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Thermodynamic perturbation theory and glass transition in simple fluids
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Physica A 372 (2006) 307–315 www.elsevier.com/locate/physa
Thermodynamic perturbation theory and glass transition in simple fluids M. Lo´pez de Haro, M. Robles Centro de Investigacio´n en Energı´a, U.N.A.M., Temixco, Morelos 62580, Mexico This paper is dedicated to Prof. Alberto Robledo on the occasion of his sixtieth birthday Available online 5 September 2006
Abstract Using the liquid state thermodynamic perturbation theory and the hard-sphere fluid as the reference system, the liquid–glass transition line of Lennard–Jones and hard-core Yukawa fluids is computed. The results are presented both in the reduced density vs. reduced temperature and in the reduced pressure vs. reduced temperature planes. A comparison with available simulation data and a discussion of the merits and limitations of this approach are also provided. r 2006 Elsevier B.V. All rights reserved. Keywords: Graphical uses interface; Molecular dynamics; Crystal defects; Dislocations
1. Introduction Despite the vast amount of literature already devoted to its study [1], the nature of the glass transition still remains as an unsolved and challenging problem. In fact, up to now there is no single theory that can account for all the features present in the rich and complex phenomenology that has been observed in supercooled liquids, amorphous solids and glasses. Since the attainment of the glassy state and the definition of the glass transition temperature depend on experimental time scales, there is little doubt that the glass transition is a nonequilibrium phenomenon. Nevertheless, in view of the so-called entropy crisis (the entropy of the glass must remain positive), there is a line of thought that goes back to the work of Adam, Gibbs and Di Marzio [2] (and has also been taken up more recently in the interpretation of the simulation work of Speedy [3–5] and in the theoretical approach of Me´zard and Parisi [6]), in which the (experimentally observed) dynamic glass transition is regarded as a kinetically controlled manifestation of an underlying (presumably second order) thermodynamic transition to an ideal glass. Although this kind of reasoning is not the only one available (take, for instance, the mode coupling approach) and is neither free from controversy, in this paper we will also adopt such a thermodynamic point of view. Interestingly enough, the work of Sastry [7] indicates that the glass transition lines determined either through thermodynamic or dynamic criteria seem to agree very well with each other.
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E-mail addresses: [email protected], [email protected] (M.L. de Haro), [email protected] (M. Robles). 0378-4371/$ - see front matter r 2006 Elsevier B.V. All rights reserved. doi:10.1016/j.physa.2006.08.013
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The idea of using the thermodynamic perturbation theory of liquids in connection with the glass transition, as we will do in what follows, is not new either. In fact, many years ago Hudson and Andersen [8], Went and Abraham [9] and Abraham [10] addressed the location of the glass transition in monoatomic liquids taking this approach. In particular, in the case of the Lennard–Jones (LJ) fluid, they used the Weeks–Chandler– Andersen (WCA) perturbation theory of liquids [11] and the hard-sphere (HS) fluid as the reference system in the perturbative scheme. In any case, the main point to be stressed here about this work is that it relies heavily on the existence of a glass transition in the HS fluid. Whether a glass transition takes place in the HS fluid has been a debatable issue for a long time, but evidence from various sources, including rather recent work, seems to suggest that this is indeed the case [3,5,12–17]. Therefore, since we will also take here the HS fluid as the reference fluid in the application of the liquid state perturbation theory, in order to study the glass transition of simple fluids we will take it for granted that a glass transition occurs in this reference system at a given packing fraction to be specified below. Further, it should be noted that, in order to get the free energy of an actual system within the thermodynamic perturbation theory of liquids, the free energy and structural properties of the reference system (in our case the HS fluid) are required. The free energy of the HS fluid of course follows from the choice of an equation of state and for this latter we will consider the Pade´ [4,3] of van Rensburg–Sanchez [18,19]. As far as the structure of the HS fluid is concerned, we will rely on an approximate (analytical method) introduced by Yuste and Santos [20] (see also Refs. [21,22]), referred to as the rational function approximation (RFA) method, to get reasonably accurate and thermodynamically consistent values of the structural properties of the HS. In particular, when used in connection with the van Rensburg–Sanchez equation of state, the RFA method leads to a glass transition in the HS fluid [13]. The paper is organized as follows. In the next section and in order to make the paper self-contained, we provide a summary of the RFA method and of the conditions leading to a glass transition in the HS fluid. Section 3 is devoted to a rather brief account of the liquid state perturbation theory and its application to a hard-core Yukawa (HCY) fluid and an LJ fluid. In Section 4 we present the results for the glass transition line for the HS, HCY and LJ fluids. The paper is closed in Section 5 with some final comments and concluding remarks.
2. The RFA method for the HS fluid In this section we recall the key aspects of the RFA method [20–22]. The purpose of this analytical-algebraic approach is to determine (approximate) reliable expressions for the radial distribution function (rdf) gHS ðrÞ of a HS fluid. The main idea is to retain the form given by Wertheim’s [24] solution to the Percus–Yevick equation and use a rational function assumption (hence the name of the procedure) for a function related to the Laplace transform GðtÞ L½rgHS ðrÞ, namely one takes GðtÞ to be given by GðtÞ ¼
t 1 , 12Z 1 et FðtÞ
(1)
where Z ¼ ðp=6Þrd 3 is the packing fraction, r the number density, d the HS diameter (units will be taken in which the diameter of the HSs d has a value of 1, so that all distances r are measured in units of the d) and FðtÞ a rational function of the form FðtÞ ¼
1 þ S 1 t þ S 2 t2 þ S 3 t3 þ S 4 t4 . 1 þ L1 t þ L2 t2
(2)
The coefficients S i and Li are algebraic functions of Z determined by imposing the two following physical restrictions to the rdf: (1) The first integral moment of hHS ðrÞ ¼ gHS ðrÞ 1 is well defined and nonzero. (2) The second integral moment of hHS ðrÞ must guarantee the thermodynamic consistency of the compressibility factor Z HS ¼ p=ðrkB TÞ (p being the pressure, kB the Boltzmann constant and T the temperature) and the isothermal susceptibility wHS ¼ ðdðZZ HS Þ=dZÞ1 .
