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Critical fluctuations and noncritical relaxations of the nitrobenzene–isooctane system near its consolute point
Chemical Physics Letters 458 (2008) 76–80
Contents lists available at ScienceDirect
Chemical Physics Letters journal homepage: www.elsevier.com/locate/cplett
Critical fluctuations and noncritical relaxations of the nitrobenzene–isooctane system near its consolute point P.K. Khabibullaev a, S.Z. Mirzaev a,b, U. Kaatze b,* a b
Heat Physics Department, Uzbek Academy of Sciences, Katartal 28, Tashkent 7000135, Uzbekistan Drittes Physikalisches Institut, Georg-August-Universität, Friedrich-Hund-Platz 1, 37077 Göttingen, Germany
a r t i c l e
i n f o
Article history: Received 12 February 2008 In final form 15 April 2008 Available online 18 April 2008
a b s t r a c t Ultrasonic attenuation spectra between 20 kHz and 3 GHz of the nitrobenzene–isooctane mixture of critical composition have been analyzed to show that they contain noncritical relaxation terms in addition to the critical term. The parameter values of the noncritical contributions obtained thereby are used in a reevaluation of smallband attenuation data from the literature. These data, measured at a large number of temperatures near the critical, are most suitable for the determination of the scaling function in the critical dynamics. The procedure allows to verify the empirical scaling function of the Bhattacharjee–Ferrell dynamic scaling theory without an adjustable parameter. Ó 2008 Elsevier B.V. All rights reserved.
1. Introduction Critical concentration fluctuations of binary fluids near their demixing point have been intensively investigated using ultrasonic spectrometry. The method has the potential to verify or disprove theories modeling the long wavelength fluctuations in the local concentrations and thus the conceptions of universality, scaling and renormalization [1] on which theories rely. Measurements of the sonic attenuation coefficient a of the fluid as a function of temperature T and frequency m yields the scaling function F(X) of the system as a ratio of the critical contributions ack to the attenuation-per-wavelength ak ¼ ak [2]: FðXÞ ¼ ack ðm; TÞ=ack ðm; T c Þ:
ð1Þ
Here X ¼ 2pm=CðeÞ is a scaled frequency with the relaxation rate C of order parameter fluctuations following power law CðeÞ ¼ C0 eZ0 ~m ;
ð2Þ
where Z0 and ~m are universal critical exponents, C0 is an amplitude characteristic of the system under consideration, and e ¼ jT T c j=T c
ð3Þ
is the reduced temperature. Tc denotes the critical temperature and k is the wavelength of the sonic field within the liquid. Hence the scaling conception reveals the critical dynamics of the binary fluid in terms of a universal function if the individual parameters critical temperature Tc and amplitude C0 of the relaxation rate are known. A prominent binary liquid, the ultrasonic attenuation coefficient data of which have been discussed several times in the past, is the
* Corresponding author. Fax: +49 551 39 7720. E-mail address: [email protected] (U. Kaatze). 0009-2614/$ - see front matter Ó 2008 Elsevier B.V. All rights reserved. doi:10.1016/j.cplett.2008.04.066
nitrobenzene–isooctane mixture of critical composition with a convenient upper critical temperature Tc = 302.29 K [3]. Annantaraman et al. reported ultrasonic attenuation coefficient data for the nitrobenzene–isooctane system at five frequencies between 4.5 and 16.5 MHz [4]. These authors discussed their results in terms of Fixman’s mode-coupling theory [5]. They found reasonable but not exact agreement between the theory and the experimental data. Later Garland reconsidered the attenuation coefficient data by Anantaraman et al. [6]. He also noticed considerable deviations from the predictions of Fixman’s theoretical model. Fenner performed ultrasonic attenuation measurements for the critical composition mixture of nitrobenzene and isooctane in the relatively broad frequency range from 1 to 91 MHz and in the wide temperature interval 0.007 6 T Tc 6 30 K [7]. He concluded that the experimental data are well fitted by the Kawasaki version [8] of the mode–mode-coupling theory. Bhattacharjee and Ferrell used Fenner’s experimental data to test their dynamic scaling theory [2,9]. The Woermann group has measured the ultrasonic attenuation coefficient of the nitrobenzene–isooctane critical mixture as a function of frequency m between 9 and 45 MHz and of the temperature interval T Tc between 0.3 and 21.8 K [10]. Later on the group re-evaluated the data in the light of the Bhattacharjee–Ferrell dynamic scaling theory using an integral version Z 3X 1 x2 dx FðXÞ ¼ ð4Þ p 0 ð1 þ x2 Þ3=2 x2 þ x3 þ X2 and alternatively the empirical form [2] of the scaling function [11] 1=2 2 FðXÞ ¼ ½1 þ 0:414ðXBF ; 1=2 =XÞ
ð5Þ
with scaled half-attenuation frequency XBF 1=2 ¼ 2:1: Treating the amplitude C0 of the relaxation rate as an adjustable parameter
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P.K. Khabibullaev et al. / Chemical Physics Letters 458 (2008) 76–80
when fitting the experimental data to the theoretical functions the empirical form Eq. (5) was found superior in that the resulting C0 value (=37.0 109 s1) agreed significantly better with the one (C0 = 43.8 109 s1) from static and dynamic light scattering [11]. The agreement between measurements and theory, however, is incomplete as the experimental data reveal systematic deviations from Eq. (5), likely reflecting inadequate assumptions on the noncritical background contributions to the total ultrasonic attenuation coefficient. Obviously, the frequency band of measurements in the above nitrobenzene–isooctane studies is too small to appropriately account for the background attenuation in addition to the critical part in the spectra, the latter extending over vast ranges of frequency. For this reason we discuss the ultrasonic attenuation data of the mixture of critical composition again, including the experimental determination of the noncritical contribution from broadband spectra. 2. Conception, original results
