1438
J. Phys. Chem. B 2007, 111, 1438-1442
Dynamic Scaling and Slowing Down in Chemical Reactions of the Critical Triethylamine-Water System I. Iwanowski and U. Kaatze* Drittes Physikalisches Institut, Georg-August-UniVersitt, Friedrich-Hund-Platz 1, 37077 Goettingen, Germany
J. Phys. Chem. B 2007.111:1438-1442. Downloaded from pubs.acs.org by UNIV OF SOUTH DAKOTA on 09/15/18. For personal use only.
ReceiVed: October 26, 2006; In Final Form: December 20, 2006
Between 100 kHz and 1 MHz, special ultrasonic attenuation measurements of the triethylamine-water mixture of critical composition have been performed for the determination of the Bhattacharjee-Ferrell scaling function. The experimental data are evaluated considering two noncritical Debye-type relaxation terms as revealed by broadband ultrasonic spectra. Shear viscosity and dynamic light scattering data from the literature are reevaluated to yield the relaxation rate of order parameter fluctuations of the critical system as a function of temperature. The power law behavior found for the relaxtion rate fits to the scaling function in the ultrasonic spectra. The relaxation times of the noncritical Debye terms display a non-Arrhenius temperature dependence, pointing at effects of slowing in the chemical reactions associated with the relaxations.
1. Introduction The critical dynamics of a variety of binary mixtures has been shown to follow the Bhattacharjee-Ferrell dynamic scaling theory.1,2 Recent examples are particulary such without additional relaxations in their ultrasonic spectra.3 However, systems for which contributions from noncritical relaxations indeed exist and can be well identified and can thus be separated from the critical part in the spectra; these have also been consistently described in terms of the Bhattacharjee-Ferrell model.4,5 So far, however, no satisfactory agreement between theory and experiment has been reached for the triethylamine-water mixture of critical composition. Previous ultrasonic measurements resulted in scaling function data that do not fall on one curve and that, in addition, deviate substantially from the theoretical form.6,7 More recent broadband ultrasonic attenuation spectrometry revealed two noncritcal background relaxation terms, in addition to the always existing asymptotic highfrequency background term.8 Consideration of these terms in the evaluation of the ultrasonic spectra resulted in a much better, but still imperfect, agreement with theory. Unfortunately, a distinctly smaller value (Γ0 ) 45 × 109 s-1) for the amplitude of the relaxation rate of concentration fluctuations was obtained from the ultrasonic spectra than from dynamic light scattering and shear viscosity measurements (Γ0 ) 96 × 109 s-1). Furthermore, attempts failed to derive the critical ultrasonic amplitude from a thermodynamic relation. This latter finding has been taken as an indication that presumptions of the Bhattacharejee-Ferrell theoretical model are not fulfilled with the triethylamine-water system. In particular, the amplitude in the heat capacity of the system is not smaller as compared with the background part but rather exceeds the background part.9 Triethylamine-water is thus no suitable system to verify or disprove the dynamic scaling theory predictions for the critical amplitude of the sonic spectra. We have, however, performed special measurements of the ultrasonic attenuation coefficient and have re-evaluated dynamic light scattering and shear viscosity data in order to show that the relaxation rate of * Corresponding author. Fax: +49-551-39-7720. E-mail: uka@physik3. gwdg.de.
