7740
J . PhJs. Chem. 1989, 93,1140-7744
Second Hyperpolarlzablnties and Statlc and Optlcai-Frequency Polarizabltty Anisotropies of Benzene, 1,3,5-TrIfluorobenzene, and Hexafluorobenzene Ian R. Gentlela and Geoffrey L. D. Ritchie*iIb School of Chemistry, University of Sydney, New South Wales 2006, Australia, and Department of Chemistry, University of New England, New South Wales 2351, Australia (Received: January 24, 1989; In Final Form: May 22, 1989)
Measurements of the electrwptical Kerr effects at 632.8 nm of gaseous benzene, 1,3,5-trifluorobenzene,and hexafluorobenzene over a range of temperature (~315-490K) and pressure (=10-90 kPa) are reported. Analyses of the temperature dependences show that for all three molecules the temperature-independent contribution to the effect, which arises from distortion of the electronic structure by the electric field, is negligibly small in comparison with the temperature-dependent contribution from molecular orientation. Reliable values of the static polarizability anisotropies are also obtained, and these are compared with optical-frequency polarizability anisotropies derived from Rayleigh depolarization ratios. Because of the oppositely signed vibrational contributions in benzene and hexafluorobenzene,the ratio of the static to the optical-frequency anisotropy is less than unity for benzene but greater than unity for hexafluorobenzene. Such an outcome effectivelyvitiates the common assumption that the ratio of the anisotropies can be approximated by the ratio of the mean static and optical-frequency polarizabilities.
Introduction Electric-field-induced birefringence in fluids has long been recognized as a valuable source of information in relation to molecular polarizability anisotropies.* Due in large measure to the work of the Le FEvres and their collaborators in Sydney, the Kerr effects of several hundred molecules, particularly aromatic species such as benzene and substituted benzenes, as solutes in nondipolar solvents, usually at 298 K, have been r e p ~ r t e d . ~The analysis of such results to yield polarizability anisotropies is, however, seriously complicated by three main factors: the consequences of solute-solvent interactions (the "solvent e f f e ~ t " ) the ; ~ contribution, usually unknown, to the observed effect from the Kerr hyperpolarizabilitie$ and, normally, a lack of information as to the precise relationship between components of the optical-frequency and static polarizability tensors. The first of these difficulties is avoided if the measurements can be performed on low-density gases or vapors, rather than on dilute solutions; the second is largely overcome if such observations are made over a range of temperatures, so that the variously temperature-dependent contributions from the polarizability and hyperpolarizabilities can be separated; and the third may not arise if the optical-frequency polarizabilities are independently available from, for instance, the depolarization ratio for Rayleigh scattered light. Over the past 20 years studies of the temperature dependence of the Kerr effect in gases and vapors have provided reliable polarizability anisotropies and Kerr hyperpolarizabilities for a h i t e d range of small molecules.6 However, benzene is the only
( I ) (a) University of Sydney. (b) University of New England. (2) Stuart, H. A . Molekulstruktur; Springer-Verlag: Berlin, 1967; Chapter 8, pp 393-438. (3) The references which follow are representative reviews of the vast literature of this topic. (a) Le Ftvre, C. G.; Le Ftvre, R. J. W. Reu. Pure Appl. Chem. (Australia) 1955, 5, 261-318. (b) Le Ftvre, C. G.; Le Ftvre, R. J. W . The Kerr Effect. In Techniques of Chemistry, Part IIIC; Weissberger, A., Ed.; Wiley-Interscience: New York, 1972; Vol. 1 , Chapter VI, pp 399-452. (c) Aroney, M. J. Angew. Chem. 1977, 16, 663-673. (4) (a) Buckingham, A. D.; Stiles, P. J.; Ritchie. G. L. D. Trans. Faraday SOC.1971, 67, 577-582. (b) Vrbancich, J.; Ritchie, G.L. D. Chem. Phys. L e f t . 1983, 94, 63-68. (5) Bogaard, M. P.; Orr, B. J. Electric Dipole Polarizabilities of Atoms and Molecules. In MTP International Reuiew of Science; Buckingham, A . D., Ed.; Phys. Chem. Ser. 2; Butterworths: London, 1975: Vol. 2, pp 149-194.
