J . Phys. Chem. 1986, 90, 1700-1706
1700
f I I
lished (we would have needed to know n-C14properties from 91.5 to 455.3 OC and from 0.35 to 176 bars). As for n-C4, we do not have to extrapolate so much and thus the representation is mofe satisfactory. Table IX gives for relation 4 an average deviation less than 8% on 440 points.
ICP
DECANE
I
0
500
1000
Figure 9. Comparison of our results with the Lohrentz-Bray-Clark model in the case of n-decane: +, our experimental data.
C x i M , to define the equivalent molecular weight. The values obtained are indicated in Table VII. Tables VI11 and IX indicate that, here again, the agreement is satisfactory when eq 4 is used. Therefore, eq 4, which is obtained from pure alkanes, with n-C14 as reference, seems to work satisfactorily with alkane mixtures. W e have also used the following relations to determine T, and
P,: T, = Cx,T,,, P, = c x , P , ,
(8)
An effect of T,, P, definitions has been noticed concerning the obtained values. From Table 111, relation 4, and eq 8, we have obtained for the 63 experimental points 16.2% average deviation (10.6% with eq 7) and 24.3% maximum deviation (16.6% with eq 7). Finally, we have indicated in Tables VI11 and IX the results obtained for all the points. Table VI11 shows that when the reference values are not extrapolated, the average deviation reaches 7.8% with a maximum deviation of 29.2% (on 239 points). If we extrapolate the reference, Table IX shows that n-C, is the only one with a bad representation, although the n-C, average deviation reaches 12.5% from relation 4 (yet, with a maximum of 52.2%). This comes from the large extrapolation affecting n-Cl4 a t temperatures very far from the ones for which relation 1 was estab-
Conclusions In this study, we have given values for the viscosity v(P,T)of the alkanes (from n-C,,, to n-C,*),some typical mixtures of alkanes, and three alkylbenzenes within the range of pressures and temperatures usually found in oil fields. This data completes recent works made on the following samples: n-C,, n-C,, n-C12,n-C,,, alkane blends, cyclohexane, and aromatic hydro~arbons.l'-'~ From a phenomenological point of view, the results are very well represented by the empirical relation 1; this relation, however, unfortunately requires 11 constants which have to be determined for each sample. W e have also shown the interest and the limits of a relation based on corresponding states and requiring a reference. The advantage lies in the fact that we just need to know well the properties of the body chosen as a reference, as well as the critical temperatures and pressures of the studied bodies. It is interesting to see that taking the alkane n-C14as a reference gives satisfactory results even if the corresponding states relation is applied to alkylbenzenes. However, reduced temperatures and pressures should be used carefully since it can lead to extrapolations for the reference (for example application of relation 4 to methane). Finally, we have noted that the corresponding-states method applies to mixtures as well, though in this case there are blending laws (Lobe's law and cubic law) which can be as precise. Acknowledgment. We thank the SNEA(P) Co. for its financial support as part of a n Industry/University contract. Registry No. n-C,,, 124-18-5;n-C,,, 112-40-3:n-C,,, 629-59-4; n-C,,, 629-62-9; n-C,,, 544-76-3; n-C,,, 593-45-3: OCBZ, 21 89-60-8: HEBZ, 1077- 16-3; BUBZ, 104-51-8. (17) Dymond, J. H.; Young, K. J.; Isdale, J. D. I n t . J. Thermophys. 1980, l , 345-373.
(18) Dymond, J. H.; Robertson, J.: Isdale, J. D. I n t . J . Thermophys. 1981, 2, 133-154. (19) Kashiwagi, H.: Makita, T. I n t . J. Thermophys. 1982, 3, 289-305.
A Dynamlc Light Scattering Study of the terf -Butyl Alcohol-Water System Thomas M. Bender and R. Pecora* Department of Chemistry, Stanford University, Stanford, California In Final Form: November 21, 1985)
94305 (Received: August 9, 1985;
The hypersonic speed of sound (C,) has been measured in the tert-butyl alcohol-water system (TBA/water) from 0.0 to 0.16 mole fraction of TBA at 10-45 "C by Brillouin scattering. Considerable dispersion in the C, as compared to that found in the zero frequency and ultrasonic range is observed. The isentropic compressibilities in the hypersonic range are also presented for IO, 20, and 25 " C . Photon correlation spectroscopy (PCS) gave no evidence of the presence of oligimers in the system and depolarized interferometry measurements detected no rotational diffusion by species in the system. The acoustic relaxation times calculated at mole fraction of TBA equal to 0.105 were found to be in agreement with literature values of both NMR and dielectric relaxation measurements on this system. The results obtained were interpreted from the viewpoint that the relaxation time observed was due to a structural relaxation.
Introduction The system of terr-butyl alcohol-water (TBA/water) has been the subject of numerous investigations in the past 50 years, In the water-rich region, this system shows a wide variety of seemingly anomalous physical properties. In the 0.05-0.10 mole fraction
of TBA (XTBA) range a t 25 O C the partial molar volume ( Vm) of TBA goes through a minimum while the partial molar heat capacity (Cp,,) of the system goes through a maximum.'-s Is( I ) Nakanishi, K . Bull. Chem. SOC.Jpn. 1960. 33, 793.
