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J. Phys. Chem. B 2009, 113, 15572–15580
On the Asymptotic Properties of a Hard Sphere Fluid Isaac C. Sanchez* and Jang S. Lee Chemical Engineering Department, UniVersity of Texas, Austin, Texas 78712 ReceiVed: February 4, 2009; ReVised Manuscript ReceiVed: September 23, 2009
An analysis of the expected divergences in thermodynamic properties at the close-pack density (ηcp ) π2/ 6) along with the known virial coefficients up to 10th order suggests a weak logarithmic singularity in the excess fluid entropy. The corresponding equation of state (EOS) also possesses a singularity at ηcp. The new EOS accurately describes extant molecular dynamics data up to the fluid-solid transition (η = 0.494) with differences of less than 1 part per thousand. This accuracy is maintained into the metastable fluid regime up to η = 0.52. In terms of accuracy, the new EOS is no better than Pade´ approximants, but the new EOS, unlike the Pade´ approximants, diverges at ηcp. In addition, a new order parameter is defined that enables all system configurations to be classified as either disordered or ordered. Monte Carlo simulations are used to determine this order parameter in the metastable fluid range. Using this new order parameter, evidence is presented to support a thermodynamic glass transition at η = 0.54. With respect to this transition, congruence is found with the traditional ideas espoused by Gibbs and DiMarzio and Adam and Gibbs. It is the rapid disappearance of disordered (random) configurations with increasing density that drives the glass transition and slows the dynamics. I. Introduction Recently, 9th order virial coefficients for hard disks and spheres1 and 9th and 10th virial coefficients for hard spheres and hyperspheres up to 8 dimensions have been calculated.2 Additionally, some previously calculated lower-order virial coefficients3-10 were recalculated more accurately.11 On the basis of the first eight virial coefficients, this author in a previous publication concluded that a singularity in the hard sphere (HS) equation of state (EOS) occurs at the close-packed packing fraction, ηcp ) π2/6 ) 0.74..., but the singularity type was not identified.12 In view of these new and updated virial coefficients, this paper reexamines the HS fluid from the perspective of insertion probabilities. For HS systems, the insertion probability (P) can be defined as the probability that a randomly inserted sphere into the fluid will find a cavity large enough to accommodate it. Calculation of insertion probabilities plays a central role in scale particle theory.13-18
βµ ) ln(λ3F/P) ) βµig(F) - ln P ) βµig(βP) - ln ZP (3) where the subscript ig denotes an ideal gas property, β ) 1/kT, and λ is the well-known thermal wavelength, λ ) h/(2πmkT)1/2. All other symbols have their customary meanings. From eq 3, the entropy per particle s is seen to be
3 + Z - βµ 2 s/k ) sig(F)/k + Z - 1 + ln P
s/k )
or the excess entropy (sE) relative to an ideal gas at the same density as the HS fluid is
sE /k ) Z - 1 + ln P )
1 F
∫0F ln PdF′ ) - ∫0F Z F′- 1 dF′ (5)
The EOS is related to the insertion probability by
1 Z ≡ P/FkT ) 1 - ln P + F
∫0 ln PdF′ F
(1)
and the inverse is given by
where the last two relations follow from eqs 1 and 2. As the density (η ) πσ3F/6) of a HS fluid approaches the close-packing fraction (ηcp ) π2/6), divergences in certain properties are expected
lim Z f +∞
(6)
lim ln P f -∞
(7)
lim sE f -∞
(8)
ηfηcp
-ln P ) Z - 1 +
∫0F Z F′- 1 dF′
(2)
where F ) N/V is the number density. These relations follow from the chemical potential, µ19 * To whom correspondence should be addressed.
(4)
ηfηcp
ηfηcp
These divergences can guide the construction of an appropriate insertion probability from the known virial coefficients.
10.1021/jp901041b CCC: $40.75 2009 American Chemical Society Published on Web 11/04/2009
On the Asymptotic Properties of a Hard Sphere Fluid
J. Phys. Chem. B, Vol. 113, No. 47, 2009 15573 ∞
One aspect of the asymptotic behavior is the possibility of a glass transition. In this regard, a new order parameter will be defined that enables all system configurations to be classified as either disordered or ordered. II. Results A. Virial Representations. The virial EOS for a HS fluid has several equivalent representations that depend on what density variable is chosen: ∞
Z-1)
∑ Bn+1F
∞
n
)
n)1
∑ B˜n+1(B2F)
n
∑ bn+1η
)
n)1
)
n)1
∞
∑ tn+1θn
n)1
K(0, R) ) 0
and
K>0
∞
∑
κn+1θn ) (1 - θ)R
n)1
where
∞
(n + 1)bn+1 n η ) n n)1
∞
(n + 1)tn+1 n θ ) n n)1
∑
∞
γn+1 n θ n n)1
∑
(11)
and along with eq 5 yields the excess entropy ∞
-sE /k )
γn+1 θn ) n(n + 1) n)1
∑
∞
tn+1 n θ ) n n)1
∑
(17)
∞
bn+1 n η n n)1 (12)
∑
∞
γn+1 n θ n n)1
∑
2 K(θ, R) ) γ2θ + (γ3 /2 - Rγ2)θ +
(10)
is the fraction of occupied space relative to the close-packed state. Substituting eq 9 into eq 2 yields the insertion probability
∑
0 3/2, the central HS is considered a member of an “ordered” cluster.