concentration fluctuations and the scaling function can be consistently represented by the Bhattacharjee-Ferrell model. Evidence is additionally obtained for a slowing down of both noncritical relaxation processes when the temperature approaches the critical. 2. Experimental Triethylamine (abbreviated TEA) has been used as delivered (>99 % Fluka). Millipore water was used to prepare TEAH2O mixtures by weighing appropriate amounts of the constituents into suitable flasks. In conformity with many previous studies, of which only some are mentioned here,8,10-16 the mass fraction of amine of the mixture of critical composition was taken as Yc ) 0.321. We are aware that more recently the somewhat smaller value Yc ) 0.317 had been preferred.9,17 The visually determined critical temperature, according to the equal volume criterion, was 18.21 °C, which compares to literature data between 18.085 and 18.3 °C.7,8,12,13,15,17-19,21 At temperatures between 10.7 and 18.20 °C, three runs of acoustical attenuation coefficient measurements for the determination of the scaling function have been performed. The measurements covered the frequency range between 100 kHz and 1 MHz where the critical part in the ultrasonic attenuation spectra predominates. A resonator method was applied in which the path length of interactions of the acoustical wave with the liquid sample is virtually enhanced by multiple reflections.22 During each run, the sample was always kept within the cell in order to avoid disturbances by refilling procedures. Unwanted effects by mechanical stress and slight disadjustments of the cell were reduced by utilizing the focusing action of a planoconcave cavity resonator.22 The attenuation coefficient R of the TEA-H2O mixture of critical composition was determined relative to water, used as reference liquid with a well-known attenuation coefficient and with matched sound velocity cs as well as almost identical density F. Higher order modes of the cavity field have been carefully considered by recording the complete transfer function of the liquid filled resonator around a resonance peak and by subsequent analysis of the transfer function in terms of the principal and higher order modes. The temperature of the cavity resonator was controlled and measured
10.1021/jp067031m CCC: $37.00 © 2007 American Chemical Society Published on Web 01/24/2007
Scaling/Slowing Down in Reactions of TEA-H2O System
Figure 1. Ultrasonic attenuation coefficient per ν2 versus frequency ν for the TEA-H2O mixture of critical composition at 290.15 K. The subdivision of the spectrum into a critical part (“c”) and non-critical background contributions (“B′”,“D1”,“D2”) is indicated by dashed lines. The full line in the graph is the complete spectral function (eq 1). At some reduced temperatures, , the specially measured low-frequency part of the spectrum, is shown in the inset.
to within 0.02 K. The error in the attenuation coefficient data was smaller than ∆R/R ) 0.05. Broadband ultrasonic spectra, static, and high-frequency shear viscosity data, as well as results from dynamic light scattering measurements, are taken from the literature8,23 where details of the measurements are also given.
J. Phys. Chem. B, Vol. 111, No. 6, 2007 1439
Figure 2. Shear viscosity ηs of the TEA-H2O mixture of critical composition versus temperature T.8,23 The full line shows the viscosity function (eq 9); the dashed line represents the background part ηb (eq 10). In the inset, a complex plane representation of the frequency dependent complex viscosity (eq 11) is given. The full line is the graph of the relaxation spectral function defined by eq 12. The dotted line indicates a possibly existing second relaxation term. The dashed line shows the sum of both terms.
3. Results and Discussion 3.1. Ultrasonic Spectra. In Figure 1, the ultrasonic attenuation spectrum of the TEA-H2O mixture of critical composition is displayed at 17 °C. In order to accentuate the low-frequency part of the spectrum, the frequency normalized representation, in which R/ν2 is plotted versus frequency ν, is used. The finding of the broadband TEA-H2O spectra to be composed of different contributions is indicated by dashed lines. Careful analysis of the experimental data8 has revealed the existence of two Debyetype relaxation terms (D1, D2) in addition to the critical term (c) and the frequency independent contribution B′. Assuming the Bhattacharjee-Ferrell theory to apply to the critical part Rc in the sonic attenuation coefficient, the ultrasonic spectra have been analytically represented by the function
R ν2
) Sν
-(1+δ)
FBF +
2
ADn
n)1
2 1 + ω2τDn
∑
Figure 3. Relaxation rate Γ of order parameter fluctuations of the TEA-H2O mixture of critical composition as a function of reduced temperature . Symbols represent data as obtained from the combined evaluation of shear viscosity and dynamic light scattering results. The line is the graph of the power law (eq 3) with the theoretical critical exponent Z0 ) 1.903 and the amplitude Γ0 ) 96 × 109 s- 1.
