Langmuir 1997, 13, 5465-5469
5465
Determination of the Optical Properties of Monolayers on Water Brian D. Casson and Colin D. Bain* Physical and Theoretical Chemistry Laboratory, Oxford University, South Parks Road, Oxford OX1 3QZ, U.K. Received May 5, 1997. In Final Form: July 25, 1997X The optical dielectric constants, e and o, have been determined by ellipsometry for a monolayer of dodecanol on aqueous solutions of KBr. In the thin-film limit, ellipsometry yields only one independent parameter, η, which is insufficient to define uniquely the three unknowns e, o, and the thickness of the monolayer, d. For a monolayer of dodecanol on water, d is known independently from X-ray diffraction. Varying the concentration of KBr changes the dielectric constant of the subphase and allows the calculation of both e and o. The roughness of the interface and the depletion layer at the surface of the salt solution must be taken into account in the analysis of the ellipsometric data. We obtain values of e ) 2.23 and o ) 2.12 at an area per molecule of 20.9 Å2.
Introduction Quantitative analysis of spectra of thin organic films requires a knowledge of the optical dielectric constants of the film. The optical constants are also needed to interpret data obtained by ellipsometry1-4 and surface plasmon resonance,5-8 two techniques that are widely used in the characterisation of thin films. The explosive growth of research on self-assembled monolayers and the revival of interest in Langmuir-Pockels and Langmuir-Blodgett (LB) films9,10 make the need for good optical constants more urgent. These films are densely packed and are structurally anisotropic. They are usually assumed to be optically uniaxial with the optic axis normal to the interface. This assumption will hold if the symmetry axis of the molecule is normal to the interface or, in the case of films with a collective tilt, the domains are much smaller than the analysis area and the tilts of the domains are randomly distributed in the plane of the interface. Uniaxial films can be characterized by the ordinary and extraordinary dielectric constants, o and e. In this paper, we describe a simple scheme for obtaining e and o for a monolayer of dodecanol on water. Ellipsometry is one of the most sensitive techniques for measuring the optical properties of thin films. Ellipsometry measures the ratio of the complex reflection coefficients for p- and s-polarized light, rp/rs. The amplitude and phase of light reflected from an interface are modified by the presence of a monolayer. The change in reflectivity is determined by the thickness and refractive index (n ) 1/2) of the monolayer. There are two principal ways in which ellipsometric measurements are reported. In the X Abstract published in Advance ACS Abstracts, September 15, 1997.
(1) Beaglehole, D. In Fluid Interfacial Phenomena; Croxton, C. A., Ed.; Wiley: New York, 1986; p 523. (2) Meunier, J. In Light Scattering by Liquid Surfaces and Complementary Techniques; Langevin, D., Ed.; Marcel Dekker: New York, 1992; Chapter 17. (3) Azzam, R. M.; Bashara, N. M. Ellipsometry and Polarized Light; North Holland: New York, 1987. (4) Lekner, J. Theory of Reflection; Nijhoff: Dordrecht, 1987. (5) Welford, K. Opt. Quantum Electron. 1991, 23, 1. (6) Pockrand, I.; Swalen, J. D.; Gordon, J. G., II; Philpott, M. R. Surf. Sci. 1977, 74, 237. (7) Barnes, W. L.; Sambles, J. R. Surf. Sci. 1986, 177, 399. (8) de Bruijn, H. E.; Altenburg, B. S. F.; Kooyman, R. P. H.; Greve, J. Opt. Commun. 1991, 82, 425. (9) Langmuir-Blodgett Films; Roberts, G. G., Ed.; Plenum: New York, 1990. (10) Ulman, A. Ultrathin Organic Films; Academic Press: San Diego, CA, 1991.