On the Asymptotic Properties of a Hard Sphere Fluid
J. Phys. Chem. B, Vol. 113, No. 47, 2009 15575
asymptote to the global density (see Figure 2). For the special case of n spheres in contact with the central sphere (d˜i ) 2 for all i), eq 26 becomes
ηlocal )
˜ ,2) 1 + nf(D 3 ˜ D
(27)
with
{
φ¯ (η) )
[
]
˜ > 3 D
Geometry requires that the smallest possible ordered cluster will involve a central sphere plus at least 10 neighbor spheres in near contact. For n ) 10, ηlocal, given by eq 27, falls to a ˜ = 3/2 and then rises to maximum at D ˜ = 2, minimum at D where it reaches a value of about 0.63. This coordination number of 10 should be compared to simulation results where HS random closest packing produces an effective contact coordination number of about 6.33-36 To summarize, if ηlocal > ηrcp for ˜ beyond D ˜ > 3/2, the central HS is considered a member any D ˜ > 4 are probably of an “ordered” cluster (searches beyond D unnecessary). At a given global density η, every system configuration of N hard spheres will have No spheres that belong to an ordered cluster (0 e No e N). Therefore, every system configuration of N hard spheres in a volume V can be characterized by the following order parameter φ
0eφe1
∫0φ F(φ, η)dφ c
and
Ωo(η) ≡ Ω(η)
∫φ1 F(φ, η)dφ c
(31)
˜ e3 1eD
(28)
φ ≡ No /N
(30)
This average value vanishes as η f 0, but it remains small even at the fluid-solid transition density, and it is conjectured to be on the order of 10-3 or less (based on the assumption that homogeneous crystal nuclei are still relatively rare). If the total number of system configurations at any density is Ω(η), then
Ωd(η) ≡ Ω(η)
1 ˜ ˜ -3)2 ˜ -3) - 3 (D (D-1)2 1 - (D ˜ 4 8 f(D,2) ) 1
∫01 φF(φ, η)dφ
(29)
Now let F(φ,η)δφ be the probability (normalized density of states) that any system configuration of global density η will have an order parameter between φ and φ + δφ. The ensemble average value of this order parameter is given by
where Ωd and Ωo are the number of “disordered” (φ < φc) and “ordered” (φ > φc) system configurations, respectively; φc is an arbitrarily chosen cutoff value of the order parameter, say 10-4. This particular cutoff value guarantees that a disordered configurational state will have at most 1 in 104 hard spheres participating in ordered clusters (crystalline-like nuclei). Intuitively, there should be little difference in the two subensembles for any cutoff value of the order parameter (φc) less than 10-4. There is a caveat worth mentioning here; the proposed definition of an ordered configuration is based on a short-range property (ordered cluster) and not long-range order. For example, at any density, placement of spheres on a lattice creates a configuration with long-range order. Under the present definition, these rare configurations would be classified as disordered for global densities less than 0.64. Using the above classification, the entire ensemble of system configurations Ω of N hard spheres can be grouped into two subensembles, disordered, Ωd, and ordered, Ωo. At densities less than the fluid-solid transition density, it is expected that Ωd . Ωo. Since each configuration has the same global density, the subensembles also have the same global density η and the same free volume fraction, 1 - η/ηcp. However, the subensemble of ordered configurations will have a larger average insertion probability than the disordered ensemble, Po > Pd. Clustering (or local ordering) in one part of the system opens space in another part of the system to conserve the global density, as illustrated in Figure 3. Thus, at fixed density, highly clustered system configurations will have higher insertion probabilities. A dramatic example of this effect occurs at the fluid-solid transition where density increases from 0.494 to 0.545,37 whereas the insertion probability, instead of decreasing, increases by 10% (see eq 3)
Ps /Pf ) Zf /Zs ) ηs /ηf ) 1.10
Figure 2. Schematic of the local density around a central HS of diameter σ in the metastable fluid regime as a function of the distance R from the center of the sphere (see Figure 1). One HS is a member of an ordered cluster (red), while the other is not (blue). Geometric considerations require that the central sphere be in near-close contact with at least 10 neighboring spheres for ηlocal > ηrcp = 0.64. Every HS in the system must be tested; the order parameter is defined as the average fraction of spheres that belong to ordered clusters. For more details, see the text and Figure 1.