ð6Þ
of the critical contribution ack and a noncritical background contribution abk . According to the Bhattacharjee–Ferrell dynamic scaling theory [2,9], the critical part can be expressed as ack ðm; TÞ ¼ cðTÞAðTÞFðXÞ
ð7Þ
with sound velocity c (=km) and amplitude factor A depending on temperature T but depending only weakly on frequency m. If the weak temperature dependencies of c and A are taken into account the scaling function follows from Eq. (7) as [12] FðXÞ ¼ F ack ðm; TÞ=ack ðm; T c Þ
ð13Þ
In Eqs. (11) and (12) ~m and m are universal critical exponents. The theoretical value of the former is ~m ¼ 0:630 [14]. For the latter m ¼ 0:671 has been used [11]. Static light scattering yielded n0 = (0.249 ± 0.060) nm [11], in nice agreement with n0 = (0.242 ± 0.004) nm for the system nitroethane–isooctane [15] and with n0 = (0.261 ± 0.007) nm for nitrobenzene-n-octane [16]. The amplitude also fairly agrees with the value from the twoscale-factor universality relation [17,18] n30 ¼ kB X=ða0 C pc Þ:
ð14Þ
In this relation kB is Boltzmann’s constant, X = (19.66 ± 0.17) 103 according to renormalization group calculations [19] and X = (18.8 ± 0.15) 103 according to series calculations [18], a0 = 0.11 [17] is the critical exponent of heat capacity and, Cpc is the amplitude in the singular part of the specific heat at constant pressure, Cp, assumed to be given by C p ðeÞ ¼ C pc ea0 þ C pb
The fundamental Eq. (1), which is the center of interest here, is based on the assumption that the total ultrasonic attenuation-perwavelength, ak, of critical systems can be considered a linear superposition ak ðm; TÞ ¼ ack ðm; TÞ þ abk ðm; TÞ
C0 ¼ 2D0 =n20 :
ð8Þ
ð15Þ 1
1
with background part Cpb. With C pc = (128 ± 4) J kg K and q(Tc) = 888 kg m3 [11] n0 = 0.279 nm follows from Eq. (14). The amplitude of the diffusion coefficient was determined as D0 = (1.36 ± 0.002) 109 m2 s1 from dynamic light scattering [11]. With this value C0 = (43.8 ± 0.07) 109 s1 followed from Eq. (13). This amplitude is significantly smaller than that of other binary systems containing nitro compounds: C0 = 187 109 s1, n-pentanol-nitromethane [20]; C0 = 156 109 s1 nitroethanecyclohexane [12]; C0 = 125 109 s1, nitroetnane-3-methylpentane [21]. It is, however, on the same order as the amplitudes of the critical mixtures methanol-n-hexane (C0 = 44 109 s1 [22]) and methanol–cyclohexane (C0 = 26 109 s1 [23]). Using C0 fixed at 43.8 109 s1 and treating B in the background part of the ultrasonic spectra (Eq. (10)) as an adjustable parameter, the scaling function data shown in Fig. 1 result. Obviously the experimental data follow the general trends of the theoretically predicted function. However, systematic frequency
with ratio F ¼ cðT c ÞAðT c Þ=ðcðTÞAðTÞÞ:
ð9Þ
For simplicity the small temperature effects in the sound velocity c and amplitude factor A are neglected near Tc so that F 1 and thus the original Bhattacharjee–Ferrell relation (Eq. (1)) follows. A major problem in the determination of the scaling function F(X) from smallband ultrasonic spectra is the separation of the background contribution abk from the total attenuation-per-wavelength ak in order to obtain the critical contribution. Frequently it is assumed that the critical part dominates ak and that the background contribution can be represented by an asymptotic high frequency term abk ¼ Bm;
ð10Þ
ð11Þ
and
DðeÞ ¼ D0 em
ð12Þ
has been assumed for the fluctuation correlation length n and the mutual diffusion coefficient D. On these assumption the dynamic scaling hypothesis C ¼ 2D=n2 [13,14] yields
Fig. 1. Lin-log plot of scaling function data according to Eq. (1) versus reduced frequency X for the nitrobenzene–isooctane mixture of critical composition. The ack ðm; TÞ=ack ðm; T c Þ values have been calculated from a-data between 15.289 and 45.122 MHz and at 20 temperature intervals T – Tc between 0.27 and 21.81 K [10], using the amplitude C0 = 43.8 109 s1 from static and quasielastic light scattering [11]. The 9.476 MHz data have been neglected as they do not fit to the other ones [11]. Symbols indicate different temperature differences (T Tc): s, 0.27 °C; 4, 0.67 °C; j, 1.22 °C; ., 1.64 °C; h, 2.12 °C; d, 2.61 °C; J, 3.11 °C; 5, 3.83 °C; e, 4.83 °C; , 5.85 °C; , 6.84 °C; g, 7.84 °C; N, 8.87 °C; , 9.85 °C; ., 11.80 °C; , 13.76 °C; , 15.95 °C; , 17.92 °C; , 19.86 °C; l, 21.81 °C. The line is graph of the empirical Bhattacharjee–Ferrell function Eq. (5). f
nðeÞ ¼ n0 e
~m
g
with B independent of frequency. This term includes the classical attenuation due to viscous friction and thermal conductivity. If the background term can be determined from the ultrasonic spectra the only unknown parameter in Eq. (1) is the amplitude C0 in the relaxation rate. Belkoura et al. have determined this parameter from static and dynamic light scattering experiments [11]. Neglecting crossover effects, power law behaviour
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dependent deviations from the theoretical form emerge at each temperature of measurement. This is particularly true at temperatures close to the critical, corresponding with large reduced frequencies X. These systematic trends in the ack ðm; TÞ=ack ðm; T c Þ data can be neither explained by an incorrect relaxation rate amplitude nor by a mistakenly constant F ¼ 1. Rather they may indicate an ill-defined background part in the evaluation of the ultrasonic attenuation spectra. 3. Broadband spectra, noncritical relaxations
Fig. 3. Sketch of suggested nitrobenzene isomer equilibrium.