S0743-7463(97)00471-X CCC: $14.00
approach adopted here, we make measurements at the Brewster angle, θB, defined as the angle where Re(rp/rs) ) 0. The experimentally determined quantity is the coefficient of ellipticity, Fj ) Im(rp/rs), at θB. When measurements are made at other angles of incidence, it is conventional to report two angles, Ψ and ∆, where rp/rs ) tan Ψei∆. If the thickness of the film, d, is very much less than the wavelength, λ, of the light used in the measurement, the ellipsometric response is determined by a single parameter, η, whichever method is employed. η is often called the ellipsometric thickness and is determined by the refractive index profile through the interface. For a homogeneous, transparent, uniaxial film there are three unknowns, e, o, and d,11 which cannot be determined uniquely from η alone. This problem is wellrecognized and various groups have attempted to overcome it by varying experimental parameters such as the incident angle or the wavelength of the light, the thickness or density of the film, or the optical properties of the incident medium or substrate. Invariably, one or more assumptions must be made about the structure of the interfacial film. It is also possible to predict e and o in monolayers from the properties of alkanes determined by scattering experiments or calculated ab initio.12,13 This approach, however, assumes a theoretical framework for the local field calculations and does not involve direct experimental measurements on monolayers. We briefly discuss previous experiments to determine e and o by ellipsometry for comparison with the approach taken in this paper. Optics. A change in the angle of incidence has been shown experimentally14 and theoretically15 not to provide any extra information about the film. Multiple-angle ellipsometry is only useful for thin (d , λ), transparent films as a means of reducing errors. Multiple wavelength ellipsometry provides sufficient data to characterize a thin, anisotropic, transparent film uniquely only when both the incident medium and substrate are sufficiently (11) e and o are continuum quantities that are used to represent the linear response of the film to an applied field, averaged over the monolayer. (12) den Engelson, D.; de Koning, B. J. Chem. Soc. Faraday Trans. 1974, 70, 1603. (13) den Engelson, D. Surf. Sci. 1976, 56, 272. (14) Ayoub, G. T.; Bashara, N. M. J. Opt. Soc. Am. 1978, 68, 978. (15) Dignam, M. J.; Moskovits, M.; Stobie, R. W. Trans. Faraday Soc. 1971, 67, 3306.
© 1997 American Chemical Society
5466 Langmuir, Vol. 13, No. 20, 1997
dispersive;16 i.e., the three unknowns, o, e, and d can be found only when both 1 and 2 can be varied independently of e and o. Such an experiment requires that the interfacial film itself is not dispersive, which in practice is a hypothetical situation. Film. A change in the thickness or density of the film has proved the most effective route for determining the optical properties of thin films by ellipsometry. LB films can easily be made with different thicknesses. If they are sufficiently thick (such that terms of order (d/λ)2 become significant), two independent parameters are obtained from an ellipsometry measurement. Measurements on more than one film, of different thicknesses, can then provide sufficient information to find o, e, and d (the thickness of a monomolecular layer).17-19 This approach needs to be applied with caution as LB multilayers may have a collective tilt in the dipping direction resulting in biaxial, not uniaxial, symmetry.7 Furthermore, the optical properties are an average over the chain, head group, and counterions (if present) and may not be appropriate for the first monolayer. The thickness of a film at the air-water interface can be varied either by changing the length of the chain20,21 or by changing the area per molecule.12,22 In densely packed monolayers of a homologous series of fatty acids20 or alcohols,21 η is a linear function of the length of the hydrocarbon chain. In our previous work, the thickness of a series of isostructural alcohol monolayers was obtained from surface X-ray diffraction data and the mean polarizability of a CH2 unit from molar refractivity measurements.23 Data from a range of chain lengths then allowed us to eliminate the contribution of roughness to the ellipticity and to deduce the optical constants e and o. Riegler and co-workers exploited the rich phase behavior of Langmuir-Pockels monolayers to determine o and e for the S and CS phases of behenic acid on water.24 They used X-ray reflection and diffraction data to provide the extra information required to determine the optical parameters of the monolayers uniquely. X-ray diffraction showed that the molecules are upright in both phases and gave an area per molecule for each phase. As the orientation of the molecules in the two phases was the same, it was assumed that the anisotropy in did not change across the transition. The Lorentz-Lorenz relation was used to relate the refractive indices in the two phases, providing enough constraints to determine both e and o. Substrate. The final experimental variable is the dielectric constant of the substrate, 2. Varying 2 is mathematically equivalent to multiple-wavelength ellipsometry with a dispersive substrate, but allows bigger changes in 2 in practice. This method can only succeed if the structure of the film remains invariant when the substrate is changed. For a uniaxial monolayer in which the optical axis is oriented normal to the surface, the ellipsometric thickness, ηd, of the monolayer is given by25,26 (16) Antippa, A. F.; Leblanc, R. M.; Ducharme, D. J. Opt. Soc. Am. 1986, 3, 1794. (17) den Engelson, D. J. Opt. Soc. Am. 1971, 61, 1460. (18) Geiss, G.; Hickel, W.; Lupo, D.; Prass, W.; Scheunemann, U. Ber. Bunsenges. Phys. Chem. 1991, 11, 1345. (19) Zhu, R.; Lin, C.; Wei, Y. Thin Solid Films 1991, 203, 213. (20) Kim, M. W.; Sauer, B. B.; Yu, H.; Yazdanian, M.; Zografi, G. Langmuir 1990, 6, 236. (21) Casson, B. D.; Braun, R.; Bain, C. D. Faraday Discuss. 1996, 104, 209. (22) Ducharme, D.; Max J.-J.; Salesse, C.; Leblanc, R. M. J. Phys. Chem. 1990, 94, 1925. (23) Vogel, A. I. J. Chem. Soc. 1946, 133. (24) Paudler, M.; Ruths, J.; Riegler, H. Langmuir 1992, 8, 184.