aromatic species which has been so examineda and, in view of the assumptions necessarily made in the analysis of solution-phase data, there is a need to extend the enquiry to substituted benzenes and other relatively large molecules. To this end, we have constructed greatly improved equipment for measurements of the temperature and pressure dependence of electric birefringence in gases7 and, most particularly, in the vapors of volatile liquids and solids. The investigation described here is a rigorous examination, over a wide range of temperature, of benzene, 1,3,5-trifluorobenzene, and hexafluorobenzene, three aromatic molecules whose electric and magnetic properties are of great interest.* Experimental Section Improved equipment and procedures for measurements of the temperature dependence of electric-field-induced birefringence in gases have recently been described.7 In order to introduce vapors of substances that are liquids at room temperature into the cell, a vapor-handling system, shown in Figure 1, similar to that used by Lukins, Buckingham, and Ritchie* was constructed. One side of a glass U-tube which contains the sample is pressurized with argon, so that the liquid is forced to rise into the heated side of the tube and therefore to vaporize into the cell. The main advantage of this arrangement lies in the fact that the pressure of the vapor in the cell can be determined indirectly from the argon pressure and the difference in liquid heights in the U-tube; the need to heat the pressure transducer to avoid condensation is therefore eliminated. A pair of two-way ball valves facilitates rapid interconversion, without disassembly, between the systems used for the study of gases, on the one hand, and vaporized liquids, on the other. To confirm the long-term constancy of the calibration of the detection system, the birefringence induced in a sample of carbon (6) (a) Buckingham, A. D.; Orr, B. J. Trans. Faraday SOC.1969, 65, 673-681. (b) Buckingham, A. D.; Orr, B. J. Proc. R . Soc. London, Ser. A 1968, 259-269. (c) Buckingham, A . D.; Bogaard, M. P.; Dunmur, D. A.; Hobbs, c. P.; Orr, B. J. Trans. Faraday SOC.1970, 66, 1548-1553. (d) Bogaard, M. P.; Buckingham, A. D.; Ritchie, G. L. D. Mol. Phys. 1970,18, 575-576. (e) Buckingham, A . D.; Sutter, H. J . Chem. Phys. 1976, 64, 364-369. (0 Bogaard, M. P.; Orr, B. J.; Buckingham, A. D.; Ritchie, G. L. D. J. Chem. SOC.,Faraday Trans. 2 1978,74, 1573-1578. (g) Bogaard, M. P.; Buckingham, A. D.; Ritchie, G . L. D. J . Chem. Soc., Faraday Trans. 2 1981, 7, 1547-1551. (h) Bogaard, M. P.; Buckingham, A. D.; Ritchie, G. L. D. Chem. Phys. Lett. 1982, 90,183-187. (7) Gentle, I. R.; Laver, D. R.; Ritchie, G. L. D. J . Phys. Chem. 1989,93, 3035-3038. (8) Lukins. P. B.; Buckinnham, A . D.; Ritchie, G. L. D. J . Phys. Chem. 1984,'88, 2414-2418.
0022-3654/89/2093-1140$01.50/00 1989 American Chemical Society
The Journal of Physical Chemistry, Vol. 93, No. 22, 1989 7741
0
Figure 1. Vapor-handling system: M, manifold; R, liquid reservoirs; AR, argon cylinder; B, two-way ball valves; PT,pressure transducer head; U, glass U-tube; S,gas sample cylinder; KC, Kerr cell; F and C, fine and coarse bellows valves; T, liquid nitrogen trap; VP, vacuum pump.
