J. Phys. Chem. B 2001, 105, 11737-11742
11737
Estimates of the Configurational Entropy of a Liquid† Robin J. Speedy‡ 504/120 Courtenay Place, Wellington 6001, New Zealand ReceiVed: May 2, 2001; In Final Form: July 9, 2001
The entropy difference between a liquid and one of the glasses that it samples is called the configurational entropy, Sc, and it may provide an important link between dynamic and thermodynamic measures of the fragility of liquids. Most previous estimates of Sc have implicitly assumed that all glasses of the same substance have the same entropy, but that assumption can lead to estimates of Sc and its temperature dependence that may be wrong by a factor of 2 or more. New estimates of Sc for toluene and ethylbenzene are used to test relations between thermodynamic and kinetic measures of fragility implied by Adam and Gibbs theory.
1. Introduction The motion of the molecules in a liquid can be separated into (a) local vibrational motions within a static configuration and (b) transitions between structurally distinct configurations.1,2 The entropy of the liquid can then be expressed as the sum of vibrational and configurational components. The vibrational entropy is the entropy of a static configuration, or a glass, and the configurational entropy measures the number of structurally distinct glasses that the liquid samples. The separation of the total entropy into the two components offers a unified description of the thermodynamics of liquids and glasses and the glass transition.2 The configurational entropy3 is also related to structural relaxation times, because structural change involves transitions between distinct configurations. The main obstacle to progress with these ideas is that estimates of the configurational entropy, from calorimetric measurements on liquids and glasses, can vary widely because they inevitably require assumptions and extrapolations that are difficult to test. The viscosity increases dramatically when a liquid is cooled toward its glass transition temperature, and the rate of increase has been correlated with thermodynamic properties.4-7 Adam and Gibbs3 argued that the viscosity, η, of a liquid varies on an isobar as
η ) η0 exp(C/TSc)
(1)
where η0 and C are approximately constant and T is the temperature. They express the configurational entropy as
Sc(T) )
∆CP(T) dT T
∫TT 0
(2)
where ∆CP(T) is the difference between the isobaric heat capacities of the liquid and a glass and T0 is the temperature where Sc extrapolates to zero. In practice5-9 ∆CP(T) is usually approximated by the excess heat capacity of the liquid relative to its crystal, and T0 is taken to be the Kauzmann10 temperature, where the entropy of the liquid extrapolates to that of the crystal. Equations 1 and 2 imply a correlation between the change in heat capacity at a glass transition and the apparent activation †
Part of the special issue “Howard Reiss Festschrift”. ‡ E-mail:
[email protected].
energy for viscosity. These are thermodynamic and dynamic measures, respectively, of the fragility4,5 of liquids, and the implied correlation is supported by experimental evidence for a broad range of liquids,4-7 although this has been questioned.8,9 Controversy seems inevitable because eq 2 implicitly assumes that all glasses of the same substance have the same heat capacity. If they do not, then eq 2 does not provide a unique definition of Sc. More plausible assumptions are shown here to lead to quite different estimates of Sc and its temperature dependence. The properties of an equilibrated, stable or metastable, phase are determined by its composition and the imposed external variables, pressure and temperature, but a glass is not equilibrated in that sense.2 A glass retains a memory of the rate, and the pressure, at which the liquid was cooled to make it, and one or more internal parameters,2,11-13 x, are needed to distinguish between glasses with different histories. Davies and Jones2 show that two internal parameters are needed when both temperature and pressure variations are considered. This work considers only temperature variations on an isobar, so the pressure is not explicitly indicated, and it is assumed that a single internal parameter suffices. The volume of a liquid, Vl(T), and its enthalpy, Hl(T), do not fluctuate significantly, and they do not change when the liquid becomes glassy, so the liquid at temperature T samples glasses of type x(T) that have the same volume, Vg(x(T),T) ) Vl(T), and enthalpy, Hg(x(T),T) ) Hl(T), as the liquid.2 The separation of the entropy of a liquid into vibrational and configurational parts implies that Sc can be defined by
