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Langmuir 2001, 17, 5872-5881
Evaluating Optical Techniques for Determining Film Structure: Optical Invariants for Anisotropic Dielectric Thin Films E. K. Mann Department of Physics, Kent State University, Kent, Ohio 44242-0001 Received December 13, 2000. In Final Form: June 12, 2001
A model-free method, based on optical invariants, of analyzing the reflectivity of thin adsorbed dielectric films is extended to dielectric waveguide techniques such as optical waveguide lightmode spectroscopy (OWLS). The reliability of film parameters determined from three general types of techniques (OWLS, scanning angle reflectometry, and scanning angle ellipsometry) is compared in the context of uniaxially anisotropic adsorbed films, with major axis perpendicular to the film. Expressions for the invariants for stratified anisotropic films in terms of simple moments of the layer optical distribution are presented. These general methods hold out the promise of more extended analysis of the optical response of thin films adsorbing on dielectric waveguides, while avoiding the pitfalls of specific optical models.
I. Introduction An increasing number of techniques have been developed for characterizing the thin films that almost invariably form at interfaces. Through selective deuteration, neutron reflectivity can be an unparalleled tool for determining the average density distribution of different moieties of adsorbed molecules at fluid/solid or fluid/fluid interfaces. Local probes such as the atomic force microscope (AFM) can determine the lateral distribution of molecules or particles at fluid/solid interfaces. The data from optical techniques, where the probe has wavelength λ . L, the thickness of the adsorbed film, entangle all these effects, along with optical anisotropy. Such techniques remain extremely useful by providing in situ characterization of the films at a laboratory scale. They have allowed exploration of an immense variety of surface films, from proteins1 to polymers2,3 to ordinary surfactants4,5 to colloidal-sized particles,6 governed by a large number of parameters including bulk concentration, ionic strength, the surface, and the history of the system. A number of different methods have been developed to improve the sensitivity of optical techniques to films with L , λ. Two different methods utilize the special properties of the Brewster angle: light polarized parallel to the incident plane (p-polarized) does not reflect at this angle. Ellipsometry measures the change in polarization of light after reflection, typically near the Brewster angle.7 Scanning angle reflectometry measures the reflectivity of p-polarized light around the Brewster angle.3,6,8 A third (1) Buijs, J.; van den Berg P. A. W.; Lichtenbelt, J. W. T.; Norde, W.; Lyklema, J. J. Colloid Interface Sci. 1996, 178, 594. Norde, W. Cell Mater. 1995, 5, 97. Malmsten, M.; Siegel, G., Wood W. G. J. Colloid Interface Sci. 2000, 224, 338. Nylander, T.; Tiberg, F.; Wahlgren, N. M. Int. Dairy J. 1999, 9, 313. (2) Eskilsson, K.; Grant, L. M.; Hansson, P.; Tiberg, F. Langmuir 1999, 15, 5150. (3) Leermakers, F. A. M.; Gast, A. P. Macromolecules 1991, 24, 718. (4) Tiberg, F.; Jonsson, B.; Lindman, B. Langmuir 1994, 10, 3714. Tiberg, F.; Jonsson, B.; Tang, J.; Lindman, B. Langmuir 1994, 10, 2294. (5) Velegol, S. B.; Fleming, B. D.; Biggs, S.; Wanless, E. J.; Tilton, R. D. Langmuir 2000, 16, 2548.; Pagac, E. S.; Prieve, D. C.; Tilton, R. D. Langmuir 1998, 14, 2333. Charron, J. R.; Tilton, R. D. J. Phys. Chem. 1996, 100, 3179. (6) Koper, G. J. M. Colloids Surf. A 2000, 165, 39. (7) Azzam, R. M. A.; Bashara, N. M. Ellipsometry and Polarized Light; North-Holland: Amsterdam, 1989.