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After some cumbersome but straightforward algebra (for details see Ref. [22]), these two conditions lead to L1 ¼ L2 ¼ S1 ¼ S2 ¼ S3 ¼ S4 ¼
1 Z þ 12ZL2 þ 2 24ZS 4 , 2 2Z þ 1 3ðZ HS 1ÞS4 , 3 1 þ 4L2 8S4 Z , 2 2Z þ 1 1 Z þ 8ZL2 þ 1 2L2 24ZS4 , 2 2Z þ 1 1 2Z Z2 þ 12ZL2 ðZ 1Þ 1 72Z2 S 4 , 12 ð2Z þ 1ÞZ " 1=2 # 1Z Z HS 1=3 wHS 1 1þ 1 , 36ZðZ HS 1=3Þ ZHS Z PY wPY
ð3Þ ð4Þ ð5Þ ð6Þ ð7Þ ð8Þ
where, Z PY ¼ ð1 þ 2Z þ 3Z2 Þ=ð1 ZÞ2 and wPY ¼ ð1 ZÞ4 =ð1 þ 2ZÞ2 . Note that the only input required is an expression for Z HS . The rdf is then given by gHS ðrÞ ¼
1 1 X j ðr nÞYðr nÞ, 12Zr n¼1 n
(9)
with YðrÞ the Heaviside step function and jn ðrÞ ¼ L1 ½t½FðtÞn . Explicitly, using the residue theorem, jn ðrÞ ¼
4 X i¼1
e ti x
n X m¼1
AðiÞ mn rnm , ðn mÞ!ðm 1Þ!
(10)
where AðiÞ mn
m1 d ¼ lim t½ðt ti Þ=FðtÞn , t!ti dt
(11)
ti being the roots of 1 þ S1 t þ S 2 t2 þ S3 t3 þ S4 t4 ¼ 0. In order to guarantee that gHS ðrÞ vanishes inside the core, S4 must be negative. However, according to the form of S 4 (cf. Eq. (8)) it may well happen that once ZHS has been chosen there exists a certain packing fraction above which S 4 is no longer negative. This was interpreted in Refs. [13,21] as an indication that, at the packing fraction Z0 where S 4 vanishes, the system ceases to be a fluid and a glass transition in the HS fluid occurs. We will adopt the same view here and also assume that the compressibility factor of the glass has the empirical form proposed by Speedy [3], namely Z glass Aglass =ð1 Z=Zrcp Þ where Aglass is a constant and Zrcp is the random close-packing fraction. For the time being these two parameters remain as unspecified quantities. In line with the expected character of the transition, it may be assumed further that while the pressure is continuous at Z0 both for the liquid and the glass, it exhibits a change in slope in going from the fluid phase to the glass. Thus, at Z ¼ Z0 , one must have Z glass ðZ0 Þ ¼ Z HS ðZ0 Þ. On the other hand, since Zglass does not represent the fluid phase, the RFA method implies that S 4 cannot be negative in the range Z0 oZoZrcp . This means that within that range S 4 is either positive or a complex number. Moreover, since a change in the structure of the system occurs for Z ¼ Z0 one would expect that this should manifest itself in a corresponding change in the qualitative character of S 4 . Thus, when Z approaches Z0 from above, either S 4 changes from positive to negative or it changes from complex to (positive) real. At Z ¼ Z0 , the first change leads to the requirement wglass ðZ0 Þ ¼ wPY ðZ0 Þ (with wglass ðqðZZ glass Þ=qZÞ1 ) while the second one implies wglass ðZ0 Þ ¼ wPY ðZ0 ÞðZ PY ðZ0 Þ 13Þ=ðZ HS ðZ0 Þ 13Þ. In view of Eq. (8), only the second condition is compatible with the assumption that the pressure exhibits a change in slope on going from the fluid phase to the glass. Therefore, in order to find the unknown quantities Z0 , Aglass and Zrcp in a self-consistent way once ZHS is specified we use the above information in the following way. First, we determine Z0 from the condition wHS ðZ0 Þ ¼ wPY ðZ0 Þ that is required to make S4 vanish (cf. Eq. (8)).
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Specifically this leads to wHS ðZ0 Þ ¼
ð1 Z0 Þ4 . ð1 þ 2Z0 Þ2
(12)
Once Z0 has been determined from Eq. (12), using the form chosen for Z glass and after a few easy algebraic steps one gets Aglass ¼
2Z 2HS ðZ0 Þð1 Z0 Þ2 3Z HS ðZ0 Þ 1
(13)
and Zrcp ¼
Z0 ð1 3Z HS ðZ0 ÞÞ . 1 Z HS ðZ0 Þð1 þ 4Z0 2Z20 Þ
(14)
Hence, all that is required within this approach to determine Z0 , Aglass and Zrcp is again only an explicit expression for Z HS . Note that wHS ðZ0 Þ as given in Eq. (12) is different from wglass ðZ0 Þ 1=Aglass ð1 Z0 =Zrcp Þ2 in agreement with the characteristic expected for a liquid–glass transition. 3. The liquid state thermodynamic perturbation theory Let us consider a system defined by a pair interaction potential fðrÞ split into a known (reference) part f0 ðrÞ and a perturbation part f1 ðrÞ. The usual perturbative expansion [25] for the Helmoltz free energy to first order in b 1=kB T leads to Z 1 A A0 ¼ þ 2prb f1 ðrÞg0 ðrÞr2 dr þ Oðb2 Þ, (15) NkB T NkB T 0 where A0 and g0 ðrÞ are the free energy and the rdf of the reference system, respectively, and N R Zis the number of particles. If the reference system is taken to be the HS fluid, then of course AHS =NkB T ¼ 0 ðZ HS 1Þ=Z0 dZ0 and g0 ðrÞ ¼ gHS ðrÞ. Note that given fðrÞ, one may in principle use the liquid state perturbation theory to obtain explicitly an approximation to the free energy of the system within the RFA method simply by choosing Z HS and substituting the ensuing analytical expressions into Eq. (15). However, depending on the form of the perturbation part of the pair interaction potential, simplified schemes that use the HS fluid as the reference fluid have been introduced in which one usually takes an effective (in general density and temperaturedependent) diameter. In Ref. [26] we have provided a unifying framework for all the classical perturbative schemes [27,28,11]) within the RFA method and illustrated it for the case of an LJ fluid. Note that the right-hand side of Eq. (15) represents always an upper bound for the value of the free energy of the real system. We remark that in the Mansoori–Canfield/Rasaiah–Stell thermodynamic perturbation theory scheme, which we will later use for the LJ fluid, the effective diameter is chosen to yield the least upper bound. 3.1. The LJ fluid For the LJ fluid the reference system is a HS fluid, i.e., one sets ( 1; rpd eff ; f0 ðrÞ ¼ 0; r4d eff ;
(16)
where d eff is an effective diameter and thus the Helmholtz free energy of the LJ fluid may be approximated by Z 1 ALJ AHS r þ 2prb fLJ ðrÞgHS eff r2 dr, (17) NkT NkT d d eff
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where fLJ ðrÞ ¼ 4ðs12 =r12 s6 =r6 Þ is the usual LJ potential (with s the distance at which fLJ ðrÞ ¼ 0 and the depth of the well). Introducing the Laplace transform of rgHS ðrÞ given in Eqs. (1)–(8), Eq. (17) may be rewritten as Z ALJ AHS 48Z 1 p þ CLJ ðtÞGðtÞ dt, (18) NkB T NkB T T 0 where CLJ ðtÞ ¼
