J. Phys. Chem. 1994, 98, 9170-9174
9170
Critical Behavior of Ionic Fluids? T. Narayanan and Kenneth S. Pitzer' Department of Chemistry and Lawrence Berkeley Laboratory, University of California, Berkeley, California 94720 Received: February 23, 1994; In Final Form: May 3, 1994"
W e report the critical behavior of osmotic compressibility (XT) that was deduced from turbidity, for two ionic fluid mixtures. The measurements covered the reduced temperature, 2, range 7 X It I7 X lo-*. The mechanism underlying the liquid-liquid phase separation is, predominantly, Coulombic in one case and solvophobic for the other. W e find that the criticality of Coulombic phase separation is better described in terms of a crossover from mean field to Ising critical exponents, rather than true mean field exponents, as the critical temperature ( Tc) is approached. In addition, we find that the critical phenomenon associated with the solvophobic phase separation in an ionic fluid is characterized by nonclassical critical exponents.
Introduction Recently, it has been shown that the critical behavior of ionic fluid mixtures, where the liquid-liquid phase separation is driven, primarily, by Coulombic interactions, is described by classical critical exponents (CEs)-even very close to the critical temperature ( Tc).l,2 This finding is in sharp contrast to that observed in neutral fluids and fluid mixtures whose CEs belong to Ising universality class.394 In neutral fluids, the intermolecular forces are short ranged, whereas in ionic fluids, the interionic interactions are long ranged.2 The restricted primitive model (RPM) that represents the ionic fluid as a distribution of charged hard spheres in a dielectric continuum, predicts a phase separation driven by Coulombic forces.' Approximate treatments of this model forecast this phase separation to occur at a dimensionless concentration C*=ca3 and temperature T* = 4 ~ ~ e a k e T / Z Zwherec e ~ , is the number density of ions, a is the hard-sphere diameter, e is the dielectric constant of the solvent, Z is the valence of the ion and other symbols have their usual meaning. The most recent Monte-Carlo-based estimate5 provides C*, N 0.030 and P,N 0.057. Hence, for 1:l electrolytes in solvents of low e (-4), a phase separation driven by Coulombic interactions is possible to observe near room temperature.lq2.6 In reality, the above phase separation is most often preempted by either the crystallization of the solute or the freezing of the solvent. But it has been possible to identify several electrolytesolvent systems that exhibit a phase separation corresponding to that predicted by RPM.1-6 These systems are constituted, mainly, by complex organic 1:1 electrolytes of low melting point in solvents of low e. The exact location of the critical point in this case is system dependent due to the presence of finite short-ranged interactions. On the other hand, the liquid-liquid coexistence often found in aqueous (and other solvents of high e) solutions of these complex electrolytes is predominantly due to increasing dislike of the ionic solute for the solvent (solvophobic phase separation),1,2+8 and it is analogous to the phase separation in neutral fluid mixtures or liquid-vapor coexistence in fluids. Ising critical behavior is expected to prevail in the second c a ~ e . ~ J While the theoretical efforts concerning Coulombic phase separation9J0 are still evolving, the experimental evidence is less complete as compared to the case of neutral fluids and fluid mixtures.3.4 Singh and Pitzer11 reported a parabolic (CE j3 = 0.5) coexistence curve for the ionic system, triethyl-n-hexylammonium triethyl n-hexyl boride (N2226B2226) in diphenyl ether, f
Dedicated to C. N. R. Rao for his 60th birthday. published in Advance ACS Abstracts, August 15, 1994.