data that followed from that analysis can indeed be well represented by the power law dependence
Γ() ) Γ0-Z0V˜ + B′
(1)
In this relation, S is the amplitude of the Bhattacharjee-Ferrell theory and δ ) R0/(Z0) is the universal exponent, with R0 denoting the specific heat critical exponent, Z0 the dynamic critical exponent, and the critical exponent of the fluctuation correlation length. FBF(Ω) is the Bhattacharjee-Ferrell scaling function, for which the empirical form2
FBF(Ω) ) [1 + 0.414(Ω1/2/Ω)1/2]-2
(2)
has been used. Here, Ω ) ω/Γ is the reduced frequency, given by the angular frequency ω ) 2πν and the relaxation rate Γ of the concentration fluctuations. Parameter Ω1/2 is the scaled halfattenuation frequency, defined by FBF(Ω1/2) ) 0.5. In conformity with experimental evidence,24 this parameter has been fixed at the theoretically predicted value2 Ω1/2 ) 2.1. In eq 1, ADn and τDn, n ) 1, 2, are the amplitudes and relaxation times, respectively, of the Debye relaxation terms. The parameters that had been obtained from a nonlinear regression analysis of the measured broadband attenuation spectra are presented in Table 1 of ref 8. The relaxation rate
(3)
upon reduced temperature ) |T - Tc|/Tc. As mentioned in the introduction, however, the resulting amplitude Γ0 ) 45 × 109 s-1 is substantially smaller than the amplitude from the shear viscosity and dynamic light scattering experiments, using the dynamic scaling hypothesis25-28
Γ() ) 2D()/ξ2()
(4)
to calculate the relaxation rate from the mutual diffusion coefficient D and the fluctuation correlation length ξ. The analysis of the broadband spectra yields Debye term relaxation times, which, at odds with Arrhenius or Eyring behavior, increase with temperature. The effect is small with τD1 () 3.1 ns at 13 and 4.1 ns at 18 °C) but is remarkably large with τD2 () 15 ns at 13 and 26 ns at 18 °C). Due to the large number of unknown parameters in the regression analysis of TEA-H2O ultrasonic spectra and because of rather similar relaxation times of the two Debye terms, the uncertainty in the parameter values obtained for the fitting procedure is rather high.8 We therefore have alternatively evaluated the attenuation coefficient data, including those especially measured in the low-frequency range of spectra (inset of Figure 1), where the critical contributions predominates.
1440 J. Phys. Chem. B, Vol. 111, No. 6, 2007
Iwanowski and Kaatze 400) K, qc ) 60 × 109 m-1, qD ) 0.9 × 109 m-1. In Figure 2, the shear viscosity data are displayed along with the graph of eq 9 to show that the experimental ηs values are well represented by theory. Also given by the dashed line is the graph of the background part ηb corresponding with the above parameters. The attempt has been made to verify the background part by extrapolation of high-frequency shear viscosity data to low frequencies.23Assuming the critical concentration of fluctuations to do not contribute to ηs at high frequencies, measurements between 20 and 130 MHz of the complex viscosity (i2 ) -1)
Figure 4. Scaling function data for the TEA-H2O system as calculated according to eq 15. Data from three runs of special low frequency measurements and from previous broadband spectrometry8 are indicated by figure symbols. The line is the graph of the empirical form of the Bhattacharjee-Ferrell scaling function.
