simulation). Here the parameters fpi g stand for fi , T i , and H^ i .

by the singular function given by Eq. (9), while above pc the fitting data start to deviate from the singular function. This suggests an existence of non-singular behavior near the glass transition recently discussed by the present author [5]. In fact, the characteristic times fti g do not diverge at the glass transition (see Figs. 16, 33, and 41). This would be a common feature of the glass transition. Thus, we have shown by comparing different glass transitions with each other that the present mean-field theory provides an useful tool to understand the dynamics near the glass transition. In order to show an existence of a supercooled state near pc , we have also discussed the dynamics of suspension of hard spheres at the artificial point A (f ¼ 0:585; l ¼ 41) and the dynamics of hard-sphere fluid at the artificial point B (f ¼ 0:585, l ¼ 70) shown in Figs. 22 and 44, respectively. One can calculate M 2 ðtÞ at the points A and B formally by using Eqs. (14) and (29) and thus show a typical dynamical behavior of a supercooled liquid near the singular point fc (see Figs. 19 and 42). This mathematical situation is always true as long as the long-time self-diffusion coefficient DLS ðpÞ is described by a singular function as DLS ðpÞg . Thus, this also suggests a possibility of the re-entrant melting (transition from crystal to supercooled liquid) and the glass transition. This will be discussed elsewhere. By comparing the experiments and simulations, we have confirmed that the hydrodynamic interactions play an important role near the glass transition for the suspension of neutral colloids. In fact, the simulation for the suspension of hard spheres does not show any glass transition without hydrodynamic interactions up to 6% polydispersity. As shown in Table 1, the hydrodynamic interactions turn out to play an important role in reducing fs and fc to those predicted by experiment. Hence we emphasize that in order to explain the experiment qualitatively and quantitatively, the theory and the simulation must include the many-body correlation effects not only due to the collision interactions but also due to the hydrodynamic interactions consistently. In this paper, we have focused only on the dynamics of the mean-square displacement near the glass transition and discussed a universal feature of the glass transition from a unified point of view. In order to understand the glass transition, however, the slow dynamics of spatial heterogeneities has to be studied in more detail. This will be discussed elsewhere together with a theoretical attempt of finding a non-singular function for the long-time self-diffusion coefficient. Acknowledgements We are grateful to Y.H. Hwang, Y. Terada, and E.R. Weeks for fruitful discussions and also to T. Shimura for technical assistance. This work was partially supported by Grants-in-aid for Science Research with No. 14540348 from Ministry of Education, Culture, Sports, Science and Technology of Japan. References [1] M. Tokuyama, I. Oppenheim (Eds.), Proceedings of the International Symposium on Slow Dynamics in Complex Systems, CP708 AIP, New York, 2004. [2] U. Bengtzelius, W. Go¨tze, A. Sjo¨lander, J. Phys. C 17 (1984) 5915.

ARTICLE IN PRESS 62 [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22]

M. Tokuyama / Physica A 364 (2006) 23–62 E. Leutheusser, Phys. Rev. A 29 (1984) 2765. M. Fuchs, W. Go¨tze, M.R. Mayr, Phys. Rev. E 58 (1998) 3384. M. Tokuyama, Physica A 315 (2002) 321. B.R.A. Nijboer, A. Rahman, Physica (Amsterdam) 32 (1966) 415. P.N. Segre`, P.N. Pusey, Phys. Rev. Lett. 77 (1996) 771. W. van Megen, T.C. Mortensen, S.R. Williams, J. Mu¨ller, Phys. Rev. E 58 (1998) 6073. M. Tokuyama, Phys. Rev. E 62 (2000) R5915. M. Tokuyama, Physica A 289 (2001) 57. M. Tokuyama, H. Yamazaki, Y. Terada, Phys. Rev. E 67 (2003) 062403; M. Tokuyama, H. Yamazaki, Y. Terada, Physica A 328 (2003) 367. M. Tokuyama, I. Oppenheim, Phys. Rev. E 50 (1994) R16; M. Tokuyama, I. Oppenheim, Physica A 216 (1994) 85. M. Tokuyama, Y. Terada, I. Oppenheim, Eur. Phys. J. E 9 (2002) 271. M. Tokuyama, Y. Terada, I. Oppenheim, Physica A 307 (2002) 27. M. Tokuyama, Y. Enomoto, I. Oppenheim, Physica A 270 (1999) 380. W. van Megen, S.M. Underwood, J. Chem. Phys. 91 (1989) 552. M. Tokuyama, Nihon Reoroji Gakkaishi 33 (2005) 113. M. Tokuyama, Y. Terada, J. Phys. Chem. (2005), submitted for publication. Y.H. Hwang, X.-l. Wu, Phys. Rev. Lett. 74 (1995) 2284. P. Gallo, R. Pellarin, M. Rovere, Phys. Rev. E 67 (2003) 041202. W. Kob, H.C. Andersen, Phys. Rev. Lett. 73 (1994) 1376; W. Kob, H.C. Andersen, Phys. Rev. E 51 (1995) 4626. P. Bartlett, P.W. Warren, Phys. Rev. Lett. 82 (1999) 1979.