0022-3654/86/2090-1700$01.50/0 0 1986 American Chemical Society
Dynamic Light Scattering of T B A / H z O entropic compressibilities measured a t 25 "C by both ultrasonic and hypersonic methods and Ks,h,respectively) and also the isothermal compressibilities ( K T ) a t 25 "C have a pronounced minimum in the same mole fraction range.6-15 Light scattering measurements of the concentration fluctuations for this system show a strong deviation a t approximately 0.05XTBA from the predicted ideal concentration dependence of the fluctuations.16 Also, a recent set of dynamic light scattering measurements of mutual diffusion coefficients in this system has given apparent hydrodynamic radii for TBA molecules at room temperature which show a weak maximum near 0.15XTBA.17 The experimental results on this system in the water-rich region have usually been interpreted in one of three fashions. One theory is that, in dilute solution, the TBA is forming hydrated oligimeric species similar to the solid clathrates (e.g. (HZO),,TBA) formed by the TBA/water system.8~16~18-z0 In this approach, a t higher concentrations of TBA the system is believed to then form clusters of these hydrated oligimers and also simple solutesolute oligimers. The second common viewpoint is that a t ordinary experimental temperatures and pressures the system is making a close approach to a critical point in liquid-liquid equilibrium phase Because of this close approach to a critical point, one would expect the physical properties of the system to vary quite strongly with composition. The third commonly held viewpoint is to consider the addition of TBA to water as promoting a stronger water structure.z3 In this approach, the TBA works as a structure maker which enhances the water-water structure in the system and also perhaps forms domains in which short-lived clustering of the alcohol molecules occurs. In this paper we present the results of new dynamic light scattering measurements on this system. W e report hypersonic speeds of sound in solution as measured by Brillouin scattering from 10 to 45 " C along with isentropic compressibilities between 10 and 25 "C a t the same hypersonic frequency. From the dispersion in the speed of sound in the system we calculate the approximate relaxation times for the system a t XTBA equal to 0.105. Also we present the results of depolarized interferometry measurements along with diffusion coefficients measured by photon correlation spectroscopy a t 10 and 20 "C.
(2) Mishra, A. K.; Ahluwalia, J. C. J . Chem. SOC.,Faraday Trans. I 1981, 77, 1469. (3) Hvidt, A.; Moss, R.; Nielsen, G. Acta Chem. Scand., Sect. B 1978, 32, 274. (4) DeVisser, C.; Perron, G.; Desnoyers, J. E. Can. J . Chem. 1977,55, 856. ( 5 ) Arnaud, R.; Avedikian, L.; Morel, J.-P. J . Chim. Phys. 1972, 69, 45. (6) Burton, C. J. J . Acoust. SOC.Am. 1948, 20, 186. (7) Baumgartner, E. K.; Atkinson, G. J . Phys. Chem. 1971, 75, 2336. (8) Tamura, K.; Maekawa, M.; Yasunaga, T. J . Phys. Chem. 1977, 81, 2122. (9) Patil, K. J.; Raut, D. N . Ind. J . Pure Appl. Phys. 1980, 18, 499. ( I O ) Endo, H.; Nornoto, 0. Bull. Chem. SOC.Jpn. 1965, 46, 3004. (11) Lara, J.; Desnoyers, J. E. J . Solution Chem. 1977, IO, 491. (12) Stone, J.; Pontinen, R. E. J . Chem. Phys. 1967, 47, 2407. (13) Asenbaum, A. Z . Naturforsch. A 1976, 31, 201. (14) Nakagawa, M.; Inubushi, H.; Moriyoshi, T. J . Chem. Thermodyn. 1981, 13, 171. (15) Kartsev. V. N.; Zabelin, V. A.; Samoilov, 0.Ya. Zh. Fiz. Khim. 1979, 53, 1774. (16) Iwasaki, K.; Fujiyama, T. J . Phys. Chem. 1977, 81, 1908. (17) Euliss, G. W.; Sorensen, C. M . J . Chem. Phys. 1984, 80, 4767. Fujiyama, T . Bull. Chem. SOC.Jpn. 1981, (18) [to, N.; Saito, K.; Kato, 7.; 54, 991. (19) Bale, H. D.; Shepler, R. E.; Sorgen, D. K. Phys. Chem. Liquids 1968, 1 , 181. (20) Goldammer, E. V.; Hertz, H. G. J . Phys. Chem. 1970, 74, 3734. (21) Vuks, M . F.; Shurapova, L. V. Opt. Commun. 1972, 1 1 , 150. (22) Vuks, M. F. Mol. Fiz. Biofir, Vodn. Sist. 1973, I , 3. (23) Tanaka, H.; Nakanishi, K.; Touhara, H. J . Chem. Phys. 1984, 81, 4065.
