J. Phys. Chem. B 2004, 108, 6719-6722
6719
Nonergodicity for a van der Waals Glass Model† Kyozi Kawasaki Electronics Research Laboratory, Fukuoka Institute of Technology, Fukuoka, Japan ReceiVed: October 3, 2003; In Final Form: January 25, 2004
We examine the question of how glassiness can remain finite in the van der Waals model of structural glass in situations where the Ginzburg criterion near the spinodal line of fluid instability against crystallization is satisfied. For d < 4, d being the dimensionality of space, finite glassiness demands certain conditions for the long-range interaction potential. For 4 < d < 6, no such condition is required, whereas for d > 6, the Ginzburg criterion becomes irrelevant. We comment on the relationship between dynamic and static calculations of the nonergodicity parameter.
1. Introduction
2. Mode Coupling Equation
One reason for difficulties in dealing with problems of structural glasses from the first principles is that of the difficulty of finding a satisfactory method of treating dynamical processes involving small (microscopic) length scales but very long time scales.1 A way to evade this dificulty is to investigate a model system in which interesting glassy behavior is associated with long spatial scales where no reference to microscopic properties is needed. In particular, we have proposed to study a model where long-range interaction potential is superimposed between different volume elements of continuum medium with no structure which we called the reference fluid.2 Approaches with the same spirit have been known for equilibrium aspects of fluid since the time of van der Waals. A good example can be found in a paper by Andersen and Chandler.3 A particularly convenient choice for this long-range potenatial is that introduced by Kac:4
In our previous work,2 we derived the mode coupling equation for the density correlator φk(t) normalized such that φk(0) ) 1 for our glass model. This equation reads
U(r) ) l-dU*(r/l )
(1.1)
with finite ∫∞0 dx U*(x), l being the force range. This interaction potential was used to derive the van der Waals equation of state as the limit of the infinite l.4,5 When the form of potential is such that Fourier compontents Uk of U(r) has negative parts, the model is found to be a good candidate of structural glass.6 We thus choose the form for Uk which behaves for k near some wavenumber km ∼ O(l-1) as
Uk ) - u2s + b(k - km)2 + ...
(1.2)
with u2s and b ∼ O(l2) some positive quantities where the ellipsis stands for higher order terms in (k - km). The purpose of this paper is to present further considerations of the model introduced in2 by taking into account the Ginzburg criterion of mean field theory and also to comment on static and dynamic calculations of the nonergodicity parameter. In particular, we will be concerned with the feedback freezing behavior of the MCT- like equations (1) and (2) below obtained in a certain limit of large l as we change the dimensionality of space d. We here mention related earlier works in which interplay of usual phase transitions and glassy behavior was studied for fluid models with long-range interactions.7,8 †
Part of the special issue “Hans C. Andersen Festschrift”.
(
)
∂2 + ωk2 φk(t) ) ∂t2
∫0t ds Mk(t - s) ∂s∂ φk(s)
(2.1)
where Mk(t) is the memory kernel defined by
Mk(t) )
F0 2mk2kBT
∫q|(k·q)Uq + (k·(k - q))Uk-q|2SqS|k-q| × φq(t)φ|k-q|(t) (2.2)
Here m, F0, Sk, and ωk are respectively the mass of a particle (one of the atoms or the molecules constituting the reference fluid), the average particle number density, the static structure factor, and the sound wave frequency, the last of which squared is given by the following:
(
ωk2 ≡ k2 c02 +
F0 U m k
)
(2.3)