This alternative evaluation procedure involves the use of the relaxation rate Γ() as a quantity known from shear viscosity and dynamic light scattering measurements. 3.2. Relaxation Rate. The hydrodynamic limit shear viscosity ηs, mutual diffusion coefficient D, and fluctuation correlation length ξ are related to one another by the suggestive StokeEinstein-Kawasaki-Ferrell equation29-32
DKF ) kBT/(6πηξ)
(5)
where kB is the Boltzmann’s constant and subscript KF indicates the definition of the diffusion coefficient according to eq 5. Close to Tc, the fluctuation correlation length
ξ() ) ξ0-V˜
(6)
diverges and the diffusion coefficient observed in the dynamic light scattering experiment depends explicitly on the amount q of the wavevector selected by the scattering geometry.33,34
{
D ) DKF RΩK(x)(1 + b2x2)Zη/2 3πηs + (1 + x2)[(qDξ)-1 - (q˜ cξ)-1] 16ηb
}
(8)
(9)
to also depend on these wavenumbers and on the fluctuation correlation length. Quantity Zη is the universal critical exponent of the viscosity38-40 and the background viscosity
ηb ) Aη exp[Bη/(T - Tη)]
ηs(ν) ) ηs(∞) + [ηs(0) - ηs(∞)]/[1 + iωτη]
(12)
is also shown. Extrapolation due to this function leads to ηs(0) values that are somewhat smaller than the background viscosity ηb.23 It is likely that the lowest frequency in the complex shear viscosity measurements of the TEA-H2O system was still too high for this extrapolation because, in correspondence with the second Debye relaxation in the acoustical spectra, another shear viscosity relaxation term might exist at frequencies below 20 MHz. The effect of such a term is indicated by the dashed line in the complex plane plot. The amplitude ξ0 ) 0.107 nm of the fluctuation correlation length agrees with the one obtained for the previous evaluation of ηs and D data,8 in which the background part in the diffusion coefficient had been assumed to be negligibly small and in which the viscosity had been represented by the equation28,33
(13)
(7)
is the Kawasaki function.35 In eq 7, q˜ c is given by q˜ c-1 ) qc-1 + (2qD)-1 where qc and qD denote cutoff wavenumbers. Dynamic scaling theory predicts the shear viscosity in the crossover region33,34,36,37
ηs ) ηb(T) exp[ZηH(ξ,qD,qc)]
(11)
have been performed for this purpose. A complex plane representation of the ηs data at 18 °C is given in the inset of Figure 2, where the graph of a Debye-type relaxation function
ηs() ) ηb(T)(Q0η)Zη
Here, R ) 1.03, b ) 0.55, x ) qξ, and
ΩK ) 3/(4x2)[1 + x2 + (x3 - x-1) arctan x]
ηs ) η′s(ν) - iη′′s(ν)
(10)
is a function of the temperature with the system specific parameters Aη, Bη, and Tη. The analytical form of crossover function H(ξ,qD,qc) is given in the literature.33,34,36 Assuming Tη ) 0 in eq 10, the combined regression analysis of the shear viscosity and dynamic light scattering data yielded the following values for the adjustable parameters: ξ0 ) (0.107 ( 0.01) nm, Aη ) (0.1 ( 0.01) × 10-6 Pa s, Bη ) (4394 (
The value is also in excellent agreement with ξ0 ) 0.110 nm as had been calculated from the heat capacity data using the two-scale factor universality relation.8 As illustrated by Figure 3, the relaxation rate Γ() nicely follows the power law (eq 3) with the universal critical exponent Z0ν˜ ) 1.903. The amplitude Γ0 ) 96 × 109 s-1, like the amplitude of the fluctuation correlation length, agrees with the value from previous evaluation of ηs and D data.8 3.3. Scaling Function. According to the BhattacharejeeFerrell dynamic scaling theory, the critical part Rcλ(ν,T) in the ultrasonic attenuation per wavelength can be expressed by the scaling function F(Ω) and the attenuation per wavelength at the critical temperature2
Rcλ(ν,T) ) F(Ω)Rcλ(ν,Tc)
(14)
where Rλ ) Rλ, λ ) cs/ν, and cs ) sound velocity. Hence, the scaling function can be experimentally determined from the ratio of sonic data at temperature T and corresponding data at the critical temperature. However, because the critical part in the attenuation coefficient is not known a priori, it has to be derived from the total attenuation by subtraction of the noncritical
Scaling/Slowing Down in Reactions of TEA-H2O System
Figure 5. Relaxation time ratio τDn(T)/τDn (18 °C), n ) 1, 2, of the noncritical relaxation terms displayed versus temperature T.