The Journal of Physical Chemistry. Vol. 90, No. 8, 1986 1701 Experimental Section The TBA used in this study was a Baker Certified analytical reagent. The water was prepared by passing commercial deionized water through a mixed ion exchange resin and then distilling. In all of the light scattering experiments the samples were contained in 1.00-cm2 glass fluorimetry cells. These cells were cleaned between uses with a chromic acid cleaning solution. T o prepare samples, the clean cells were first rinsed with water which was passed by a peristaltic pump through a 0.22-pm filter. The sample cell washes were periodically checked for dust by passing a 300-mW 488-nm laser beam through the sample and viewing the beam path with a 12X microscope. When the dust motes were reduced to what experience has shown to be a suitably low number, the cell was considered to be cleaned. The solution was then delivered to the clean cell by filtering it from a syringe through a 0.22-km filter into the cell. This form of checking for dust motes is effective in water because of the low Rayleigh scattering of water a t room temperature. Any dust motes present scattered blue light which was easily observed against the weak red Raman shifted light that the water emitted. All work in this paper was performed with Spectra-Physics Model 165 argon-ion lasers operating a t 488 nm. In the interferometric experiments the laser was operated in single mode by using an oven-stabilized etalon. The interferometer apparatus was used a t a fixed angle of 90" and is the same as that previously describedz4with the exception that the current apparatus uses a Burleigh interferometer with a Burleigh DASl stabilization/data acquisition unit. The scattering vector length is defined as
where n is the refractive index of the sample, h is the wavelength of the incident light in vacuo, and 0 is the scattering angle. Therefore, a typical value of q for these experiments is 2.5 X IO7 m-'. The free spectral range used in all experiments was 20.30 GHz with an average finesse of 55 for half of the 10 and 20 "C measurements and an average finesse of 75 for all other measurements. The free spectral range was calibrated from literature values for the speed of sound in water a t 20 0C.2s.z6 The photon correlation experiments were performed a t a lab angle of 25" with appropriate refractive index corrections made to obtain the true scattering angle (typically 26.3"-27.5") observed from the square cells. The sample cells were held in a n indexmatching bath which also served as a thermostat for the cells. The total apparatus used has been described by Flamberg and P e ~ o r a . ~ , The correlation functions were taken with a Brookhaven BI2020 72 channel, 4-bit correlator. All measurements were made by multiplexing to obtain 128 channels. Tests with 910-A diameter polystyrene spheres showed that the light was being detected in a homodyne fashion.** The actual concentration of TBA in the samples was determined by measuring the refractive indices of the samples at 20 " C and comparing to a constructed calibration curve for the TBA/H,O system a t 20 "C. The refractive index measurements a t this temperature and a t all the experiment temperatures were performed using a Bausch-Lomb Abbe refractometer. The required viscosity data for this system were extrapolated from literature values. The data handling was performed on a Charles River Data System 68000 series computer equipped with a floating point processor. The interferometric data were fit to the following equation:29 (24) Stanton, S . G.; Pecora, R.; Hudson, B. S. J . Chem. Phys. 1983. 78, 3365. (25) Ostwald, J.; Pazold, W.; Weis, 0. Appl. Phys. 1977, 13, 351. (26) Choi, P.-K.; Takagi, K. Jpn. J . Appl. Phys. 1933, 22, 890. (27) Flamberg, A.; Pecora, R. J . Phys. Chem. 1984, 88, 3026. (28) Berne, B.; Pecora, R. Dynamic Light Scattering; Wiley: New York, 1976. (29) Masiano, G.; Migliardo, P.; Aliotta. F.; Vasi, C.; Wanderlingh, F.; D'Arrigo, G. Phys. Reu. Lett. 1984, 52, 1025.
1702 The Journal of Phj3sical Chemistry, Vol. 90. No. 8, 1986
A,
+ A , ex.(
-2)-r,’ +
+
rl
r2
i.2
+
r2’+ [ u - (us2- r22)i:2]2
Bender and Pecora TABLE I: Hypersonic Speed of Sound in the System TBA/WaterQ C,. m / s .YiB&
I O ‘C
20 “C
25 “C
35 ‘C
45 ‘C
0.00 0.021 0.033 0.042 0.049 0.056 0.066 0.094 0.125 0.159
1445
1484 1560 1598 1640 1659 1665 1673 1632 1614 1557
1488
1496
1524
I599
1592
1582
1630
1607
1582
1629
1589
1557
1560
1504
I456
1159
1094
1035
1550 1601 1671 1734 1767 1773 1746 1725 1699 1315
1 .o
1215
“ W a v e vector = (2.43-2.55) X IO’ m - ’ . Estimated error h0.35R.
where u and are parameters that characterize the Rayleigh is the Brillouin line half-width, wB is the Rayleigh to line, Brillouin spacing, and the Ai’s are linear parameters which were floated in the nonlinear least-squares program used to fit eq 2 to the data. The instrumental line shape was determined to be a Voigt function by using 9 10-b; diameter polystyrene spheres. In all the data fits the value of u obtained from the fitting of the instrumental line to a Voigt function was given as a constant to be used in that data fit. Because of the complexities of the “exact” form of S(q,w) for a binary system with a relaxation,30 this approximate form for S(q,w) was used. Of the applicable approximate models for S(q,o) in the literature, eq 2 was found to consistently give the lowest fit variances and the most nonsystematic residuals. I n order to obtain the true line width of the Brillouin lines observed, it was necessary to deconvolve the instrumental line width out of the experimental data. The instrumental line width as mentioned above was obtained by fitting the scattered light from 910-A diameter polystyrene spheres with a Voigt function. Approximate deconvolution of the Brillouin line widths was then accomplished by convolving the instrumental line shape with Lorentzians of known half-widths and then fitting the resultant “Brillouin lines” with a Lorentzian model. The resultant correction curve was used to deconvolve the data. In view of the approximate model used to fit the data, the uncertainty in these calculated line widths may be as high as 20%. The autocorrelation data were fit with CONTIN, a constrained and conditioned inverse Laplace transform (ILT) p r ~ g r a m . ~The ’ strength of C O N T I N lies in that it requires no initial guesses as to the number of relaxation modes present in the data. Also, the autocorrelation data were fit to DISCRETE, a program which fits the data to sums of discrete exponentials!’ and requires no initial specifications of the number of exponentials present. The DISCRETE results were compared for consistency with those given by C O N T I N .