set of techniques follows the change in the resonance wavenumber for optical waveguides as the thin film forms on its surface.9-11 This paper focuses on the interpretation of data from such linear optical techniques, comparing the ambiguities that optical anisotropy induces for the different techniques. All three types of techniques can be sensitive to very thin films. The difficulty lies in interpreting the optical data in terms of film parameters: the film thickness and average density, as well as the distribution of molecules or particles within the film. Many different film configurations can lead to the same optical signal. Typically, a very specific film model, the uniform isotropic film, is assumed in order to analyze the data in terms of a thickness and optical density. Often a further assumption, as to the optical density, is made to deduce an “optical thickness”. The use of optical invariants12,13 to analyze optical data avoids specifying a model to this degree. Optical invariants are the set of optical parameters that determine all reflectivity measurements on thin planar films. The optical response of the thin film is treated by introducing a fictitious flat, abrupt Fresnel interface in the interfacial region (see Figure 1). In analogy with the Gibbs interface of surface thermodynamics, the difference between the optical response of the Fresnel interface and the true optical response is given in terms of surface excess quantities, here surface excess polarization densities.12,14 A multipole expansion of these polarization densities yields two such parameters, one parallel and one perpendicular to the surface, at each multipole order.15 However, any measurable quantities must be independent of the placement of the fictitious Fresnel interface: the optical response thus depends only on those combinations of the (8) Schaaf, P.; De´jardin, Ph.; Schmitt, A. Langmuir 1987, 3, 1131. (9) Tiefenthaler, K.; Lukosz, W. J. Opt. Soc. Am. B 1989, 6, 209. (10) Bernard, A.; Bosshard, H. R. Eur. J. Biochem. 1995, 230, 416. (11) Ramsden, J. J. Q. Rev. Biophys. 1993, 27, 4. Ramsden, J. J. J. Stat. Phys. 1993, 73, 853. (12) Bedeaux, D.; Vlieger, J. Physica 1973, 67, 55. (13) J. Lekner Theory of Reflection; Martinus Nijhoff Publishers: Dordrecht, The Netherlands, 1987. (14) Haarmans, M. T.; Bedeaux, D. Thin Solid Films 1995, 258, 213. (15) This ignores any long-range anisotropy parallel to the surface.
10.1021/la001746d CCC: $20.00 © 2001 American Chemical Society Published on Web 08/24/2001
Optical Techniques for Determining Film Structure
Figure 1. (a) Schematic of the Fresnel interface, in analogy with the Gibbs interface. The real interface at left is replaced by the perfect abrupt interface at right, plus additional excess dipolar polarization densities γ˜ e,β and quadrupolar densities τ,δ at the interface. (b) Schematic for introducing in two different ways the Fresnel interface for an optical waveguide. Note the difference in the assumed waveguide thickness. The choice z0(1), shown on the upper right, is made both for simplicity and for minimization of the influence of the waveguide on the excess polarization densities.
polarizabilities that are invariant with respect to a displacement of the fictitious interface. These combinations are the optical “invariants”. For thin dielectric films between dielectric media, the optical response can be written in terms of at most three independent invariants, one to first order in the characteristic film thickness L/λ, the wavelength of light, and two to second order in L/λ.12-14,16 Optical invariants allow a determination of the maximum amount of information available from the data with a minimum amount of ambiguity. They are directly based on the information that is experimentally accessible for thin films. Using full optical equations in terms of the physical parameters of a particular model can be deceptive when these parameters do not appear independently in the optical data. However, interpreting the invariants may be difficult: they are not directly related to any simple optical quantity, but rather to combinations of them as entangled by the optical measurement themselves. Earlier work17 addressed this difficulty and demonstrated the utility of the invariant formulation for both particulate18,19 and protein20 films, analyzing experimental data from scanning angle reflectometry and model data from both this technique and ellipsometry. In both cases, a crucial question was the relation between “optical” thicknesses determined through a simple model and the true physical thickness of the film. The invariants were (16) Nondielectric materials can also be analyzed using optical invariants,12,14 but this article will be confined to the dielectric case for simplicity. (17) Mann, E. K.; Heinrich, L.; Voegel, J. C.; Schaaf, P. Prog. Colloid Polym. Sci. 1998, 110, 296. (18) Mann, E. K.; Heinrich, L.; Semmler, M.; Voegel, J. C.; Schaaf, P. J. Chem. Phys. 1998, 108, 7416. Mann, E. K.; Zeeuw, E. A. v. d.; Koper, G. J. M.; Schaaf, P.; Bedeaux, D. J. Phys. Chem. 1995, 99, 790. (19) van Duijvenbode, R. C.; Koper, G. J. M. J. Phys. Chem. 2000, 104, 9878. (20) Mann, E. K.; Heinrich, L.; Voegel, J. C.; Schaaf, P. J. Chem. Phys. 1996, 105, 6085. Heinrich, L.; Mann, E. K.; Voegel, J. C.; Schaaf, P. Langmuir 1997, 13, 3177.