(32)
At first, this seems counterintuitive because the semiordered solid phase is denser than the fluid phase with less free volume. However, the higher degree of local ordering (clustering) in the denser solid phase creates larger cavities than those available in the less dense, disordered fluid phase.38 Insertion probabilities can also be written in terms of subensemble averages:
∫0φ P(η, φ)F(η, φ)dφ Pd(η) ) ∫0φ F(η, φ)dφ Ω(η) φ ) ∫ P(η, φ)F(η, φ)dφ Ωd(η) 0 c
c
c
(33)
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J. Phys. Chem. B, Vol. 113, No. 47, 2009
Po(η) )
Ω(η) Ωo(η)
∫φ1 P(η, φ)F(η, φ)dφ c
Sanchez and Lee
(34)
and the average insertion probability for the entire ensemble is seen to be given by a weighted average of the subensemble averages
P)
∫01 P(η, φ)F(η, φ)dφ ≡ Ω1 [ΩdPd + ΩoPo] (35)
As discussed above, clustering causes Po(η) > Pd(η) for all densities (see Figure 3). A consequence of the ordered subensemble having a higher insertion probability at all densities is that the two ensembles, if they could be separated, would exist at the same density at different pressures (different βP); the disordered ensemble will always have a higher pressure, that is, Zd - Zo > 0. This can be seen from eq 1
Zd - Zo ) ln(Po /Pd) -
1 η
∫0η ln(Po/Pd)dη′ > 0 (36)
The inequality follows because the ratio Po/Pd increases monotonically with density, and thus
ln(Po /Pd) >
1 η
∫0η ln(Po/Pd)dη′ > 0
(37)
In words, the value of ln(Po/Pd) averaged over a density range will always be less than the value of ln(Po/Pd) evaluated at the highest density where it reaches its maximum value. This effect has been observed in computer simulations where ordering (local crystallization) in the metastable liquid regime lowers the pressure.26,39,40 A caveat needs be added to the above argument that leads to eq 36. The assumption is that the subensembles can be sampled, but this is only possible if a constraint (bias) is imposed on the
Figure 3. This two-dimensional schematic illustrates how clustering affects insertion probabilities (insertion probability ) target area/total area). Although both have the same global density, the right-hand configuration (RHC) is more clustered than the left-hand configuration (LHC). Dotted (target) areas can accommodate the center of a randomly inserted disk without overlapping neighbors. Clustering in one part of the system opens space in another part of the system to conserve the global density. As illustrated, this results in a higher insertion probability for the RHC (larger target area). Configurations that are more uniform in density, or, equivalently, configurations with a more finely divided free volume distribution, will always have a smaller insertion probability. Increased clustering can be affected by adding attractive interactions; computer simulations confirm that the insertion probability increases with increased clustering.38
Figure 4. Monte Carlo simulation data to determine the new order parameter defined as the average fraction of spheres that are members of an ordered “cluster”. A HS is a member of an ordered cluster if the local packing density exceeds 0.64 (see Figure 2). Both random onlattice (triangles) and off-lattice (circles) initial states were created to check possible initial state bias, and no obvious systematic differences were observed. The line through the data is to guide the eye only.