e Abstract
0022-3654/94/2098-9170$04.50/0
over the reduced temperature, t = I(T- Tc)/Tcl,range 1 t 2 It I10-I. They12 further demonstrated that the refractive index difference between the coexisting phases, which behaves like the order parameter in fluids and fluid mixtures, is satisfactorily described by an exponent, j3 = 0.476 (close to mean field value of j3 = 0.5),in the t range l e 5It I1 k 2 . However, their results12 did not exclude the possibility of a crossover to Ising j3 (=0.325) for t I10-4. Weingartner et al.13 measured the correlation length ( F ) of concentration fluctuations, in the ionic mixture of tetra-nbutylammonium picrate and 1-tridecanol, using dynamic light scattering. An asymptotic power law fit to the 5 (=[of-", where EO and Y are critical amplitude and exponent, respectively) data yielded v = 0.523, which is close to its mean field value (=0.5). With the inclusion of first correction-to-scaling term, the 5 data were equally consistent with the mean field and Ising values of Y (0.5 and 0.63, respectively). They also examined the behavior of osmotic compressibility, XT ( = X O ~ where , xo and y are critical amplitude and exponent respectively) and reached an identical conclusion as that in the case of 5. Their analysis strongly dispelled the possibility of spherical model C E s (v = 1 and y = 2). Note that mean-field and spherical model values of C E j3 are the same and hence could not be discerned in the work of Singh and Pitzer.12 Subsequently, Zhang et al.14 deduced the C E y from turbidity measurements in the same system (N222682226 in diphenyl ether) that was investigated by Singh and Pitzer.lIJ2 They obtained y = 1.01 f 0.01 over the t range 10-4 Ir I10-1 and discarded the possibility of both king and spherical model y. However, they did not examine the role of extended scaling terms. Thus, it is unclear from the earlier investigations7~*11-14 whether the critical phenomenon associated with Coulombic phase separation belongs to a mean-field type universality class or a crossover to Ising behavior occurs very close to T,. In fact, such a crossover is observed in Na-NH3 solution.15 In this article, we attempt to answer the above question; we have investigated the critical behavior of the following ionic systems: (i) tetra-nbutylammonium picrate (TBAP) + 1-dodecanol (DD), (ii) TBAP 1,Cbutanediol (BD), hereafter referred to as TPDD and TPBD, respectively. DD is a low dielectric solvent (e 4.6 a t Tc)and hence the phase separation is predominantly due to Coulombic interactions. This system is very similar to that investigated by Weingartner et al.13 except for the marginally higher e. Whereas the e of BD is sufficiently high (-26 at Tc) that the phase separation in this case is caused, primarily, by the solvophobic effects. The choice of these systems was guided by the fact that they have very similar values of Tc and critical concentration, thus allowing a direct comparison of the results. In addition, DD
+
0 1994 American Chemical Society
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Critical Behavior of Ionic Fluids and BD are completely miscible in all proportions and hence are suitable to make a mixed solvent where e can be varied continuously.I6 The turbidity of the critical mixtures was measured as Tc is approached from the one-phase region. C E y of susceptibility was deduced from turbidity.
Experimental Section Tetra-n-butylammonium picrate (TBAP) was synthesized by the slow neutralization of tetra-n-butylammonium hydroxide with picric acid. The picrate salt was purified by repeated recrystallization from methanol solution. The resulting salt was dried over vacuum for several days. The melting point of the TBAP was 91 f 1 O C , and it was reproducible from batch to batch. Special care was taken to eliminate fibers and other foreign particles from the salt. Solvents DD (98%, Aldrich) and BD (99%, Aldrich) were filtered prior to using in the sample preparation. The critical concentration of the mixtures were determined using the equal-volume coexistence criterion. Their values interms of mole fraction of TBAP are 0,147 f 0.002 and 0.145 f 0.002 (corresponding to weight fractions 0.304 and 0.469) for TPDD and