dioxide at =670 kPa (5000 mmHg) and each chosen temperature was compared with results obtained in the definitive temperature dependence study of this compound;' over the course of the investigation reported here deviations of less than 1% were normally observed. Samples used were as follows: benzene (Merck analytical reagent, 199.7%), dried over sodium wire and fractionally distilled, purity >99.9%; 1,3,5-trifluorobenzene (Aldrich, >98%), fractionally distilled from phosphorus pentoxide, purity >99.95%; and hexafluorobenzene (Bristol Organics Ltd), purified by contact with oleum for 5 h at room temperature, washed four times with water, dried over calcium sulfate, and fractionally distilled from phosphorus pentoxide, purity >99.9%. The purities were determined by gas chromatography; immediately prior to use each sample was subjected to several freeze-pump-thaw cycles and vacuumdistilled into the apparatus. The definition of the molar Kerr constant, ,K, is6a ,K = 6nVm[(n2+ 2 ) 2 ( ~+ r 2)*]-'[(n11- n,)F2]+o
(1) where n and cr are the refractive index and relative permittivity of the medium in the absence of the field; rill - n, is the fieldinduced birefringence for light polarized parallel and perpendicular to the uniform electric field, F ;and V, is the molar volume. To take account of molecular interactions, the Kerr constant can be expressed in terms of the molar volume as ,K = AK BK~,-' ... (2)
+
+
in which AK and BK are the first and second Kerr virial coefficients, respectively. Measurements of the vapor-state electric-field-induced birefringences of benzene, 1,3,5-trifluorobenzene, and hexafluorobenzene were made at eight or more temperatures within the available span (=3 15-490 K) and, at each temperature, over a range of pressures (=IO-90 kPa) up to about 70% of the equilibrium vapor pressures. The observed nulling voltages were used to establish the birefringences of the vapors under the specified conditions; from these, values of ,KO = (2/27)(q - n,)VmF2 were derived and fitted to the relatiodb = AK
+ [BK + AK(2Ae + f/ZAR)]V,-'
(3)
10 -1
20 -3
30
V, /mol m
Figure 2. Density dependence of the Kerr effect of 1,3,5-trifluorobenzene at 424.2 K. TABLE I: Temperature Dependence of the Vapor-State Kerr Effects of Benzene, 1,3,5-Trifluorobenzene, and Hexafluorobenzene at 632.8 nm i03Ti/ no. of 106~1m3 1027,4,/m5
T/K
K-I
490.1 462.4 423.8 396.1 381.2 367.8 349.2 330.6 315.5
2.040 2.163 2.360 2.525 2.623 2.719 2.864 3.025 3.170
490.1 457.6 424.2 395.9 381.7 349.4 331.0 315.8
2.040 2.185 2.357 2.526 2.620 2.862 3.021 3.167
8 8 7 7 7 8 7 6
490.5 489.8 461.0 457.5 424.3 423.6 395.8 395.6 381.5 381.4 367.7 330.9 330.8 315.4 315.4
2.039 2.042 2.169 2.186 2.357 2.361 2.527 2.528 2.621 2.622 2.720 3.022 3.023 3.171 3.171
8 6 8 7 7 7 6 7 8 8 9 7 7 5 5
pressures
7
IO 4 5 6 5 7 7 5
mol-l
V-* mol-l
-450 -514 -627 -730 -801 -877 -991 -1135 -1280
8.70 f 0.10 9.25 f 0.08 10.2 f 0.3 10.82 f 0.11 11.29 f 0.07 11.85 f 0.10 12.15 f 0.08 12.75 f 0.08 13.4f 0.3
-502 -604 -731 -873 -963 -1234 -1450 -1660
12.56 f 0.07 13.27 i 0.05 14.51 f 0.08 15.60 f 0.11 16.05 f 0.05 17.45 f 0.04 18.39 f 0.08 19.6 f 0.4
-553 -555 -666 -682 -836 -840 -1015 -1017 -1143 -1144 -1259 -1771 -1773 -2056 -2056
15.05 i 0.10 15.06 f 0.10 16.58 f 0.10 15.84 f 0.10 17.63 f 0.15 17.49 f 0.10 18.70 f 0.10 18.73f 0.10 19.53 f 0.10 19.30 f 0.10 20.11 f 0.11 21.33 f 0.22 22.34 f 0.18 23.1 f 0.7 22.4 f 0.6
p/kPa Benzene
14.9-84.4 12.5-87.7 37.6-82.5 20.7-64.0 11.9-71.7 12.0-52.9 18.7-70.5 12.5-44.7 13.5-20.9
1,3,5-Trifluorobenzene
21.5-85.3 21.1-85.4 22.9-84.2 19.6-82.3 21.7-83.2 18.5-66.0 19.1-45.3 11.9-21.4
Hexafluorobenzene
10.9-84.1 20.9-88.4 20.0-88.2 11.3-80.5 11.0-82.4 30.4-88.1 23.1-80.4 17.8-82.5 11.7-84.4 11.7-82.9 17.7-75.6 12.0-43.1 11.5-42.7 9.3-19.1 10.8-19.3
in which A, and AR are the low-density molar dielectric polarization and refraction, respectively. Density virial coefficients, B, were used to obtain molar volumes, V,, from the vapor temperatures and pre~sures,~ and in all cases the corrections for nonideality were < 1.5%. Figure 2 shows, as a typical example, the dependence of ,KO on V,-' for 1,3,5-trifluorobenzene at 424.2 K. The results are summarized in Table I, where the errors quoted are precisions taken as the standard deviations obtained from the
least-squares fitting of straight lines to the density-dependence data; with calibration errors and, at the lower temperatures, uncertainties in the pressures, the overall accuracy of the measurements is estimated as f 2 % . Because of the relatively low pressures at which the observations were made, the second Kerr virial coefficients, BK,calculable from eq 3, were too poorly determined to merit further consideration here. Although the Kerr effects of benzene, 1,3,5-trifluorobenzene, and hexafluorobenzene as solutes in nondipolar solvents have been much studied,I0 only benzene has previously been examined in
(9)Dymond, J. H.;Smith, E. B. The Virial Coefficients of Pure Gases and Mixtures; Clarendon Press: Oxford, U.K., 1980. Values for 1,3,5-trifluorobenzene were estimated as the means of the corresponding values for benzene and hexafluorobenzene.