Sc(T) ) Sl(T) - Sg(x(T),T)
(3)
where Sl(T) is the entropy of the liquid and Sg(x(T),T) is the vibrational entropy of a glass of type x(T). The configurational entropy measures the number of structurally distinct glasses, Ng(x) (or configurations,2,3 energy minima,14 or inherent structures15), of type x(T)
Sc(T) ) kB ln{Ng(x(T))}
(4)
where kB is the Boltzmann constant. Viscosity is related to the number of glasses that the liquid samples because flow depends on transitions between them. Adam and Gibbs3 recognized that
10.1021/jp0116656 CCC: $20.00 © 2001 American Chemical Society Published on Web 10/05/2001
11738 J. Phys. Chem. B, Vol. 105, No. 47, 2001
Speedy
this suggests a relation between viscosity and Sc and that flow ceases if Sc goes to zero. The entropy of model glasses can be measured directly16-19 in simulation experiments, and the few simulation studies20-24 that have tested Adam and Gibbs’ theory using the Sc defined by eq 3 have supported eq 1. With a minor modification, eq 1 describes the temperature dependence of diffusivity in a hard disk mixture from the ideal gas to the glass transition.24 It is important to note that eqs 2 and 3 are inconsistent and they can lead to quite different estimates of Sc. The inconsistency is apparent if we note that the difference between the heat capacities of a liquid and one of the glasses that it samples is
∆CP(T) ) CP,l(T) - CP,g(x(T),T)
(5)
and the heat capacity of an individual glass (in which x is fixed) is2 CP,g(x,T) ) T(∂Sg(x,T)/∂T)x,P. Equations 3 and 5 then give
∆CP(T) ) T(∂Sc(T)/∂T)P + T(∂Sg(x(T),T)/∂x)T,P(∂x(T)/∂T)P (6) Equation 2 assumes that ∆CP(T) ) T(∂Sc(T)/∂T)P, which neglects the last term in eq 6. Goldstein emphasized this point many years ago, and his analysis of heat capacity data for glasses with different histories suggests that the last term is significant and that for some substances it dominates. If CP,g(x,T) is assumed to be independent of x, then the last term in eq 6 is zero and eqs 2 and 3 are consistent, but that assumption is clearly too simple.25,26 A better assumption, that CP,g(x,T) varies linearly with x, is developed in section 3. Section 4 shows that the assumption is consistent with experimental data27 for liquid and glassy toluene and ethylbenzene. It provides a way to estimate the terms in eq 6 from experimental data, and it suggests that neglecting the last term can double the apparent temperature dependence of Sc near the glass transition temperature and double the apparent value of Sc in the equilibrium liquid. In section 4 the revised expressions for Sc are used to test the relation between thermodynamic and kinetic measures of fragility implied by eq 1. 2. Basic Relations An individual glass of type x at temperature T has enthalpy
Hg(x,T) ) Hg(x,0) +
∫0 CP,g(x,T) dT T
(7)
where Hg(x,0) is the enthalpy at absolute zero and CP,g(x,T) is the isobaric heat capacity. For concreteness the internal parameter is identified26 here as x ) Hg(x,0), but if we assume that all Ng(x) structurally distinct glasses with the same history have essentially the same bulk properties,17 then any other property that varies with x could serve as well. In practice, glasses behave reversibly only at low temperatures, and the heat capacity at higher temperatures, where x may change irreversibly with time, needs to be estimated by extrapolation.27 Equation 7 also applies to the crystal, and the enthalpy of the crystal at absolute zero is taken as the reference point for enthalpy so that x ) 0 for the crystal. The vibrational entropy of an individual glass is
Sg(x,T) )
∫0T
CP,g(x,T) dT T
(8)
This entropy is zero at absolute zero. It is the entropy corresponding to the phase space that the particular glass can access,11,16,28 and it is not the same as the usual calorimetric
Figure 1. Ratio of the heat capacities of glassy and crystalline ethylbenzene27 (squares, right axis) and ratio of the heat capacity of a glass formed by vapor-depositing pentene to one formed by cooling liquid pentene30 (circles, left axis), plotted against the reciprocal temperature. From the slopes shown, and using the enthalpy differences27,30 at low temperature for x - xr in eq 11, θR ) 0.03 for pentene and 0.035 for ethylbenzene. These are larger than the best-fit values for toluene and ethylbenzene in Table 1.