TPBD, respectively. The TC's varied from batch to batch of TBAP (58-63 OC for TPDD and 59-61 O C for TPBD) but their temporal drift was less than 1 mK/day for both systems. Samples were contained in cylindrical quartz turbidity cells with polished flat windows. The beam path length ( I ) of the cells used are 1 and 5 cm and had gas-tight Teflon stoppers made from hi-vac Teflon stopcocks. It was nearly impossible (in our arrangement) to mix and homogenize the sample (inside the bath) without completely melting the lower salt-rich phase. Hence, the cell along with the mount was heated uniformly well above 92 OC and shaken vigorously to ensure thorough mixing. This homogenized sample was quickly placed into a well-stirred water bath which is maintained at a temperature much above T,. The temperature of the bath was controlled to better than f 2 mK. The bath temperature was measured using a calibrated thermistor. The bath water was filtered periodically. The sample cell mount is supported from the bath housing by means of micropositioners and hence the cell inside the bath can be moved in any desired direction or rotated about a vertical axis. The optical arrangement consisted of a He-Ne laser ( h = 632.8 nm) of power 1 mW and other essential optical components. The incident beam was split partially (30%) using a beam splitter to monitor the laser intensity fluctuations. The resulting beam that was weakly focused at the cell position was made to pass through the polished bath windows a t normal incidence. Spatially filtered incident and transmitted intensities were measured using photodiodes in conjuction with the conventional electronic circuitary. The intensity-voltage linearity of these circuits have been tested. The output signals were sent through low-pass filters (- 1 s) and amplified further. To eliminate the effect of incident laser intensity fluctuations, the instantaneous voltage referring to the transmitted intensity was divided by that due to incident intensity using a precision analog voltage divider. This voltage ratio (VT) multiplied by the scale factor of the divider was measured using a 61/2 digit multimeter. Normally, VTwas stable to 0.1% after the thermal equilibration of the sample, and it was measured with the same precision. At each temperature, the transmission loss due to the bath and reflection losses a t the bath windows were corrected by dividing VT with the sample cell by that without the cell (which can be moved away from the beam by means of themicrometer arrangement), and this ratioisdenoted by IT. To further eliminate the contributions from the noncritical back ground turbidity ( T B ) and the reflection losses at the cell windows, IT at any temperature was normalized by IT at a reference temperature (=IR) which was the farthest temperature of measurement ( T - T, 30 K) where the critical part of the turbidity ( T , ) is below the accuracy of the measurement. Thus the resulting quantity represented by IT/IR is the critical part of
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The Journal of Physical Chemistry, Vol. 98, No. 37, 1994 9171
TABLE 1: Transmitted Intensity Ratio (ZT/ZR) as a Function of Temperature (1) for TBAP + DD Mixture ( I = 5 cm)' T/K
IT/IR
T/K
IT/IR
354.2851 352.6985 35 1.26 16 349.8961 348.6039 347.3554 346.17 13 345.0476 343.9665 342.9301 341.933 1 340.9698 340.0500 339.1587 338.2971 337.4673 337.0604 336.6606 336.2674 335.8796 335.8808 335.5018 335.1265
0.988 63 0.987 53 0.986 69 0.986 16 0.985 35 0.983 48 0.982 71 0.981 14 0.979 95 0.977 83 0.975 12 0.972 19 0.969 67 0.966 01 0.961 26 0.955 06 0.951 73 0.946 56 0.941 52 0.935 28 0.935 58 0.928 31 0.920 74
334.7583 334.3935 334.0391 333.6831 333.4737 333.2663 333.0595 332.8557 332.7216 332.5883 332.4558 332.3185 332.2540 332.2191 332.1886 332.1527 332.1212 332.1027 332.0821 332.0681 332.0551 332.0421
0.907 97 0.892 68 0.873 94 0.845 66 0.822 90 0.792 86 0.753 21 0.696 63 0.645 23 0.578 07 0.485 21 0.355 34 0.274 66 0.227 77 0.179 85 0.133 29 0.086 15 0.059 78 0.037 10 0.024 37 0.014 01 0.006 65
TheuncertaintiesinTandIT/IRare k 2 mKand -0.001, respectively. The noncritical background turbidity, TB = -I-' In IR,is about 0.003 57 cm-I for this sample.