Casson and Bain
ηd )
(e - 1)(e - 2) d + (o - e)d e
(1)
where 1 is the dielectric constant of the incident medium. The coefficient of ellipticity Fjd is related to ηd by27
Fjd )
π x1 + 2 d η λ 1 - 2
(2)
Differentiating eq 1 with respect to 2 gives
( )
1 ∂ηd ) -1 d ∂2 e
(3)
If d is known from independent measurements, a plot of ηd against 2 yields e. o can then be found from any individual value of ηd. We adopt this approach here. We have investigated monolayers of 1-dodecanol on KBr solutions of different concentrations by ellipsometry. We will provide evidence that the structure of the dodecanol monolayer is independent of the KBr concentration. The structure of these monolayers has been determined by Berge and co-workers by X-ray diffraction,28 which allows us to estimate d and hence to calculate e from eq 3. We will show that interpretation of the ellipticity solely in terms of the optical properties of the monolayer leads to errors and that roughness and depletion-layer effects need to be considered. Experimental Section 1-Dodecanol (Fluka, >99.5%) was used as received. Potassium bromide (BDH AnalaR, 99.5%) was roasted in an oven at 500 °C to burn off organic impurities. The concentrations, [KBr], are quoted in percentage mass of KBr in the solution. The water was ultrahigh purity (Elga UHQ). We checked the cleanliness of all glassware by measuring the ellipticity of ultrapure water placed in the glassware. The ellipsometer was built by Beaglehole Instruments (Wellington, NZ) and incorporated photoelastic modulation of a HeNe laser (633 nm) at 50 kHz and lock-in detection of the reflected light at 50 and 100 kHz. The surface tension measurements were performed on a Kru¨ss tensiometer by the du Nou¨y ring method. All measurements were taken at 20 °C. Monolayers of dodecanol on water or KBr solutions were prepared by placing a small drop of liquid dodecanol onto the clean surface of the water or salt solution. The dodecanol spread rapidly to form a complete monolayer in equilibrium with the bulk droplet. This situation is indistinguishable from a monolayer adsorbed from a saturated solution of the alcohol.29,30 At any given temperature the area per molecule is fixed and welldefined as shown by X-ray diffraction measurements.28
Results and Discussion Determination of Optical Constants. Monolayers of medium-chain alcohols (C9-C14) on water have been studied extensively. They show first-order phase transitions from a crystalline solid to a liquid state at welldefined temperatures, Tm(2D). These phase transitions have been investigated by ellipsometry,21,29,30 surface tension,29,31,32 sum-frequency spectroscopy,21,30 X-ray dif(25) Beaglehole, D. J. Phys. (Paris) 1983, supplement to volume 44, C10-147. (26) Bercegol, H.; Gallet, F.; Langevin, D.; Meunier, J. J. Phys. (Paris) 1989, 50, 2277. (27) Drude, P. Ann. Phys. Chem. (Leipzig) 1891, 43, 126. (28) Legrand, A.; Renault, J. F.; Konovalov, O.; Chevigny, E.; AlsNielson, J.; Gru¨bel, G.; Berge, B. Thin Solid Films 1994, 248, 95. (29) Berge, B.; Renault, A. Europhys. Lett. 1993, 21, 773. (30) Braun, R.; Casson, B. D.; Bain, C. D. Chem. Phys. Lett. 1995, 245, 326. (31) Ross, J. J. Phys. Chem. 1958, 62, 531. (32) Trapeznikov, A. Acta Physicochim. URSS 1945, 20, 589.