c0 being the sound speed of the reference fluid. We have also used the notation ∫q‚‚‚ to denote 1/(2π)d ∫ dq‚‚‚. The main difference with the original mode coupling equation1 is that Fourier components of the direct correlation functions that enter the latter equations are here replaced by -Uk/kBT. The derivation of (1) and (2) retains the leading order terms in the expansion in powers of the smallness parameter r0/l, r0 being a characteristic microscopic length of the reference fluid, and does not rely from the outset on the uncontrolled factorization approximations that entered the original derivation.1 As a matter of fact, in the derivation of (1) and (2) (see section 4.2 of ref 2), the Gaussian assumption of the density fluctuations plays the central role. This permits us to factorize the fourbody correlation function of density fluctuations and to neglect the three-body correlation function. In section 4.4. of ref 2 and the reference cited there, the corrections to the Gaussian assumption for our van der Waals model system are argued to be of higher orders in r0/l than those retained. We will be concerned with the behavior of (1) with (2) in the limit of long force range l f ∞ and ask in what sense
10.1021/jp036972l CCC: $27.50 © 2004 American Chemical Society Published on Web 02/27/2004
6720 J. Phys. Chem. B, Vol. 108, No. 21, 2004
Kawasaki
glassiness still remains. This is the relevant question since glassiness is considered to disappear in the limit l f ∞ (J. Ja¨ckle, private communication, and ref 9). As a first step to address to this question, we try to scale out l from (1), (2), and (3). Inspection of these equations suggests us to put, with κ ≡ lk
(
l2
)
∂ +ω ˜ κ2 φ˜ κ(t) ) -l-d ∂t2
∫0 dt′ M˜ κ(t - t′) ∂t′∂ φ˜ κ(t′) t
(2.5)
with the scaled memory kernel given by
M ˜ κ(t) ≡
F0 2mκ2kBT
∫κ′ ((κ·κ′)U˜ (κ′) +
If we focus on the long time behavior near possible freezing, φ˜ κ(t) will be dominated by its nonoscillating component and the inertial term in (5) can be neglected to obtain
∫0t dt′ M˜ κ(t - t′) ∂t′∂ φ˜ κ(t′)
(3.1)
u2 - us2 b
(3.2)
ξ-2 )
where u2 is the inverse of the isothermal compressibility of the ref reference fluid χref T , that is, u2 ≡ 1/χT . [χT(F0) in subsection 3.3 ref of ref 2 should be replaced by χT .] We have also excluded the region u2 < us2 where the fluid is unstable against crystallization. Since u2 - us2 ∼ l0 and b ∼ l2 as far as the l dependence is concerned, we can set b ) l2B. Thus, we have ξ-2 ∼ l-2. We then put ξ ) lξ˜ and km ) l-1κm to get
S˜ (κ) ) Sk )
κ·(κ - κ)′U ˜ (|κ - κ′|))2S˜ κ′S˜ κ-κ′φ˜ κ′(t)φ˜ κ-κ′(t) (2.6)
ω ˜ κ2φ˜ κ(t) ) -l-d
kBT kB T 1 1 ≈ F0 u2 + Uk F0b ξ-2 + (k - k )2 m
(2.4)
eq 2.1 now looks as 2
Sk ) with
Uk ) U ˜ (κ), ωk2 ) ω ˜ κ2/l2, Mk(t) ) l-d-2M h κ(t) S˜ κ ≡ Sκ/l, φ˜ κ(t) ≡ φκ/l(t)
spinodal the mean field form for Sk is6,2
(2.7)
It looks as though the rhs vanishes in the limit l f ∞. However, we need to note that the pseudo spinodal line becomes infinitely sharp in this limit6 requiring careful analyses of singularities in S˜ κ. We end this section by the following comments. The mode coupling theory (MCT)- like equation derived here differs from the original MCT equation1 in an important way as to its implications on glass transitions. Here possible singularity in Sk near spinodal brings in a second length greater than the force range l, which is needed to produce frustration effects giving rise to glassiness. This is evident in static considerations of glassiness.9 On the other hand, the original MCT theory assumes Sk to be an input with no sigularity. Alternatively, we can avoid Sk completely from (5), (6), and (7) if we decide to use unnormalized density-density correlator Sk(t). However, Sk enters as the initial condition; Sk(0) ) Sk, which will be needed in considering the nonergodicity parameter below. Here we comment on the word “mean field theory”. Originally, this word was used to describe the van der Waals theory for fluids and the Weiss theory for magnets where the force range is taken to be infinite from the outset, and hence, no spatial degrees of freedom enter. On the other hand, the mode coupling equations (1) and (2) or the similar ones in1 are sometimes also referred to losely as mean field theories. However, this is misleading. The MCT equations contain fluctuations in an essential way to produce nonlinear feedback effects. Such feedback effects vanish if the force range l is taken to be infinite from the outset as is evident in the scaled eq 7. In this paper, we will be primarily concerned with these fluctuation effects, where the dimensionality of space enters in an important manner as in other fluctuation effects.