background contribution. In terms of eq 1, considering Rλ ) (R/ν2)csν, the scaling function may be obtained as
F(Ω) ) 2
S(Tc) S(T)
R/ν2(T) -
2 ADn(T)/[1 + ω2τDn (T)] - B′(T) ∑ n)1
(15)
2
R/ν2(Tc) -
2 (Tc)] - B′(Tc) ∑ADn(Tc)/[1 + ω2τDn
n)1
Assuming, in the temperature range between 10.7 and 18.21 °C, the amplitude S to be independent of temperature and taking the relaxation rate Γ() as following from the shear viscosity and dynamic light scattering measurements, the experimental scaling function data have been fitted to the empirical FBF(Ω) function defined by eq 2. In this fitting procedure, parameters AD1 and τD1 of the high-frequency Debye term as well as the asymptotic high-frequency background parameter B′ have been fixed at values obtained by inter- and extrapolation of data from the broadband spectra.8 Hence, only τD2 and AD2 were adjusted to reach optimum agreement of scaling function data (eq 15) with the analytical form (eq 2). The result of the regression analysis is shown in Figure 4. Because the critical contribution resulted from subtraction of the noncritical parts from the total attenuation, the scatter in the F(Ω) data is rather large. Within the limits of this scatter, however, the data from different runs and different temperatures of measurement fall on one curve and agree with the empirical scaling function (eq 2). 3.4. Slowing of Noncritical Relations. The resulting values for the relaxation time τD2 are somewhat larger than those from the previous analysis.8 Within the limits of experimental errors, the broadband attenuation spectra are nevertheless well represented by the relaxation function (eq 1). The temperature dependence τD2(T)/τD2 (13 °C) is unbiased by the slight readjustment of parameters. The relaxation times of both Debye terms increase with T (Figure 5), at odds with Arrhenius or Eyring characteristics. The high-frequency Debye relaxation term has been assigned8 to the protolysis reaction
TEA + H2O H TEAH+ + OH-
(16)
and the low-frequency term to the rotational isomerization of TEA ethyl groups
TEA H TEA*
(17)
J. Phys. Chem. B, Vol. 111, No. 6, 2007 1441
Figure 6. Ultrasonic attenuation spectrum in the frequency normalized format for a TEA-H2O mixture of noncritical composition (Y ) 0.17 ) 0.53Yc) at 10 °C () Tc - 8.21 K). The data have been taken from Figure 1 in ref 46. Dashed lines indicate the subdivision of the spectrum into the noncritical terms (“B′”,“D1”,“D2”) and a Bhattacharjee-Ferrell term (“c”). The full line represents the sum of these terms.
