616
The Journal of Physical Chemistry, Vol. 82, No. 5, 1978
t
solution dielectric constant Debye parameter (see eq 4), cm-l r/2a-1 solution density, g/cm3 solution density in the absence of any particles (C O), g/cm3 solution viscosity, g cm-l s-l solution viscosity in the absence of any particles (C O), g cm-' s-l
x
p
pf pfo
7
-
v0
Acknowledgment. We appreciate the very helpful advice given to us by Professor John Quinn. Partial support of the work was received through NSF Grant ENG75-13440. The experimental work reported here was performed at the Department of Chemical Engineering, Cornel1 University, and we are grateful for the support given to us I
George A. Miller (13) (14) (15) (16) (17) (18) (19) (20) (21) (22) (23) (24) (25) (26) (27) (28) (29)
References and Notes M. J. Stephen, J. Chem. Phys., 55, 3878 (1971). D. L. Ermak, J . Chem. Phys., 62, 4189, 4197 (1975). J. L. Anderson and C. C. Reed, J . Chem. Phys., 64, 3240, 4336 (1976). J. L. Anderson and C. C. Reed, "Diffusion of Interacting Brownian Particles", submitted for publication. G. K. Batcheior, J. Nuid Mech., 74, 1 (1976). T. Raj and W. H. Fiygare, Biochemistry, 13, 3336 (1974). P. Doherty and G. B. Benedek, J . Chem. Phys., 61, 5426 (1974). S.B. Dubin, J. H. Lunacek, and G. B. Benedek, Proc. Nafl. Acad. Sci. U.S.A., 57, 1164 (1967). S. Aipert and G. Banks, Biophys. Chem., 4, 287 (1976). G. D. J. Phiilies, G. B. Benedek, and N. A. Mazer, J . Chem. Phys., 65, 1863 (1976). K. H. Keiier, E. R. Canales, and S. I. Yum, J . Phys. Chem., 75, 379 (1971). G. I.Taylor, Proc. Phys. Soc., London, Sect. 6 ,67, 857 (1954).
(30) (31)
(32)
(33) (34)
R. A. Wooding, Proc. R . Soc. London, Ser. A, 252, 120 (1959). R. A. Wooding, J . Fluid Mech., 13, 129 (1962). C. H.Lin, Ph.D. Thesis, University of Illinois, Urbana, 1971. C. H. Lin and J. A. Quinn, submitted for publication. T. Raj, personal communication. C. Tanford and J. G. Buzzeii, J. Phys. Chem., 60, 226 (1956). J. Happei and H. Brenner, "Low Reynolds Number Hydrodynamics", Prentice-Hail, Engiewood Cliffs, N.J., 1965. C. Tanford, "Physical Chemistry of Macromolecules", Wiiey, New York, N.Y., 1961. M. L. Wagner and H. H. Scheraga, J. Phys. Chem., 60, 1066 (1956). A. J. Goidman, R. G. Cox, and H. Brenner, Chem. Eng. Sci., 21, 1151 (1966). D. Stitger and T. L. Hill, J . Phys. Chem., 63, 551 (1959). G. Scatchard, I.H. Scheinberg, and S.H. Armstrong, J. Am. Chem. Soc.. 72. 535 (1950). C. Tanforb, S.A. Swanson, and W. S.Shore, J . Am. Chem. Soc., 77, 6414 (1955). V. L. Viiker, Ph.D. Thesis, Massachusetts Institute of Technoioav, -. Cambridge, 1975. R. Hogg, T. W. Heaiy, and D.W. Fuerstenau, Trans. Faraday Soc., 62, 1838 (1966). J. G. Brodnyan and E. L. Keiiey, J. Pokm. Sci. C, No. 27, 263 (1969). H. J. Van den Hull and J. W. Vanderhoff, "Clean Monodisperse Latexes as Model Colloids", Polym. Colloids, Proc. Symp., 1970. A. A. Kozinski and E. N. Lightfoot, AIChEJ., 17, 81 (1971). This estimate was obtained from Ermak's work using the resuks shown in Table IV for M = 0 (no coions present) and assuming that [&'/Do - 11 is proportional to lqI3Z2/a3,where Z is the valence of the counterions (unity for potassium). This value is obtained by multiplying the coefficient 2.5 by the partial specific volume of polystyrene latex which equals about 0.94 A 0.01. Our slightly higher experimental values may be the resuk of adsorption of stabilizing surfactant molecules to the particle surface. Standard error was computed as the root-mean-square error from the mean value of the three results, normalized by the mean. H@)= 2(f0/f,) -I-(fo/fii)- 3, where ,f is the steady friction coefficient for each of two identical, freely rotating spheres translating at equal velocities perpendicular to the line connecting their centers, and fii is the analogous coefficient when the two spheres are translating parallel to the line of centers.