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found to give directly both the total adsorbed mass and the thickness of the layer, both for particulate and for isotropic nonuniform films; an additional parameter yielded information about the uniformity of the film.17,18,20 The limits of both ellipsometry and reflectometry in determining the invariants, and thus the film structure, were clear, directly from the expressions for the optical response in terms of the invariants.17-20 This article adapts the invariant formulation of optical data analysis to dielectric waveguide techniques such as optical waveguide lightmode spectroscopy (OWLS). The invariant formulation allows a discussion of the critical impact of film anisotropy on parameters deduced with data from waveguides, in comparison with both ellipsometry and reflectivity. The use of waveguides provides a sensitive and increasingly popular method for studying adsorption at the liquid/solid interface.9,11,21 Light propagates along the waveguide, described in more detail in section II, at discrete resonance frequencies. The presence of a film at the interface between the waveguide and a solution changes the resonance frequency of the waveguide. The shift in this frequency can be measured with great precision, which allows the determination of the amount of adsorbed material with corresponding precision, provided one can translate the frequency shift into the desired film parameters. OWLS and similar techniques have allowed exquisitely precise determinations of deposition kinetics for the protein onto solid surfaces.11 The quality of these experiments was sufficient to demonstrate that, depending on the protein and the surface, the increase in total adsorbed mass with time does not follow the simple Langmuir model, but is in agreement with a random sequential adsorption model.11 When the uniform, isotropic film model was used to deduce both the thickness and the total mass from the shift in wavevector of two resonances, the results were more ambiguous. Guemouri et al.22 looked at two different proteins with different physical character: transferrin, which is modeled as an oblate spheroid with major and minor axes of 9.3 and 3.2 nm, respectively, and fibronectine, which is a highly elongated and flexible protein of approximate dimensions 120 × 2 nm. With transferrin, data analysis by the isotropic film model showed a constant thickness of ∼3.5 nm during adsorption. This thickness is consistent with the short dimension of the molecule and may indicate that this molecule adsorbs preferentially on its side. The fibronectine film, using the same model to analyze the optical data, was found to continuously increase in thickness during deposition, from 0.2 to 0.6 nm. Such thicknesses are unphysically small. The protein may unfold significantly at the surface, particularly when much surface is available, at the beginning of the deposition. A simple polymer, shown to lie flat at both solid and liquid interfaces, has a layer thickness of about 0.7 nm liquid surfaces, as shown by neutron reflectivity,23 and also about 0.7 nm on solid surfaces, as shown by X-ray reflectivity.24 A protein film is unlikely to be so thin, and it is very unlikely indeed to be thinner. Guemouri et al.22 suggest that the very thin deduced thickness indicates that the (21) Tien, P. K. Rev. Mod. Phys. 1977, 49, 361. (22) Guemouri, L.; Ogier, J.; Ramsden, J. J. J. Chem. Phys. 1998, 109, 3265. (23) Lee, L. T.; Mann, E. K.; Langevin, D.; Farnoux, B. Langmuir 1991, 7, 3076. Lee, L. T.; Mann, E. K.; Guiselin, O.; Langevin, D.; Farnoux, B.; Penfold, J. Macromolecules 1993, 26, 7046. (24) Daillant, J.; Benattar, J. J., Leger, L. Phys. Rev. A 1990, 41, 1963.
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Mann
Figure 2. Schematics of different optical techniques. Heavy dashed lines indicate typical light paths. (a) OWLS and other dielectric waveguide experiments. The refractive index of the waveguide g > w, s, the dielectric constants of the solution and of the support, respectively. θg is greater than the angle of total reflection at both interfaces. Light will propagate down the waveguide with little loss if the total phase change zigzagging across the guide and back is an integer multiple of 2π, leading to constructive interference. Typically, Lg ) 200 nm and g ) 1.8, allowing excitation of only the lowest TE and TM modes; this Lg value optimizes the sensitivity of these two modes to the adsorbed film. Here, light is coupled in and out of the guide with a grating coupler, but a prism may also be used for coupling. (b) Ellipsometry and reflectometry. The angle of incidence θ s may be varied.