entropy, which is evaluated by integrating from the ideal gas limit. They differ because the glass transition is not a reversible process16 in the thermodynamic sense. Kivelson and Reiss28 note that Sc(x) corresponds to the thermodynamic work required to constrain24,29 a liquid to one of its glasses. Equations 3 and 8 imply that Sc(x) is the residual calorimetric entropy of a glass of type x at absolute zero. The Gibbs free energy of the liquid is expressed in terms of its glasses as26
Gl(T) ) Gg(x(T),T) - TSc(x(T))
(9)
where Gg(x(T),T) ) Hg(x(T),T) - TSg(x(T),T) is the contribution from an individual glass and the last term is the contribution from the Ng (x) glasses that the liquid samples. The equilibrium liquid samples glasses of the type that minimize its free energy so the equilibrium value of x(T) is implicitly determined2 by the condition
(∂Gl(T)/∂x)T,P ) 0
(10)
3. History Dependence of Glass Properties Experiments12,25,30 and simulations17,22,24 show that the properties of glasses depend on their history. For instance, the entropy of glasses formed by a simulated tetravalent network model17 increases by about half the entropy of melting when the rate at which the fluid is compressed to form a glass is slowed by 5 orders of magnitude. For practical reasons such large variations have not been measured for real glasses, but Figure 1 shows that a glass formed by vapor-depositing pentene30 has a higher heat capacity than one formed by cooling the liquid. Goldstein25 reviews experimental evidence for six other substances, and he suggests that the history-dependent entropy variations that can be measured experimentally are only a few times larger than the uncertainties. The rest of this section develops the two heuristic assumptions (i) that CP,g(x,T) varies linearly with x and (ii) that the magnitude of the variation is inversely proportional to temperature. Assumption i is guided by Sastry’s finding22 that the entropy of simulated Lennard-Jones glasses varies linearly with x and assumption ii by the experimental observation that the ratio of the heat capacity of glasses to that of their crystals varies
Configurational Entropy of a Liquid
J. Phys. Chem. B, Vol. 105, No. 47, 2001 11739
(approximately) as 1/T at low temperature and is close to unity at high temperature, as shown in Figure 1. Assumptions i and ii relate the thermal properties of a glass of type x to those of a reference glass of type xr. They imply that
CP,g(x,T) ) CP,g(xr,T){1 + (θ/T)(x - xr)}
(11)
where θ is a material-specific constant. Substituting eq 11 into eqs 7 and 8 gives
Hg(x,T) ) Hg(xr,T) + (x - xr){1 + θSg(xr,T)}
(12)
and
Sg(x,T) ) Sg(xr,T) + θ(x - xr)Wg(xr,T)
(13)
where
Wg(xr,T) )
∫0 {CP,g(xr,T)/T2} dT T
(14)
Plots of the lowest temperature measurements of CP,g(xr,T)/T2 for glassy o-terphenyl,31 toluene,27 and ethylbenzene27 show that it tends toward zero as T f 0. If each glass is regarded as a collection of many independent parts, the central limit theorem suggests the normal distribution26
Ng(x) ) exp{N[R - γ(xm - x)2]}
(15)
where R, γ, and xm are material-specific constants and N is the number of molecules. Equation 15 is consistent with simulation studies17,18,20,22,24 for a variety of models, and it is shown to be consistent with the experimental data for toluene and ethylbenzene in the next section. Equations 4 and 15 give the molar configurational entropy as
Sc ) R{R - γ(xm - x)2}
(16)
where R ) NkB is the gas constant. Substituting eqs 12, 13, and 16 into eqs 9 and 10 determines the equilibrium value of x(T)
x(T) ) xm -
1 + θ{Sg(xr,T) - TWg(xr,T)} 2γRT
(17)
The last two equations provide a compelling argument that CP,g(x,T) varies with x because they yield
T(∂Sc(x)/∂x)T,P ) 1 + θ{Sg(xr,T) - TWg(xr,T)}
(18)
which depends on the properties of the substance, as it should. If CP,g(x,T) does not vary26,32 with x, then θ ) 0 and eqs 9 and 10 imply26 that T(∂Sc(x)/∂x)T,P ) ∂Gg(x,T)/∂x)T,P ) 1, which might be a good approximation for some materials but is not valid in general. 4. Estimates for Toluene and Ethylbenzene Heat capacity measurements27,33 for crystalline, liquid, and glassy toluene and ethylbenzene were integrated numerically to calculate the properties of the liquids and glasses. A caveat is that, to suppress freezing, the glass heat capacity measurements were made on mixtures containing 10% benzene and contributions to the measured heat capacity were assumed to be additive.27 This assumption is justified27 at high temperatures, where the pure liquids can be studied, but the present analysis is sensitive to any uncertainty in the heat capacity of glasses at
Figure 2. Properties of the reference glasses that appear in eqs 1220 versus temperature for ethylbenzene (solid lines) and toluene (dashed lines).