with TEA* denoting a structural conformer of TEA. A Debyetype relaxation in the spectra of pure triethylamine41,42 has been also discussed in terms of a rotational isomerization of TEA. The relaxation time of that Debye term, however, was noticeably smaller than the τD2 values of the TEA-H2O mixture of critical composition. We suggest a collective redistribution of solvent molecules associated with structural isomerization of TEA in aqueous solution to be the reason for the enlargement of relaxation times. The non-Arrhenius behavior of the ultrasonic relaxation times associated with the protolysis (eq 16) and structural isomerization (eq 17) may be taken to indicate slowing down of the chemical reactions near the critical point. Proccacia et al.43,44 have predicted effects of slowing of such reactions in which both constituents of a binary fluid participate. Following those predictions, the unimolecular reaction (eq 17) should be unaffected by critical fluctuation. However, considerable effects of slowing had been also observed in the Debye term relaxation times of butyric acid water mixtures, reflecting the equilibria between monomers and single hydrogen-bonded chain like dimers as well as between the chain-like dimers and double hydrogen-bonded cyclic dimers.45 Obviously, an intrinsic effect of slowing down near the critical point exists also with the latter (unimolecular) reaction. The non-Arrhenius behavior of relaxation times associated with structural isomerizations in critical binary mixtures probably indicates the second constituent (water), via rearrangements of solvation shells, to be involved in the reactions. 4. Conclusions Ultrasonic attenuation spectra of the TEA-H2O mixture of critical composition can be well represented by a BhattacharjeeFerrell term, considering the critical dynamics, and three background contributions. The latter include two Debye-type relaxation terms, reflecting noncritical chemical reactions, and an asymptotic high-frequency part. Within the limits of experimental errors, the scaling function derived from the sonic attenuation data agrees with the theoretical predicted form if the Bhattacharjee-Ferrell amplitude is assumed to be independent of temperature and if the relaxation rate of concentration fluctuations is fixed to the power law resulting from dynamic light scattering and shear viscosity measurements. The relaxation times of both Debye relaxation terms indicate effects of slowing down near the critical point, even though one term is assigned to a structural isomerization of TEA. This finding may be taken to indicate water to play a noticeable role in the essentially unimolecular isomerization. Water probably participates due
1442 J. Phys. Chem. B, Vol. 111, No. 6, 2007 to an extensive rearrangement of hydration shells associated with the isomerization of TEA. Unfortunately, it is difficult to study the behavior of the Debye-type relaxations without interferences by the critical term in the attenuation spectra. As shown by the literature data for a noncritical TEA-H2O system46 in Figure 6, even substantially apart from the critical point, noticeable effects from the Bhattacharjee-Ferrell contributions emerge. Acknowledgment. We thank R. Behrends and S. Z. Mirzaev for spirited discussions. Financial support by the Deutsche Forschungsgemeinschaft, Bonn, Germany, is gratefully acknowledged. References and Notes (1) Bhattacharjee, J. K.; Ferrell, R. A. Phys. ReV. A 1981, 24, 1643. (2) Ferrell, R. A.; Bhattacharjee, J. K. Phys. ReV. A 1985, 31, 1788. (3) Iwanowski, I.; Leluk, K,; Rudowski, M.; Kaatze, U. J. Phys Chem. A 2006, 110, 4313. (4) Iwanowski, I.; Sattarow, A.; Behrends, R.; Mirzaev, S. Z.; Kaatze, U. J. Chem. Phys. 2006, 124, 144505. (5) Iwanowski, I.; Mirzaev, S. Z.; Kaatze, U. Phys. ReV. E 2006, 73, 061508. (6) Yun, S. S. J. Chem. Phys. 1970, 52, 5200. (7) Garland, C. W.; Lai, C.-N. J. Chem. Phys. 1978, 69, 1342. (8) Behrends, R.; Telgmann, T.; Kaatze, U. J. Chem. Phys. 2002, 117, 9828. (9) Flewelling, A. C.; DeFonseka, R. J.; Khaleeli, N.; Partee, J.; Jacobs, D. T. J. Chem. Phys. 1996, 104, 8048. (10) Bloemen, E.; Thoen, J.; Van Deal, W. J. Chem. Phys. 1980, 73, 4628. (11) Sanchez, G.; Meichle, M.; Garland, C. W. Phys. ReV. A 1983, 28, 1647. (12) Furrow, G. P.; Greer, S. C. J. Chem. Phys. 1983, 79, 3474. (13) Fast, S. J.; Yun, S. S. J. Chem. Phys. 1985, 83, 5888. (14) Chaar, H.; Moldover, M. R.; Schmidt, J. W. J. Chem. Phys. 1986, 85, 418. (15) Pe´pin, C.; Bose, T. K.; Thoen, J. Phys. ReV. Lett. 1988, 60, 2507.
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