TABLE 2 Transmitted Intensity Ratio (ZT/ZR) as a Function of Temperature (1)for TBAP + BD Mixture ( I = 5 cm)J T/K h/IR T/K h/IR 351.3516 349.5881 348.3649 347.7539 346.5757 345.4461 344.3632 343.3268 342.3316 341.3714 340.4485 339.5557 338.6941 337.8616 337.0590 336.6658 336.2792 335.9015 335.5273 335.1542 334.7907 334.4328 334.0803 333.8687
0.992 85 0.991 45 0.989 51 0.987 94 0.987 56 0.984 86 0.981 76 0.981 54 0.980 22 0.975 87 0.974 02 0.969 02 0.964 10 0.957 29 0.949 43 0.947 07 0.941 40 0.932 43 0.921 98 0.909 42 0.894 59 0.874 69 0.844 28 0.821 26
333.6635 333.5258 333.3868 333.2541 333.1167 333.0510 332.9845 332.9170 332.8497 332.7837 332.7180 332.6840 332.6524 332.6174 332.5848 332.5663 332.5456 332.5250 332.5054 332.4935 332.4794 332.4664 332.4599
0.794 04 0.765 80 0.733 53 0.691 46 0.636 95 0.603 97 0.564 24 0.518 81 0.466 67 0.402 32 0.329 23 0.284 62 0.240 18 0.194 16 0.145 77 0.11803 0.090 57 0.063 92 0.040 64 0.027 93 0.017 23 0.009 10 0.006 05
TheuncertaintiesinTand ITIIRarek2mKand -0.001, respectively. The noncritical background turbidity, T B = -I-' In IR, is about 0.0263 cm-l for this sample. the transmitted intensity and T , (from now on referred to as turbidity) is calculated using the following expression:17J8
A set of representative IT/IR data for TPDD and TPBD is provided in Tables 1 and 2. Data very close to Tc ( t < 10-4) are excluded from the tables as they are not used in the analysis for deducing the leading CEs due to the following reasons: (i) Below t 10-4, IT may have been affected (in the level of the precision of measurement) by laser heating. (ii) The expression used to describe T , (eqs 2 and 3) need not hold good for such small values o f t ( 0.01
0.152 f 0.066 5.94 f 0.21 4.99 0.13 5.43 f 0.09
t < 7 x 10-3
TPBD
332.428 f 0.002
*
26.3
1.o 1.24 1.24 1.24
7 0 x 105 1.556 f 0.027 5.604 0.161 1.818 0.076 1.717 f 0.035 1.791 f 0.103
71
2.92 f 0.21 0 0 0.71 h 0.18 0
XVZ 0.16 0.04 0.13 0.19 0.27
s:'7 were fixed at their experimental values. The standard deviation of T, = 0.0002(1 + 0.581yI). CEs 7 and v were fixed at their theoretical values with v = 0.63 for y = 1.24 and v = 0.5 for y = 1.0. The error bars of the parameters correspond to 1 standard deviation obtained from the fit.
An accurate determination of Tcis crucial in obtaining reliable estimates of CEs. For very thick samples (1 = 5 cm), in the neighborhood of T,, most of the scattered intensity in the forward direction will beattenuated. Hence thespindalring that indicates the onset of phase separation18 could not be observed for I = 5 cm samples but was observed for I = 1 cm samples. For two samples of the same concentration, rC should have the same value at their respective Tis. Hence, Tis of I = 5 cm samples can be deduced by comparing the turbidity values to those of the corresponding I = 1 cm samples. All Tis listed in Table 3 are assigned in this manner.