kBT 1 F0B ξ˜ -2 + (κ - κ )2 m
(3.3)
Let us now return to (7) and estimate the scaled memory kernel M ˜ , (6), appearing on its rhs for κ ≈ κm. Since S˜ (κ) is peaked around κ ) κm with the width K ≡ ξ˜ -1, the integral ∫ dκ′ is dominated by the region κ′ ≈ κm and |κ - κ′| ≈ κm. Hence we have κ′ d-2 π κ˜ d-1 ∫ dκ′ ) ∫d-2 m ∫ dθ sin (3 + θ) ∫ d∆κ′
(3.4)
where ∆κ′ ≡ κ′ - κm, θ the angular deviation from 60° of the κˆ ′ is the angle between the directions of κ and κ′, and ∫d-2 integral over the solid angle in the space of the remaining d 2 degrees of freedom. [The Jacobian Jd of the transformation of d-dimensional Cartesian coordinates x1, x2,‚‚‚, xd to the polar coordinates r,θ1,θ2,‚‚‚,θd-1 is Jd ) rd-1sind-2θd-1 sind-3θd-2 × ‚‚‚ × sin θ2. Hence we find the relation Jd ) r sind-2θd-1Jd-1.] As long as the integrals converge, we can estimate ∫ dκ′ ∼ K2κd-2 because |θ| is restricted to the region of the size K. m For ∆κ ∼ ∆κ′ ∼ K, we have S˜ (κ) ∼ S˜ (κm) ∼ K-2. Therefore we estimate the rhs of (5) or (7) as
˜ k ∼ l-dK2K-2K-2 ∼ l-dK-2 l-d M
(3.5)
where κm∼O(1). 4. Nonergodicity Parameter A measure of glassiness is given by the nonergodicity parameter ˜fκ defined by ˜fκ ≡ limtf∞φ˜ κ(t). Its equation is obtained from (5) or (7) by replacing φ˜ κ(t) by ˜fκ1
˜f κ )F ˜ κ({f˜}) 1 - ˜f κ
(4.1)
with
F ˜ κ({f˜}) ≡ l-d
1 M ˜ κ(φ˜ f ˜f ) ω ˜ κ2
(4.2)
We now recall (3) which is written as 3. Estimation of the Mode Coupling Strength In this section, we will estimate the strength of mode coupling represented by the rhs of (5) or (7) in the large l region. Near
S˜ (κ) )
kBT 1 BF0 K2 + ∆κ2
(4.3)
Nonergodicity for a van der Waals Glass Model
J. Phys. Chem. B, Vol. 108, No. 21, 2004 6721
with ∆κ ≡ κ - κm. Now, S˜ (κ) is peaked at κ ) κm; that is, the integrand on the rhs of (2) is peaked at κ′ ) κm and |κ - κ′| ) κm. Then we write κ′ ) κm + ∆κ′ and |κ - κ′| ) κm + ∆κ" with ∆κ" ≈ κm([2 - 2 cos(π/3 + θ)]1/2 - 1) + 1/2 (∆κ + ∆κ′) ˜ (κ) ≈ -us2 where the ≈ κm x3/2 θ + 1/2 (∆κ + ∆κ′) and U angle between κ and κ′ was taken to be π/3 + θ. Then we have for the integral in the memory kernel (6)
∫κ′ ‚‚‚ ) ≈
κd-1 m Ωd-1 d (2π)
κd-2 m (2π)
d
∫ d∆κ′ ∫ dθ sind-2 (π3 + θ)‚‚‚
(x32)
Ωd-1
d-3
∫ d∆κ′ ∫ d∆κ′′ ‚‚‚
(4.4)
where Ωd ) πd/2 d/Γ(1 + d/2) is the surface area of a unit hypersphere in the d-dimensional space and integrations over ∆κ′ and ∆κ′′ are for the interval (-∞, +∞) since they are dominated by the region ∆κ′, ∆κ′′ j K. We thus obtain
Uk ) - us2 + b(k - km)2 + ‚‚‚ ) -us2 + B∆κ2 + ‚‚‚ (4.12) with ∆κ ≡ l (k - km). Hence, the requirement 1 , W ∼ us2/B is in fact that Uk must have a deep minimum -us2 with small curvature (B) around it. Aside from this special case, glassiness tends to diappear for d < 4. This result is consistent with that of7 where the ergodic- nonergodic transition in the disordered sytem was found to be absent for d ) 3. 4.2. 4 < d < 6. Writing d ) 4 + ϑ with 0 < ϑ < 2 (10) becomes
() l r0
F ˜κ ∼ W
(4+ϑ)ϑ/(2-ϑ)
p-2/(2-ϑ)
(4.13)
The Ginzburg criterion p . 1 now is
F ˜ (f˜) ≈ l-d Z
4.1. d < 4. To obtain a finite feedback effect from the MC mechanism with the Ginzburg criterion satisfied, we must require F ˜ κ ∼ 1 or W ∼ (Vth/c0)2 us2/B . 1 for d < 4. Let us remind that Vth ∼ c0. We look at the expansion
()
∫ d∆κ′ ∫ d∆κ′′ K2 +1 ∆κ′2 K2 +1∆κ′′2 ˜f κ +∆κ′˜f κ +∆κ′′ m
l r0
m
(4.5)
( ) ( )
d-3
Ωd-1 (2π)d
()
u2 - us2 l d 3-d/2 , p≡ s r0 u2
(4.7)
where r0 ∼ r0-1/d is a microscopic length characterizing the reference fluid such as the average neighbor distance of molecules constituting the reference fluid. The quantity is the dimensionless distance from the spinodal. The Ginzburg criterion for spinodal of crystallization is given by10
p.1
(4.8)
[The Ginzburg criterion given in footnote 5 of ref 2 is not relevant here since it is concerned with the liquid-gas transition.] Therefore, we can write