TABLE 1: Best-Fit Values of the Constants in Eqs 12-17 Chosen To Fit the Enthalpy, H, and Entropy, S, of Liquid Ethylbenzene (EB) and Toluene (Methylbenzene, MB) at 10 or 20 K Intervals between the Glass Transition Temperature and the Highest Temperature, Th, of the Data Used in the Fita substance Th/K EB EB MB MB MB MB
θR
300 0.0165 300 0 360 -0.005 360 0 200 0.008 200 0
xm/RK
γ(RK)2
σH/RK
R
σS/R
1100 1950 1663 1343 1043 1370
5.18 × 3.12 × 10-6 3.53 × 10-6 4.23 × 10-6 5.04 × 10-6 4.11 × 10-6
8.3 33 8.8 16 4.3 7.6
2.73 7.1 6.90 5.04 3.22 5.09
0.04 0.14 0.05 0.07 0.03 0.04
10-6
a σ and σ are the standard deviations in the enthalpy and entropy, H S respectively. For comparison, results are included for the case when θ ) 0 is forced.
low temperatures, where the assumption has not been verified. Another caveat is that the glass heat capacities are extrapolated to high temperatures. Yamamuro et al.27 fit the heat capacity of the glasses with a combination of Debye and Einstein functions, and they suggest that their extrapolations to high temperature are accurate to within 5%. In the present work their27 heat capacity data, both measured and extrapolated, were fitted to polynomials in the inverse temperature, as described in detail previously,26 to calculate the properties of the reference glass in eqs 12-14. The glasses studied by Yamamuro et al.27 are the reference glasses. For ethylbenzene xr ) 552 RK and for toluene xr ) 331 RK relative to the crystals27 at absolute zero. The energy unit is RK ) 8.314 J/mol. The terms involving the properties of the reference glass that appear in eqs 12-18 are approximately linear in temperature as shown in Figure 2. The constants θ, γ, and xm were determined by comparing the enthalpy of the liquid to the enthalpy of one of the glasses that it samples, from eq 12, with x(T) from eq 17. After the values of θ, γ, and xm that best fit the enthalpy were found, the other unknown, R, was chosen to fit the entropy of the liquids, using eqs 3, 13, and 16 with x(T) from eq 17. The results are listed in Table 1, and the quality of the fit for ethylbenzene is illustrated in Figure 3. Table 1 shows that the entropy is fitted precisely with just one free parameter, R, which justifies the assumption of eq 15. For ethylbenzene the best-fit value of θ is positive, consistent with the trends shown in Figure 1. Figure 4 shows the configurational entropy calculated from eqs 16 and 17 with the constants in Table 1. It shows that if Sc is calculated from eq 2, with ∆CP(T) chosen as the heat capacity of the liquid relative to the crystal, or relative to the experimental glass (θ ) 0), the slope near the glass transition temperature and the magnitude
11740 J. Phys. Chem. B, Vol. 105, No. 47, 2001
Figure 3. Enthalpy of liquid ethylbenzene (squares), relative to the crystal at 0 K, and entropy (circles) plotted against temperature. The lines show values calculated with the constants from the first row in Table 1.
Speedy
Figure 5. Configurational entropy, Sc, versus temperature for toluene. The top curve shows the excess entropy, Sex, of the liquid relative to the crystal (extrapolated26 above the melting temperature, Tm ) 178 K). It locates the Kauzmann temperature near 100 K. The other lines, which cross at the experimental glass transition temperature27 (Tg )117 K), are from eqs 16 and 17 with the constants from Table 1. They are labeled with the values of θR from Table 1.