Results The turbidity of the sample is the integral of the scattered light intensity (I,) over all angles per unit length.17 Accurate measurement of turbidity provides reliable estimates of CEs y and v of susceptibility (XT) and correlation length (4) respectively.17J8 In addition, the quantity measured is the unscattered intensity which is devoid of the potential errors such as multiple scattering.18 However, lack of direct measurement of 6 as well as the temperature dependence of noncritical background turbidity (.e) can complicate the data analysis. Taking Ornstein-Zernike (OZ) form18 of I,, 7, is given by the following expression due to Puglielli and Ford:17
10-4
10-3 1o-2 (T-TJT,"
10-1
Figure 1. Background-corrected turbidity of a critical mixture of TBAP and DD. The turbidity is calculated using the intensity data given in Table 1. For thesakeofclarityofpresentationthefittedlinecorresponding to mean field CEs close to T, is truncated.
with
whereflcu) is the OZ correction factor, n is the concentration ( x ) dependent refractive index of the mixture and AO is the vacuum wavelength of the incident light. For CE q = O,f(a) takes the following form:l7
The temperature dependence of thefla) factor is significant only in the vicinity of T,. Hence, a can be adequately described by the simple scaling relation:
a = 2 ( 2 ~ n 5 0 t - u / ~= 0 )aot-2v 2
(4)
where tois the prefactor of correlation length and t (=I(T- Tc)/ Tcl) is the reduced temperature. The numerical value offla) is not expected to change significantly for 7 (=0.03)# 0. For x = xc (the critical concentration), XT has the following scaling form:18
xT = Xot?(1
+ XltAl + ~
+ ...)t
~
~
(~5 )
where xo is the critical amplitude, X I ,~ 2 ..., , etc., are correctionto-scaling amplitudes of XT and A1 (-0.5) is the correction-toscaling e~ponent.3-~Using eqs 2 and 5 , the temperature dependence of 7, can be described by the following expression:
l
and 71 = XI, 7 2 = x2, ..., etc. The behavior of T~ (using 1 = 5 cm sample) for TPDD mixture over the t range 7 X 2 t 1 10-4 is displayed in Figure 1. An unambiguous deduction of the value of y is possible only iffla) is properly determined, which in the absence of direct measurements of [ requires the use of data very close to T,. All data points were appropriately weighed using the conventional propagation of errors19 which in this case stem, mainly, from the uncertainties in IT, I,, and T . T~ values up to t 1 10-4 were used in the nonlinear least-squares analysis (similar to that described in ref 19) and Tcwas fixed a t its experimental value. The broad conclusions are not altered even if Tc is allowed to vary up to 10 mK. In addition, the parameter a0 was fixed at its lower limit and then increased in small steps of 0.5% until the reduced x2 (x>)19 passes through a minimum and a0 corresponding to minimum x> is deemed as its best value. The analysis showed that a simple scaling expression, eq 6 with 71, 72. ..., etc., = 0, is inadequate to describe all the data. This is manifested by the strong preference for Ising values of y and v (1.24 and 0.63 respectively) by the data near Tc(dotted line in Figure 1) and a mean field effective y (= 1.O) for (T- T,)1 3 K (dashed line in Figure 1). Equation 6 with the first correction term ( ~ 2 ..., , etc. = 0) and Ising CEs adequately describes all the data as shown by the solid line in Figure 1. Higher order correction terms (72, ..., etc.) were not required by the data. The main results of the data analysis are provided in Table 3. The turbidity data of TPBD mixture is depicted in Figure 2. The measurements covered the t range 6 X 2 t 2 lo". rC
Critical Behavior of Ionic Fluids
The Journal of Physical Chemistry, Vol. 98, No. 37, 1994 9173 2000
1600
-5 1200
-
i-
8
c
\
-
10-3
E
% .-c
Et
800
Experimental (I=lcm)
I1-5cml . .. .0. . ., Experimental iYnp
-1singrmrrecson 1o
-~
10.3
-
1o-2
10.'