In Fig. 2 the ultrasonic attenuation spectrum of a nitrobenzene– isooctane mixture of critical compositions, measured at frequencies between 20 kHz and 3 GHz, is displayed in the frequency normalized format a=m2 versus m. This representation of attenuation coefficients is preferred here because the a=m2 data will approach a constant B0 ¼ B=c
ð16Þ
at high frequencies. The a values have been determined with an error of less than 10% using three methods of measurements [24]. The temperature of the sample was controlled to within 0.01 K. Within the frequency range of measurements the a=m2 data decrease with m and, even in the GHz region, do not approach a constant B0 . In principle, such broad range of negative slope in the frequency normalized attenuation data is predicted by the Bhattacharjee–Ferrell scaling function. We therefore tried to analyze the attenuation-per-wavelength spectrum, that corresponds with the a=m2 data of Fig. 2, also in terms of the function defined by Eq. (6), with its critical part according to Eq. (7) and with its noncritical background part assumed to simply follow Eq. (10). In this analysis of experimental data the relaxation rate was fixed at the value following from Eq. (2) with amplitude C0 = 43.8 109 s1 as determined by the static and dynamic light scattering measurements. This relaxation function, however, was found inadequate for a representation of the experimental data within the limits of their errors. Rather a spectral function with two additional relaxation terms ak ðmÞ ¼ cAFðXÞ þ
A1 xs1 A2 xs2 þ þ Bm 1 þ x2 s21 1 þ x2 s22
ð17Þ
was required for a suitable analytical description of the broadband spectrum. In Eq. (17) the Ai and si, i = 1, 2, are the relaxation amplitudes and relaxation times, respectively. For the critical composition mixture at T Tc = 0.2 K, A1 = 32 103 and s1 = 10 ns as well as A2 = 37 103 and s2 = 0.35 ns is found. Quite remarkable the relaxation frequency ð2ps1 Þ1 ¼ 16 MHz of the former term is located within the frequency range of all smallband attenuation coefficient measurements mentioned above. Disregarding this relaxation term in the analysis of the data thus likely affects the results for the critical contribution ack ðm; TÞ. Spectra of the nitrobenzene–isooctane critical composition mixture at T Tc = 5, 10 and 20 K also reveal the noncritical relaxation terms. Both, relaxation amplitudes and relaxation times, decrease with T Tc, thus with T. At T Tc = 20 K, for example, A1 = 15 103 and s1 = 6 ns, A2 = 24 103, and s2 = 0.28 ns. At temperatures more distant from Tc an additional relaxation term emerges in the spectra, with relaxation frequency at the lower end of the frequency range of measurements. Probably due to slowing close to Tc this term does not exist in the measuring range (Fig. 2). The term with relaxation time s2 between 0.28 and 0.35 ns is located at the upper end of the frequency range of measurements (ð2ps2 Þ1 ¼ 0:45 ; . . . ; 0:56 GHz). A high frequency relaxation term with relaxation frequency 5.3 GHz at 20 °C has been also reported for pure nitrobenzene and has been assigned to association [25]. Because of the dilution of nitrobenzene by isooctane the relaxation frequency for an association process should indeed be smaller in the critical mixture than in pure nitrobenzene. Nevertheless the high frequency relaxation term with s2 between 0.28 and 0.35 ns does not seem to be due to nitrobenzene association. Noncritical mixtures with mole fractions X = 0.2 and X = 0.8 of nitrobenzene reveal relaxation times s2 = 0.15 ns and s2 = 0.21 ns, respectively, at variance with the characteristics of an association mechanism. We suggest this relaxation term to reflect an isomerization of the nitrobenzene molecule. Likely, rotational isomers as sketched in Fig. 3, with different angels of the –NO2 dipole moment relative to the benzene ring, are conformers, which are well separated by an activation energy barrier due to the different interactions between the oxygen lone pair electrons and the delocalized ring electron system. Presently we do not know the molecular process leading to the relaxation term with relaxation time s1. The only idea of the mechanism behind this term is the conclusion from SANS, SAXS, and dynamic light scattering studies of n-alkanes in nitrobenzene, which indicated aggregation of the alkanes within the mixtures [26]. The possible existence of an isooctane aggregate formation process in mixtures with nitrobenzene, that is not included in the Bhattacharjee–Ferrell critical dynamics, needs further investigations. 4. Critical contribution, scaling function
Fig. 2. Frequency normalized ultrasonic attenuation spectrum of the nitrobenzene– isooctane mixture of critical composition at T Tc = 0.2 K [24]. The dotted line indicates the critical contribution with scaling function according to Eq. (5). The dashed lines are graphs of the relaxation terms with relaxation frequencies ð2ps1 Þ1 and ð2ps2 Þ1 , respectively. The full curve shows the total attenuation function (Eq. (17)).