-I
16501
I
rz
Results
The hypersonic speeds of sound (ch)in the water-rich region of TBA/water systems at 10-45 “C are given in Table I. Also given is the c h in pure TBA. Although TBA freezes in the range of 25.0-25.3 “C. it is relatively easy to supercool. The sample used to measure the pure TBA ch may have contained traces of water, but the refractive index of the sample at 25 O C as compared to literature values was in good agreement. The estimated error in the Ch measured is 0.35%. The only literature values with which to compare this work are those of Stone et al.l? and Asenbaum.I3 Comparison of our work with that of Stone et al. is only possible (30) Fishman, L.; Mountain, R. D. J . Phys. Chern. 1970, 7 4 , 2178. (31) Provencher, S. W. CONTIN Users Manual; European Molecular Biology; Laboratory Technical Report EMBL-DAO2, Heidelberg, Nov 1980. Provencher, S. W. Compt. Phys. Cornmiin. 1982, 27,213. Provencher. S. W. Makromol. Chem. 1979. 180, 201. ( 3 2 ) Provencher, S. W . J . C h e m Phyc. 1976, 64. 2772
1600
?E
1
1
--i
:
v
1550iI7 / 1500
t
I
’ I
0 00
0 05 0.10 0 15 MOLE FRACTION TEA Figure 1. T h e hypersonic speed of sound in TBA/water solutions a t 25 “ C as a function of X,,,. Because of the variation of the solution refraction index with change in TBA concentration, q varies from 2.43 X IO’ to 2.55 X IO’ m-’; ref 13, 0 ;this work, A;results from ref 13 after normalization to the C, of water obtained in this work, 0 .
qualitatively because no raw data was published in their article; yet the qualitative agreement between their work and this work appears to be good. Asenbaum’s work was done a t 25 O C and is compared to this work in Figure 1. The difference between the results may arise from the difference in our respective wave vectors. For the case of pure water, Asenbaum’s 4 is 2.30 X 10’ m-l while the 4 in this work is equal to 2.43 X lo7 m-l. It has been shown, however, that the dispersion in the c h of water a t so the difference probably room temperature is quite sma11,25~26 is due to a systematic error in determining the free spectral range (FSR) used to calculate the ch. Since the agreement of our Ch of water at 25 OC is good compared with values from the literature, we have confidence in our reported value for the C,. As the points labeled with open squares in Figure 1 show, upon normalizing h of water from this work, one obtains Asenbaum’s data to the c values for Ch that agree well with the results from this work. I f one defines a quantity called the excess speed of sound (Chx) for a binary solution as ’1‘
=
Ch.ubai
-
xTB.4chTBA - ( 1 - ,yTBA)Ch“o
(3)
where the Ch’are the values for the hypersonic speed of sound in the superscripted pure components, one should have a first-order representation of the nonideality of the speed of sound in these is shown solutions. A plot of Chxisotherms as a function of XTBA in Figure 2. The approximate error in the Chx shown is 7.5 m/s. At all concentrations and temperatures studied except for the
The Journal of Physical Chemistry, Vol. 90, No. 8, 1986 1703
Dynamic Light Scattering of T B A / H 2 0 I
I
C ~
320-
240 -
-
.
f
h
v)
E
Y
*.c
?C,O-
0
0.00
25°C
0.05
0.10
0.1 5
MOLE FRACTION TEA
80 35°C
0
i
0.00
' 45°C 1
I
I
0.05
0.10
0.15
MOLE FRACTION TEA
.;
Figure 2. The excess speeds of sound in TBA/water solutions a t temperatures from 10 to 45 O C as a function of A',,,: 10 O C , 0 ;20 O C , 25 OC,A; 35 OC, e; 45 O C , V.
highest concentration a t 45 "C, the system shows positive deviations from ideality. Also, an attempt to fit the c h to Rao's law
where Q is a temperature-independent constant and V, is the molar volume of the mixture, led to large systematic errors as the mole fraction of TBA in the system increased. The thermodynamic identity
0.00
0.05
0.10
0.15
MOLE FRACTION TEA
along with literature values for Cp,m,4Vm,334expansivities (e),33 and isothermal compressibilities KTI43l5were used to obtain the zero frequency isentropic compressibilities. The isentropic compressibility (Ks,o)obtained by using eq 5 is termed a zero frequency compressibility because all the constants used to evaluate it were taken with thermodynamic methods. Ultrasonic speed of sound data for this system also gives K,, but it is a K,,u obtained from a C measured by driving an ultrasonic wave through the solutions. The K, from hypersonic data is obtained by observing the light scattered by thermal phonons in the system in order to obtain C,. If relaxation processes are present near the frequency range used to study the system, these K,'s will not be the same at a given concentration because of the dispersion in the speed of sound. Figure 3a gives a comparison of the K,'s obtained at 25 OC in various frequency ranges. The Ks,uvalues are from ref 8, 9, and 11 while the hypersonic compressibilities were calculated from the density values in ref 3. The error in the calculated zero frequency Ks,ois too large to say for certain whether the Ks,ois greater than the K,,", but there is a definite difference in the Ks,hand the other K,'s. Several workers have observed two relaxations by ultrasonics and predicted the presence of a higher frequency The work of Stone et a1.I2 also showed considerable dispersion in the c h measured at two different hypersonic frequencies. Preliminary experiments in our laboratory a t different values of the wave vector have also confirmed the presence of large dispersion in Ch a t hypersonic frequencies. Unfortunately, complete comparisons of the different K,'s at other (33) Desrosiers, N.; Desnoyers, J. E. Can. J . Chem. 1976, 54, 3800. (34) Blandamer, M. J.; Clarke, D. E.; Hidden, N. J.; Symons, M. C. R. Trans. Faraday Sor. 1968, 64, 2691.