optical boundary does not coincide with the physical boundary. However, analysis by optical invariants suggests that the optical thickness of an isotropic, stratified layer corresponds to an average thickness as defined in the usual manner by the first and zero moment of the optical distribution.17,20 Such a thickness would not be much less than the thickness of a compact layer. Clerc and Lukosz, using an anisotropic uniform thin film model to fit their data from similar experiments,25 suggest that a very small degree of anisotropy may be responsible for anomolous thickness readings in protein layers. Such anisotropy may be expected if the proteins are modeled as ellipses aligned at the interface. Using a Maxwell-Garnett argument, Lukosz25 suggests that the anisotropy can easily be ∼2% of the refractive index increment due to the protein. Clerc and Lukosz25 suggest that their data on adsorbed avidin layers support a model for the protein as ellipsoids standing on end. Here, direct analytical expressions for the effect of anisotropy on the determined layer parameters will be developed using the invariant formulation. It is not necessary to invoke the uniform film model, but such a model will be used in comparing directly the sensitivity of the waveguide techniques, ellipsometry, and reflectometry. Section II of this paper will discuss waveguide data in terms of optical invariants. Section III will discuss the effect of optical anisotropy on invariants within in the framework of the stratified layer model. This framework is expected to be appropriate for the adsorption of proteins or surfactants. Section IV will compare the sensitivity of the determination of mass and thickness for adsorbed layers to their anisotropy, for the three different techniques, using both the invariants and model data. Appendix A presents the invariants for a stratified anisotropic layer in terms of moments of the optical density distribution. Appendix B presents recipes for determining invariants from optical data and film parameters from those invariants. It also discusses in detail the prospect of determining more information than presently available from such optical techniques as OWLS.
index film, mounted on a base of lower refractive index and put in contact with a liquid, also of lower refractive index. A prism or grating couples the waveguide to light incident from outside. Beyond a certain incident angle, light is totally reflected with respect to both waveguide walls. Light within the waveguide can be described in terms of multiple reflections along the guide. Light propagates down the waveguide if, after zigzagging across the guide, it remains in phase with all light following other paths to that point. The phase change, with respect to incoming light, from one round trip across the guide is 2k0Lg(g - N2)1/2; Lg is the thickness of the waveguide, k0 ) 2π/λ is the wavenumber for light in a vacuum, g is the dielectric constant within the waveguide, and
N ) g1/2 sin θg
(1)
where θg is the angle with respect to the normal to the waveguide of the wavevector within the interior of the waveguide (see Figure 2a).26 The phase also shifts as the light is reflected at each interface, by a value Φgs at the guide/substrate interface and a value Φgaw at the waveguide/adsorbed layer/fluid interface. The condition for propagation, or resonance, is thus
2k0Lg(g - N2)1/2 + Φgs + Φgaw ) 2πm
(2)
where m is any integer. The phase change Φgs can be calculated from the standard Fresnel coefficients, if the guide/substrate interface can be considered flat and abrupt:
[( ) (
Φgs ) -2 arctan
g s
σ
N2 - s2
)]
g2 - N2
1/2
(3)
The OWLS waveguide system is schematically shown in Figure 2a. The waveguide consists of a high-refractive
where σ ) 0 for waves polarized perpendicularly to the incident plane (s or TE waves) and σ ) 1 for parallel polarization (p or TM waves). Φgac can be calculated by standard methods if the structure of the adsorbed film is known.9 In practice, the frequency shift for different resonance modes of the waveguide is virtually always calculated by assuming that the adsorbed film is a uniform, homogeneous layer with a given thickness and optical density. Given a measurement of the actual frequency shifts for at least two
(25) Lucosz, W. Biosens. Bioelectron. 1997, 12, 175. Clerc, D.; Lukosz, W. Biosens. Bioelectron. 1997, 12, 185.
(26) For convenience, we will most often use dielectric constants rather than refractive indices n, but ) n2 as usual.