system to separate the two subensembles. The thermodynamic contribution of constraints on a HS system has been outlined,41 but it is unclear how it would affect the above argument. In practice, a bias can be imposed in a molecular dynamics simulation of a metastable HS fluid by suppressing nucleation and growth of ordered clusters. As mentioned above, and consistent with eq 36, this biased sampling of predominantly disordered configurations yields a higher pressure than otherwise expected from the EOS. In Figure 4, some Monte Carlo (MC) results illustrate the behavior of the new order parameter as a function of density. The number of spheres in each simulation varied from 800 to 900. Low-density initial configurations (η ≈ 0.2) were created either by random insertion or by randomly placing spheres on a simple cubic lattice. Periodic boundary conditions were employed. Both off-lattice and on-lattice configurations were randomized for 106 MC cycles. This initial randomization was then followed by an implementation of the “squash method”.40,42 This algorithm separates the center-to-center distance between all spheres to a certain predetermined minimum distance, Smin. Afterward, all N sphere diameters are increased to Smin to yield a final density of η ) N(πS3min/6)/L3, where L3 is the box volume. The system was then randomized again for another 107 MC cycles. The local density around each of the N spheres was then determined, and the fraction of system spheres that exceeded a local density of 0.64 was calculated. As can be seen, the results shown in Figure 4 do not reveal any obvious systematic difference between on-lattice and off-lattice initial configurations. Each data point was determined from a unique initial configuration, either on- or off-lattice. III. Discussion A. Numerical Results. In Table 2, the new EOS, eq 21, is compared with recent molecular dynamics data in Table 2. As can be seen, the differences between the MD results and those calculated from eq 21 are less than 1 part per thousand (ppt) up to η = 0.52. Up to this density, the estimated errors in the MD data are less than 0.1 ppt.43 This order of magnitude difference
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TABLE 2: Comparison of Molecular Dynamics (MD) Simulation Data of Kolafa, Labik, and Malijevsky43 and Kolafa67 with a Virial Expansion Correct to 10th Order and Equation 21a Z ) P/FkT
densities σ3F
η
θ ) η/ηcp 10th virial
0.20 0.40 0.60 0.65 0.70 0.75 0.78 0.80 0.83 0.85 0.88 0.90 0.91 0.92 0.93 0.94
0.10472 0.20944 0.31416 0.3403 0.3665 0.3927 0.4084 0.4189 0.4346 0.4451 0.4608 0.4712 0.4765 0.4817 0.4869 0.4922
0.14142 0.28284 0.42426 0.4596 0.4950 0.5303 0.5515 0.5657 0.5869 0.6010 0.6222 0.6364 0.6435 0.6505 0.6576 0.6647
0.95 0.96 0.97 0.98 0.99 1.00 1.01 1.02 1.03 1.04 1.05 1.06 1.07 1.08 1.09
0.4974 0.5027 0.5079 0.5131 0.5184 0.5236 0.5288 0.5341 0.5393 0.5445 0.5498 0.5550 0.5603 0.5655 0.5707
0.6717 0.6788 0.6859 0.6930 0.7000 0.7071 0.7142 0.7212 0.7283 0.7354 0.7425 0.7495 0.7566 0.7637 0.7707
1.5536 2.5216 4.2931 4.9452 5.7171 6.6345 7.2675 7.7288 8.4861 9.0388 9.9478 10.612 10.964 11.328 11.707 12.100
MD 1.5536 2.5216 4.2950 4.9497 5.7270 6.6551 7.2990 7.7701 8.5476 9.1185 10.065 10.763 11.134 11.522 11.926 12.349
Metastable Fluid 12.508 12.791 12.932 13.254 13.372 13.739 13.829 14.249 14.304 14.784 14.797 15.347 15.309 15.939 15.847 16.559 16.395 17.201 16.97 17.85(2) 17.57 18.59(4) 18.19 19.41(8) 18.83 20.32(10) 19.50 21.30(15) 20.20 22.15(30)
eq 21
ppt diff.
1.5536 2.5216 4.2951 4.9493 5.7277 6.6545 7.2982 7.7716 8.5497 9.1200 10.067 10.770 11.144 11.528 11.935 12.359
0 0 0 -0.1 0.1 -0.1 -0.1 0.2 0.2 0.2 0.2 0.6 0.9 0.5 0.7 0.9
12.797 13.260 13.744 14.251 14.773 15.328 15.910 16.511 17.150 17.82 18.53 19.26 20.04 20.86 21.72
0.4 0.4 0.4 0.1 -0.7 -1.2 -1.8 -2.9 -3.0 -1.6 -3.4 -7.9 -14 -27 -20
a The last column expresses the difference between eq 21 and the MD data in parts per thousand (ppt).