(IO) (a) Aroney, M. J.; Cleaver, G.; Pierens, R. K.;Le E w e , R. J. W. J . Chem. Soc., Perkin Trans. 2 1974,3-5. (b) Ritchie, G.L. D.; Vrbancich, J. Aust. J . Chem. 1982,35, 869-880.
7742 The Journal of Physical Chemistry, Vol. 93, No. 22, 1989 25
7
I
Gentle and Ritchie TABLE II: Analysis of the Temperature Dependence of the Zero-Density Kerr Effects of Benzene, 1,3,5-Trifluorobenzene, and Hexafluorobenzene at 632.8 nm C6H6
lo2' X i n t e r c e d m 5 V2 0.3 f 0.3
r
.
C6F6
C6H3F3
0.40 f 0.13
0.6 f 0.6
4.15 f 0.12
5.96 f 0.05
7.16 f 0.25
0.4 f 0.4 34.2 f 1.0 -6.23 f 0.19 -5.48 f 0.23 0.88 f 0.05
0.5 f 0.2 49.0 f 0.4 -6.51 f 0.20 -7.54 f 0.24 1.16 f 0.05
0.7 f 0.7 58.9 f 2.1 -7.06 f 0.21 -8.3 f 0.4 1.18 f 0.06
I
mol-! a X siope/m5 V-2 mol-l K" 1omyK/C m4 v-j 1080AaA~o/C2 m4 V-2 1OmAa/C m2 V-I 1OmAa0/C m2 V-I
'>
In
E . Y
rcq N
hao/
z
Intercept and slope of plot of A K against T 1(eq 4). ref 19; see text for explanation.
2.6
2.2
3.0
TABLE 111: Estimated Electronic and Vibrational Contributions to tbe Static Polarizability Anisotropies of Benzene, 1,3,5-Trifluorobenzene, and Hexafluorobenzene
3.4
103T -'I K-'
Figure 3. Temperature dependence of the zero-density molar Kerr constants of benzene, 1,3,5-trifluorobenzene, and hexafluorobenzene.
the vapor phase. A single-temperature measurement by Stuart and Volkmannll 55 years ago yielded a bulk Kerr constant, K, at 546 nm, 386.8 K, and 101.3 kPa of 5.56 X esu, which corresponds to a molar Kerr constant, ,K, of -14.5 X m5 mol-' at 546 nm or, with a correction for the wavelength dependence of the polarizability anisotropy, ~ 1 3 . X7 m5 V2 mol-' at 632.8 nm. Interpolation from the present measurements indicates that the value at 632.8 nm, 386.8 K, and 101.3 kPa should be =11.9 X m5 V2mol-'; the original value, obtained under difficult conditions with a visual rather than a photometric detection system, was therefore only about 15% too high. More recently, an antecedent of our apparatus was used to examine the temperature and pressure dependence of the Kerr effect of benzene.6d However, the constraints of these measurements were such that the density dependence of the effect, as expressed by eq 3, could not be detected, and it is now clear that the putative zero-density molar Kerr constants, A K , used in the subsequent analysis were about 10% too high. In addition, the temperature extrapolation ratio R = T,,,/(T,, - Tmin)was 4.8, as compared to the much more favorable value of only 1.8 in the present investigation.