Since, within the frequency range of the scaling function data in Fig. 1, the latter relaxation term interferes with the critical term in the spectra, we have re-calculated the critical part in ak using the relation
P.K. Khabibullaev et al. / Chemical Physics Letters 458 (2008) 76–80
ack ðm; TÞ ¼ ak ðm; TÞ
A1 xs1 ~ Bm: 1 þ x2 s21
ð18Þ
~ considers all contributions proportional to m in the Parameter B attenuation-per-wavelength spectra, corresponding with independency upon m in the a=m2 format. As illustrated by Fig. 2, the high frequency relaxation term is constant in the relevant frequency range, so that ~ ¼ B þ 2ps2 A2 B
ð19Þ
~ at the values obcan be used. We fixed the parameters A1 ; s1 and B tained by inter- and extrapolation of the data from the broadband spectra at T Tc = 0.2, 5, 10, and 20 K (Section 3) when calculating ack ðm; TÞ from the smallband measurements at 20 temperatures (0.27 6 T Tc P 21.81 K [10]) discussed before (Fig. 1). The scaling function data resulting without any adjustable parameter is displayed in Fig. 4 to show that the systematic deviations of the experimental data from the empirical form of the Bhattacharjee–Ferrell theory are missing now. Small variations in the values of the background term parameters would further enhance the agreement between experiment and theory. Such variations are likely justified by the fact that results from different laboratories are combined, in particular by the use of different samples for the determination of the noncritical contributions and the measurements of the scaling function. Consideration of the noncritical relaxation term has consequently resulted in a critical term amplitude factor A smaller than in the previous evaluation of data. The amplitude factor is related to the amplitude !d BF p2 dC pc X1=2 C0 cg 2 S¼ ð20Þ 2T c 2p C 2pb
79
follows. This value is noticeably smaller than |g| = 0.5 obtained without taking the relaxation term in the frequency range around 16 MHz into account. It differs the more from g = (0.56 ± 0.01) that had been derived from the thermodynamic data [11] using Eq. (22). We have re-evaluated the ultrasonic attenuation spectrum at T Tc = 0.2 K (Fig. 2) assuming an amplitude S = 5.5 105 s0.94 m1 of the critical term, corresponding with |g| = 0.5. On such requirement the experimental spectrum can indeed be well represented by the relaxation spectral function defined by Eq. (18). The resulting relaxation rate of concentration fluctuations (C = 9 105 s1), however, deviates substantially from the value (C = 3.2 104 s1) predicted from static and dynamic light scattering measurements. Deviations of the coupling constants determined from the amplitude S of the critical term in the sonic attenuation spectra and from the thermodynamic relation Eq. (22) have been occasionally found in the past. For the critical binary mixture methanol– cyclohexane, for instance, |g| = 0.14 followed from the amplitude S and g = 0.21 from the pressure variation of the critical temperature along the critical line and the thermal expansion coefficient [22]. These deviations have been assigned to a large error in the g-values obtained from the delicate difference between two terms in the thermodynamic relation defined by Eq. (22). For the nitrobenzene–isooctane mixture, however, the dTc/dP is negative [11] so that the contributions from both terms in Eq. (22) sum up without enhancing the error in the resulting g-value noticeably. A substantial temperature dependence in the coupling constant near Tc emerged with the system nitroethane–cyclohexane [12]. With the present nitrobenzene–isooctane mixture of critical composition, however, g was found independent of T within the relevant temperature range Tc + 0.05 K 6 T 6 Tc + 0.2 K [11]. We regret having presently no explanation for this discrepancy between the coupling constants from different parameters.
of the Bhattacharjee–Ferrell theory according to S ¼ Amd :
ð21Þ
Here d ¼ a0 =ðZ 0 mÞ = 0.06 and g is the adiabatic coupling constant that can be calculated as dT c T c ap ð22Þ g ¼ qðT c ÞC p ðT c Þ dP qðT c ÞC p ðT c Þ from the slope dTc/dP in the pressure dependence of the critical temperature and the thermal expansion coefficient, ap , at constant pressure. Our analysis of the ultrasonic spectra yielded S = 2 105 s0.94 m1. Using c = 1186 ms1 and Cpb = 1.635 Jg1 K1 from Ref. [11] and the other parameters as given before |g| = 0.27
5. Conclusions Broadband ultrasonic attenuation spectra of the nitrobenzene– isooctane mixture of critical composition reveal noncritical relaxation terms in addition to the critical term. The proper assignment of these relaxations to molecular mechanisms requires further investigations, including a broad range of mixtures and also more specific experimental methods. Consideration of the relaxation terms in the evaluation of smallband literature data at a large number of temperatures near Tc yields a superior agreement of the critical part in the ultrasonic attenuation with the empirical scaling function of the Bhattacharjee–Ferrell theory. From the amplitude of the critical part follows a reasonable amount |g| = 0.27 of the adiabatic coupling constant. It is, however, unclear presently why this value differs from g = 0.56 as derived from thermodynamic parameters. Acknowledgements S.Z. Mirzaev gratefully acknowledges a fellowship granted by the Alexander von Humboldt Foundation, Bonn, Germany. References [1] [2] [3] [4] [5] [6]
Fig. 4. Scaling function data as in Fig. 1 but with critical contribution calculated according to Eq. (18). The full curve represents again Eq. (5) with C0 = 43.8 109 s1 (Fig. 1).
[7] [8] [9]
H.E. Stanley, Rev. Mod. Phys. 71 (1999) 388. R.A. Ferrell, J.K. Bhattacharjee, Phys. Rev. A 31 (1985) 1788. A. Zielesny, D. Woermann, J. Chem. Soc. Faraday Trans. 90 (1994) 2215. A.V. Anantaraman, A.B. Walters, P.D. Edmonds, C.J. Pings, J. Chem. Phys. 44 (1966) 2651. M. Fixman, J. Chem. Phys. 36 (1962) 1961. C.W. Garland, in: W.P. Mason, R.N. Thurston (Eds.), Physical Acoustics, vol. 7, Academic Press, New York, 1970, p. 51. D.B. Fenner, Phys. Rev. A 23 (1981) 1931. K. Kawasaki, Phys. Rev. A 1 (1970) 1750. J.K. Bhattacharjee, R.A. Ferrell, Phys. Rev. A 24 (1981) 1643.
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[10] G. Jaschull, H. Dunker, D. Woermann, Ber. Bunsenges. Phys. Chem. 88 (1984) 630. [11] L. Belkoura, F.P. Harnisch, S. Kölchens, T. Müller-Kirschbaum, D. Woermann, Ber. Bunsenges. Phys. Chem. 91 (1987) 1036. [12] R. Behrends, I. Iwanowski, M. Kosmowska, A. Szala, U. Kaatze, J. Chem. Phys. 121 (2004) 5929. [13] L.P. Kadanoff, J. Swift, Phys. Rev. 166 (1968) 89. [14] P.C. Halperin, B.I. Hohenberg, Rev. Mod. Phys. 49 (1977) 435. [15] Y. Garrabos, G. Zalczer, D. Beysens, Phys. Rev. A 25 (1982) 1147. [16] M. Souto-Caride, J. Troncoso, J. Peleteiro, E. Carballo, L. Komani, Chem. Phys. 324 (2006) 483. [17] A.J. Lui, M.E. Fisher, Physica A 156 (1989) 35. [18] P.F. Rebillot, D.T. Jacobs, J. Chem. Phys. 109 (1998) 4009.