Figure 3. (a, top) T h e frequency dispersion of the isentropic compressibility of TBA/water solutions a t 25 O C as a function of X,,,: zero frequency, 0;ultrasonic frequency (ref 8,9, 1 l ) , 0 ;hypersonic frequency, A. (b, bottom) The isentropic compressibility of TBA/water solutions 10 O C , 0 ; 20 O C , m; 25 O C , a t 1 0 , 2 0 , and 25 O C as a function of A',,: A.
temperatures is not possible due to limited thermodynamic data on this system. Figure 3b shows the isentropic compressibilities as measured by hypersound at 10, 20, and 25 O C . The three isotherms in Figure 3b seem to have a common intersection a t approximately 0.025 mole fraction. This behavior has also been observed in the work of Stone et al. and in ultrasound measurements on this system.I0 The sound damping coefficient a (units of inverse distance) may be obtained from the Brillouin data by the following a/f =
8r3r2
-
G3q2 where f is the frequency displacement of the Brillouin line from the Rayleigh line and r2is the Brillouin line half-width. As may be seen from eq 6, division of the obtained damping coefficients by the speed of sound in the solution gives the reduced damping coefficient. Because of the technique we used to deconvolve the instrumental line width from the Brillouin doublets, the absolute values we measured for a are subject to up to 20% error, but even so, the values we measured for a/f a t all temperatures studied clearly show a monotonic increase with X,,,. This same trend was also observed by Asenbaum" a t 25 OC. (35) Fablinskii, I . L. Molecular Scaftering o f l i g h t ; Plenum Press: New York, 1968. (36) Flygare, W. H. Molecular Structure and Dynamics; Prentice-Hall: Englewood Cliffs, 1978.
1704
The Journal of Physical Chemistry, Vol. 90, No. 8, 1986
The dispersion of the Ch in this system should contain information about the characteristic time 7 for the relaxation process that caused the dispersion. Unfortunately the interpretation of the data is difficult for several reasons. Our results do not give the dispersion in the system as a function of wavelength, hence little is known about the dispersion curve. Also, the error in the a/f values make them of little practical use in trying to calculate the relaxation time associated with the dispersion. Perhaps the worst barrier to obtaining the relaxation time is that there i< not available a complete set of the thermodynamic constants needed to calculate the zero frequency speed of sound in the system. Nonetheless, if several assumptions are made it is possible to obtain approximate values for the relaxation time of the process u hich causes the dispersion. The dispersion of the speed of sound may be used to determine the relaxation time as shown in the following equation,37
Bender and Pecora TABLE 11: Literature and Calculated Relaxation Times ( 7 ) and Activated State Thermodynamic Constants in 0.105XTBA Solutions at 25 OC a. Literature and Calculated Values 7.
AG* is the Gibbs free energy of activation for the relaxation process. The AG* from eq 8 has a very interesting interpretation if the relaxation times observed are due to a structural relaxation process. Hirai and E ~ r i n have g ~ ~suggested that in a structural relaxation the rearrangement of molecules is a cooperative process which involves the movement of man] molecules around a hole in the system. In this model, the rearrangement of molecules about a hole consists of several elemental processes, each of which is very similar to the processes giving rise to shear viscosity. Because of the similarity in the postulated elemental processes which occur in viscous and structural relaxation processes, rate theory predicts identical functional forms for the temperature dependences of' the relaxation times for those p r o c e s ~ e s . ~Using ~ , ~ ~this model, one would expect the enthalpies of activation for dielectric and structural relaxation to be very similar, with the AH* of the structural process being slightly larger due to the need to provide energy to overcome both rotational and translational barriers."' In using eq 7 the initial assumption we make is that the dispersion can be characterized by a single relaxation time. Next it is necessary to determine the Co for the system in order to calculate the actual dispersion in the Ch values. Because of the approximate nature of this calculation, we chose to calculate a relaxation time a t X,,, = 0.105 where the dispersion in the Cs is greater than that a t lower X,,, (see Figure 3a). The values of Coused were calculated by using eq 5 along with the previously referenced literature values for the parameters in the equation. Because of the lack of hypersonic measurements on binary liquid mixtures, little information is available as to the actual temperature dependence of C,. In the case of several diols and triols which have been studied by ultrasonics, 1/C, (dC,/dT) has been shown to be approximately a factor of 2.5-3.5 times larger than 1 / C , (dC0/dT).40 It is not clear whether or not this temperature dependence is an artifact of not correcting for a high-frequency ( 3 7 ) Samios, D.; Karayannis, M.; Dorfmuller. T. A&. M o / . Relax. Inleracf. Processes 1978, 12, 3 13 (38) Herzfeld, K. F.; Litovitz. T. A . Absorption arid Dispersion qf'L'ltmsonic Waces: Academic Press: New York. 1959. (39) Hirai, N.; Eyring. H . J . Appl. Phys. 1958, 29, 810. (40) Hennelly, E. J.; Heston. Jr., W . M.:Symth. C. P. J . A m . Chem. Soc. 1948, 70. 4102.
