J. Phys. Chem. 1985,89, 3409-3411
3409
Cotton-Mouton Effect, Anisotropic Polarizability, and Magnetic Hyperpolarizability of Sulfur Dioxide P. B. Lukins and C. L. D. Ritchie* School of Chemistry, University of Sydney, New South Wales 2006, Australia (Received: February 6, 1985)
Measurements of the Cotton-Mouton effect of sulfur dioxide over a range of temperature (=298-463 K) and pressure (=120-320 kPa) are reported. The temperaturedependent contribution to the effect is combined with known values of the magnetizabilities, the mean polarizability, and the polarizability tensor component ratio derived from the pure rotational Raman spectrum to obtain all three elements of the optical-frequency polarizability of the molecule. Results (lO'"'a,/C m2 V-I, etc., x direction perpendicular to molecular plane, y and z directions in plane, z coincident with C2axis) emerge as 3.16 & 0.19,6.03 & 0.29, and 3.79 f 0.12 at 632.8 nm.
Introduction The anisotropic electric dipole polarizability of the sulfur dioxide molecule is currently of interest.'+ Although the mean optical-frequency1 and static5 polarizabilities are accurately known from measurements of the refractive index and relative pkrmittivity of the gas, the diagonal elements of the polarizability have not yet been unambiguously determined by experiment. Of most interest in relation to the optical-frequency polarizability are the results of Murphy? who determined the polarizability tensor component ratio, Rzo,required to simulate the pure rotational contour in the Raman spectrum of sulfur dioxide at 514.5 nm. Combination of this result with the previously reported' mean polarizability and Rayleigh depolarization ratio permitted the derivation of alternative sets of physically reasonable polarizabilities for the molecule. The two possibilities differ in the sign, but not the magnitude, of the difference, azz- a, between the polarizability in the direction of the molecular dipole moment and the mean value. In order to choose between these sets, Murphy argued that the anisotropy, a, - a,is almost certainly negative, because the observed Kerr c o n s t a d of sulfur dioxide is negative. Qualitative support for this view was drawn from the intuitive expectation that the molecule would be most polarizable along its bonds, and least so perpendicular to the molecular plane. Interpretation of the single-temperature Kerr constant in this way is, however, complicated by lack of knowledge of the contribution which the first hyperpolarizability, PK,makes to the observed effect. One way to remove the uncertainty is to exploit the temperature dependence of the Kerr effect in order to establish the sign and magnitude of both azz- a and OK,as was demonstrated in the similarly ambiguous case of hydrogen ~ u l f i d e . ~ Alternatively, the temperature dependence of the Cotton-Mouton effect can be used.s We recently showed, with fluorobenzene as an example,sh that the latter effect, in conjunction with the refractive index and Rayleigh depolarization ratio, can provide (1) Bogaard, M. P.; Buckingham, A. D.; Pierens, R. K.;White, A. H. J . Chem. SOC.,Faraday Trans. I 1978, 74, 3008-3015. (2) Patel, D; Margolese, D.; Dyke, T. R. J . Chem. Phys. 1979, 70, 2740-2747. (3) Murphy, W. F. J. Raman Spectrosc. 1981, 11, 339-345. (4) Bacskay, G. B. J. Chem. Phys. 1983, 79, 2090-2092. (5) Le FEvre, R. J. W.; Ross, I. G.; Smythe, B. M. J. Chem. SOC.1950, 276-283. (6) (a) Szivessy, G. Z . Phys. 1924,26,323-342. (b) Le Wvre, R. J. W.; Ritchie, G. L. D. J. Chem. SOC.1965, 3520-3528. (7) Bogaard, M. P.; Buckingham, A. D.; Ritchie, G. L.D. Chem. Phys. Lett. 1982, 90, 113-187. (8) (a) Bogaard, M. P.; Buckingham, A. D.; Corfield, M. G.; Dunmur, D. A.; White, A. H. Chem. Phys. Lett. 1972, 12, 558-559. (b) Geschka, H.; Pferrer, S.;HBussler, H.; HUttner, W. Ber. BunFenges. Phys. Chem. 1982,86, 790-795. (c) Kling, H.; Dreier, E.; HUttner, W. J . Chem. Phys. 1983, 78, 4309-4314. (d) Kling, H.; Geshka, H.; Hiittner, W. Chem. Phys. Lett. 1983, 96,631-635. (e) Lukins,P. B.; Buckingham, A. D.; Ritchie, G. L.D. J . Phys. Chem. 1984,88,2414-2418. (f) Kling, H.; HUttner, W. Chem. e b y s . 1984, 90,207-214. ( 8 ) Lukins, P. B.; Laver, D. R.; Buckingham, A. D.; Ritchie, G. L. D. J . Phys. Chem. 1985,89, 1309-1312. (h) Lukins, P. B.; Ritchie, G. L. D. J. Phys. Chem. 1985,89, 1312-1314.