[19] C. Bervillier, C. Gordrèche, Phys. Rev. B 21 (1980) 5427. [20] I. Iwanowski, R. Behrends, U. Kaatze, J. Chem. Phys. 120 (2004) 9192. [21] I. Iwanowski, K. Leluk, M. Rudowski, U. Kaatze, J. Phys. Chem. A 110 (2006) 4313. [22] I. Iwanowski, A. Sattarow, R. Behrends, S.Z. Mirzaev, U. Kaatze, J. Chem. Phys. 124 (2006) 144505. [23] R. Behrends, U. Kaatze, M. Schach, J. Chem. Phys. 119 (2003) 7957. [24] S.S. Aliev, L.A. Davidovich, D.F. Pulatova, P.K. Khabibullaev, Sov. Phys. Acoust. 22 (1976) 64. [25] S.S. Aliev, L.E. Kvasova, L.V. Lanshina, K. Parpiev, P.K. Khabibullaev, Sov. Phys. Acoust. 16 (1970) 250. [26] A.P. Radlinski, L. Barrè, D. Espinat, J. Mol. Struct. 383 (1996) 51.
Contents lists available at ScienceDirect
Chemical Physics Letters journal homepage: www.elsevier.com/locate/cplett
Critical fluctuations and noncritical relaxations of the nitrobenzene–isooctane system near its consolute point P.K. Khabibullaev a, S.Z. Mirzaev a,b, U. Kaatze b,* a b
Heat Physics Department, Uzbek Academy of Sciences, Katartal 28, Tashkent 7000135, Uzbekistan Drittes Physikalisches Institut, Georg-August-Universität, Friedrich-Hund-Platz 1, 37077 Göttingen, Germany
a r t i c l e
i n f o
Article history: Received 12 February 2008 In final form 15 April 2008 Available online 18 April 2008
a b s t r a c t Ultrasonic attenuation spectra between 20 kHz and 3 GHz of the nitrobenzene–isooctane mixture of critical composition have been analyzed to show that they contain noncritical relaxation terms in addition to the critical term. The parameter values of the noncritical contributions obtained thereby are used in a reevaluation of smallband attenuation data from the literature. These data, measured at a large number of temperatures near the critical, are most suitable for the determination of the scaling function in the critical dynamics. The procedure allows to verify the empirical scaling function of the Bhattacharjee–Ferrell dynamic scaling theory without an adjustable parameter. Ó 2008 Elsevier B.V. All rights reserved.
1. Introduction Critical concentration fluctuations of binary fluids near their demixing point have been intensively investigated using ultrasonic spectrometry. The method has the potential to verify or disprove theories modeling the long wavelength fluctuations in the local concentrations and thus the conceptions of universality, scaling and renormalization [1] on which theories rely. Measurements of the sonic attenuation coefficient a of the fluid as a function of temperature T and frequency m yields the scaling function F(X) of the system as a ratio of the critical contributions ack to the attenuation-per-wavelength ak ¼ ak [2]: FðXÞ ¼ ack ðm; TÞ=ack ðm; T c Þ:
ð1Þ
Here X ¼ 2pm=CðeÞ is a scaled frequency with the relaxation rate C of order parameter fluctuations following power law CðeÞ ¼ C0 eZ0 ~m ;
ð2Þ
where Z0 and ~m are universal critical exponents, C0 is an amplitude characteristic of the system under consideration, and e ¼ jT T c j=T c
ð3Þ
is the reduced temperature. Tc denotes the critical temperature and k is the wavelength of the sonic field within the liquid. Hence the scaling conception reveals the critical dynamics of the binary fluid in terms of a universal function if the individual parameters critical temperature Tc and amplitude C0 of the relaxation rate are known. A prominent binary liquid, the ultrasonic attenuation coefficient data of which have been discussed several times in the past, is the
* Corresponding author. Fax: +49 551 39 7720. E-mail address: [email protected] (U. Kaatze). 0009-2614/$ - see front matter Ó 2008 Elsevier B.V. All rights reserved. doi:10.1016/j.cplett.2008.04.066
nitrobenzene–isooctane mixture of critical composition with a convenient upper critical temperature Tc = 302.29 K [3]. Annantaraman et al. reported ultrasonic attenuation coefficient data for the nitrobenzene–isooctane system at five frequencies between 4.5 and 16.5 MHz [4]. These authors discussed their results in terms of Fixman’s mode-coupling theory [5]. They found reasonable but not exact agreement between the theory and the experimental data. Later Garland reconsidered the attenuation coefficient data by Anantaraman et al. [6]. He also noticed considerable deviations from the predictions of Fixman’s theoretical model. Fenner performed ultrasonic attenuation measurements for the critical composition mixture of nitrobenzene and isooctane in the relatively broad frequency range from 1 to 91 MHz and in the wide temperature interval 0.007 6 T Tc 6 30 K [7]. He concluded that the experimental data are well fitted by the Kawasaki version [8] of the mode–mode-coupling theory. Bhattacharjee and Ferrell used Fenner’s experimental data to test their dynamic scaling theory [2,9]. The Woermann group has measured the ultrasonic attenuation coefficient of the nitrobenzene–isooctane critical mixture as a function of frequency m between 9 and 45 MHz and of the temperature interval T Tc between 0.3 and 21.8 K [10]. Later on the group re-evaluated the data in the light of the Bhattacharjee–Ferrell dynamic scaling theory using an integral version Z 3X 1 x2 dx FðXÞ ¼ ð4Þ p 0 ð1 þ x2 Þ3=2 x2 þ x3 þ X2 and alternatively the empirical form [2] of the scaling function [11] 1=2 2 FðXÞ ¼ ½1 þ 0:414ðXBF ; 1=2 =XÞ
ð5Þ
with scaled half-attenuation frequency XBF 1=2 ¼ 2:1: Treating the amplitude C0 of the relaxation rate as an adjustable parameter
77
P.K. Khabibullaev et al. / Chemical Physics Letters 458 (2008) 76–80
when fitting the experimental data to the theoretical functions the empirical form Eq. (5) was found superior in that the resulting C0 value (=37.0 109 s1) agreed significantly better with the one (C0 = 43.8 109 s1) from static and dynamic light scattering [11]. The agreement between measurements and theory, however, is incomplete as the experimental data reveal systematic deviations from Eq. (5), likely reflecting inadequate assumptions on the noncritical background contributions to the total ultrasonic attenuation coefficient. Obviously, the frequency band of measurements in the above nitrobenzene–isooctane studies is too small to appropriately account for the background attenuation in addition to the critical part in the spectra, the latter extending over vast ranges of frequency. For this reason we discuss the ultrasonic attenuation data of the mixture of critical composition again, including the experimental determination of the noncritical contribution from broadband spectra. 2. Conception, original results