in the expected errors between the new EOS and the MD data is not surprising. The new EOS is based on only 10 virial coefficients of limited accuracy; the 8th virial coefficient is only known to 4 significant figures, and there is uncertainty in the 3rd significant figure of the 10th virial coefficient (see Table 1).2 Published [5/4] and [4/5] Pade´ approximants2 based on these virial coefficients also yield comparable differences (less than 1 ppt) between calculated and MD data up to densities of 0.52. Therefore, in terms of accuracy, the new EOS is no better than the Pade´ approximants, but asymptotically, the new EOS is superior because the approximants incorrectly remain finite at ηcp. B. Extension of EOS into the Metastable Fluid Regime. Can the EOS for the stable fluid be extended into the metastable fluid regime? It can be argued that the radius of convergence of the virial series should not extend beyond the transition density of 0.494. In contrast, Reiss41,44,45 and others20,46,47 have argued that the virial series contains only geometric information about the close-packed state with a convergence radius only limited by the close-packed density. This conclusion is also consistent with an argument summarized in Huang’s textbook,48 attributed to Mayer and Mayer,49 that concludes with the statement, “The virial expansion of the EOS does not contain all the information about the EOS.” From the latter perspective, the virial EOS is incapable of predicting the fluid-solid transition, but nevertheless, it still reflects all possible configu-
rational states of the system, including highly clustered (ordered) states. The divergences mentioned in eqs 6-8 presuppose that the EOS can be extended into the metastable regime and that the geometric character of the EOS becomes manifest at the close-packed density. In other words, a virial EOS passes smoothly from the fluid-solid transition density to the crystalline close-packed density without interruption; at the very highest densities, it is more descriptive of the semiordered crystalline phase than a disordered, metastable fluid phase. In this regard, the predicted logarithmic divergence of the entropy is consistent with the studies of others22-25 who conclude that the excess entropy of the HS crystalline phase also diverges logarithmically as η f ηcp. At the fluid-solid transition, it is expected that in the fluid state, the number of disordered configurations will dominate
Ωd(0.494) . Ωo(0.494)
(38)
Thus, at the f f s transition, the insertion probability for the fluid phase will be dominated by the disordered ensemble insertion probability, P = Pd. Now, let the fluid density exceed the fluid-solid transition density so that it is on the metastable fluid branch. The EOS will continue to accurately describe the metastable fluid as long as Ωd . Ωo. In a computer simulation of a HS fluid on the metastable fluid branch, nucleation and growth of ordered clusters is suppressed. In a MD simulation, this can be achieved by a pressure jump (βP jump), say from the stable fluid branch to a density on the metastable branch. Subsequent configurations generated from this initial metastable state will be biased toward the disordered ensemble. However, as long as the ensemble average, based on the EOS, yields an insertion probability controlled by the disordered subensemble, the extension of the EOS into the metastable region will be accurate. As density increases, the fraction of configurations belonging to the ordered subensemble will increase. As a consequence, the calculated insertion probability for the entire ensemble will become more biased toward the value of the insertion probability associated with the ordered subensemble, P f Po > Pd, (see eq 35). When this happens, the virial EOS, which properly reflects both ordered and disordered states, will begin to underestimate the pressure or, equivalently, overestimate the density. Therefore, at high enough densities on the metastable branch, the imposed simulation bias for the disordered subensemble will cause the extension of the EOS onto the metastable region to exhibit a negative deviation (underestimated pressure). Inspection of Table 2 indicates that significant negative deviations from the EOS start to occur beyond η = 0.54. C. Random Packing and the Glass Transition. What is the maximum density achievable for disordered configurational states satisfying φ < φc? This restriction effectively eliminates crystal-like nuclei and suggests that the maximum density achievable for the disordered ensemble is very near or at ηrcp = 0.64. This view is consistent with the conjecture of Kamien and Liu32 that “ηrcp represents a special well-defined divergent end point of a set of metastable branches of the pressure.” It is also consistent with Speedy’s conclusion that dense HS glassy states are very reproducible.40 In addition, as η f ηrcp, the disordered subensemble should include as members “maximally random jammed states”.27 Is there a glass transition? The theoretical concept of an “ideal glass transition” has been put forth35,50-52 but also criticized when applied to a HS system.26,51,53 The idea behind the ideal glass transition is that it occurs in real liquids at a unique
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Sanchez and Lee
{
-Zd η < ηK 1 ∂ ln Ω ) -Zo η > ηK N ∂ ln η
Figure 5. Theoretical configurational entropy diagram of the total (ln Ω), disordered (ln Ωd), and ordered (ln Ωo) state entropies. The Kauzmann density (ηK) occurs when the ordered state entropy overtakes the disordered state entropy. Also note that the disordered and ordered entropies are nonadditive, (ln Ω * ln Ωd + ln Ωo).