# 6 " 6 6 s t 1 t4 . 10! 4! d eff d eff s
(19)
Within the Mansoori–Canfield/Rasaiah–Stell scheme [28], the temperature- and density-dependent effective diameter d d eff ðr ; T Þ (where r ¼ rs3 and T ¼ kB T= are the reduced density and the reduced temperature, respectively) that minimizes the free energy is in turn determined from "Z ! # Z 6 10 ðp=6Þr q Z HS 1 4 1 6 1 t 1 t t4 dZ þ dt ¼ 0, (20) qd 0 Z T d 1 et FðtÞ d 10! 4! 0 where it is understood that the packing fraction appearing in both ZHS and FðtÞ (this latter given in Eq. (2)) must be expressed in terms of r , i.e., using r ð6=pÞZ. Once ALJ has been determined with the aid of Eqs. (18)–(20), the compressibility factor of the LJ fluid readily follows through the relation q ALJ Z LJ ¼ Z . (21) qZ NkT T;N This is all that is needed to compute the liquid–glass transition in the LJ fluid as done below. 3.2. The HCY fluid Now we examine a different case that of a HCY fluid. This is a system whose molecules interact via the pair potential ( 1; rps0 ; fHCY ðrÞ ¼ (22) s0 0 r exp½zðr s0 Þ=s0 ; r4s0 ; where s0 is the hard-core diameter, 0 is the depth of the potential well at r ¼ s0 and z is the inverse range parameter. It is usually taken as z ’ 2 in order for the properties of this fluid to be similar to the ones of the LJ fluid. Clearly, if z ! 0 this potential leads to Coulombic interaction and in the limit z ! 1 it reduces to the HS interaction where the diameter of the spheres is s0 . We now rewrite fHCY ðrÞ in the form required by the liquid state perturbation theory as fHCY ðrÞ ¼ f0 ðrÞ þ f1 ðrÞ, where the reference potential f0 will again be a HS pair potential (where the diameter of the spheres will be d eff s0 ) and the perturbed part f1 is given by ( 0; rps0 ; f1 ðrÞ ¼ (23) s0 0 r exp½zðr s0 Þ=s0 ; r4s0 : Substituting Eq. (23) into Eq. (15) one gets the approximation to the free energy of the HCY fluid, namely Z 1 AHCY AHS ¼ 2prbs0 0 exp½zðr s0 Þ=s0 gHS ðrÞr dr, (24) NkT NkT s0 or, equivalently, AHCY AHS ¼ 12Zb0 ez GðzÞ, NkT NkT
(25)
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where GðzÞ Gðt ¼ zÞ is again the Laplace transform of rgHS ðrÞ given in Eqs. (1)–(8) and thus, once z has been fixed, a function only of the packing fraction Z. Therefore, in this instance one gets an explicit analytical result for the free energy of the HCY fluid within the thermodynamic perturbation theory. Further, in this case one may also readily determine the compressibility factor as q AHCY q ðZGðzÞÞ. (26) ZHCY ¼ Z ¼ ZHS ðZÞ b0 ez qZ NkT T;N qZ Since it will be used later on, we now give the explicit form of the equation of state of the HCY fluid in terms of the dimensionless variables P0 ¼ Ps30 =0 , r0 ¼ rs30 and T 0 ¼ kT=0 , namely p q P0 ¼ r0 T 0 Z HS r0 r0 ez 0 ðr0 GðzÞÞ, (27) 6 qr where it is again understood that the packing fraction appearing in both ZHS and GðzÞ must be expressed in terms of r0 , i.e., using r0 ð6=pÞZ. We remark that in contrast with the case of the LJ fluid in which the effective diameter of the reference HS system turns out to be dependent on both temperature and density, for the HCY fluid it simply coincides with the hard-core diameter. This difference will have an influence on the determination of the liquid–glass transition line as discussed in the next section. 4. Results In order to proceed and perform actual calculations, one needs a choice for ZHS . Since this quantity is unfortunately not known exactly, as mentioned earlier we will consider the Pade´ [4,3] of van Rensburg–Sanchez [18,19] given by ZHS Z 43 ¼
1 þ 1:024385Z þ 1:104537Z2 0:4611472Z3 0:7430382Z4 , 1 2:975615Z þ 3:00700Z2 1:097758Z3
(28)
which is known to be rather accurate both in the stable and metastable fluid regions. This leads (cf. Eqs. (12)–(14)) within the RFA method to a glass transition in the HS fluid at Z0 ¼ 0:5604 to Aglass ¼ 2:765 and to Zrcp ¼ 0:6448. It should be remarked that all these values are in very good agreement with other estimates in the literature [13]. Further, for this choice of Z HS one gets wHS ðZ0 Þ ¼ 0:0083 and wglass ðZ0 Þ ¼ 0:0062. We can now determine the location of the liquid–glass transition line for both the LJ fluid and the HCY fluid. We begin with the LJ system where this line in the r –T plane may be easily derived from the simple relationship p 3 r d ðr ; T Þ ¼ Z0 , (29) 6 where d is determined using Eq. (20). The outcome [23] is shown in Fig. 1 where we have also included the recent simulation results of Di Leonardo et al. [29]. As clearly seen in the figure, not only the qualitative trend observed in the simulations is reproduced, but also the quantitative agreement is rather good. In the case of the HCY fluid, the results are derived from the equality p 0 3 r s 0 ¼ Z0 (30) 6 so that the liquid–glass transition line in the r0 –T 0 plane is simply a line parallel to the T 0 -axis whose abscissa is r00 ¼ ð6=pÞZ0 =s30 . This result is due to the fact that we are restricting ourselves to the first order term in the thermodynamic perturbation expansion. Of course one should expect the finite Yukawa part of the potential to have an influence on the temperature dependence of the liquid–glass transition, but this is not captured by the present approximation. The liquid–glass transition line may also be shown in different thermodynamic planes, particularly the P –T plane with the aid of the equation of state. In Fig. 2 we display the liquid–glass transition lines for the HS fluid, the LJ fluid and the HCY fluid with z ¼ 2:5 in the reduced temperature vs. reduced pressure plane. To our knowledge, no simulation data are available for the
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1.4 1.2
T∗
1 0.8 0.6 0.4 0.2 0 0.9
1
1.2
1.1 ∗
Fig. 1. Liquid–glass transition line of the Lennard–Jones fluid as obtained with the Mansoori–Canfield/Rasaiah–Stell perturbation scheme using ZHS ¼ Z43 . Also included are the simulation data of Di Leonardo et al., in Ref. [29], and their vertical size accounts for the error bars.