ð6Þ
of the critical contribution ack and a noncritical background contribution abk . According to the Bhattacharjee–Ferrell dynamic scaling theory [2,9], the critical part can be expressed as ack ðm; TÞ ¼ cðTÞAðTÞFðXÞ
ð7Þ
with sound velocity c (=km) and amplitude factor A depending on temperature T but depending only weakly on frequency m. If the weak temperature dependencies of c and A are taken into account the scaling function follows from Eq. (7) as [12] FðXÞ ¼ F ack ðm; TÞ=ack ðm; T c Þ
ð13Þ
In Eqs. (11) and (12) ~m and m are universal critical exponents. The theoretical value of the former is ~m ¼ 0:630 [14]. For the latter m ¼ 0:671 has been used [11]. Static light scattering yielded n0 = (0.249 ± 0.060) nm [11], in nice agreement with n0 = (0.242 ± 0.004) nm for the system nitroethane–isooctane [15] and with n0 = (0.261 ± 0.007) nm for nitrobenzene-n-octane [16]. The amplitude also fairly agrees with the value from the twoscale-factor universality relation [17,18] n30 ¼ kB X=ða0 C pc Þ:
ð14Þ
In this relation kB is Boltzmann’s constant, X = (19.66 ± 0.17) 103 according to renormalization group calculations [19] and X = (18.8 ± 0.15) 103 according to series calculations [18], a0 = 0.11 [17] is the critical exponent of heat capacity and, Cpc is the amplitude in the singular part of the specific heat at constant pressure, Cp, assumed to be given by C p ðeÞ ¼ C pc ea0 þ C pb
The fundamental Eq. (1), which is the center of interest here, is based on the assumption that the total ultrasonic attenuation-perwavelength, ak, of critical systems can be considered a linear superposition ak ðm; TÞ ¼ ack ðm; TÞ þ abk ðm; TÞ
C0 ¼ 2D0 =n20 :
ð8Þ
ð15Þ 1
1
with background part Cpb. With C pc = (128 ± 4) J kg K and q(Tc) = 888 kg m3 [11] n0 = 0.279 nm follows from Eq. (14). The amplitude of the diffusion coefficient was determined as D0 = (1.36 ± 0.002) 109 m2 s1 from dynamic light scattering [11]. With this value C0 = (43.8 ± 0.07) 109 s1 followed from Eq. (13). This amplitude is significantly smaller than that of other binary systems containing nitro compounds: C0 = 187 109 s1, n-pentanol-nitromethane [20]; C0 = 156 109 s1 nitroethanecyclohexane [12]; C0 = 125 109 s1, nitroetnane-3-methylpentane [21]. It is, however, on the same order as the amplitudes of the critical mixtures methanol-n-hexane (C0 = 44 109 s1 [22]) and methanol–cyclohexane (C0 = 26 109 s1 [23]). Using C0 fixed at 43.8 109 s1 and treating B in the background part of the ultrasonic spectra (Eq. (10)) as an adjustable parameter, the scaling function data shown in Fig. 1 result. Obviously the experimental data follow the general trends of the theoretically predicted function. However, systematic frequency
with ratio F ¼ cðT c ÞAðT c Þ=ðcðTÞAðTÞÞ:
ð9Þ
For simplicity the small temperature effects in the sound velocity c and amplitude factor A are neglected near Tc so that F 1 and thus the original Bhattacharjee–Ferrell relation (Eq. (1)) follows. A major problem in the determination of the scaling function F(X) from smallband ultrasonic spectra is the separation of the background contribution abk from the total attenuation-per-wavelength ak in order to obtain the critical contribution. Frequently it is assumed that the critical part dominates ak and that the background contribution can be represented by an asymptotic high frequency term abk ¼ Bm;
ð10Þ
ð11Þ
and
DðeÞ ¼ D0 em
ð12Þ
has been assumed for the fluctuation correlation length n and the mutual diffusion coefficient D. On these assumption the dynamic scaling hypothesis C ¼ 2D=n2 [13,14] yields
Fig. 1. Lin-log plot of scaling function data according to Eq. (1) versus reduced frequency X for the nitrobenzene–isooctane mixture of critical composition. The ack ðm; TÞ=ack ðm; T c Þ values have been calculated from a-data between 15.289 and 45.122 MHz and at 20 temperature intervals T – Tc between 0.27 and 21.81 K [10], using the amplitude C0 = 43.8 109 s1 from static and quasielastic light scattering [11]. The 9.476 MHz data have been neglected as they do not fit to the other ones [11]. Symbols indicate different temperature differences (T Tc): s, 0.27 °C; 4, 0.67 °C; j, 1.22 °C; ., 1.64 °C; h, 2.12 °C; d, 2.61 °C; J, 3.11 °C; 5, 3.83 °C; e, 4.83 °C; , 5.85 °C; , 6.84 °C; g, 7.84 °C; N, 8.87 °C; , 9.85 °C; ., 11.80 °C; , 13.76 °C; , 15.95 °C; , 17.92 °C; , 19.86 °C; l, 21.81 °C. The line is graph of the empirical Bhattacharjee–Ferrell function Eq. (5). f
nðeÞ ¼ n0 e
~m
g
with B independent of frequency. This term includes the classical attenuation due to viscous friction and thermal conductivity. If the background term can be determined from the ultrasonic spectra the only unknown parameter in Eq. (1) is the amplitude C0 in the relaxation rate. Belkoura et al. have determined this parameter from static and dynamic light scattering experiments [11]. Neglecting crossover effects, power law behaviour
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P.K. Khabibullaev et al. / Chemical Physics Letters 458 (2008) 76–80
dependent deviations from the theoretical form emerge at each temperature of measurement. This is particularly true at temperatures close to the critical, corresponding with large reduced frequencies X. These systematic trends in the ack ðm; TÞ=ack ðm; T c Þ data can be neither explained by an incorrect relaxation rate amplitude nor by a mistakenly constant F ¼ 1. Rather they may indicate an ill-defined background part in the evaluation of the ultrasonic attenuation spectra. 3. Broadband spectra, noncritical relaxations
Fig. 3. Sketch of suggested nitrobenzene isomer equilibrium.