temperature54,55 (Kauzmann temperature) where the entropy of the metastable liquid equals the entropy of the solid. Within the context of the newly defined order parameter, this condition would be satisfied in a HS fluid at or near the Kauzmann density (ηK)
ln Ωd(ηK) ) ln Ωo(ηK)
(39)
This condition is illustrated in Figure 5, where the total, disordered, and ordered entropies are plotted as a function of density. The Kauzmann density, defined by eq 39, is somewhat arbitrary because the definition of disordered and ordered states is somewhat arbitrary. However, in principle, the order parameter could be tuned so that the condition is satisfied. Identifying a Kauzmann density does not imply that there will be an accompanying thermodynamic phase transition. This has been the primary focus of previous criticism.26,51,53 This issue can be investigated within the context of the new order parameter as follows. The total configurational entropy is given by
ln Ω ) ln[Ωd + Ωo] ) ln Ωd + ln[1 + Ωo /Ωd] ) Nsd /k +
and for large N
ln Ωo > ln Ωd
{
ln Ωd η < ηK ln Ωo η > ηK
(41)
Now using the thermodynamic relation (∂S/∂V)T ) (∂P/∂T)V or (∂ ln Ω/∂V)U ) βP yields
∂(S/Nk) 1 ∂ ln Ω ≡ ) -Z/η ∂η N ∂η
(42)
(44)
or η > ηK, which in turn implies
ηK < 0.545
(45)
A HS glass transition can only be observed if extraordinary measures are taken to avoid partial solidification of the metastable fluid. Consistent with this and the above inequality is the observation that a HS fluid rapidly solidifies within a few million collisions (N ) 256) for η = 0.55.58 The Kauzmann density can also be estimated directly by using eq 20 for the excess entropy and the entropy of fusion given by
(46)
where the above entropies are at the same value of βP
sig(βP) ) sig(η) + k ln Z ln Ω )
(43)
where the inequality, Zd > Zo follows from eq 36. The conclusion is that there will be a jump discontinuity in the slope of the entropy at ηK. A discontinuity in the slope signals a second-order phase transition characteristic of the Gibbs-DiMarzio (GD) theory of the glass transition.56,57 If another pair of subensembles is defined by adjusting the cutoff value of the order parameter (φc) that allows significant amounts of short-range order in the disordered ensemble, then a new and higher transition density would be defined, and the strength of the transition would diminish. If no distinction is made between disordered and ordered states, there is no transition. The latter is described by the virial EOS. As mentioned previously, a virial EOS passes continuously from the fluid-solid transition density to the crystalline close-packed density without interruption; at the very highest densities, it describes a semiordered solid phase. In a simulation, a natural distinction between disordered and ordered states arises when efforts are made to avoid crystallization of a metastable fluid. The above argument helps to place an upper bound on the Kauzmann density. At the fluid-solid transition, the fluid at a density of 0.494 is in equilibrium with a solid phase at 0.545.37 This implies that when the metastable fluid phase reaches a density of 0.545, the inequality implied in eq 41 has already been reached, that is
(sf - ss)/k ) Zf - Zs ) Zf[1 - ηf /ηs] ) 1.2
ln[1 + exp[-N(sd - so)/k]] (40)
or
Zd > Zo
(47)
From eq 20, the excess entropy of the fluid at the transition is -4.87k. The metastable fluid entropy decreases to -4.9 - 1.2 ) -6.1k at η ) 0.54 (θ ) 0.73). This again supports the conclusions expressed in eqs 44 and 45. As mentioned in the previous section, significant negative deviations from the EOS begin to appear for η > 0.54, indicative of the onset of a glass transition. Most proposed HS glass transition densities fall in the range 0.54 e η e 0.60 (0.73 e θ e 0.80).35,40,46,59-63 It is also interesting to note that “random loose-packing” of hard spheres occurs at ηrlp = 0.555.30,64 Using special algorithms,40 one can push into the glassy region of phase space as shown in Figure 5. Notice that both the glass
On the Asymptotic Properties of a Hard Sphere Fluid
J. Phys. Chem. B, Vol. 113, No. 47, 2009 15579