(Figure 2). When θ ) 0, the maximum shifts to infinite temperature.26 5. Measures of Fragility Two indicators of thermodynamic fragility4,5 are the rate at which the liquid climbs up the energy landscape near the glass transition temperature, Tg
Figure 4. Configurational entropy, Sc, versus temperature for ethylbenzene. The top curve shows the excess entropy, Sex, of the liquid relative to the crystal (extrapolated26 above the melting temperature, Tm ) 178 K). It locates the Kauzmann temperature near 89 K. The other two lines, which cross at the experimental glass transition temperature27 (Tg )115 K), are from eqs 16 and 17 with the constants from Table 1. They are labeled with the values of θR from Table 1. The line labeled 0 is the entropy of the liquid relative to that of the experimental glass, which is the Sc of Yamamuro et al.,27 and it goes to zero near 101 K. The best-fit line (θR ) 0.0165) goes to zero near 94 K.
of Sc at higher temperatures are more than twice those obtained with the parameters that best fit the properties of the liquid. This is because the last term in eq 6 is more than half of ∆CP(T), which is consistent with some of Goldstein’s 25 examples. Figure 4 also shows that Sc(θR ) 0.0165) goes through a weak maximum near 240 K and is essentially independent of temperature above the melting temperature, so that eq 1 predicts an Arrhenius temperature dependence for the viscosity of the equilibrium liquid. For toluene the best-fit value of θ is small and sensitive to the temperature range of the data used in the fitting exercise. The properties of the liquid have been measured33 to 360 K, and when all the data are used, the best-fit θ is small and negative (Table 1). However, the fit is quite good with θ ) 0, and the negative value may be spurious in view of the caveats noted at the start of this section. A positive value of θ is obtained when the fit is limited to temperatures below 200 K (Table 1), but again the fit is quite good with θ ) 0. Figure 5 shows the configurational entropy for toluene calculated from eqs 16 and 17 with the constants in Table 1. The maxima in Sc when θ is positive, shown in Figures 4 and 5, occur when the right side of eq 18 goes through zero. This occurs because the term in braces in eq 18 is negative
(∂x(T)/∂T)P,Tg )
1 + θSg(xr,Tg) 2γRTg2
(19)
and the rate of change of Sc with temperature
T(∂Sc(x(T))/∂T)P,Tg ) [1 + θ{Sg(xr,Tg) - TgWg(xr,Tg)}](∂x(T)/∂T)P,Tg (20) In a previous paper26 I focused on the total number of glasses that a liquid can form, exp(RN), as an indicator of thermodynamic fragility. Sastry22 has since shown that when the density of a Lennard-Jones mixture is increased, R decreases but the kinetic fragility increases. He suggests that the fragility is primarily determined by the width of the distribution of glasses in eq 15, which is proportional to γ-1/2, and this in accord with eqs 19 and 20. Martinez and Angell7 suggest that fragility arises mainly from an increase in the vibrational entropy of glasses with x. To examine this point, we need to distinguish between the way the number of glasses varies with x (eq 15) and the way that the entropies of the glasses vary with x (eq 13). These two factors, which are logically independent, both contribute to the thermodynamic fragility defined by eqs 19 and 20, the first through γ and the second through θ. However, the term θSg(xr,T) in eq 19 is only 0.17 for ethylbenzene near Tg, and less for toluene, so it is clear that γ is the prime determinant of fragility. This conclusion is likely to be generally valid because toluene is one of the most fragile7 liquids and the term θSg(xr,T) makes, at most, a very small contribution to its fragility. Both measures of fragility (in eqs 19 and 20) increase with decreasing γ, but they vary with θ in the opposite sense. An increase in θ increases (∂x(T)/∂T)P, because Sg(xr,T) is positive, but it decreases T(∂Sc(x(T))/∂T)P, because the term in braces in eq 20 is negative (Figure 2).