K-2 )
()
B B l ) s s u2 u2 r0
D(d)d
p-D(d)
(4.9)
with D(d) ≡ 1/3 - d/2. Then, from (5) we obtain using κm∼O (1)
F ˜κ ∼ W
() r0 l
[D(d)-1]d
p-D(d)
(4.10)
with
W≡Z
()
kBTus2 Vth 2 us2 B ∼ ∼ c0 B rd0us2 mc02F0rd0B
p∼
(4.6)
We now introduce two dimensionless quantities and p as
≡
1-(ϑ/2) ) p . 1
(4.14)
Thus, in order for the MC term to give sufficient feedback for nonergodicity, that is, F ˜ ∼ 1, we need to take
with s 2 3 F0(u2) kBT 2 d-2 x3 κm Z≡ 2 2c02 mkBT BF0
4+ϑ
(4.11)
Here Vth is the molecular speed of thermal movement and is of the order of (kBT/m)1/2 where we have used F0rd0 ∼ 1. We now examine consequences of the above results in various regions of spatial dimensionlity d.
() l r0
[(4+ϑ)ϑ]/2
.1
(4.15)
In contrast to the case d < 4 this is satisfied with W∼1. 4.3. d > 6. The Ginzburg criterion p . 1 is here always satisfied right around the stability threshold ) 0, and F ˜ κ can get trivially large. On the other hand, for d f ∞, we note from (6) and (11) that W has the d-dependence as
W∼
( ) x3 2
d-3
Ωd-1 (2π)
) d
( ) x3 2
d-3
π(d-1)/2 (d - 1) d+1 (2π)dΓ 2
(
)
(4.16)
where Γ(x) ≈ (2π/x)1/2 e-xxx for x . 1. W can thus be made quite small and even a smallest mode coupling strength can give rise to nonergodicity. 5. Discussion The nonergodicity problem has also been considered from a static point of view via computation of the configurational entropy Sc. For three-dimensional systems with long-range forces it is found that Sc/V ∝ l-3 with V the system volume9 which vanishes in the van der Waals limit. A finite result is obtained if the system volume is rescaled as V f V/l3, that is, the configurational entropy per rescaled unit volume (∝l3) remains finite. The reason for the different conclusion from dynamical consideration of this paper is yet to be accounted for except to point out the totally different ways in which nonlinearity responsible for glassiness enters the static and dynamic approaches, which is here repeated.2,11 Static approaches (e.g., ref 12) relies on nonlinearities that appear in free energy functionals such as that in the ideal gas contribution to the Ramakrishnan-Yussouf density functional. On the other hand, an extra density factor entering the dynamical density functional theory13 is responsible for nonlinearity in dynamic calculations. Note in particular that the apparent nonlinearity of the ideal gas contribution of the Ramakrishnan-Yussouf density functional “miraculously” disppears here. This is what
6722 J. Phys. Chem. B, Vol. 108, No. 21, 2004
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is expected because ideal gas should not give rise to any nonlinearity in dynamics after all. Another point that can be made is that dynamics based on MCT is for the regime without activated processes whereas the static calculation of configuration entropy9 is for the regime dominated by activated processes over free energy landscape. For a recent discussion on this relationship, see ref 14. We should also mention that in the limit of infinite spatial dimensionality the MCT-like dyamic calculation is argued to give the same nonergodicity as the static one.17 Here we mention possible relevance of looking at the situation at infinite spatial dimensnionality as touched upon at the end of section 4 where certain simplifications on static properties are noted.15 [We note that the hydrodynamic interaction behaves as the distance raised to the power 2 - d. Thus, the long-range nature of this interaction which is nagging for three-dimensional colloidal glass becomes short-ranged at higher dimensionalities.] Finally, we make two remarks on the very interesting work16 to be referred to as KT. Stationarity condition for metastability in static approach is
δH{F} )0 δF(r)
(5.1)