II. Optical Waveguide Techniques in the Invariant Formulation
Optical Techniques for Determining Film Structure
resonance modes, the thickness L and relative optical density ∆w of the film can be calculated. The local mass density within the film c can then be calculated by assuming c ) ∆w/(d/dc),27 to give the total adsorbed mass M ) Lc. The deduced parameters must however be considered “optical averages” for the film parameters, in that they depend in uncontrolled ways on the particular model chosen to calculate the optical response. The invariant method for the analysis of optical data for the thin films states that all the information on the film (to second order in L/λ) is entangled in three invariants, as discussed in the Introduction. The case of the waveguide is slightly different, because the thin film of interest coats a thicker film, the waveguide, which is typically 200 nm thick (see Figure 2). With ordinary light, λ ∼ 600 nm, the invariant approximation to second order in L/λ is not valid for the complete film/waveguide system. However, the multipole expansion for the excess polarizability densities, combined in the invariants, still validly describes the reflectivity of the adsorbed film alone. The optical response of the waveguide can again be described in terms of propagation within the waveguide and reflection at the two interfaces, but with the reflective coefficient of the thin waveguide/adsorbed film/water interfacial region given through surface excess polarization densities. Note however that we can no longer argue that the optical response depends only on “invariants” with respect to a translation of the fictitious Fresnel interface. The response of the waveguide depends on whether the adsorbed layer is in fact on the waveguide surface or at some distance away from that surface. In an alternate view, it remains true that the overall optical response of the waveguide cannot depend on the position of the fictitious Fresnel interface used to calculate the reflective coefficient of the thin adsorbed film. However, it is no longer just the optical invariants discussed above that enter into the optical response of the complete system. The position of the border of the central, homogeneous part of the waveguide is defined by the same Fresnel interface (Figure 1b illustrates two choices for the fictitious Fresnel interface): If we move the position of the fictitious Fresnel interface, we must also reconsider the thickness of the waveguide. If the waveguide interface itself can be considered as flat and abrupt, it is simplest to take the fictitious Fresnel surface as the real one, at the waveguide surface (choice z0(1) in Figure 1b). Experimentally, care should be taken to ensure that this is a reasonable approximation. Otherwise, the transition layer at the waveguide surface should be treated together with the thin adsorbed layer. In what follows, the waveguide surface is assumed to be sufficiently Fresnel-like, and all excess polarization densities will be given with respect to that surface. Whether we consider that the fictitious Fresnel surface is no longer arbitrary in position or that the thickness of the waveguide must be adjusted with this fictitious surface, it is clear that the optical response will not be constrained by the optical invariants as in simpler experimental systems. At least one polarization density, whose noninvariant part would be compensated for by variations in the assumed thickness of the waveguide itself, must play a role. In what follows, we will describe the waveguide measurements in terms of the first-order optical invariant (27) Demonstrated in bulk: De Feijter, J. A.; Benjamins, J.; Veer, F. A. Biopolymers 1978, 17, 1759. dc/d ) 2.0 g/cm3 for a large variety of proteins.
Langmuir, Vol. 17, No. 19, 2001 5875
J1 (first order with respect to L/λ), the two independent second-order invariants J21 and J22, plus γ˜ e, the dipolar term of the total excess polarization density parallel to the interface (the only dipolar term for the TE mode). The interpretation of these quantities will be discussed in section III in the framework of the stratified layer model, expected to be appropriate for the adsorption of proteins or surfactants. The phase change Φgaw at the waveguide/fluid interface can be readily calculated from the reflectivity coefficients in terms of the surface excess polarization density, as given by Bedeaux et al.14 For TE waves,
[
]
(g - Ns2)1/2 rs(θ) ) exp -2iγ˜ e × w - g (g - Ns2)1/2(1 + J21) - i(Ns2 - w)1/2(1 - J21) (g - Ns2)1/2(1 + J21) + i(Ns2 - w)1/2(1 - J21)
(4a)
while for TM waves,
rp(θ) ) exp[-2iγ˜ e(g - Np2)1/2/(w - g)] ×
[
(g - Np2)1/2(1 - (J21 - J22Np2)) -
i
(
g Np2 J1 + (Np2 - w)1/2(1 + (J21 - J22Np2)) w w