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J. Phys. Chem. B, Vol. 105, No. 47, 2001 11741
To relate26 kinetic and thermodynamic measures of fragility, the parameter C in eq 1 needs to be eliminated because it contains material-specific information about the activation barriers between glasses. C is eliminated by writing eq 1 as
TgSc(Tg)
ln{η(T)/η0} ) ln{η(Tg)/η0}
(21)
TSc(T)
For the purpose of correlating kinetic and thermodynamic measures of fragility for a range of liquids, Tg is usually defined as the temperature where η(Tg) has the same value,4,5,7 typically 1013 P (or structural relaxation times of 100 s). The extrapolated viscosity in the high-temperature limit (η0 in eq 1) is about 10-4 P for molecular liquids, and Angell34 suggests that log{η(Tg)/ η0} ≈ 17 is roughly the same for all substances when Tg is chosen so that η(Tg) ) 1013 P. A common4,5,34 measure of kinetic fragility is the slope, m, of a plot of the logarithm of the viscosity against Tg/T near the kinetic glass transition temperature Tg. If eq 1 is correct, this slope can be expressed as
m)
(
)
d log{η(T)/η0} d(Tg/T)
)
(
P,Tg
log{η(Tg)/η0)}
)
∂ ln{TSc(T)} ∂ ln{T}
(22)
P,Tg
which relates the kinetic fragility to thermodynamic properties. For ethylbenzene the thermodynamic derivative in eq 22 has the value 4.8 near the calorimetric glass transition temperature, Tg ) 115 K, and eq 22 predicts that m ) 17 × 4.8 ) 82. I have not found a direct measure of the kinetic fragility for ethylbenzene, but it is likely to lie between values for its nearest relatives toluene and propylbenzene. Hinze and Sillescu35 measured the relaxation time for molecular reorientation in toluene and showed that their results along with other measures of relaxation times, τ, spanning 10 orders of magnitude, are represented by the Vogel-Tamman-Fulcher equation
τ ) τ0 exp
(
)
B T - T0
(23)
where τ0, B, and T0 are fitted constants. Their results yield35 m ) 83 for toluene (at Tg ) T(τ ) 100 s) ) 117 K). A fit of dielectric relaxation times for toluene36 to eq 23 yields m ) 105 at Tg ) 115 K, where the dielectric relaxation time is 100 s. The calorimetric glass transition temperature27 is closer to 117 K. Measured dielectric relaxation times for propylbenzene37 in the temperature range 127 K < T < 160 K were fitted to eq 23 to get m ) 72 at Tg )125 K. Thus, the estimate of m ) 82 from the right side of eq 22 for ethylbenzene is intermediate between kinetic estimates of m for toluene and propylbenzene. For toluene the thermodynamic estimates obtained from the data in Table 1, m ) 250 (if θR ) -0.005) or 142 (if θR ) 0.008), are significantly higher than the kinetic estimates m ) 83 or 105 mentioned above. The measured27 configurational entropy of toluene is about half that of ethylbenzene near Tg, and this is the main reason the value of the thermodynamic derivative in eq 22 is much higher for toluene. 6. Conclusions The main conclusion is that estimates of Sc(T) from calorimetric data are very sensitive to the generally unknown dependence of the heat capacity of glasses on their history. In
Figure 6. Excess entropy of liquid ethylbenzene relative to the crystal, Sex, and configurational entropy, Sc (for θR ) 0.0165), versus reciprocal temperature.