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a valuable route to the principal polarizabilities of molecules of C, or similar symmetry for which the corresponding magnetizabilities are known. Moreover, in cases where Rm has been derived from rotational Raman intensities, the ambiguity arising from use of the Rayleigh depolarization ratio, which gives a quadratic equation, can be obviated and the unique set of polarizabilities determined. To this end, we examined the magnetic birefringence of sulfur dioxide over a wide range of pressure and temperature. The results, which are reported here, allowed us to resolve the uncertainty associated with Murphy's findings and to quantify the contribution which the magnetic hyperpolarizability makei to the Cotton-Mouton effect of this molecule.
Theory The low-density molar Cotton-Mouton constant, ,,.,C is, , in S I units@
where eq l a is a measure of the refractive index difference, nll - n,, induced in the gas by the magnetic induction B, and eq l b is the theoretical relationship between the observed birefringence ~ and fundamental molecular proper tie^.^ In eq lb, A7 (= v - 1/377aa$g) is the anisotropy in the magnetic hyperpolarizability; axx,ayy,azzand xxx, x,,,, xzzare the diagonal elements of the optical-frequency polarizability and the magnetizability, respectively; and x (= 1/3xaa)is the mean magnetizability. It remains to recall that the isotropic refractive index, n, the polarizability tensor component ratio, RB and the Rayleigh depolarization ratio, po, are expressible in terms of the polarizability aslJ
[(n2- l)/(n2 + 2)IVm = (N~/9eo)(a, + a y y + a z z ) (2) and
so that simultaneous solution of an appropriate combination of eq lb-4, in conjunction with the known magnetizabilities and the mean polarizability, a (= 1/3aaa), yields all three components of the anisotropic polarizability. Experimental Section Apparatus and procedures as recently describedsc-gwere used to observe the magnetic field induced birefringence of sulfur (9) Buckingham, A. D.; Pople, J. A. Proc. Phys. SOC.,London, Sect. B 1956, 69, 1133-1138.
0 1985 American Chemical Society
~
,
~
3410
The Journal of Physical Chemistry, Vol. 89, No. 15, 1985
Lukins and Ritchie
TABLE I: Cotton-Mouton Effect of sulfur Dioxide at 632.8 MI T/K 462.7 430.3 403.0 376.1 348.0 325.2 298.1
103~/~-l 2.161 2.324 2.481 2.659 2.874 3.075 3.355
no. of pressures
P/wa
measurements
10 11 11 12 10 10 11
177-281 128-284 130-305 125-243 149-268 120-239 135-322
88 84 96 114 90 91 99
no. of 106B/m3mo1-I -133 -163 -195 -229 -279 -328 -41 1
IO2',C/mS A-2 mol-I 0.627 f 0.037 0.585 f 0.057 0.611 f 0.025 0.702 f 0.035 0.798 0.031 0.911 f 0.026 0.898 f 0.020
*
dioxide. A gas chromatographic analysis of Matheson anhydrous-grade sulfur dioxide confirmed the purity of the sample as >99.98%. Measurements of the birefringence a t 632.8 nm were made at seven temperatures ( ~ 2 9 8 - 4 6 3 K) and, at each temperature, over a range of pressure (up to ~ 3 2 kPa). 0 From an operational viewpoint, the definition of the molar Cotton-Mouton constant is
in which SB2dl = 3.289 f 0.045 TZm; q5 = (27rl/X)(nu - nl) is the phase retardation in light of wavelength X after traversal of the path length, I; and values of V,-', the reciprocal of the molar volume, were calculated from the pressures and recorded density virial coefficients.I0 Values of 4 were typically 10-5-104 rad, and the limiting sensitivity of the detection system was typically f5 X rad, under favorable circumstances f 2 X lW7 rad. The experiments therefore involved the evaluation, at each temperature, of the slope of a plot of 4' = 9n(n2 + 2)-24 against Vm-I. Results are summarized in Table I, where the uncertainties shown are based on the standard deviations derived from the least-squares fitting of straight lines to the density dependence data, together with appropriate allowance for systematic errors. It is noteworthy that the Cotton-Mouton constant of sulfur dioxide is similar in magnitude to that of cyclopropanesg and only about 1/50 as large as that of benzenese at the same temperature; the reliable determination of the temperature dependence of such small effects was well beyond the capability of earlier versions of the equipment.8a
2.0
~~
~
~~
~
~
(10) Dymond, J. H.; Smith, E. B. "The Virial Coefficients of Pure Gases and Mixtures"; Clarendon Press: Oxford, 1980. (j 1) Carusotto, S.;Iacopini, E.; Palacco, E.; Scuri, F.; Stefanini, G.; Zavattini, E. J . Opt. SOC.Am. E., Opt. Phys. 1984, 1, 635-640. (!2) Coonan, M. H.; Laver, D. R.; Ritchie, G. L. D.; Turner,P. J., unpublished results.