Discussion Temperature Dependence of Kerr Effects. For a nondipolar molecule with an axis of 3-fold or higher symmetry (labeled with subscript 2 ) the classical statistical mechanical expression for the zero-density molar Kerr constant, A K , is,12 in SI ~ n i t s ' ~ - ' ~ AK = (N*/405to)[5yK
+ (kT)-'AaAaO]
Data from
(4)
in which y K = [3y,8,8(-w;w,0,0) - yaa88(-w;w,0,0)]/10 is the second Kerr hyperpolarizability;5 and A a (=a,,- a x xand ) Aao (=cuoZz - 'yoxx) are the anisotropies in the optical-frequency and static polarizabilities, respectively. The observed Kerr constant trierefore represents the sum of two contributions, a temperature-independent term from distortion of the electronic structure by the field and a temperature-dependent term from molecular orientation; and AK should exhibit a linear dependence on TI. Figure 3, which shows plots of A K against TI,confirms this expectation; and Table I1 contains the intercepts and slopes of (11) Stuart, H. A.; Volkmann, H. Ann. Phys. 1933, 18, 121-149. (12) Buckingham, A. D.; Pople, J. A. Proc. Phys. SOC.A 1955, 68, 905-909. (1 3) Lukins, P. B.; Laver, D. R.; Buckingham, A. D.; Ritchie, G. L. D. J . Phys. Chem. 1985, 89, 1309-1 3 12. (14) The procedure outlined in ref 13 gives the relationships for interconversion of polarizabilities, a,and hyperpolarizabilities, y, from SI units to cgs electrostatic units (esu) as (a'/esu) = 106(4sc0)-1(ac m2 v-l) = 0.8988 X 1016((u/Cm2 V-l) and (y'lesu) = l@(4sq,)-2(r/C m/V-l) = 0.8078 x 102S(y/C m4 V-3).
10'oAao,I/C m2 V-' 1O4AaoVib/Cm2 V-I 1OmAa0,,/C m2 V-l 10mAaoe,,,/C m2 V-I
-5.75 -6.01 +0.86 -4.89 -5.48 f 0.2 -7.54 f 0.24
-6.43 -1.11 -7.54 -8.34 f 0.38
the weighted-fit least-squares straight lines, together with the derived second Kerr hyperpolarizabilities and polarizability anisotropies. In the case of benzene, the quantities 1060yK/Cm4 V-3 and 1OE0AaAao/C2m4 V2obtained by Bogaard, Buckingham, and Ritchie (0.8 f 0.4; 36.0 f 1.0)6dand in the present work (0.4 f 0.4; 34.2 f 1.0) are in good agreement, although a much narrower temperature range was available in the earlier investigation. It is also immediately obvious from the intercepts that for all three molecules the temperature-independent contribution to the Kerr constant is very small, indeed almost negligible, in comparison to the temperature-dependent contribution at normal temperatures. At 300 K the percentages of AK which originate in yKare C6H6 2 f 2%, C6H3F32 f 1%, and C6F6 2 f 3%; and we consider that such results are likely to be typical of aromatic or other highly anisotropic molecules. Despite the unavoidably large uncertainties, the present results suggest that yKincreases with the number of fluorine substituents, as perhaps might be expected. Values of closely related second hyperpolarizabilities of benzene have also been obtained from studies of other nonlinear optical phenomena (third-harmonic generation,I5 three-wave mixing,16and electricfield-induced second-harmonic generati~n'~); however, comparisons with the result reported here are complicated by dispersion and other factors. A knowledge of the optical-frequency polarizability anisotropies, Acu, from the depolarization ratios for Rayleigh scattered makes possible the extraction of the static polarizability anisotropies, Amo, from the slopes of the plots of AK against TI.In the cases of 1,3,5-trifluorobenzene and hexafluorobenzene such data have been reportedI9 for wavelengths of 488.0 and 514.5 nm, but not for 632.8 nm. However, for optical frequencies, w, small relative to electronic absorption frequencies, A a is a linear function of w2, so that short extrapolations provided the required information. Values of Acu, Aao, and the ratio Acu0/Acu are shown in (15) Hermann, J. P. Opr. Commun. 1973, 9, 74-79. (16) (a) Levenson, M. D.; Bloembergen, N. J . Chem. Phys. 1974, 60, 1323-1327. (b) Levenson, M. D.; Bloembergen, N. Phys. Reu. B 1974, IO, 4447-4463. (17) (a) Levine, B. F.; Bethea, C. G. J. Chem. Phys. 197CC63.2666-2682. (b) Ward, J. F.; Elliott, D. S. J . Chem. Phys. 1978, 69, 5438-5440. (c) Shelton, D. P. J . Opt. SOC.Am. B 1985, 2, 1880-1882. (d) Pantinakis, A.; Dean, K. J.; Buckingham, A. D. Chem. Phys. LeU. 1985, 120, 135-139. (e) Shelton, D. P. Chem. Phys. Letr. 1985, 121, 69-72. (18) Alms, G. R.; Burnham, A. K.; Flygare, W. H. J . Chem. Phys. 1975, 63, 3321-3326. (19) Bogaard, M. P.; Buckingham, A. D.; Pierens, R.K.; White, A. H. J . Chem. SOC.,Faraday Trans. I 1978, 74, 3008-3015.