In Fig. 2 the ultrasonic attenuation spectrum of a nitrobenzene– isooctane mixture of critical compositions, measured at frequencies between 20 kHz and 3 GHz, is displayed in the frequency normalized format a=m2 versus m. This representation of attenuation coefficients is preferred here because the a=m2 data will approach a constant B0 ¼ B=c
ð16Þ
at high frequencies. The a values have been determined with an error of less than 10% using three methods of measurements [24]. The temperature of the sample was controlled to within 0.01 K. Within the frequency range of measurements the a=m2 data decrease with m and, even in the GHz region, do not approach a constant B0 . In principle, such broad range of negative slope in the frequency normalized attenuation data is predicted by the Bhattacharjee–Ferrell scaling function. We therefore tried to analyze the attenuation-per-wavelength spectrum, that corresponds with the a=m2 data of Fig. 2, also in terms of the function defined by Eq. (6), with its critical part according to Eq. (7) and with its noncritical background part assumed to simply follow Eq. (10). In this analysis of experimental data the relaxation rate was fixed at the value following from Eq. (2) with amplitude C0 = 43.8 109 s1 as determined by the static and dynamic light scattering measurements. This relaxation function, however, was found inadequate for a representation of the experimental data within the limits of their errors. Rather a spectral function with two additional relaxation terms ak ðmÞ ¼ cAFðXÞ þ
A1 xs1 A2 xs2 þ þ Bm 1 þ x2 s21 1 þ x2 s22
ð17Þ
was required for a suitable analytical description of the broadband spectrum. In Eq. (17) the Ai and si, i = 1, 2, are the relaxation amplitudes and relaxation times, respectively. For the critical composition mixture at T Tc = 0.2 K, A1 = 32 103 and s1 = 10 ns as well as A2 = 37 103 and s2 = 0.35 ns is found. Quite remarkable the relaxation frequency ð2ps1 Þ1 ¼ 16 MHz of the former term is located within the frequency range of all smallband attenuation coefficient measurements mentioned above. Disregarding this relaxation term in the analysis of the data thus likely affects the results for the critical contribution ack ðm; TÞ. Spectra of the nitrobenzene–isooctane critical composition mixture at T Tc = 5, 10 and 20 K also reveal the noncritical relaxation terms. Both, relaxation amplitudes and relaxation times, decrease with T Tc, thus with T. At T Tc = 20 K, for example, A1 = 15 103 and s1 = 6 ns, A2 = 24 103, and s2 = 0.28 ns. At temperatures more distant from Tc an additional relaxation term emerges in the spectra, with relaxation frequency at the lower end of the frequency range of measurements. Probably due to slowing close to Tc this term does not exist in the measuring range (Fig. 2). The term with relaxation time s2 between 0.28 and 0.35 ns is located at the upper end of the frequency range of measurements (ð2ps2 Þ1 ¼ 0:45 ; . . . ; 0:56 GHz). A high frequency relaxation term with relaxation frequency 5.3 GHz at 20 °C has been also reported for pure nitrobenzene and has been assigned to association [25]. Because of the dilution of nitrobenzene by isooctane the relaxation frequency for an association process should indeed be smaller in the critical mixture than in pure nitrobenzene. Nevertheless the high frequency relaxation term with s2 between 0.28 and 0.35 ns does not seem to be due to nitrobenzene association. Noncritical mixtures with mole fractions X = 0.2 and X = 0.8 of nitrobenzene reveal relaxation times s2 = 0.15 ns and s2 = 0.21 ns, respectively, at variance with the characteristics of an association mechanism. We suggest this relaxation term to reflect an isomerization of the nitrobenzene molecule. Likely, rotational isomers as sketched in Fig. 3, with different angels of the –NO2 dipole moment relative to the benzene ring, are conformers, which are well separated by an activation energy barrier due to the different interactions between the oxygen lone pair electrons and the delocalized ring electron system. Presently we do not know the molecular process leading to the relaxation term with relaxation time s1. The only idea of the mechanism behind this term is the conclusion from SANS, SAXS, and dynamic light scattering studies of n-alkanes in nitrobenzene, which indicated aggregation of the alkanes within the mixtures [26]. The possible existence of an isooctane aggregate formation process in mixtures with nitrobenzene, that is not included in the Bhattacharjee–Ferrell critical dynamics, needs further investigations. 4. Critical contribution, scaling function
Fig. 2. Frequency normalized ultrasonic attenuation spectrum of the nitrobenzene– isooctane mixture of critical composition at T Tc = 0.2 K [24]. The dotted line indicates the critical contribution with scaling function according to Eq. (5). The dashed lines are graphs of the relaxation terms with relaxation frequencies ð2ps1 Þ1 and ð2ps2 Þ1 , respectively. The full curve shows the total attenuation function (Eq. (17)).