3.0
3.5
IO~T-*/K-~
Figure 1. Temperature dependence of the vapor-state Cotton-Mouton effect of sulfur dioxide. TABLE 11: Analysis of the Temperature Dependence of the Cotton-Mouton Effect of Sulfur Dioxidea nromrtv r - - r - - -,
1027(interceut~/mS A-2 mol-l
Discussion
From eq l b it can be seen that the molar Cotton-Mouton constant should exhibit a linear dependence on the reciprocal of , against the absolute temperature. Figure 1 shows a plot of C T',and Table I1 contains the intercept and slope of the weighted-fit least-squares straight line, together with the molecular polarizabilities derived from eq 1b-4. In respect of the temperature-independent contribution represented by the intercept, it may be noted, for comparison, that the values of Aq (quoted here as IOS0Aq/C m2 V-' T2)and the contribution which this quantity makes to the Cotton-Mouton constant at 298 K are very similar for sulfur dioxide (-15 f 30, -5 f 15%) and cyclopropane (-20 f 20, -10 f lo%), which was recently examined.@ Notwithstanding the availability of experimental results for a range of molecules (Ar, Kr, Xe;" 02;" N,, CO, "0;" OCO, OCS, SCS;8f*'2C2H2, C2H4, C2H6;8dC3H6'8 C,H6, C,H3F3, C6F6sb9e)few, if any, trends in Aq are as yet discernible. Indeed the relatively low precision of values obtained from observations of the temperature dependence of the Cotton-Mouton effect, and the intrinsic complexity of Aq, may be such as to preclude the development of, for example, a bond-
2.5
value _._.
-0.06 f 0.13 0.292 f 0.046 -15 f 30 -5.08 f 0.08 3.09 f 0.07 1.99 f 0.05 4.326 f 0.043 0.30 f 0.05 3.16 f 0.19 6.03 f 0.29 3.79 f 0.12 -0.54 f 0.13 3.056 3.11 (2.93) 5.71 (4.53) 4.16 (5.52)
Directions of molecular axes: x perpendicular to plane; y and z in plane, z coincident with C2 axis. Reference 13. Reference 1, 632.8 nm; assumed uncertainty of &l%. dReference 3, 514.5 nm. 'Equations lb-3. f~~ = 5p0 (3 - 4p0)-'; ref 1, 632.8 nm. %Equations lb, 2, and 4; alternative solution of quadratic equation in parentheses (see text).
additivity model of this property. Simultaneous solution of eq lb-3, in conjunction with the anisotropic magnetizability, xa@,13the mean optical-frequency polarizability, a,and the polarizability tensor component ratio, R20,3yields the principal polarizabilities of sulfur dioxide shown in Table 11. The justification for use of the only available value of Rzo,determined at 514.5 nm, lies in the fact that this quantity, like the anisotropy parameter, K ~ is, a ratio of polarizabilities which (13) Ellenbroek, A. W.; Dymanus, A. Chem. Phys. Lett. 1976, 42, 303-306.
J . Phys. Chem. 1985,89, 3411-3417 is unlikely to be much affected by a relatively small change in wavelength. Furthermore, trial calculations showed that the polarizabilities so derived are remarkably insensitive to small variations in Rzoand that physically reasonable extreme values give results which are well within the ranges indicated by the uncertainties. The conclusions of Murphy are therefore validated: the molecule is most polarizable along its bonds and least so in a direction perpendicular to the molecular plane (Le., oiyy > aZ2 > axx),and the anisotropy, aZz- a,is negative, as inferred from the negative Kerr constant of the molecule. It is of interest that, on the basis of the polarizabilities here determined, the dipole moment-polarizability anisotropy contribution to the Kerr constant of this molecule, i.e., the term (NA/270tokZP)r2(a,2 - a),’is predicted to give rise to approximately 130% of the observed effect6 a t 293 K. In contradistinction to inferences drawn from refractivity virial data,I4 the first Kerr hyperpolarizability, OK, of sulfur dioxide is therefore almost certainly positive, not negative, in sign, and of very much smaller magnitude than s u g g e ~ t e d . ’ ~ Comparison of the polarizabilities (expressed here as 1O4OaXx/C mZV-’, etc.) now found (632.8 nm; 3.16 f 0.19, 6.03 f 0.29, 3.79 f 0.12) with those preferred by Murphy (514.5 nm; 3.35 f 0.04, 5.92 f 0.01, 3.91 f 0.05) shows that, when account is taken of errors, dispersion is insignificant in the range 514.5-632.8 nm. The polarizabilities of sulfur dioxide can alternatively be obtained through simultaneous solution of eq lb, 2, and 4; the choice between the sets, shown in Table 11, which emerge from the
+
(14) Blythe, A. R.; Lambert, J. D.; Petter, P. J.; Spoe.1, H. h o c . R. Soc. London, A 1960, 255,427-433.