where H{F} is the appropriate free energy density functional with F(r) the density. This is to be compared with the dynamic one which requires vanishing of the body force density f (r) ) -F(r)∇δH{F}/δF(r), which becomes, for nonvanishing density, i.e., F(r) > 0
δH{F} )0 δF(r)
(5.2)
δH{F} ) µ{F} δF(r)
(5.3)
∇ This is equivalent to
Here µ{F} is a chemical potential coming from constancy of the total mass ∫ drF(r) and can be a functional of the density but does not depend on r. Hence, condition (1) is stronger than (2) or (3). In fact, in ref 16, (3) was used since the conservation law of the total mass was imposed. In ref 16, the equation for nonergodicity parameter was derived for a simple TDGL-type model with Brazovskii Hamiltonian with cubic nonlinearity via mode coupling like treatment. A similar equation was derived by a purely static means for the same Hamiltonian by a method that picks up extrema of the Hamiltonian. The latter may be reproduced by
employing the real replica method of Monasson as explaind in ref 9. Such equivalence was not found for our van der Waals model.2 The discrepancy is primarily due to the fact that KT's model Hamiltonian has cubic nonlinearity which is responsible for nonergodicity both in static and dynamic calculations. On the other hand, in our case, we use reversible mode couplng in dynamic calculation whereas our statc calculation uses quartic nonlinearity in Hamiltonian.9 [If a Hamiltonian with quartic nonlinearity were used in ref 16, the equation for nonergodicity parameter would be similar to the static one obtained here by static means.] This exemplifies confusion surrounding our understanding of glassy behavior. See section 6.1 of ref 2 for an explicit discussion of this problem. We conclude this paper by reminding that the MCT-like equation of this paper was obtained by retaining the leading order terms of the expansion in powers of the smallness parameter r0/l, thus permitting us to investigate corrections to this equation by examining higher order terms of this expansion in a controlled manner. Acknowledgment. It is my great pleasure to contribute to this special issue of J. Phys. Chem. B honoring H. C. Andersen, with whom I overlapped at MIT almost four decades ago. References and Notes (1) Go¨tze, W. In Liquids, Freezing and Glass Transition; Hansen, J., Levesque, D., Zinn-Justin, J., Eds.; North-Holland; Amsterdam, 1991. (2) Kawasaki, K. J. Stat. Phys. 2003, 110, 1249. Here are the corrections to a few typographical errors in this paper. The F in eq 5.11 and in the unnumbered equation next to (5.14) should read F. The denominator of the first factor in this same unnumbered equation should read G-1 - ΣF. The K in eq 5.15 should read K. (3) Andersen, H. C.; Chandler, D. J. Chem. Phys. 1970, 53, 547. (4) Kac, M.; Uhlenbeck, G. E.; Hemmer, P. C. J. Math. Phys. 1963, 4, 216 and 229. (5) Lebowitz, J. L.; Mazel, A.; Presutti, E.; Cond. Mater./9809144 and Cond. Mater./9809145. (6) Klein, W.; Gould, H.; Ramos, R. A.; Clejan, I.; Mel’cuk, A. I. Physica A 1994, 205, 738. (7) Schilling, R. Z. Physik 1997, 143, 463. (8) Aksenov, A. L.; Plakida, N. M.; Schreiber, J. J. Phys. C 1987, 20, 375. (9) Loh, K.-K. et al.; Cond. Mater./0206494. (10) Klein, W, Cond. Mater./0104342. (11) Kawasaki, K.; Miyazima, S. Z. Physik B 1997, 103, 423. (12) Wu, S.; Schmalian, J.; Kotliar, G.; Wolynes, P, Cond. Mater./ 0305404. (13) Fuchizaki, K.; Kawasaki, K. J. Phys. Cond. Matter 2002, 46, 12203 and the rearlier references therein. (14) Lubachenko, V.; Wolynes, P. G. Cond. Mater./0307089. (15) Frisch, H. L.; Rivier, N.; Wyler, D. Phys. ReV. Lett. 1985, 54, 2061. (16) Klein, W.; Frisch, H. L. J. Chem. Phys. 1986, 84, 968. (17) Kirkpatrick, T. R. J. Chem. Phys. 1986, 85, 3515. (18) Kirkpatrick, T. R.; Thirumalai, D. J. Phys. A: Math. Gen. 1989, 22, L149. (19) Kirkpatrick, T. R.; Wolynes, P. G. Phys. ReV. A 1987, 35, 3072.