and solid approach their limiting densities with infinite negative slopes. This follows from eq 42 and the fact that Z diverges positively at both ηcp and ηrcp. The divergence at ηrcp does not imply that ln Ωd(ηrcp) ) 0 nor does it imply that this divergence is similar to that at ηcp, although it is known that the divergence properties of the HS glass and crystal are similar.35,40 Speedy40,65 has estimated the residual entropy of a dense glass at 0.2k; the absence of a zero on the vertical axis of Figure 5 is intentional. In summary, in simulations that sample configurations with little or no short-range order in the metastable fluid regime, a thermodynamic-like transition should be expected, as illustrated in Figure 5. The transition should appear near η ) 0.54. Although not a true thermodynamic stability limit with an infinite compressibility, the glass transition resembles a stability limit. At the transition, the disordered fluid becomes entropically less favorable than the ordered phase. Pushing the disordered phase beyond this point will manifest as a significant negative departure from the EOS (higher pressure or lower density than expected). This effect is seen in Table 2. The physics of the HS glass transition is congruent with the view of GD.56,57 In the GD model, the number of ways that chain molecules can be randomly packed rapidly diminishes as the density increases. Similarly for a HS system, random packing becomes increasingly difficult with increasing density. There is a minor difference between the GD picture and the present one. In the GD model, which assumes that the system is random, the thermodynamic transition occurs when the configurational entropy for random configurations becomes vanishingly small. The GD model does not allow ordered states; therefore, a transition to an ordered system (liquid crystal or crystal) is impossible. As described herein, a HS fluid glass transition occurs at the Kauzmann density where the entropy of the disordered states falls below the entropy of ordered states. An extension of GD ideas is the well-known Adams-Gibbs (AG) relation66 for the relaxation time τ for cooperative motions
[ ] C TSconf
τ ∼ exp
(48)
where C is a constant. Originally, this relation was proposed for the configurational entropy (Sconf) which vanishes at a unique nonzero temperature (T0) associated with a second-order phase transition. Within the context of the ideas expressed herein, a proposed modified version of the AG relation for HS systems would be
[ ]
τ ∼ exp
AN ln Ωd
(49)
where A is a dimensionless constant and temperature has been omitted because there is no temperature scale for a HS system (the only relevant variable is βP). There will be a dramatic slowing down of motion for a metastable HS fluid that is prevented from crystallizing. The basic idea of the GD and AG models is preserved in that it is the rapid disappearance of disordered (random) configurations that slows down the dynamics. IV. Conclusions • A virial-coefficient-based analysis suggests a weak logarithmic singularity for the excess entropy of a HS fluid at the close-packed density (ηcp ) π2/6). This logarithmic diver-
gence is consistent with logarithmic singularities in the excess entropy predicted for HS crystals.22-25 • A new approximate EOS has been developed based on the 10 known virial coefficients that diverges at ηcp. When compared to published and accurate MD data, the new EOS is no better than existing Pade´ approximants up to η = 0.52, but the Pade´ approximants incorrectly remain finite at ηcp. • A new order parameter is defined that allows every system configuration to be classified as either “disordered” or “ordered.” At any given density, the configurational states of a HS system can be classified as belonging to either the disordered (random) subensemble or the ordered subensemble. The Kauzmann density is defined as the density where the configurational entropies of these two subensembles become equal. • A thermodynamic glass transition (second order) is predicted at the Kauzmann density (η = 0.54). • The physics of the HS glass transition harmonizes with the classical ideas of Gibbs and DiMarzio56,57 and Adam and Gibbs.66 It is the rapid disappearance of disordered (random) configurations with increasing density that drives the glass transition and slows the dynamics. References and Notes (1) Labik, S.; Kolafa, J.; Malijevsky, A. Phys. ReV. E 2005, 71. (2) Clisby, N.; McCoy, B. M. J. Stat. Phys. 2006, 122, 15. (3) Kratky, K. W. Physica A 1976, 85, 