Since, within the frequency range of the scaling function data in Fig. 1, the latter relaxation term interferes with the critical term in the spectra, we have re-calculated the critical part in ak using the relation
P.K. Khabibullaev et al. / Chemical Physics Letters 458 (2008) 76–80
ack ðm; TÞ ¼ ak ðm; TÞ
A1 xs1 ~ Bm: 1 þ x2 s21
ð18Þ
~ considers all contributions proportional to m in the Parameter B attenuation-per-wavelength spectra, corresponding with independency upon m in the a=m2 format. As illustrated by Fig. 2, the high frequency relaxation term is constant in the relevant frequency range, so that ~ ¼ B þ 2ps2 A2 B
ð19Þ
~ at the values obcan be used. We fixed the parameters A1 ; s1 and B tained by inter- and extrapolation of the data from the broadband spectra at T Tc = 0.2, 5, 10, and 20 K (Section 3) when calculating ack ðm; TÞ from the smallband measurements at 20 temperatures (0.27 6 T Tc P 21.81 K [10]) discussed before (Fig. 1). The scaling function data resulting without any adjustable parameter is displayed in Fig. 4 to show that the systematic deviations of the experimental data from the empirical form of the Bhattacharjee–Ferrell theory are missing now. Small variations in the values of the background term parameters would further enhance the agreement between experiment and theory. Such variations are likely justified by the fact that results from different laboratories are combined, in particular by the use of different samples for the determination of the noncritical contributions and the measurements of the scaling function. Consideration of the noncritical relaxation term has consequently resulted in a critical term amplitude factor A smaller than in the previous evaluation of data. The amplitude factor is related to the amplitude !d BF p2 dC pc X1=2 C0 cg 2 S¼ ð20Þ 2T c 2p C 2pb
79
follows. This value is noticeably smaller than |g| = 0.5 obtained without taking the relaxation term in the frequency range around 16 MHz into account. It differs the more from g = (0.56 ± 0.01) that had been derived from the thermodynamic data [11] using Eq. (22). We have re-evaluated the ultrasonic attenuation spectrum at T Tc = 0.2 K (Fig. 2) assuming an amplitude S = 5.5 105 s0.94 m1 of the critical term, corresponding with |g| = 0.5. On such requirement the experimental spectrum can indeed be well represented by the relaxation spectral function defined by Eq. (18). The resulting relaxation rate of concentration fluctuations (C = 9 105 s1), however, deviates substantially from the value (C = 3.2 104 s1) predicted from static and dynamic light scattering measurements. Deviations of the coupling constants determined from the amplitude S of the critical term in the sonic attenuation spectra and from the thermodynamic relation Eq. (22) have been occasionally found in the past. For the critical binary mixture methanol– cyclohexane, for instance, |g| = 0.14 followed from the amplitude S and g = 0.21 from the pressure variation of the critical temperature along the critical line and the thermal expansion coefficient [22]. These deviations have been assigned to a large error in the g-values obtained from the delicate difference between two terms in the thermodynamic relation defined by Eq. (22). For the nitrobenzene–isooctane mixture, however, the dTc/dP is negative [11] so that the contributions from both terms in Eq. (22) sum up without enhancing the error in the resulting g-value noticeably. A substantial temperature dependence in the coupling constant near Tc emerged with the system nitroethane–cyclohexane [12]. With the present nitrobenzene–isooctane mixture of critical composition, however, g was found independent of T within the relevant temperature range Tc + 0.05 K 6 T 6 Tc + 0.2 K [11]. We regret having presently no explanation for this discrepancy between the coupling constants from different parameters.
of the Bhattacharjee–Ferrell theory according to S ¼ Amd :
ð21Þ
Here d ¼ a0 =ðZ 0 mÞ = 0.06 and g is the adiabatic coupling constant that can be calculated as dT c T c ap ð22Þ g ¼ qðT c ÞC p ðT c Þ dP qðT c ÞC p ðT c Þ from the slope dTc/dP in the pressure dependence of the critical temperature and the thermal expansion coefficient, ap , at constant pressure. Our analysis of the ultrasonic spectra yielded S = 2 105 s0.94 m1. Using c = 1186 ms1 and Cpb = 1.635 Jg1 K1 from Ref. [11] and the other parameters as given before |g| = 0.27
5. Conclusions Broadband ultrasonic attenuation spectra of the nitrobenzene– isooctane mixture of critical composition reveal noncritical relaxation terms in addition to the critical term. The proper assignment of these relaxations to molecular mechanisms requires further investigations, including a broad range of mixtures and also more specific experimental methods. Consideration of the relaxation terms in the evaluation of smallband literature data at a large number of temperatures near Tc yields a superior agreement of the critical part in the ultrasonic attenuation with the empirical scaling function of the Bhattacharjee–Ferrell theory. From the amplitude of the critical part follows a reasonable amount |g| = 0.27 of the adiabatic coupling constant. It is, however, unclear presently why this value differs from g = 0.56 as derived from thermodynamic parameters. Acknowledgements S.Z. Mirzaev gratefully acknowledges a fellowship granted by the Alexander von Humboldt Foundation, Bonn, Germany. References [1] [2] [3] [4] [5] [6]
Fig. 4. Scaling function data as in Fig. 1 but with critical contribution calculated according to Eq. (18). The full curve represents again Eq. (5) with C0 = 43.8 109 s1 (Fig. 1).
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