(T-TJT;' Figure 2. Background-corrected turbidity of critical mixtures of TBAP and BD obtained using two different sample lengths (1 and 5 cm). The inset is provided to illustrate the distinction between continuous and dotted lines. Data far away from Tofor I = 1 cm sample are not shown as they obscure the clarity of the figure due to their randomness.
values near Tc are very similar to that of TPDD mixture. The main difference between the two mixtures is that TPBD has a larger value of TB (=0.0263 cm-I) as compared to TPDD (TB = 0.003 57 cm-I). Hence T~ values, for t L 0.06 of TPBD, have very large uncertainties and are unsuitable to use in the analysis. Unlike in the case of TPDD, the entire data can be fitted to eq 6 corresponding to Ising y and v, without correction terms (71, 72, ...,etc. = 0). Although the inclusion of the correction term is not warranted (as indicated by the merging of dotted and solid lines in Figure 2), it improved the fit marginally. Thus the critical behavior of turbidity in this system is well characterized by the Ising CEs. The outcome of data analysis is given in Table 3.
Discussion The results presented in the previous section reveal that the CEs of TPBD mixture belong to Ising category. As BD is a high-t solvent, the primary factor responsible for phase separation is the solvophobic effect-whose criticality is expected to be Ising like.738 On the other hand, the critical behavior of turbidity in TPDD mixture cannot be described by a simple power law. Note that DD is a low-c solvent, and hence the Coulombic forces could have a major role in the phase separation.1p2,6This system shows a strong preference for Ising CEs very close to Tc and has a mean-field-like effective value of y far away from Tc. Alternatively, the evolution of critical behavior in this case can be perceived as a crossover of CEs from their mean field to the Ising limit as Tcis neared. This aspect can be more readily visualized using Figure 3. From eq 6 with T I , 72, ..., etc. = 0, it is straightforward to show that the inverse of OZ corrected turbidity is linear in T-1 for the mean field value of y. This feature is illustrated far away from Tc in Figure 3. The inset depicts the preference for Ising y near T,. This type of crossover phenomenon is observed in the case of metal-ammonia solutions15 and in polymer blends.20-21 Earlier investigations,l2-l4 involving Coulombic phase separation, did not exclude a crossover to Ising CEs in the neighborhood of Tc but found no evidence requiring such a behavior. On the other hand, we find a definite departure from true mean field behavior and C E s assume Ising values as t 0. However, our investigation differs in having more precise measurements very close to T,. In addition, due to a marginally higher value of c (=4.6) the crossouer region in TPDD may have shifted farther
-
400
0 2.80
2.84
2.88
2.92
2.96
3.00
1o3 T-' (K-')
Figure 3. Reciprocal of Ornstein-Zernike corrected Ha))turbidity as a function of inverse temperature for the TPDD mixture. f ( a ) refers to the best-fit value using eq 6 for Ising CEs. The error bars correspond to the standard deviation Of f ( a ) T C - l .
away from Tc and hence became more readily observable in the experiment. This argument is consistent with the finding that the entire experimental t range for TPBD is described by Ising CE's. In other words, the crossover region in TPBD is pushed beyond the experimental t range as a result of the large value of c (-26). The best-fit values of (YO yielded 60 0.32 and 0.35 nm for TPDD and TPBD, respectively. Hence, a mere comparison of the magnitude of 50 is inadequate to explain the smaller size of the asymptotic region in TPDD. Indeed, the value of 60 is very sensitive to the choice of v. Furthermore, microemulsions22 and polymer mixtures23have 50 an order of magnitude larger than simple liquid mixtures but have similar size of the asymptotic region.22~23 We conclude that the critical behavior of a Coulombic fluid is better described in terms of a crossover from mean field to Ising CEs rather than either true mean field or Ising CEs. Our finding is in agreement with the earlier theoretical predictionlo that RPM should develop an Ising critical point. We do not find any evidence to support the theme24that an ionic critical point is a higher order critical point. The essential difference between systems that are predominantly ionic and systems dominated by nonionic forces is in their behavior away from T,. The former show mean-field-like effective CEs (except very close to Tc)and the latter display near-Ising effective CEs over most of the temperature range accessible to experiments. We are currently examining systems with intermediate c to determine if the crossover region shifts away from T, but is still present within the range of measurement.