607. (4) Kratky, K. W. Physica A 1977, 87, 584. (5) Kratky, K. W. J. Stat. Phys. 1982, 27, 533. (6) Kratky, K. W. J. Stat. Phys. 1982, 29, 129. (7) Ree, F. H.; Hoover, W. G. J. Chem. Phys. 1964, 40, 939. (8) Ree, F. H.; Hoover, W. G. J. Chem. Phys. 1967, 46, 4181. (9) Kim, S. a.; H, D. Phys. Lett. A 1968, 27, 378. (10) van rensburg, E. J. J. Phys. A: Math. Gen. 1993, 26, 4805. (11) Vlasov, A. Y.; You, X. M.; Masters, A. J. Mol. Phys. 2002, 100, 3313. (12) Sanchez, I. C. J. Chem. Phys. 1994, 101, 7003. (13) Reiss, H.; Frisch, H. L.; Helfand, E.; Lebowitz, J. L. J. Chem. Phys. 1960, 32, 119. (14) Reiss, H.; Frisch, H. L.; Lebowitz, J. L. J. Chem. Phys. 1959, 31, 369. (15) Lebowitz, J. L.; Helfand, E.; Praestgaard, E. J. Chem. Phys. 1965, 43, 774. (16) Siderius, D. W.; Corti, D. S. J. Chem. Phys. 2007, 127, 19. (17) Siderius, D. W.; Corti, D. S. Ind. Eng. Chem. Res. 2006, 45, 5489. (18) Heying, M.; Corti, D. S. J. Phys. Chem. B 2004, 108, 19756. (19) Widom, B. J. Phys. Chem. 1982, 86, 869. (20) Wang, X. Z. Phys. ReV. E 2002, 66. (21) Wang, X. Z. J. Chem. Phys. 2004, 120, 7055. (22) Alder, B. J.; Hoover, W. G.; Young, D. A. J. Chem. Phys. 1968, 49, 3688. (23) Speedy, R. J. J. Phys.: Condens. Matter 1998, 10, 4387. (24) Velasco, E.; Mederos, L.; Navascues, G. Mol. Phys. 1999, 97, 1273. (25) Finken, R.; Schmidt, M.; Lo¨wen, H. Phys. ReV. E 2001, 65, 016108. (26) Rintoul, M. D.; Torquato, S. J. Chem. Phys. 1996, 105, 9258. (27) Torquato, S.; Truskett, T. M.; Debenedetti, P. G. Phys. ReV. Lett. 2000, 84, 2064. (28) Steinhardt, P. J.; Nelson, D. R.; Ronchetti, M. Phys. ReV. B 1983, 28, 784. (29) Berryman, J. G. Phys. ReV. A 1983, 27, 1053. (30) Song, C.; Wang, P.; Makse, H. A. Nature 2008, 453, 629. (31) Jaeger, H. M.; Nagel, S. R. Science 1992, 255, 1523. (32) Kamien, R. D.; Liu, A. J. Phys. ReV. Lett. 2007, 99, 4. (33) Rintoul, M. D.; Torquato, S. Phys. ReV. E 1998, 58, 532. (34) Bernal, J. D.; Mason, J. Nature 1960, 188, 910. (35) Parisi, G.; Zamponi, F. J. Chem. Phys. 2005, 123. (36) Silbert, L. E.; Ertas, D.; Grest, G. S.; Halsey, T. C.; Levine, D. Phys. ReV. E 2002, 65, 6. (37) Hoover, W. G.; Ree, F. H. J. Chem. Phys. 1968, 49, 3609. (38) Stone, M. T.; In ’t Veld, P. J.; Lu, Y.; Sanchez, I. C. Mol. Phys. 2002, 100, 2773. (39) Rintoul, M. D.; Torquato, S. Phys. ReV. Lett. 1996, 77, 4198. (40) Speedy, R. J. J. Chem. Phys. 1994, 100, 6684. (41) Schaaf, P.; Reiss, H. J. Chem. Phys. 1990, 92, 1258. (42) Jodrey, W. S.; Tory, E. M. Phys. ReV. A 1985, 32, 2347. (43) Kolafa, J.; Labik, S.; Malijevsky, A. Phys. Chem. Chem. Phys. 2004, 6, 2335.
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J. Phys. Chem. B, Vol. 113, No. 47, 2009
(44) Reiss, H.; Hammerich, A. D. J. Phys. Chem. 1986, 90, 6252. (45) Kegel, W. K.; Reiss, H.; Lekkerkerker, H. N. W. Phys. ReV. Lett. 1999, 83, 5298. (46) Richard, P.; Oger, L.; Troadec, J. P.; Gervois, A. Phys. ReV. E 1999, 60, 4551. (47) Bowen, L.; Lyons, R.; Radin, C.; Winkler, P. Phys. ReV. Lett. 2006, 96. (48) Huang, K. Statistical Mechanics; Wiley: New York, 1965. (49) Mayer, J. E.; Mayer, M. G. Statistical Mechanics; Wiley: New York, 1940. (50) Angell, C. A.; Tucker, J. C. J. Phys. Chem. 1974, 78, 278. (51) Stillinger, F. H.; Debenedetti, P. G.; Truskett, T. M. J. Phys. Chem. B 2001, 105, 11809. (52) Debenedetti, P. G. Metastable Liquids; Princeton University Press: Princeton, NJ, 1996. (53) Donev, A.; Stillinger, F. H.; Torquato, S. Phys. ReV. Lett. 2006, 96. (54) Speedy, R. J. Biophys. Chem. 2003, 105, 411.
Sanchez and Lee (55) Stillinger, F. H. J. Chem. Phys. 1988, 88, 7818. (56) Gibbs, J. H.; DiMarzio, E. A. J. Chem. Phys. 1958, 28, 373. (57) DiMarzio, E. A.; Gibbs, J. H. J. Chem. Phys. 1958, 28, 807. (58) Speedy, R. J. J. Phys.: Condens. Matter 1997, 9, 8591. (59) Gotze, W.; Sjogren, L. Phys. ReV. A 1991, 43, 5442. (60) Speedy, R. J. Mol. Phys. 1998, 95, 169. (61) Robles, M.; Haro, M. L.; d.; Santos, A.; Yuste, S. B. J. Chem. Phys. 1998, 108, 1290. (62) Kim, K.; Munakata, T. Phys. ReV. E 2003, 68. (63) Eisenberg, E.; Baram, A. Proc. Natl. Acad. Sci. U.S.A. 2007, 104, 5755. (64) Onoda, G. Y.; Liniger, E. G. Phys. ReV. Lett. 1990, 64, 2727. (65) Speedy, R. J. Mol. Phys. 1993, 80, 1105. (66) Adam, G.; Gibbs, J. H. J. Chem. Phys. 1965, 43, 139. (67) Kolafa, J. Phys. Chem. Chem. Phys. 2006, 8, 464.
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