(15) Buckingham, A. D.; Orr, B. J. G e m . Soc. Rev. 1967,21, 195-212.
3411
quadratic equation is unambiguous, since it is known from the foregoing that ayyis the largest component and that a,, - a is negative in sign. However, rigorous calculation of the uncertainties is now not straightforward, and we are therefore inclined to favor the values derived by the former procedure. We note also that an ab initio S C F calculation of the infinite-wavelength electronic polarizability of sulfur dioxide has been reported4 and that molecular beam electric resonance spectroscopy has provided somewhat uncertain estimates of the anisotropy in the static polarizability of this molecule.2
Conclusion The investigation described here illustrates the usefulness of the Cotton-Mouton effect as a route to the electric dipole polarizabilities of molecules whose magnetizabilities have been determined by means of the microwave Zeeman effect. In the case of sulfur dioxide, combination of the temperature-dependent contribution to the effect with previously reported measurements of the refractive index, the polarizability tensor component ratio derived from the pure rotational Raman spectrum, and the Rayleigh depolarization ratio permitted the unambiguous evaluation of all three elements of the optical-frequency polarizability of the molecule. Acknowledgment. The award of a University of Sydney Special Project Research Scholarship (to P.B.L.) and financial support from the Australian Research Grants Scheme (to G.L.D.R.) are gratefully acknowledged. Registry No. Sulfur dioxide, 7446-09-5.
Neutron scattering Study of Micelle Structure in Isotropic Aqueous Solutions of Poly(oxyethylene) Amphiphlles M. Zulauf,* K. Weckstrom? European Molecular Biology Laboratory, 156X, F-38042 Grenoble- CZdex, France
J. B. Hayter, Institut Luue-Lungevin, 156X, F-38042 Grenoble-CZdex, France
V . Degiorgio, Dipartimento di Elettronica-Sezione
di Fisica Applicata, Universita di Pavia, I - 271 00 Pavia. Italy
and M. Corti U S E s.p.a., I-20100 Milano, Italy (Received: February 1 1 , 1985; In Final Form: March 22, 1985)
We have investigated the properties of micelles formed by the nonionic poly(oxyethy1ene) amphiphiles C,ES and ClZE8in D20as a function of temperature below the lower consolution boundary (LCB) and of concentration below the isotropichexagonal phase boundary by elastic and quasi-elastic neutron scattering. We have also performed static measurements at temperatures below the LCB on C&6 and C& solutions. We find little variation in micelle size throughout the range of conditions studied (including regions near to the mesophase boundaries), provided intermicellar interactions are taken into account by their structure factor. The latter is calculated under the assumption that the direct interaction potential is hard sphere plus a short-ranged attractive tail. Far away from the LCB the attractive part is negligible, but becomes important upon approaching the LCB. At higher concentrations the predominant interaction is hard-sphere repulsion. Our measurements give some information on micellar hydration: the number of bound water molecules per ethylene oxide group is between 1 and 2.5 depending on temperature and concentration.
Introduction There is currently a considerable debate on the structure of micelles in aqueous solutions of nonionic surfactants.’ Single-
particle properties such as micelle size and shape and micellar hydration may depend in general on temperature and total surfactant concentration, but also collective properties arising from intermicellar interactions are affected by these parameters. It is not always obvious how to separate single-particle and collective (1) Degiorgio, v. In of AmphiPhil-. vesicles and properties in the analysis of experimental data, particularly when Mi-mubim”; h g i ~ g i ov., , m,M., ~ & - ~ ~ lbem, h ~ d 1985. one is interested in intermediate or high surfactant concentrations. 0022-3654/85/2089-3411%01.50/0
0 1985 American Chemical Society