The Journal of Physical Chemistry, Vol. 93, No. 22, 1989 7743
Kerr Effects Of C6H6, C6H,F3, and C6F6
TABLE IV: Polarizations, Wavenumbers, Integrated Absorption Coefficients, and Vibrational Contributions to the Parallel and Perpendicular Components of the Polarizability of Benzene and HexafluorobenzeneO 1O1*Sj/cm mode sYm polarizn aj/cm-l F7' molecule-' (Yo,, aO1 aOribb hoibc Benzene u4
a2u
II
673
1.0413
14.64
0.9060 0.9060
V14 v13
VI2
el, el, el,
I
I I
1038 1486 3080
1.0069 1.0008 1.oooo
1.47 2.17 9.91
0.0185 0.0132 0.0141 0.0458 0.333
+0.860
Hexafluorobenzene 215
1S542
0.43
0.391 1
315 1006 1531
1.2833 1.008 1 1.0007
0.42 67.40 90.79
0.0724 0.9035 0.5216
0.391 1
I I I
1.4974 1.129
'Spectroscopic data from refs 25 and 26; polarizabilities quoted here as 10%oII/C m2 V-I, etc. Table 11; the quoted errors in the latter two quantities take into account uncertainties in the slopes and in the optical-frequency anisotropies. The results for Aa0/Aa are of particular interest because, although the ratio for benzene (0.88 f 0.05)is confirmed as less than unity, those for 1,3,5-trifluorobenzene (1.16 f 0.05) and hexafluorobenzene (1.18 f 0.06) are significantly greater than unity, an unexpected outcome. The static polarizability anisotropy differs from the optical-frequency value for two reasons: first, the frequency dependence of the dominant electronic component; and second, the fact that in a static field, as opposed to an optical-frequency field, the contribution from the vibrational states of the molecule is not negligible. In view of the present observations it is instructive to separate these two factors and attempt to estimate hao from experimental data, as follows. Electronic Contribution, Aaocl. Since vibrational states do not contribute significantly to the optical-frequency polarizability anisotropy, extrapolation to zero frequency of values of A a obtained from Rayleigh depolarization ratios for visible frequencies provides an estimate of the electronic contribution, Aaoel, to Aao. To interpret the frequency dependence of Aa, use is made of the sum-over-states formulaS
in which the superscripts 11 and I refer to directions parallel and perpendicular to the molecular z axis: C/l and Ci* are proportional to the oscillator strengths of the ith parallel and perpendicular electronic transitions; will and wiL are the frequencies of the transitions; and w is the frequency of the incident light. If the dominant transitions that are responsible for the dispersion in the parallel and perpendicular components of the electric polarizability are well separated from optical frequencies (Le., will, wiL >> w ) , the summations can be replaced by single terms for each component. Equation 5 can then be written as
where C&ll, w,&l and CcfrL,wcffL are effective oscillator strengths and frequencies for the parallel and perpendicular components, respectively. The polarizability anisotropy would therefore be expected to exhibit a linear dependence on w2 in the optical region and, as already noted, such behavior has been confirmed experimentally for benzene and a range of small molecule^.^^^^^ Additional evidence to support this contention has been obtained from recent a b initio molecular-orbital calculations,20 which indicate
+
bffOvib = 1 / 3 ( a 0 12a0,).