Optical Properties of Monolayers
Langmuir, Vol. 13, No. 20, 1997 5467
Fj ) Fjd + FjR + Fjdl
)
Figure 1. Ellipsometric parameter, η, as a function of the dielectric constant of the KBr solution, 2: values obtained directly from the measured values of Fj (open diamonds); values corrected for roughness and depletion layer effects (solid diamonds).
fraction,28,33,34 and X-ray reflection.35 X-ray diffraction shows that just below Tm(2D) all of the monolayers have a hexagonal unit cell of area 21.5 Å2 with the chains oriented normal to the surface. The sum-frequency spectra show that the molecules are highly conformationally ordered, with the hydrocarbon chains largely all trans. For a monolayer of dodecanol on pure water Tm(2D) ) 39.1 °C. Phase transitions are highly sensitive to structural changes in the monolayer. For a monolayer of dodecanol on a 20% KBr solution Tm(2D) ) 40 ( 1 °C, within experimental error of the value on pure water. The constancy of Tm(2D) is strong evidence that the structure of the monolayer changes little when water is replaced by strong salt solutions. We measured Fj for a monolayer of dodecanol on pure water and KBr solutions with concentrations up to 30% KBr. Figure 1 shows a plot of η against 2, where η has been calculated directly from the measured values of Fj (eq 2). Values of 2 were taken from ref 36, but the Brewster angle measured in the experiments agreed well with the literature values (tan θB ) 21/2). The data fall on a good straight line with a slope of -9.5 ( 0.2 Å. Since the molecular axis is normal to the surface, we chose d to be the length of the all trans C12 hydrocarbon chain (ignoring the terminal hydrogen) and assumed that the OH group is in the water and has the same dielectric constant as the water, giving d ) 12 × 1.27 Å ) 15.24 Å. As monolayers of alcohols are known to nucleate ice crystals,37 an approximate match between the optical properties of the hydroxyl group and water is not unreasonable. From eq 3 we can now calculate e ) 2.67 ( 0.10 and eq 1 then gives o ) 2.23 ( 0.03. In this calculation we have assumed that the only contribution to Fj is the dielectric profile of the monolayer. This assumption is incorrect. The roughness of the interface must be considered, and in these experiments there is also a depletion layer arising from the use of concentrated salt solutions. Equation 2 can be written more generally as (33) Berge, B.; Konovalov, O.; Lajzerowicz, J.; Renault, A.; Rieu, J. P.; Vallade, M. Phys. Rev. Lett. 1994, 73, 1652. (34) Renault, A.; Legrand, J. F.; Goldmann, M.; Berge, B. J. Phys. II 1993, 3, 761. (35) Rieu, J. P.; Legrand, J. F.; Renault, A.; Berge, B.; Ocko, B. M.; Wu, X. Z.; Deutsch, M., J. Phys. II 1995, 5, 607. (36) CRC Handbook of Chemistry and Physics, 64th ed.; CRC Press: Boca Raton, FL, 1983-1984; p D-246. (37) Popovitz-Biro, R.; Wang, J. L.; Majewski, J.; Shavit, E.; Leiserowitz, L.; Lahav, M. J. Am. Chem. Soc. 1994, 116, 1179.
π x1 + 2 d (η + ηR + ηdl) λ 1 - 2
(4)
where ηd is the ellipsometric thickness of the monolayer, ηR is a term due to the roughness of the interface, and ηdl is a term due to the depletion layer. In a previous study of monolayers on pure water, we were able to deduce a value of FjR ) 0.4 × 10-3 from the ellipticities of a homologous series of alcohol monolayers on water, C9 to C16, just below Tm(2D) where the monolayers are isostructural.21 FjR arises from capillary waves and from the intrinsic roughness of the monolayer, though we believe that the latter contribution is small for crystalline monolayers. The present case is more difficult because both FjR and Fjdl may depend on the KBr concentration. To quantify the changes in FjR and Fjdl with concentration, we measured the ellipticities and surface tensions, γ, of clean salt solutions in the absence of a monolayer (Table 1). γ increased from 73.1 mN m-1 for pure water to 79.1 mN m-1 for a 30% KBr solution. The increase in surface tension of pure salt solutions is due to the depletion of salt at the interface and is described by the Gibbs equation.38 The depletion layer arises from the repulsion between the charge and its image in the air or in the monolayer. The magnitude of the image charge, q′, induced by a charge, q, is given by classical electrostatics as39
q′ )
20 - 10 20 + 10
q
(5)