42.0 f 12.00 8.0 30.0 20.8 f 1.0
~~~~
13.8 f 0.8 9.7
12.0
b. Activated States Thermodvnamic Constants
AG* this work first approximation second approximation dielectric relaxation (ref 42) "Reference 43.
where C, is the infinite frequency speed of sound in the solution and T is the relaxation time of the process. Equation 7 can lead to an inaccurate calculation of T if Ch is known a t only one value of the wave vector q. If one makes an initial assumption as to the functional form for the change in 7 with temperature, however, one can use that function in conjunction with eq 7 to obtain a more accurate estimate of 7. One plausible function to use to represent the temperature dependence of 7 is that from the Eyring rate process theory:38
AG*C
DS ~~
this work first approximation second approximation NMR" dielectric relaxationh
9.7 13.8 12.0
AH*
1s'
25.9
54.3 40.6 46.6
bReference 42; values quoted are taken a t
O.IOOXTBA. ' S G * and AH* are given in kJ/mol and AS* in J/(moi K).
relaxation in the system or if it is authentic. In the case of 1,2,6-hexanetrioI,"' after a complete set of measurements of the relaxation times, including those in the hypersonic range, it was found that C, showed almost the same temperature dependence as C,. In our first attempt to fit the data we set 1/C, (dC,/dT) equal to either 1, 2, or 3.5 times l/Co (dC,/dT). To actually fit this data we then coupled the approximation for the temperature dependence of C, with the temperature dependence of T that is predicted by eq 7. Using the above approximation for the temperature dependence of C, and eq 8 we fit the C, for the system as a function of temperature. Regardless of the factor chosen for the relation between 1/C, (dC,/dT) and l / C o (dCo/dT),the data fits gave 1 G * between 9.5 and 10.0 kJ/mol. The summed variances in the calculated C, obtained by using this approximation for C, were large and indicated that with the data available the model could not distinguish between the various temperature factors. Upon substituting eq 8 into eq 7, rearranging, and taking the natural logarithm of the resultant equation one obtains
Equation 9 shows that if (Cm2 - Ch2)is approximately constant, a plot of 2 In [(Coqh/kT)]-In [ ( c h 2 - Co2)]vs. 1/Tshould give a straight line with -2AG*/k. The fit of the c h using eq 9 with (C,? - ch2) assumed constant gave a statistically lower summed variance in the calculated C, as compared to the previous model for C,. The Act found by using this approximation for C, was 13.8 kJ/mol. Because of the lower summed variance for the C, predicted by this fit, we believe it to better represent the data, but we report both sets of predicted results in this paper. The calculated values of AC* and 7 at 0.105XTBA are given in Table 11, a and b. In both tables the values are given for both the first approximation, in which 1/C, (dC,/dT) was set equal to a factor times l / C o (dCo/dT),and the second approximation, in which (C,' - Ch') was assumed to be constant with temperature. In the tables, comparison is made t o dielectric measurements made on this system in the 9.4-25.0-GHz range by Yastremskii et aL4? along with N M R relaxation times.43 Table IIa shows that there is approximate agreement between the AG* values from this study and those from the work of Yastremskii et al. In the case of the 7 calculated with the approximation of (C-' - Ch2)being constant, there also seems to be agreement between the 7 a t 25 'C given by the various methods. Assuming that the AH*of the acoustical (41) Dorfmuller, T. A.; Dux, H.: Fytas, G.; Mersch. W . J . Chem. Phys. 1979. 7 1 . 366.