-1.106
cAaorib= aoll- aol.
that to an excellent approximation both components are linear in w2 at wavelengths greater than -450 nm. It may also be noted that the lowest allowed transitions of each polarization in 1,3,5trifluorobenzene and hexafluorobenzene occur at energies similar to or higher than those for benzene?l so that eq 6 should provide an adequate description of the dispersion in the polarizability anisotropies of all three molecules. In the case of benzene, extrapolation of available gave the value of AaO,,shown in Table 111. However, for the other two molecules the requisite measurements have been performed at only two visible frequencies, so that extrapolations to wz = 0 could be hazardous. For these it was deemed preferable to assume that the dispersion in PaeI scales to that of benzene, an approximation which is justifiable on the basis of the similarity in energies and oscillator strengths*I of the principal electronic states of the three species. Table 111 contains the values of Aaocl so obtained. Vibrational Contribution, A a o ~ b Vibrational . contributions to aoll,aoL,and Aao can be determined from infrared intensities if the symmetries of the relevant states are taken into account. Here the treatment of Gough, Orr and Scoles,22as applied to carbon dioxide, was followed. The expression for the vibrational contribution to the parallel or perpendicular component of the polarizability is
where gj is the degeneracy of the vibrational state, uj is the wavenumber (in cm-I) of the transition from the nondegenerate ground state, and Si is the gas-phase integrated absorption coefficient (in cm/molecule). Three factors contribute to the quantity Fj:23*24 a Boltzmann factor, which accounts for induced emission; the reciprocal of the vibrational partition function; and the isotopic abundance of the main species. The vibrational partition function allows for thermal population of excited vibrational states, but for benzene and hexafluorobenzene the resolution of the infrared spectra is not sufficiently high to resolve hot-band intensities which will be included in the absorption (20) Spackman, M. A., unpublished SCF calculations with a 6-31G(+sd+sp) basis set optimized for accurate prediction of electric polarizabilities; the calculated optical-frequencymean polarizability of benzene is within 1.5% of the experimental value. (21) Philis, J.; Bolovinos, A.; Andritsopoulos, G.;Pantos, E.; Tsekeris, P. J . Phys. B At. Mol. P h p . 1981, 14, 3621-3635. (22) Gough, T. E.; Orr, B. J.; Scoles, G . J . Mol. Spectrosc. 1983, 99, 143-1 58. (23) Pugh, L. A.; Narahari Rao, K. Intensities from Infrared Spectra. In Molecular Spectroscopy: Modern Research; Narahari Rao, K., Ed.;Academic Press: New York, 1976; pp 165-227. (24) Suzuki, I. J . Mol. Spectrosc. 1980, 80, 12-22.
7744
J . Phys. Chem. 1989, 93, 7744-7745
coefficient. It is appropriate, therefore, to take the partition function as unity, as has also been done for the isotopic abundance. The necessary spectroscopic data are available for benzene and h e x a f l ~ o r o b e n z e n e ~but, ~ * ~unfortunately, ~ not for 1,3,5-trifluorobenzene. Table I11 summarizes the analysis of the vibrational contribution to the static polarizability anisotropy of these two molecules; and Table IV details the polarizations, wavenumbers, and absorption coefficients of the four infrared-active vibrations, together with the individual contributions which these make to the parallel and perpendicular components of the polarizability. A similar treatment of the infrared spectra of benzene and hexafluorobenzene has been given by Bishop and C h e ~ n g , ~ ' although their results are approximate in that the population of excited states was assumed to be negligible; polarizabilities in agreement with theirs emerge if the factors Fjare taken as unity. Comparison ofAa0,, and AaoeXpt.With the assumptions made in the extrapolation procedure and the level of reliability of the infrared intensities in mind, the agreement between the values of Aao estimated in the manner described here and those obtained from the temperature dependence of the Kerr effect is good. Tables I11 and IV also reveal the reason for the unexpectedly different values of the ratio AaO/Aa for benzene and hexafluorobenzene; for the former, AaoVibis of opposite sign to Aaoel, Aao < Aa, and AaO/Aa < 1; but for the latter, Aaoib has the same sign as Aaoel, Aao > Aa, and Aao/Aa > 1 . It is also of interest to note that the vibrational polarizability is dominated in the case of benzene by the low-wavenumber out-of-plane C-H bending ( v 4 ) mode, but in the case of hexafluorobenzene by the high-intensity in-plane stretching (Y~~.Y,~) modes. As already mentioned, infrared intensities are unavailable for 1,3,5-trifluorobenzene; however, the experimental results reported here, in particular the (25) Spedding, H.;Whiffen, D. H . Proc. R . Soc. London, Ser. A 1956, 238, 245-255. (26) Steele, D.;Wheatley, W. J . Mol. Spectrosc. 1969, 32, 265-275. (27) Bishop, D.M.;Cheung, L. M. J . Phys. Chem. ReJ Data 1982, 11, 119-133.