where 20 and 10 are the static dielectric constants of water and of the adjacent medium, respectively. Since 20 ∼ 80 . 10, the magnitude of the image charge is virtually the same for air (10 ) 1) or a hydrocarbon (10 ∼ 2) as the adjacent medium. Consequently, the effect of salt on the surface tension will be the same with or without the monolayer. Berge reported a value of γ ) 23 mN m-1 for a monolayer of dodecanol on pure water at 20 °C.29 Thus we would expect γ to increase from 23 to 29 mN m-1 as [KBr] in the subphase increases from 0 to 30%.40 The capillary wave contribution to ellipticity was first considered by Beaglehole.41 Subsequently, Meunier expressed the roughness contribution to Fj from capillary waves in the form42
FjR ) -
3π (1 - 2) 2λ x + 1 2
xπkT 6γ
(6)
We can use this equation to predict the variation of FjR with [KBr] for a monolayer of dodecanol on water. As [KBr] increases from 0% to 30%, FjR decreases by 12% due to the effect of surface tension but increases by 13% due to the change in 2. The two effects cancel almost exactly. We therefore assume a value of FjR ) 0.4 × 10-3 at all concentrations. Fjdl does depend on the concentration of KBr. Provided that the monolayer and depletion layer do not overlap (38) Defay, R.; Prigogine, I. Surface Tension and Adsorption; Longmans: London, 1966; English Edition, p 63. (39) Jackson, J. D. Classical Electrodynamics; Wiley: New York, 1975; p 148. (40) Direct measurement of γ for monolayer-coated surfaces by detachment methods is complicated by the long equilibration times of the solid monolayer. (41) Beaglehole, D. Physica 1980, 100B, 163. (42) Meunier, J. J. Phys. (Paris) 1987, 48, 1819.
5468 Langmuir, Vol. 13, No. 20, 1997
Casson and Bain
Table 1. Dielectric Constant, Surface Tension, and Ellipticity of Clean Water and KBr Solutions KBr concn/%
2a
γ/mN m-1
Fj × 103
0 10 15 20 25 30
1.774 1.8101 1.8282 1.8472 1.8660 1.8887
73.1 74.7 75.9 76.8 78.0 79.1
0.37 0.41 0.41 0.43 0.46 0.48
a
Reference 36.
Figure 2. Values of the ellipticity, Fj, as a function of concentration of KBr for a monolayer-free solution: experimentally measured values (solid circles) and calculated values of the roughness contribution to the ellipticity (open circles).
(which is a reasonable assumption), Fjdl will be the same with and without the monolayer for the reasons discussed above. Fj increased from +0.37 × 10-3 to +0.48 × 10-3 for the clean KBr solutions as the concentration increased from 0 to 30% (Table 1). The change in Fj arises from an increase in 2, a decrease in the roughness, and a growth in the depletion layer. We can account for the change in roughness using eq 6 and the measured values of γ for the clean solutions (Table 1). Subtracting the roughness contribution from the measured Fj for the monolayer-free solutions gives a measure of Fjdl. It is well-known that eq 6 overestimates ηR (e.g., for pure water eq 6 gives Fj ) +0.59 × 10-3, while we measure Fj ) +0.37 × 10-3), but the scaling with γ and 2 is believed to be correct.42 We therefore use eq 6 to calculate FjR, scaling each value by the same factor such that FjR(0% KBr) ) +0.37 × 10-3. Figure 2 shows the measured ellipticities and the calculated roughness contributions, FjR for the clean solutions as a function of [KBr]. The difference between the two lines gives Fjdl at each concentration. We can now calculate Fjd from the measured ellipticities by subtracting the contributions from roughness and depletion-layer effects
Fjd ) Fj - Fjdl - 0.0004
(7) d
Figure 1 shows a plot of the resulting values of η against 2. The slope of the linear fit to these points is -8.4 ( 0.2 Å. From eq 3, we calculate e ) 2.23 ( 0.07 and eq 1 then gives o ) 2.12 ( 0.03. The errors quoted arise from the uncertainty in the slope in Figure 1. There may also be a number of systematic errors arising from our treatment of the results. The thickness, d, we chose to be 15.24 Å, 12 times the accepted length of a CH2 group in a fully extended hydrocarbon chain. Even in crystalline hydrocarbons there are gauche defects which will tend to decrease d. A molecular dynamics simulation of a monolayer of palmitic