Fluctuation Theory of the Resonance Enhancement of Rayleigh Scattering in Absorbing Media George A. Mlller School of Chemisfry, Georgia Institute of Technology, Atlanta, Georgia 30332 (Received December 2, 1977)
The macroscopic fluctuation theory of light scattering is modified to include resonance enhancement under an absorption peak of the scattering medium. The required modification is simply to replace the fluctuation average ( (6n)2)with ( ( ~ 3 n+) (c%)~) ~ where m = n - ik is the complex refractive index.
Light scattering by transparent isotropic media is accurately described by a macroscopic fluctuation theory first set down by Einstein.1p2 The turbidity of a volume element, V', is dependent on fluctuations in the refractive index (or dielectric constant) in V' according to the relation
where Xo is the wavelength of the incident radiation in vacuo. References may be found scattered through the literature as to the inadequacy to eq 1 if the medium absorbs a t io. Thus, in one light scattering study of the aggregation of dyes in ~ o l u t i o nit, ~was decided to confine Xo to the far red where absorption was quite small. In absorbing media the refractive index is complex. It has been suggested4that the imaginary part (called here 0022-3654/78/2082-0616$01.OO/O
the absorption coefficient) must be very much smaller than the real part, if one is to be able to detect any scattered light at all, and, therefore, eq 1should be valid for practical cases. This plausible argument ignores the fact that one is dealing with fluctuations in refractive index and not the average value. In the case of solutions of dyes and the like, fluctuations in the absorption coefficient may be comparable to fluctuations in the real refractive index. It is the purpose here to modify eq 1to include the effect of absorption on the actual scattering process. The proximity of Xo to an absorption manifold of the medium causes a resonance enhancement of the scattering a t the scattering site. (The quantum theory treatment a t the molecular level is well known5 but is not capable of giving accurate results for condensed media.) There is a concurrent loss of intensity in both incident and scattered beams as they pass through the medium. This Beer's law 0 1976 American
Chemical Society
The Journal of Physical Chemistry, Vol. 82, No. 5, 1978 617
Rayleigh Scattering in Absorbing Media
correction6 is not of concern here except as it affects the feasibility of carrying out the light scattering experiment. It will be seen that eq 1 is actually valid in most practical cases, but that deviations may be observable in the case of colloidal dyes. A simple approach to the fluctuation theory of resonance enhancement may be found in the famous treatment by Mie7 of the scattering by absorbing spheres. As a first step it will be shown that Mie theory can be used to derive eq 1.
Transparent Media A nonabsorbing spherical particle of volume, V’, whose diameter is small relative to ho, has a scattering cross section
solution. The absorption coefficient is related to Beer’s law through the molar absorption coefficient, a, by k = aMho/47r where M is the molarity of the absorbing solute and Beer’s law is written as I = Ioe-dx. If the path length is of the order of x = 1cm, then aM should be of the order of unity for a reasonable fraction of light to reach the detector. This will be the starting point for estimating the conditions under which resonance enhancement would be observable. Consider the excess turbidity, T,, arising from concentration fluctuations in a solution.lvs Using the molarity scale, one would write eq 1 as
According to fluctuation theoryg one has where h is the wavelength in the medium and n’is the ratio of the particle refractive index to that of the surrounding medium. It there are N’ particles per unit volume, the turbidity is T = N’C. Suppose one applies this relation to a small fluctuation, an, in the refractive index of an element, V’, of a homogeneous medium whose average refractive index is n. Accordingly, one writes
where p2 is the chemical potential of the solute, k is the Boltzmann constant, and V’is in cm3to be consistent with the conventional units of T . For an ideal solution on the molarity scale, one has ap2/aM = RT/M, where R is the gas constant. Therefore, one obtains 7,
26 n 3n
6n