Dynamic Light Scattering of T B A / H 2 0 dispersion process is approximately equal to the AH* from the dielectric process, as has been discussed above, we calculated the AS* in Table IIb. If we assume the validity of the assumption of the equality of these AH*'s, Table IIb shows that both approximations for C, give a AS* which is in reasonable agreement with that from the dielectric measurements of Yastremskii et al.4z The similarity in the AS* measured by this work and the AS*for the dielectric relaxation measurements indicates that, if the relaxation time observed is due to a structural relaxation, it is due to some intermolecular rather than an intramolecular relaxation. For instance, if the relaxation was due to the rotation of the O H group about the C-C bond, the process would not cause a large change in the local order and would therefore have a lower AS* than that observed for dielectric relaxation. If the process involved a cooperative motion of a collection of molecules, the AS*observed by acoustics would be much closer to the A S * observed by dielectric relaxation measurement^.^^ Interferometric measurements of the depolarized light scattered from a solution containing one optically anisotropic species in a solvent of low or no anisotropy is one means of obtaining rotational diffusion coefficients for the solute.28 In the rotational diffusion approximation for a dilute solution of cylindrically symmetric molecules, it can be shown that ZvH(w),44 the spectral density at frequency change w associated with the depolarized component of the scattered field, is given by
where A is a constant, N is the total number density, X i s the mole fraction of solute, n is the solution refractive index, /3 is the optical anisotropy, and eR is the rotational diffusion coefficient about an axis perpendicular to the symmetry axis of the molecule. The depolarized light scattering relaxation time, q ,is equal to 1/68R.44 During the course of Ch measurements at 10 and 20 "C, we attempted to observe T~ for TBA solutions in the 0.02-0.16XTB, range. At all concentrations at both temperatures no depolarized signal was observed. The complete absence of a depolarized signal means that the experiment gave no evidence of the presence of optically anisotropic species in the solution. Aside from the ever present possibility that oligimers were present in too low a concentration to detect by this technique, there is another possible situation in which oligimers could have been present without our detecting them. If oligimers were formed which had a very small value for the optical anisotropy this would have prevented their detection. Ito et a1.I6 have proposed the presence of hydrated alcohol molecules in this system which have the form [ ( H 2 0 ),,TBA]. These hydrated alcohol complexes are supposed to be polyhedral in structure. If such a highly hydrated species existed in the system, it might appear optically isotropic. It is obvious then, that the results from depolarized interferometry measurements on this system do not necessarily exclude the possibility of oligimers being formed in the system. One can definez8a correlation length ( 5 ) in a binary solution as
Here D, is the mutual translation diffusion coefficient and 7 is the viscosity of the solution. By performing photon correlation spectroscopy (PCS), one can obtain the mutual diffusion coefficient D, for a solute in a binary solution. By choosing to interpret PCS data for small molecules in terms of (, one avoids making assumptions regarding the applicability of hydrodynamic friction arguments to solute molecules which are of the same relative size as the solvent molecules. Before performing a PCS experiment on this system, one may make three predictions of the possible results of the experiment. In all of the cases, we will initially assume that, if the oligimers
The Journal of Physical Chemistry, Vol. 90, No. 8,I986 1705
I
0.10
1
0.15
0.20
0.25
MOLE FRACTION TEA
Figure 4. T h e correlation length isotherms of TBA/water solutions as a function of TBA a t 10 'C, 0 ; 20 'C, W; 35 OC, +; 45 OC,V. Points a-d are experimental results from this work while the remaining points are from ref 17.
are formed, (1) they have an index of refraction which is sufficiently different from that of the solvent so as to make detection possible and (2) they are present in sufficient quantities to be detected by PCS. The first possible result would be the case where no oligimers were formed and the only relaxation time observed was due to the diffusion of the alcohol monomers. The second possible result would be that oligimers dissociated and re-formed on such a short time scale that the single relaxation observed would be an average value associated with the kinetic constants of the reaction.z8 The final possibility is that the oligimers formed would be long-lived and that they would give a relaxation time due to their diffusion which would also be observed along with the relaxation time due to the diffusion of the monomeric alcohols. Although several workers have studied this system by PCS and found only single exponential autocorrelation functions, the data handling programs used may have used algorithms which biased the results. Because of the success which our laboratory has had in using C O N T I N on PCS results, we hoped that measurements on this system, if analyzed by C O N T I N , might yield an as of yet unobserved relaxation time due to the translation of oligimers in the solution. The results of our work are shown in Figure 4 (points a-d) along with the data from Euliss et aLI7 All our results were preferentially fit to single exponentials by both the programs CONTIN and DISCRETE. Figure 4 shows that all the isotherms for E go through a maximum a t approximately 0.1SXTBA.Unfortunately, the error bars on [ in both our work (points a-d) and that of Euliss et al.', make it impossible to accurately observe the temperature trend in the results, but the maxima in the isotherms are definitely present even in consideration of this error. In light of the above discussion, there are two possible explanations for the PCS results of this work. The first possibility corresponds to case (2) above, in which the lifetime of the oligimer may have been so short that the single relaxation time observed is a complicated average over the dynamic equilibrium of species. The second possibility corresponds to case (1) above, in which no oligimers are present in the system and the relaxation time observed is due to the diffusion of the monomeric alcohol. Of course, if one allows the possibility that any oligimers formed may index match the solvent or be present in very low concentration, then any of the cases are possible. The nonexistence of long-lived oligimers in the TBA/water system has been reported by workers in a N M R but all that may safely be said from our measurements is that no oligimers were directly observed in this system by PCS. Conclusion
In this work we have found no direct evidence of long-lived oligimeric species in the water-rich region of the TBA/water system. Both depolarized interferometry and PCS have detected no signs of aggregates in the system. We were not, however, able to discount (1) the possibility of the existence of aggregates whose
(44) Bauer, D. R.; Braurnan, J. I.; Pecora, R. J . Chem. Phys. 1975, 63,
53.
(45) Uedaira, H.; Kida. J. Nippon Kuguku Kuishi 1982, 4 , 534.