1 0.8
T∗
0.6 0.4 0.2
2
4
6
8
10
12
P∗ Fig. 2. Liquid–glass transition lines of the hard-sphere fluid (continuous line), the Lennard–Jones fluid (dashed line) and the hard-core Yukawa fluid with z ¼ 2:5 (dotted line) in the reduced temperature vs. reduced pressure plane as obtained with the liquid state perturbation theory using the RFA method and taking ZHS ¼ Z43 .
liquid–glass transition lines of these three fluids in this plane, so a proper assessment of the results displayed in this figure is not possible at this stage. 5. Discussion The results of the previous section deserve further comments. To begin with, although approximate, our approach is self-consistent and constitutes a fully analytical derivation with no free parameters. It leads naturally to what we have interpreted as a glass transition in the HS fluid. This is a necessary requirement in order to apply the liquid state thermodynamic perturbation theory to the computation of the glass transition line of simple fluids when one takes the HS fluid as the reference system. Secondly, the overall agreement between the trends of the simulation results for the glass transition line of the LJ fluid and our thermodynamic results is encouraging and seems to provide further support to the notion that the glass transition may be associated to an underlying thermodynamic phase transition. In particular, we note that, in agreement with the assertion of the mode coupling approach, the comparison with the simulation results (that refer to the dynamical glass transition) indicates that, at least for the LJ fluid, the dynamical transition temperature is higher than the thermodynamic liquid–glass transition temperature.
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As far as the HCY fluid is concerned, no comparison with either experimental or simulation data was made since to our knowledge these data are not presently available. Nevertheless, in the range of application of the liquid state perturbation theory for the Helmoltz free energy to first order in b and taking as the reference system a HS fluid, the features observed for the HCY system should also hold for any other system with a hard core. That is, for such systems in the reduced density vs. reduced temperature plane the glass transition line should be a straight line parallel to the temperature axis, while in the reduced pressure vs. reduced density plane it should be a straight line (parallel to the pressure axis and coinciding with the straight line of the HS fluid). Finally, although of course the existence of a glass transition in the HS is the only requirement to apply the liquid state thermodynamic perturbation theory to the study of the glass transition in simple fluids, one may reasonably wonder whether the above statements that stem out of the results we have presented are tied to the choice of the equation of state of the HS fluid or, even more profoundly, to the form of the rational function FðtÞ taken in the RFA method. As far as the first issue is concerned, we have pointed out elsewhere [13,23] that this is not the case. The second issue is subtler and more difficult to handle. If the rational form is maintained for FðtÞ, then (cf. Eq. (2)) the next order of approximation would involve two new coefficients say L3 and S 5 (one in the numerator and one in the denominator) that would require two extra conditions to be fixed. It is conceivable that with these additions, one could still find a threshold value of the packing fraction beyond which one could not guarantee that gHS ðrÞ vanishes inside the core and this could then be interpreted again as the packing fraction corresponding to the glass transition. But the opposite, i.e., the nonexistence of such a threshold value, is possible as well. However, the main difficulty in going beyond the approximation of Eq. (2) and in deciding on the two previous possibilities resides in the fact that the extra conditions required to fix the two new coefficients do not seem to have a clear physical interpretation. Therefore, we take a pragmatic point of view and stick to the original RFA method which, although admittedly approximate, has proved to be both accurate and free from physical inconsistencies. With such a view, we are thus persuaded that the results we have derived stand and hope to encourage simulation or other theoretical approaches to assess their full value. Acknowledgements We acknowledge the financial support of DGAPA-UNAM through project IN-110406. References [1] For reviews and two recent articles discussing many aspects of this problem see: C.A. Angell, Science 267 (1995) 1924; M.D. Ediger, C.A. Angell, S.R. Nagel, J. Phys. Chem. 100 (1996) 13200; C.A. Angell, K.L. Ngai, G.B. Mackenna, P.F. McMillan, S.W. Martin, J. Appl. Phys. 88 (2000) 3113; P.G. Debenedetti, F.H. Stillinger, Nature 410 (2000) 259; D.R. Reichman, P. Charbonneau, J. Stat. Mech. P0513 (2005); F. Sciortino, J. Stat. Mech. P0515 (2005). [2] J.H. Gibbs, E.A. Di Marzio, J. Chem. Phys. 28 (1958) 373; G. Adams, J.H. Gibbs, J. Chem. Phys. 43 (1965) 139. [3] R.J. Speedy, J. Chem. Phys. 100 (1994) 6684. [4] R.J. Speedy, J. Phys. Condens. Matter 9 (1997) 8591. [5] R.J. Speedy, Mol. Phys. 95 (1998) 169. [6] M. Me´zard, G. Parisi, J. Phys. Condens. Matter 12 (2000) 6655. [7] S. Sastry, Phys. Rev. Lett. 85 (2000) 590. [8] S. Hudson, H.C. Anderssen, J. Chem. Phys. 69 (1978) 2323. [9] H.R. Wendt, F.F. Abraham, Phys. Rev. Lett. 41 (1978) 1244. [10] F.F. Abraham, J. Chem. Phys. 72 (1980) 359. [11] J.D. Weeks, D. Chandler, H.C. Andersen, J. Chem. Phys. 54 (1971) 5237. [12] B. Baeyens, H. Verschelde, Z. Phys. B 102 (1997) 255. [13] M. Robles, M.L. de Haro, A. Santos, S.B. Yuste, J. Chem. Phys. 108 (1998) 1290. [14] W. van Megen, S.M. Underwood, Nature 362 (1993) 616. [15] W. van Megen, T.C. Mortensen, S.R. Williams, J. Mu¨ller, Phys. Rev. E 58 (1998) 6073. [16] G. Parisi, F. Zamponi, J. Chem. Phys. 123 (2005) 144501. [17] D.I. Goldman, H.S. Swinney, arXiv:cond-mat/0511322, 2005. [18] E.J.J. van Rensburg, J. Phys. A 26 (1993) 4805. [19] I.C. Sanchez, J. Chem. Phys. 101 (1994) 7003.