acid [CH3(CH2)14CO2H] showed a decrease of 2% in the end-to-end distance when the area per molecule was increased from 18.5 Å2, where the chains are virtually all trans, to 21.0 Å2.43 This decrease in thickness arises principally from gauche defects near the end of the chain and from gauche-trans-gauche kinks in the middle of the chain. The area per molecule for a dodecanol monolayer on water at 20 °C is 20.9 Å2.35 A 2% decrease in d would increase the calculated values of e and o by 0.06 and 0.03, respectively. This method of determining e and o is therefore very sensitive to the value of d. The precise value of FjR has a smaller effect: increasing FjR by 0.1 × 10-3 reduces e by 0.03. Fjdl, like FjR, is subtracted from the measured ellipticity: errors in Fjdl will have a similar effect to those in FjR. Although the precision of our measurements could be improved by taking more readings, particularly at different angles, the overall accuracy of our result is limited by the accuracy with which the thickness is known. The treatment of the interfacial roughness also introduces some uncertainty into our analysis. Comparison with Literature Values. It is useful to compare the values of e ) 2.23 and o ) 2.12 determined here with experimental values for comparable systems. To compare measurements made at different densities, F (i.e. different cross-sectional areas per chain), we employ the Clausius-Mossotti relationship44
-11 ) constant +2F
(8)
The use of eq 8 implicitly assumes that the local field corrections are isotropic. Riegler and co-workers measured e ) 2.37 and o ) 2.16 for a behenic acid monolayer on water at an area per molecule of 19.3 Å2.24 For the same monolayer at an area per molecule of 20.9 Å2 eq 8 yields e ) 2.22 and o ) 2.04. In our previous work on alcohol monolayers with different chain lengths on water21 we found e ) 2.29 and o ) 2.12 for an area per molecule of 21.5 Å2. Converting these values to an area per molecule of 20.9 Å2, we find e ) 2.34 and o ) 2.16. A useful comparison can also be drawn with crystalline polyethylene, for which optical constants e ) 2.50 and o ) 2.31 and an area per chain of 18.3 Å2 have been reported.45 Correcting these dielectric constants to an area of 20.9 Å2 yields e ) 2.24 and o ) 2.09, in close agreement with the values determined in this work. These four sets of experiments involve three distinct chemical systems and make very different assumptions in the analysis. The principal assumption in this work is that the film thickness is that of a fully extended (CH2)12 chain. In our previous work with alcohol monolayers of different chain lengths, we assumed that the mean polarizabilities of liquid alkanes and alcohol monolayers were the same.21 Riegler and co-workers assumed that the two phases of a Langmuir monolayer have the same anisotropy and used X-ray diffraction measurements to reduce the number of independent unknowns.24 Comparison of monolayers with bulk polyethylene ignores the contributions from amorphous regions and the interlamellar interfaces to the optical properties of the polymer. Despite all these differences, the calculated dielectric constants differ by only 5%, and the refractive indices by less than 3%. While there is unquestionably a need to refine the data further, the agreement provides a degree (43) Karaborni, S.; Toxvaerd, S. J. Chem. Phys. 1992, 96, 5505. (44) Marion, J. B.; Heald, M. A. Classical Electromagnetic Radiation; Academic Press: San Diego, CA, 1980; p 289. (45) Polymer Handbook; Brandrup, J., Immergut, E. H., Eds.; Wiley: New York, 1989; p V/20.
Optical Properties of Monolayers
of confidence that the optical parameters calculated by diverse routes are broadly correct. Conclusion Knowledge of the optical constants of thin anisotropic organic films is essential for quantitative interpretation of data obtained from ellipsometry, surface plasmon resonance, and surface spectroscopy. Ellipsometry is a highly sensitive technique for studying monolayers but suffers from the limitation that, in the thin-film limit, only one independent parameter can be measured while there are three unknowns: e, o, and d. To circumvent this problem, we have measured the ellipticity of monolayers of dodecanol on KBr solutions of varying concentrations. For these monolayers, the thickness is known
Langmuir, Vol. 13, No. 20, 1997 5469
from X-ray diffraction measurements and variation in the substrate dielectric constant allows the determination of both e and o. We have shown that an analysis of the ellipsometric data requires a careful treatment of surface roughness and of the depletion layer. After allowing for these effects we find e ) 2.23 ( 0.07 and o ) 2.12 ( 0.03 at an area per molecule of 20.9 Å2. These values are very sensitive to the value of d used in the analysis of the data but nevertheless are in moderate to excellent agreement with optical constants determined by other methods. Acknowledgment. We thank the EPSRC and Unilever Research for a CASE award to B.D.C. LA9704718