1706
J . Phys. Chem. 1986, 90, 1706-1717
index of refraction closely matches that of the solvent or (2) the existence of short-lived aggregates which have a low optical anisotropy. N o direct evidence was taken in this work which can be used to verify or disprove the existence of a dome-shaped critical region in the phase space of the TBA/water system. It is known that the addition of small amounts of KC1 to the TBA/water system will cause the appearance of a closed-loop two-phase region in the phase diagram of the TBA/water ~ y s t e m . ” This ~ ~ ~ salting out of the system has been interpreted as indicating the nearness of the postulated critical point to the actual state of the room temperature and pressure TBA/water system. Unfortunately, salting out arguments as to the existence of a nearby critical point in a binary system are based on inferences arrived a t from observations on a ternary system. Especially in light of the numerous organic solvent systems which show maxima/minima in various bulk properties, great care should be taken in applying bulk phase transition arguments to explain such behavior. The fact that the correlation lengths measured for this system do not go through a large maximum a t any concentration or temperature currently studied is evidence that if a critical point exists in liquid-liquid phase space, it is not extremely nearby to the room temperature and pressure portion of the phase diagram which has been probed in this and another work.” The idea of TBA being a promoter of water-water structure has been presented by workers in NMR,*OS~~ dielectric relaxation:* and molecular dynamics.23 T h e molecular dynamics work a t 0.032XTBA by Tanaka et aI.23is of particular interest. The atom-atom radial distribution functions (AArdfs) for water taken from the dynamics study of Tanaka et al. show that, compared to pure water, the addition of T B A to water gives higher correlations between water molecules out to 5 A. This is indicated by the strengthening of the maxima and the sharpening of the minima of the AArdsf. Calculations by Tanaka et al. also show that the
potential energy of water decreases upon the addition of TBA. As compared to pure water, these dynamics results indicate that upon addition of TBA an increase in the ordering of the water lattice occurs along with an ensuing increase in the rigidity of the water lattice. This increasing rigidity of the solvent lattice would certainly explain the initial increase in the C, of the system upon addition of TBA, since the speed of sound is higher in solids than it is in liquids. Because of the low concentration at which the work of Tanaka et al. was performed though, it gives no insight as to why the C, go through a maximum and then decrease as the XTBA increases. Overall, the authors of this paper are of the opinion that addition of TBA to water up to XTBA = 0.05 principally has the effect of increasing the liquid structure. This viewpoint seems to be reasonably well supported by this work, the results of NMR studdielectric relaxation data,42 and molecular dynamics XTBA = 0.05, the decrease in the speed of ~ a l c u l a t i o n s .Above ~~ sound is indicative of a loss of rigidity (increase in adiabatic compressibility) of the solution. The similarities between the activated state parameters and T’S from dielectric relaxation measurements and from acoustics measurements a t O.lO%TBA indicate that, a t this concentration, the behavior of the system is compatible with a model of structural relaxation due to some cooperative motion of a collection of molecules. This structural relaxation is present a t mole fractions of TBA which are greater than where the minimum in the compressibility of the system is observed. The relaxation could be due to the presence of clathrate-type oligimers, but we were not able to directly detect the presence of these oligimers by either PCS or depolarized interferometry.
Acknowledgment. This work was supported by National Science Foundation Grants CHE82-00512 and CHE85-11178. T.M.B. also thanks the other members of R.P.’s laboratory for their help and support. Registry No. TBA, 75-65-0; H,O,7732-18-5
(46) Timrnermans, J.; Poppe. G. C. R . Acad. Sei. 1935. 201, 608.
Experimental Test of McDougall’s Theory for the Onset of Convective Instabilities in Isothermal Ternary Systems Donald G. Miller* and Vincenzo Vitaglianot Chemistry and Materials Sciences Department, Lawrence Livermore National Laboratory, Livermore, California 94550 (Receiued: August 21, 1985)
The dynamic theory of McDougall for convective instabilities induced by isothermal diffusion has been tested experimentally. These experiments, using two homogeneous layers separated initially by a sharp boundary (free diffusion), were done on the ternary system SrCI, (0.5 M)-NaCI (0.5 M)-H20, chosen because of its large cross-term diffusion coefficients. The experiments accurately verify McDougall’s theory for the onset of the fingering instability at the center of the boundary. As predicted, this instability occurs first for compositions where the system is statically stable, Le., no diffusion-induced density inversion. At boundary edges, density inversion and dynamic instability of the diffusive-overstable type begin simultaneously. The experiments are consistent with theory, although instabilities were not detected as close to the onset point as for fingering. Optical arrangements are described which are sensitive enough to detect the instabilities. Theoretical consequences of the free diffusion boundary conditions are that (1) at the edges the fingering instability condition is proportional to the static-diffusive one, and (2) the onset compositions at the edges depend only on D,,, whereas fingering at the center depends on H , as well.
1. Introduction
In recent years, the occurrence of convective instabilities has been observed or postulated in a wide variety of phenomena which involve diffusion.’,* These instabilities involve at least two diffusive components, which may involve heat, matter, magnetic field, angular momentum, etc. When there are just two, the process ‘Permanent address: Dipartimento di Chimica, Via Mezzocannone 4, Universitl di Napoli, 80134 Naples, Italy.
is called double diffusive convection (DDC). Examples of DDC are the heat-salt diffusivities which give rise to fingering in the oceans’,* and the isothermal diffusivities of two solutes which can give rise to convections which ruin measurements of diffusion coefficient^.^-^ Depending on experimental circum( 1 ) Huppert, H. E.; Turner, J. S. J . Fluid Mech. 1981, 106,299. (2) Turner, J. S. Annu. Reu. Fluid Mech. 1985, 17, 11. (3) Vitagliano, V.; Zagari, A,: Sartorio, R.; Corcione, M . J . Phys. Chem. 1972, 76, 2050.
0 1986 American Chemical Society