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Physica A 372 (2006) 307–315 www.elsevier.com/locate/physa
Thermodynamic perturbation theory and glass transition in simple fluids M. Lo´pez de Haro, M. Robles Centro de Investigacio´n en Energı´a, U.N.A.M., Temixco, Morelos 62580, Mexico This paper is dedicated to Prof. Alberto Robledo on the occasion of his sixtieth birthday Available online 5 September 2006
Abstract Using the liquid state thermodynamic perturbation theory and the hard-sphere fluid as the reference system, the liquid–glass transition line of Lennard–Jones and hard-core Yukawa fluids is computed. The results are presented both in the reduced density vs. reduced temperature and in the reduced pressure vs. reduced temperature planes. A comparison with available simulation data and a discussion of the merits and limitations of this approach are also provided. r 2006 Elsevier B.V. All rights reserved. Keywords: Graphical uses interface; Molecular dynamics; Crystal defects; Dislocations
1. Introduction Despite the vast amount of literature already devoted to its study [1], the nature of the glass transition still remains as an unsolved and challenging problem. In fact, up to now there is no single theory that can account for all the features present in the rich and complex phenomenology that has been observed in supercooled liquids, amorphous solids and glasses. Since the attainment of the glassy state and the definition of the glass transition temperature depend on experimental time scales, there is little doubt that the glass transition is a nonequilibrium phenomenon. Nevertheless, in view of the so-called entropy crisis (the entropy of the glass must remain positive), there is a line of thought that goes back to the work of Adam, Gibbs and Di Marzio [2] (and has also been taken up more recently in the interpretation of the simulation work of Speedy [3–5] and in the theoretical approach of Me´zard and Parisi [6]), in which the (experimentally observed) dynamic glass transition is regarded as a kinetically controlled manifestation of an underlying (presumably second order) thermodynamic transition to an ideal glass. Although this kind of reasoning is not the only one available (take, for instance, the mode coupling approach) and is neither free from controversy, in this paper we will also adopt such a thermodynamic point of view. Interestingly enough, the work of Sastry [7] indicates that the glass transition lines determined either through thermodynamic or dynamic criteria seem to agree very well with each other.
Corresponding author.
E-mail addresses: [email protected], [email protected] (M.L. de Haro), [email protected] (M. Robles). 0378-4371/$ - see front matter r 2006 Elsevier B.V. All rights reserved. doi:10.1016/j.physa.2006.08.013
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The idea of using the thermodynamic perturbation theory of liquids in connection with the glass transition, as we will do in what follows, is not new either. In fact, many years ago Hudson and Andersen [8], Went and Abraham [9] and Abraham [10] addressed the location of the glass transition in monoatomic liquids taking this approach. In particular, in the case of the Lennard–Jones (LJ) fluid, they used the Weeks–Chandler– Andersen (WCA) perturbation theory of liquids [11] and the hard-sphere (HS) fluid as the reference system in the perturbative scheme. In any case, the main point to be stressed here about this work is that it relies heavily on the existence of a glass transition in the HS fluid. Whether a glass transition takes place in the HS fluid has been a debatable issue for a long time, but evidence from various sources, including rather recent work, seems to suggest that this is indeed the case [3,5,12–17]. Therefore, since we will also take here the HS fluid as the reference fluid in the application of the liquid state perturbation theory, in order to study the glass transition of simple fluids we will take it for granted that a glass transition occurs in this reference system at a given packing fraction to be specified below. Further, it should be noted that, in order to get the free energy of an actual system within the thermodynamic perturbation theory of liquids, the free energy and structural properties of the reference system (in our case the HS fluid) are required. The free energy of the HS fluid of course follows from the choice of an equation of state and for this latter we will consider the Pade´ [4,3] of van Rensburg–Sanchez [18,19]. As far as the structure of the HS fluid is concerned, we will rely on an approximate (analytical method) introduced by Yuste and Santos [20] (see also Refs. [21,22]), referred to as the rational function approximation (RFA) method, to get reasonably accurate and thermodynamically consistent values of the structural properties of the HS. In particular, when used in connection with the van Rensburg–Sanchez equation of state, the RFA method leads to a glass transition in the HS fluid [13]. The paper is organized as follows. In the next section and in order to make the paper self-contained, we provide a summary of the RFA method and of the conditions leading to a glass transition in the HS fluid. Section 3 is devoted to a rather brief account of the liquid state perturbation theory and its application to a hard-core Yukawa (HCY) fluid and an LJ fluid. In Section 4 we present the results for the glass transition line for the HS, HCY and LJ fluids. The paper is closed in Section 5 with some final comments and concluding remarks.
2. The RFA method for the HS fluid In this section we recall the key aspects of the RFA method [20–22]. The purpose of this analytical-algebraic approach is to determine (approximate) reliable expressions for the radial distribution function (rdf) gHS ðrÞ of a HS fluid. The main idea is to retain the form given by Wertheim’s [24] solution to the Percus–Yevick equation and use a rational function assumption (hence the name of the procedure) for a function related to the Laplace transform GðtÞ L½rgHS ðrÞ, namely one takes GðtÞ to be given by GðtÞ ¼
t 1 , 12Z 1 et FðtÞ
(1)
where Z ¼ ðp=6Þrd 3 is the packing fraction, r the number density, d the HS diameter (units will be taken in which the diameter of the HSs d has a value of 1, so that all distances r are measured in units of the d) and FðtÞ a rational function of the form FðtÞ ¼
1 þ S 1 t þ S 2 t2 þ S 3 t3 þ S 4 t4 . 1 þ L1 t þ L2 t2
(2)
The coefficients S i and Li are algebraic functions of Z determined by imposing the two following physical restrictions to the rdf: (1) The first integral moment of hHS ðrÞ ¼ gHS ðrÞ 1 is well defined and nonzero. (2) The second integral moment of hHS ðrÞ must guarantee the thermodynamic consistency of the compressibility factor Z HS ¼ p=ðrkB TÞ (p being the pressure, kB the Boltzmann constant and T the temperature) and the isothermal susceptibility wHS ¼ ðdðZZ HS Þ=dZÞ1 .