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Acknowledgment. This research was supported by the Director, Office of Energy Research, Office of Basic Energy Sciences, Division of Chemical Sciences, of the U S . Department of Energy under Contract DE-AC03-76SF00098. We appreciated discussions with Dr. J. M. H. Levelt Sengers. References and Notes (1) Pitzer, K. S.Acc. Chem. Res. 1990,23,333 and references therein. (2) Levelt Sengers, J. M. H.; Given, J. A. Mol. Phys. 1993, 80, 899. (3) Sengers, J. V.; Levelt Sengers, J. M.H. Annu. Rev. Phys. Chem. 1986, 37, 189.
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The Journal of Physical Chemistry, Vol. 98, No. 37, 1994
(4) Kumar, A.; Krishnamurthy, H. R.; Gopal, E. S.R. Phys. Rep. 1983, 98, 57. ( 5 ) Panagiotopoulos, A. Z . Fluid Phase Equilib. 1992, 76, 97; Caillol, J.-M. J . Chem. Phys. 1994, 100, 2161. (6) Weingartner, H.; Merkel, T.; Maurer, U.;Conzen, J.-P.; Glasbrenner, H.; Kashammer, S. Ber. Bunsen-Ges. Phys. Chem. 1991, 95, 1579. (7) Japas, M. L.; Levelt Sengers, J. M. H. J . Phys. Chem. 1990, 94, 5361. (8) Schroer, W.; Wiegand, S.;Weingartner, H. Ber. Bunsen-Ges. Phys. Chem. 1993, 97, 975. (9) Fisher, M. E.; Levin, Y. Phys. Rev. Lett. 1993, 71, 3826. (10) Stell, G. Phys. Rev.A 1992, 45, 7628. (11) Singh, R. R.; Pitzer, K. S. J. Am. Chem. SOC.1988, 110, 8723. (12) Singh, R. R.; Pitzer, K. S. J . Chem. Phys. 1990, 92, 6775. (13) Weingartner, H.; Wiegand, S.; Schroer, W. J . Chem. Phys. 1992,96, 848. (14) Zhang, K. C.; Briggs, M. E.; Gammon, R. W.; Levelt Sengers, J. M. H. J. Chem. Phys. 1992, 97, 8692.
Narayanan and Pitzer (15) Chieux, P.;Sienko, J. M. J . Chem. Phys. 1970,53, 566. Chieux, P.; Jal, J.-F.; Hily, L.;Dupuy, J.; Leclerq, F.; Damay,P. J. Phys. (Paris) 1 1991, I , c5-3. (16) Narayanan, T.; Pitzer, K. S., to be published. (17) Puglielli, V. G.; Ford, Jr., N. C. Phys. Rev. Lett. 1970, 25, 143. (18) Goldburg, W. I. In Light Scattering Near Phase Transitions; Cummins, H. Z., Levanyuk, A. P., Eds.; North-Holland: Amsterdam, 1983; p 531. ( 19) Bevington, P. R. Data Reduction and Error Analysisfor the Physical Sciences; McGraw-Hill: New York, 1969. (20) Bates, F. S.; Rosedale, J. H.; Stepanek, P.; Lodge, T. P.; Wiltzius, P.; Fredrickson, G.H.; Hjelm, Jr., R. P. Phys. Rev. Leu. 1990, 65, 1893. (21) Schwahn, D.; Mortensen, K.; Madeira, H. Y. Phys. Rev. Lett. 1987, 58, 1544. (22) Aschauer, R.; Beysens, D. Phys. Rev. E 1993,47, 1850. (23) Kuwahara, N.; Sato, H.; Kubota, K. Phys. Rev. E 1993, 48, 3176. (24) Kholodenko, A. L.; Beyerlein, A. L. Phys. Lett. A 1993, 175, 366.