simulation experiments an equilibrated liquid can be rapidly quenched17-23,38 or constrained,24 so that it is trapped in one of the glasses that it samples and the Sc defined by eq 3 can be measured directly. The present estimates of Sc(T) for toluene and ethylbenzene are likely to be more reliable than earlier estimates because the assumptions (eqs 11 and 15) on which they are based are guided by the simulation results. Measured relaxation times35-37 for both toluene and propylbenzene, spanning 10-12 orders of magnitude, are well represented by eq 23. For Adam and Gibbs’ eq 1 to agree with eq 23 Sc(T) should vary in proportion to 1 - T0/T. Figure 6 shows that the excess entropy of liquid ethylbenzene relative to the crystal, Sex(T), is linear in 1/T, which means that eq 1 agrees with eq 23 if Sc(T) is replaced by Sex(T). But the new estimate of Sc(T) shows some curvature, and this offers no new support for eq 1. Acknowledgment. I thank C. A. Angell and S. Sastry for helpful discussions. This work was supported by the Marsden fund administered by the Royal Society of New Zealand. References and Notes (1) Simon, F. Ergeb. Exakten Naturwiss. 1930, 9, 222. (2) Davies, R. O.; Jones, G. O. AdV. Phys. 1953, 2, 370. (3) Adam, G.; Gibbs, J. H. J. Chem. Phys. 1965, 43, 139. (4) Angell, C. A. In Relaxations in Complex System; Ngai, K., Wright, G. B., Eds.; National Technical Information Series; U.S. Department of Commerce: Springfield, VA, 1985. (5) Angell, C. A. J. Res. Natl. Inst. Stand. Technol. 1997, 102, 171. (6) Richert, R.; Angell, C. A. J. Chem. Phys. 1998, 108, 9016. (7) Martinez, L.-M.; Angell, C. A. Nature 2001, 410, 663. (8) Rowland, C. M.; Santangelo P. G.; Ngai, K. L J. Chem. Phys. 1999, 111, 5593. (9) Ngai, K. L.; Yamamuro, O. J. Chem. Phys. 1999, 111, 10403. (10) Kauzmann, W. Chem. ReV. 1948, 43, 219. (11) Ja¨ckle, J. Philos. Mag. B 1981, 44, 533. (12) Goldstein, M. J. Appl. Phys. 1975, 46, 4153. (13) Speedy, R. J. J. Phys. Chem. B 1999, 103, 8128. (14) Goldstein, M. J. Chem. Phys. 1969, 51, 3728. (15) Stillinger, F. H.; Weber, T. A. Science 1984, 225, 978; J. Chem. Phys. 1985, 83, 4767. Stillinger, F. H. J. Chem. Phys. 1988, 88, 7818, (16) Speedy, R. J. Mol. Phys. 1993, 80, 1105. (17) Speedy, R. J.; Debenedetti, P. G. Mol. Phys. 1996, 88, 1293. (18) Bu¨chner, S.; Heuer, A. Phys. ReV. E 1999, 60, 6507. (19) Sciortino, F.; Kob. W.; Tartaglia, P. Phys. ReV. Lett. 1999, 83, 3214. (20) Speedy, R. J. Mol. Phys. 1998, 95, 169. (21) Sastry, S. Phys. ReV. Lett. 2000, 85, 5590. (22) Sastry, S. Nature 2001, 409, 164. (23) Scala, A.; Starr, F.; La Nave, F. W.; Sciortino. F.; Stanley, H. E. Nature 2000, 406, 166. (24) Speedy, R. J. J. Chem. Phys. 2001, 114, 9069. (25) Goldstein, M. J. Chem. Phys. 1976, 64, 4767.
11742 J. Phys. Chem. B, Vol. 105, No. 47, 2001 (26) Speedy, R. J. J. Phys. Chem. B 1999, 103, 4060. (27) Yamamuro, O.; Tsukushi, I.; Lindqvist, A.; Takahara, S.; Ishikawa, M.; Matsuo, T. J. J. Phys. Chem. B 1998, 102, 1605. (28) Kivelson, D.; Reiss, H. J. Phys. Chem. B 1999, 103, 8337. (29) Reiss, H. Methods of Thermodynamics; Dover Publications: New York, 1996. (30) Takeda, K.; Yamamuro, O.; Suga, H. J. Phys. Chem. 1995, 99, 1602. (31) Chang, S. S.; Bestul, A. B. J. Phys. Chem. 1972, 56, 503. (32) Stillinger, F. H. J. Phys. Chem. B 1998, 102, 2807.
Speedy (33) Scott, D. W.; Guthrie, G. B.; Messerly, J. F.; Todd, S. S.; Berg, W. T.; Hossenlopp, I. A.; McCullough, J. P. J. Phys. Chem. 1962, 66, 911. (34) Angell, C. A. Science 1995, 267, 1924. (35) Hinze, G.; Silescu, H. J. Chem. Phys. 1996, 104, 314. (36) Do¨ss, A.; Hinze, G.; Scheiner, B.; Hemberger, J.; Bo¨hmer, R. J. Chem. Phys. 1997, 107, 1740. (37) Hansen, C.; Stickel, F.; Richert, R.; Fischer, E. W. J. Chem. Phys. 1998, 108, 6408. Richert, R. Personal communication. (38) Sciortino, F.; Kob, W.; Tartaglia, P. J. Phys.: Condens. Matter 2000, 12, 6525.