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After some cumbersome but straightforward algebra (for details see Ref. [22]), these two conditions lead to L1 ¼ L2 ¼ S1 ¼ S2 ¼ S3 ¼ S4 ¼
1 Z þ 12ZL2 þ 2 24ZS 4 , 2 2Z þ 1 3ðZ HS 1ÞS4 , 3 1 þ 4L2 8S4 Z , 2 2Z þ 1 1 Z þ 8ZL2 þ 1 2L2 24ZS4 , 2 2Z þ 1 1 2Z Z2 þ 12ZL2 ðZ 1Þ 1 72Z2 S 4 , 12 ð2Z þ 1ÞZ " 1=2 # 1Z Z HS 1=3 wHS 1 1þ 1 , 36ZðZ HS 1=3Þ ZHS Z PY wPY
ð3Þ ð4Þ ð5Þ ð6Þ ð7Þ ð8Þ
where, Z PY ¼ ð1 þ 2Z þ 3Z2 Þ=ð1 ZÞ2 and wPY ¼ ð1 ZÞ4 =ð1 þ 2ZÞ2 . Note that the only input required is an expression for Z HS . The rdf is then given by gHS ðrÞ ¼
1 1 X j ðr nÞYðr nÞ, 12Zr n¼1 n
(9)
with YðrÞ the Heaviside step function and jn ðrÞ ¼ L1 ½t½FðtÞn . Explicitly, using the residue theorem, jn ðrÞ ¼
4 X i¼1
e ti x
n X m¼1
AðiÞ mn rnm , ðn mÞ!ðm 1Þ!
(10)
where AðiÞ mn
m1 d ¼ lim t½ðt ti Þ=FðtÞn , t!ti dt
(11)
ti being the roots of 1 þ S1 t þ S 2 t2 þ S3 t3 þ S4 t4 ¼ 0. In order to guarantee that gHS ðrÞ vanishes inside the core, S4 must be negative. However, according to the form of S 4 (cf. Eq. (8)) it may well happen that once ZHS has been chosen there exists a certain packing fraction above which S 4 is no longer negative. This was interpreted in Refs. [13,21] as an indication that, at the packing fraction Z0 where S 4 vanishes, the system ceases to be a fluid and a glass transition in the HS fluid occurs. We will adopt the same view here and also assume that the compressibility factor of the glass has the empirical form proposed by Speedy [3], namely Z glass Aglass =ð1 Z=Zrcp Þ where Aglass is a constant and Zrcp is the random close-packing fraction. For the time being these two parameters remain as unspecified quantities. In line with the expected character of the transition, it may be assumed further that while the pressure is continuous at Z0 both for the liquid and the glass, it exhibits a change in slope in going from the fluid phase to the glass. Thus, at Z ¼ Z0 , one must have Z glass ðZ0 Þ ¼ Z HS ðZ0 Þ. On the other hand, since Zglass does not represent the fluid phase, the RFA method implies that S 4 cannot be negative in the range Z0 oZoZrcp . This means that within that range S 4 is either positive or a complex number. Moreover, since a change in the structure of the system occurs for Z ¼ Z0 one would expect that this should manifest itself in a corresponding change in the qualitative character of S 4 . Thus, when Z approaches Z0 from above, either S 4 changes from positive to negative or it changes from complex to (positive) real. At Z ¼ Z0 , the first change leads to the requirement wglass ðZ0 Þ ¼ wPY ðZ0 Þ (with wglass ðqðZZ glass Þ=qZÞ1 ) while the second one implies wglass ðZ0 Þ ¼ wPY ðZ0 ÞðZ PY ðZ0 Þ 13Þ=ðZ HS ðZ0 Þ 13Þ. In view of Eq. (8), only the second condition is compatible with the assumption that the pressure exhibits a change in slope on going from the fluid phase to the glass. Therefore, in order to find the unknown quantities Z0 , Aglass and Zrcp in a self-consistent way once ZHS is specified we use the above information in the following way. First, we determine Z0 from the condition wHS ðZ0 Þ ¼ wPY ðZ0 Þ that is required to make S4 vanish (cf. Eq. (8)).
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Specifically this leads to wHS ðZ0 Þ ¼
ð1 Z0 Þ4 . ð1 þ 2Z0 Þ2
(12)
Once Z0 has been determined from Eq. (12), using the form chosen for Z glass and after a few easy algebraic steps one gets Aglass ¼
2Z 2HS ðZ0 Þð1 Z0 Þ2 3Z HS ðZ0 Þ 1
(13)
and Zrcp ¼
Z0 ð1 3Z HS ðZ0 ÞÞ . 1 Z HS ðZ0 Þð1 þ 4Z0 2Z20 Þ
(14)
Hence, all that is required within this approach to determine Z0 , Aglass and Zrcp is again only an explicit expression for Z HS . Note that wHS ðZ0 Þ as given in Eq. (12) is different from wglass ðZ0 Þ 1=Aglass ð1 Z0 =Zrcp Þ2 in agreement with the characteristic expected for a liquid–glass transition. 3. The liquid state thermodynamic perturbation theory Let us consider a system defined by a pair interaction potential fðrÞ split into a known (reference) part f0 ðrÞ and a perturbation part f1 ðrÞ. The usual perturbative expansion [25] for the Helmoltz free energy to first order in b 1=kB T leads to Z 1 A A0 ¼ þ 2prb f1 ðrÞg0 ðrÞr2 dr þ Oðb2 Þ, (15) NkB T NkB T 0 where A0 and g0 ðrÞ are the free energy and the rdf of the reference system, respectively, and N R Zis the number of particles. If the reference system is taken to be the HS fluid, then of course AHS =NkB T ¼ 0 ðZ HS 1Þ=Z0 dZ0 and g0 ðrÞ ¼ gHS ðrÞ. Note that given fðrÞ, one may in principle use the liquid state perturbation theory to obtain explicitly an approximation to the free energy of the system within the RFA method simply by choosing Z HS and substituting the ensuing analytical expressions into Eq. (15). However, depending on the form of the perturbation part of the pair interaction potential, simplified schemes that use the HS fluid as the reference fluid have been introduced in which one usually takes an effective (in general density and temperaturedependent) diameter. In Ref. [26] we have provided a unifying framework for all the classical perturbative schemes [27,28,11]) within the RFA method and illustrated it for the case of an LJ fluid. Note that the right-hand side of Eq. (15) represents always an upper bound for the value of the free energy of the real system. We remark that in the Mansoori–Canfield/Rasaiah–Stell thermodynamic perturbation theory scheme, which we will later use for the LJ fluid, the effective diameter is chosen to yield the least upper bound. 3.1. The LJ fluid For the LJ fluid the reference system is a HS fluid, i.e., one sets ( 1; rpd eff ; f0 ðrÞ ¼ 0; r4d eff ;
(16)
where d eff is an effective diameter and thus the Helmholtz free energy of the LJ fluid may be approximated by Z 1 ALJ AHS r þ 2prb fLJ ðrÞgHS eff r2 dr, (17) NkT NkT d d eff
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where fLJ ðrÞ ¼ 4ðs12 =r12 s6 =r6 Þ is the usual LJ potential (with s the distance at which fLJ ðrÞ ¼ 0 and the depth of the well). Introducing the Laplace transform of rgHS ðrÞ given in Eqs. (1)–(8), Eq. (17) may be rewritten as Z ALJ AHS 48Z 1 p þ CLJ ðtÞGðtÞ dt, (18) NkB T NkB T T 0 where CLJ ðtÞ ¼