value of Aa0/Aa, indicate that A a o ~isb negative for this molecule and comparable in magnitude to that of hexafluorobenzene. Summary The present investigation of the temperature dependence of vapor-state electric-field-induced birefringence has demonstrated that for benzene, 1,3,5-trifluorobenzene, and hexafluorobenzene the temperature-independent contributions to the effects at normal temperatures are negligibly small. In consequence, the assumption commonly made in single-temperature solution-phase Kerr-effect studies, to the effect that the second hyperpolarizabilities can be ignored, appears to be acceptable, at least for these and similarly anisotropic molecules. Analyses of the temperature dependences in conjunction with the known optical-frequency polarizability anisotropies have also yielded reliable static polarizability anisotropies. In the cases of benzene and hexafluorobenzene these are reasonably predicted from the dispersion in the dominant electronic component, together with appropriate allowance for the much smaller contribution (-1 5%) from molecular vibrations. It is of particular interest to note that, because of the oppositely signed vibrational contributions in these two molecules, the ratio of the static to the optical-frequency anisotropy is less than unity for benzene, as expected, but significantly greater than unity for hexafluorobenzene. Such an outcome greatly reduces the reliability and the usefulness of the assumption which usually underlies the analysis of solution-phase Kerr constants, namely that the ratio of the anisotropies can be approximated by the ratio of the mean static and optical-frequency polarizabilities. Acknowledgment. A Commonwealth Postgraduate Research Award (to I.R.G.), financial support from the Australian Research Council (to G.L.D.R.), technical assistance from Dr. D. R. Laver (University of Sydney), and helpful discussions with Professor B. J. Orr (Macquarie University) and Dr. M. A. Spackman (University of New England) are gratefully acknowledged. Registry No. Benzene, 71-43-2; 1,3,5-trifluorobenzene,372-38-3; hexafluorobenzene,392-56-3.
COMMENTS Comments on "On the Structure of Aggregates of Adsorbed Surfactants: The Surface Charge Density at the Hemimkelle/Admicelle Transition"' Sir: In a recent paper, Yeskie and Harwell' have suggested that
formation of bilayered admicelles is favored during the adsorption of ionic surfactants from aqueous solutions onto charged minerals at surface charges away from the point of zero charge of the mineral; conversely, the formation of monolayered hemimicelles is thought to be favored close to the point of zero charge. The above conclusion was based on a theoretical approach which relied solely on the calculated chemical potentials of the surfactant molecules in hypothetical hemimicellar and admicellar aggregates. Lower values of chemical potential were arrived at for the admicellar case and higher values for the hemimicellar case. They further inferred that both hemimicelles and admicelles might be formed simultaneously on a heterogeneous surface where most highly charged patches may favor bilayers and less highly charged patches may favor hemimicelles. We do not intend to discuss the validity of the assumptions involved in the derivations presented, but do want to alert caution Due to editorial error, a reply to this Comment was published previously. See: Harwell, J. H.; Yeskie, M. A. J . Phys. Chem. 1989, 93, 3372.
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before accepting generalizations based on thermodynamic treatment alone, of hypothetical aggregate systems. The evolved model should be able to contain the experimental results from both bulk and spectroscopic studies reported on these systems. We felt it necessary to clarify certain pertinent points about hemimicelles relevant to this context so that some of our recently published and forthcoming works on adsorbed layers of sodium dodecyl sulfate on alumina (same system as studied in ref 1) could be viewed in the right perspective in relation to the model we suggested earlier. In the bilayer model, a rigid parallelism has been sought between surfactant aggregation processes in solution and on the solid surface mainly to account for the micelle-like nature of the adsorbed aggregates. The aggregation number and other solution properties of micelles remain the same over a wide concentration range whereas the aggregation number of the surfactant adsorbates undergoes continuous changes. The bilayer model fails to adequately represent the evolutionary changes of the aggregates before maximum adsorption density is attained. The bilayered admicelles, if formed from the beginning of the aggregation process, would be expected to result in a similar microenvironment within the aggregates. The limitation of this model is to be understood not only in the light of the contact-angle studies and particle flotation ( I ) Yeskie, M . A.; Harwell, J. H . J . Phys. Chem. 1988, 92, 2346
0 1989 American Chemical Society