# 6 " 6 6 s t 1 t4 . 10! 4! d eff d eff s
(19)
Within the Mansoori–Canfield/Rasaiah–Stell scheme [28], the temperature- and density-dependent effective diameter d d eff ðr ; T Þ (where r ¼ rs3 and T ¼ kB T= are the reduced density and the reduced temperature, respectively) that minimizes the free energy is in turn determined from "Z ! # Z 6 10 ðp=6Þr q Z HS 1 4 1 6 1 t 1 t t4 dZ þ dt ¼ 0, (20) qd 0 Z T d 1 et FðtÞ d 10! 4! 0 where it is understood that the packing fraction appearing in both ZHS and FðtÞ (this latter given in Eq. (2)) must be expressed in terms of r , i.e., using r ð6=pÞZ. Once ALJ has been determined with the aid of Eqs. (18)–(20), the compressibility factor of the LJ fluid readily follows through the relation q ALJ Z LJ ¼ Z . (21) qZ NkT T;N This is all that is needed to compute the liquid–glass transition in the LJ fluid as done below. 3.2. The HCY fluid Now we examine a different case that of a HCY fluid. This is a system whose molecules interact via the pair potential ( 1; rps0 ; fHCY ðrÞ ¼ (22) s0 0 r exp½zðr s0 Þ=s0 ; r4s0 ; where s0 is the hard-core diameter, 0 is the depth of the potential well at r ¼ s0 and z is the inverse range parameter. It is usually taken as z ’ 2 in order for the properties of this fluid to be similar to the ones of the LJ fluid. Clearly, if z ! 0 this potential leads to Coulombic interaction and in the limit z ! 1 it reduces to the HS interaction where the diameter of the spheres is s0 . We now rewrite fHCY ðrÞ in the form required by the liquid state perturbation theory as fHCY ðrÞ ¼ f0 ðrÞ þ f1 ðrÞ, where the reference potential f0 will again be a HS pair potential (where the diameter of the spheres will be d eff s0 ) and the perturbed part f1 is given by ( 0; rps0 ; f1 ðrÞ ¼ (23) s0 0 r exp½zðr s0 Þ=s0 ; r4s0 : Substituting Eq. (23) into Eq. (15) one gets the approximation to the free energy of the HCY fluid, namely Z 1 AHCY AHS ¼ 2prbs0 0 exp½zðr s0 Þ=s0 gHS ðrÞr dr, (24) NkT NkT s0 or, equivalently, AHCY AHS ¼ 12Zb0 ez GðzÞ, NkT NkT
(25)
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where GðzÞ Gðt ¼ zÞ is again the Laplace transform of rgHS ðrÞ given in Eqs. (1)–(8) and thus, once z has been fixed, a function only of the packing fraction Z. Therefore, in this instance one gets an explicit analytical result for the free energy of the HCY fluid within the thermodynamic perturbation theory. Further, in this case one may also readily determine the compressibility factor as q AHCY q ðZGðzÞÞ. (26) ZHCY ¼ Z ¼ ZHS ðZÞ b0 ez qZ NkT T;N qZ Since it will be used later on, we now give the explicit form of the equation of state of the HCY fluid in terms of the dimensionless variables P0 ¼ Ps30 =0 , r0 ¼ rs30 and T 0 ¼ kT=0 , namely p q P0 ¼ r0 T 0 Z HS r0 r0 ez 0 ðr0 GðzÞÞ, (27) 6 qr where it is again understood that the packing fraction appearing in both ZHS and GðzÞ must be expressed in terms of r0 , i.e., using r0 ð6=pÞZ. We remark that in contrast with the case of the LJ fluid in which the effective diameter of the reference HS system turns out to be dependent on both temperature and density, for the HCY fluid it simply coincides with the hard-core diameter. This difference will have an influence on the determination of the liquid–glass transition line as discussed in the next section. 4. Results In order to proceed and perform actual calculations, one needs a choice for ZHS . Since this quantity is unfortunately not known exactly, as mentioned earlier we will consider the Pade´ [4,3] of van Rensburg–Sanchez [18,19] given by ZHS Z 43 ¼
1 þ 1:024385Z þ 1:104537Z2 0:4611472Z3 0:7430382Z4 , 1 2:975615Z þ 3:00700Z2 1:097758Z3
(28)
which is known to be rather accurate both in the stable and metastable fluid regions. This leads (cf. Eqs. (12)–(14)) within the RFA method to a glass transition in the HS fluid at Z0 ¼ 0:5604 to Aglass ¼ 2:765 and to Zrcp ¼ 0:6448. It should be remarked that all these values are in very good agreement with other estimates in the literature [13]. Further, for this choice of Z HS one gets wHS ðZ0 Þ ¼ 0:0083 and wglass ðZ0 Þ ¼ 0:0062. We can now determine the location of the liquid–glass transition line for both the LJ fluid and the HCY fluid. We begin with the LJ system where this line in the r –T plane may be easily derived from the simple relationship p 3 r d ðr ; T Þ ¼ Z0 , (29) 6 where d is determined using Eq. (20). The outcome [23] is shown in Fig. 1 where we have also included the recent simulation results of Di Leonardo et al. [29]. As clearly seen in the figure, not only the qualitative trend observed in the simulations is reproduced, but also the quantitative agreement is rather good. In the case of the HCY fluid, the results are derived from the equality p 0 3 r s 0 ¼ Z0 (30) 6 so that the liquid–glass transition line in the r0 –T 0 plane is simply a line parallel to the T 0 -axis whose abscissa is r00 ¼ ð6=pÞZ0 =s30 . This result is due to the fact that we are restricting ourselves to the first order term in the thermodynamic perturbation expansion. Of course one should expect the finite Yukawa part of the potential to have an influence on the temperature dependence of the liquid–glass transition, but this is not captured by the present approximation. The liquid–glass transition line may also be shown in different thermodynamic planes, particularly the P –T plane with the aid of the equation of state. In Fig. 2 we display the liquid–glass transition lines for the HS fluid, the LJ fluid and the HCY fluid with z ¼ 2:5 in the reduced temperature vs. reduced pressure plane. To our knowledge, no simulation data are available for the
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1.4 1.2
T∗
1 0.8 0.6 0.4 0.2 0 0.9
1
1.2
1.1 ∗
Fig. 1. Liquid–glass transition line of the Lennard–Jones fluid as obtained with the Mansoori–Canfield/Rasaiah–Stell perturbation scheme using ZHS ¼ Z43 . Also included are the simulation data of Di Leonardo et al., in Ref. [29], and their vertical size accounts for the error bars.
1 0.8
T∗
0.6 0.4 0.2
2
4
6
8
10
12
P∗ Fig. 2. Liquid–glass transition lines of the hard-sphere fluid (continuous line), the Lennard–Jones fluid (dashed line) and the hard-core Yukawa fluid with z ¼ 2:5 (dotted line) in the reduced temperature vs. reduced pressure plane as obtained with the liquid state perturbation theory using the RFA method and taking ZHS ¼ Z43 .
liquid–glass transition lines of these three fluids in this plane, so a proper assessment of the results displayed in this figure is not possible at this stage. 5. Discussion The results of the previous section deserve further comments. To begin with, although approximate, our approach is self-consistent and constitutes a fully analytical derivation with no free parameters. It leads naturally to what we have interpreted as a glass transition in the HS fluid. This is a necessary requirement in order to apply the liquid state thermodynamic perturbation theory to the computation of the glass transition line of simple fluids when one takes the HS fluid as the reference system. Secondly, the overall agreement between the trends of the simulation results for the glass transition line of the LJ fluid and our thermodynamic results is encouraging and seems to provide further support to the notion that the glass transition may be associated to an underlying thermodynamic phase transition. In particular, we note that, in agreement with the assertion of the mode coupling approach, the comparison with the simulation results (that refer to the dynamical glass transition) indicates that, at least for the LJ fluid, the dynamical transition temperature is higher than the thermodynamic liquid–glass transition temperature.
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As far as the HCY fluid is concerned, no comparison with either experimental or simulation data was made since to our knowledge these data are not presently available. Nevertheless, in the range of application of the liquid state perturbation theory for the Helmoltz free energy to first order in b and taking as the reference system a HS fluid, the features observed for the HCY system should also hold for any other system with a hard core. That is, for such systems in the reduced density vs. reduced temperature plane the glass transition line should be a straight line parallel to the temperature axis, while in the reduced pressure vs. reduced density plane it should be a straight line (parallel to the pressure axis and coinciding with the straight line of the HS fluid). Finally, although of course the existence of a glass transition in the HS is the only requirement to apply the liquid state thermodynamic perturbation theory to the study of the glass transition in simple fluids, one may reasonably wonder whether the above statements that stem out of the results we have presented are tied to the choice of the equation of state of the HS fluid or, even more profoundly, to the form of the rational function FðtÞ taken in the RFA method. As far as the first issue is concerned, we have pointed out elsewhere [13,23] that this is not the case. The second issue is subtler and more difficult to handle. If the rational form is maintained for FðtÞ, then (cf. Eq. (2)) the next order of approximation would involve two new coefficients say L3 and S 5 (one in the numerator and one in the denominator) that would require two extra conditions to be fixed. It is conceivable that with these additions, one could still find a threshold value of the packing fraction beyond which one could not guarantee that gHS ðrÞ vanishes inside the core and this could then be interpreted again as the packing fraction corresponding to the glass transition. But the opposite, i.e., the nonexistence of such a threshold value, is possible as well. However, the main difficulty in going beyond the approximation of Eq. (2) and in deciding on the two previous possibilities resides in the fact that the extra conditions required to fix the two new coefficients do not seem to have a clear physical interpretation. Therefore, we take a pragmatic point of view and stick to the original RFA method which, although admittedly approximate, has proved to be both accurate and free from physical inconsistencies. With such a view, we are thus persuaded that the results we have derived stand and hope to encourage simulation or other theoretical approaches to assess their full value. Acknowledgements We acknowledge the financial support of DGAPA-UNAM through project IN-110406. References [1] For reviews and two recent articles discussing many aspects of this problem see: C.A. Angell, Science 267 (1995) 1924; M.D. Ediger, C.A. Angell, S.R. Nagel, J. Phys. Chem. 100 (1996) 13200; C.A. Angell, K.L. Ngai, G.B. Mackenna, P.F. McMillan, S.W. Martin, J. Appl. Phys. 88 (2000) 3113; P.G. Debenedetti, F.H. Stillinger, Nature 410 (2000) 259; D.R. Reichman, P. Charbonneau, J. Stat. Mech. P0513 (2005); F. Sciortino, J. Stat. Mech. P0515 (2005). [2] J.H. Gibbs, E.A. Di Marzio, J. Chem. Phys. 28 (1958) 373; G. Adams, J.H. Gibbs, J. Chem. Phys. 43 (1965) 139. [3] R.J. Speedy, J. Chem. Phys. 100 (1994) 6684. [4] R.J. Speedy, J. Phys. Condens. Matter 9 (1997) 8591. [5] R.J. Speedy, Mol. Phys. 95 (1998) 169. [6] M. Me´zard, G. Parisi, J. Phys. Condens. Matter 12 (2000) 6655. [7] S. Sastry, Phys. Rev. Lett. 85 (2000) 590. [8] S. Hudson, H.C. Anderssen, J. Chem. Phys. 69 (1978) 2323. [9] H.R. Wendt, F.F. Abraham, Phys. Rev. Lett. 41 (1978) 1244. [10] F.F. Abraham, J. Chem. Phys. 72 (1980) 359. [11] J.D. Weeks, D. Chandler, H.C. Andersen, J. Chem. Phys. 54 (1971) 5237. [12] B. Baeyens, H. Verschelde, Z. Phys. B 102 (1997) 255. [13] M. Robles, M.L. de Haro, A. Santos, S.B. Yuste, J. Chem. Phys. 108 (1998) 1290. [14] W. van Megen, S.M. Underwood, Nature 362 (1993) 616. [15] W. van Megen, T.C. Mortensen, S.R. Williams, J. Mu¨ller, Phys. Rev. E 58 (1998) 6073. [16] G. Parisi, F. Zamponi, J. Chem. Phys. 123 (2005) 144501. [17] D.I. Goldman, H.S. Swinney, arXiv:cond-mat/0511322, 2005. [18] E.J.J. van Rensburg, J. Phys. A 26 (1993) 4805. [19] I.C. Sanchez, J. Chem. Phys. 101 (1994) 7003.
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