[
)] /
(g - Np2)1/2(1 - (J21 - J22Np2)) +
(
)]
Np2 g J + (N 2 - w)1/2(1 + (J21 - J22Np2)) i w 1 w p
(4b)
For nonadsorbing systems and the total internal reflection required by the waveguide technique, (N2 - w)1/2 as well as all the refractive indices and parameters of the film are real. It follows directly from eq 4 that, for TE waves, the phase shift on reflection from the adsorbed film is given by
(g - Ns2)1/2 Φsgaw ) -2γ˜ e w - g
(
2 arctan
)
(Ns2 - w)1/2(1 - J21) (g - Ns2)1/2(1 + J21)
(5a)
while for TM waves, the phase shift is given by
Φpgaw ) -2γ˜ e
(
2 arctan
(g - Np2)1/2 w - g
)
J1Np2 + (Np2 - w)1/2g(1 + (J21 - J22Np2)) w(g - Np2)1/2(1 - (J21 - J22Np2))
(5b) We can expand the arguments of the arctan, keeping only terms to second order in L/λ, as appropriate for this level of approximation, to find
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(
Φsgaw ) -2 arctan
Φpgaw
(
)
Ns2 - w 2
g - Ns
1/2
- 2γ˜ e 2
))
Np2 - wg
(
1/2
rp(θ) s
r (θ)
(g - Np2)1/2 +2 × g - w
J12Np4(g - Np2)1/2
)
(g2 - w2)(Np2 - wg)
(5d)
gw wg ) g + w
(6)
Notice that while we have kept all terms to second order in L/λ, there are in both cases first-order terms, containing only γ˜ e and J1, and these will dominate the phase shifts for L/λ , 1 (more exactly, for L/λ < 0.01; see appendix B.) Under these conditions, eqs 3 and the first-order terms of eqs 5c and 5d will give a direct relation between the condition for resonance (eq 2) and γ˜ e and J1. The measurement of two mode resonances, typically the firstorder TE (or s) and TM (or p) modes, will thus determine γ˜ e and J1, through the equations
g - w (g - Ns2)1/2
(
πms - k0Lg(g - Ns2)1/2 +
(
2 arctan
J1 )
Np2 - wg Np2wg /g
(
γ˜ e -
×
g - w (g - Np2)1/2
(
))
Ns2 - w g - Ns2
1/2
(7a)
πmp - k0Lg(g - Np2)1/2 +
((
2 arctan
) )))
g Np2 - w w - N 2 g p
s
Light travels with angle θi within the layer with refractive index ni ) i1/2 and the different angles are related by the Snell law, ni sin θi ) nj sin θj. The standard Fresnel coefficients for reflection and transmission at a perfect, abrupt interface, tij and rij, are given by:
(
ni cos θi
and
σ
-
)/( /(
nj cos θj
i
ts,p ij ) 2
σ
j
ni cos θi σ
nj cos θj
+
nj cos θj
i
ni cos θi ni cos θi σ
ni nj
σ
) )
+
σ
i
jσ jσ
(10)
where σ ) 0 for s polarization (indicated by a superscript s) and σ ) 1 for p polarization (superscript p). J23 is a linear combination of J21 and J22 (see eq 12 with its discussion). By inspection, we see that in principle the waveguide data contain the most information, γ˜ e, J1, and two secondorder invariants, while ellipsometry contains the least, J1 and only one second-order invariant. This conclusion must be moderated by practical considerations. Waveguides are usually optimized for the determination of adsorbed quantitites and support only the first TE and TM modes: only two resonance shifts can be determined. Fortunately, the first-order terms γ˜ e and J1 dominate all resonance shifts for sufficiently thin films, and, in principle, these can both be determined (see Appendix B). The resonances will, however, give less overall information than the less sensitive reflectometry technique, where all three low-order invariants enter at the same, second order in L/λ. How this plays out in practice will be explored in a particular case: uniaxially anisotropic films with major axis perpendicular to the interface. This case is expected to apply to surfactants and proteins adsorbing at an interface.
1/2
(7b)
where ms is the order of the s resonance with wavenumber k0Ns inside the guide, while mp is the order of the p resonance with wavenumber k0Np. The interpretation of the quantities γ˜ e and J1 for homogeneous stratified layers with uniaxial anisotropy perpendicular to the layer is discussed below. The expressions for the waveguide resonance shift can be compared with the dependence on the invariants of scanning angle reflectometry:20
|rp(θ)|2 ) (rpsw)2 2tpsw tpwsrpsw(J23 - s(sin2 θs - sin θB2)J22) + ((tpsw)2(tpws)2/4w)sin2 θs tan θs tan θwJ12 (8) and of scanning angle ellipsometry13
)
(tpsw)2 sin θs tan θs (s + w) tpsw tpws sin2 θs J23 - i J1 (9) w 2 1/2
rs,p ij )
where we define
γ˜ e ) -
(
(tpsw)3 tan2 θs sin2 θs 2 1 p rsw J1 rssw 4(ws)1/2
(5c)
(g - Np2)1/2(Np2 - w)1/2 × - 2wg (g - w)(Np2 - wg)
2(J21 - J22Np2) -
≡ tan ψei∆ )
2 1/2
J21(Ns - w) (g - Ns ) 4 g - w
(( )
J1Np2wg/g
(g - Ns2)1/2 + g - w
1/2
g Np2 - w ) -2 arctan w - N 2 g p
γ˜ e -
Mann
III. Optical Invariants for Uniaxially Anisotropic Uniform Films Here we will assume the adsorption of protein or other molecules from solution with dielectric constant w to the flat surface of a solid with dielectric constant 1. Note that, for the waveguide geometry in Figure 2a, 1 ) g, while for the ellipsometry and scanning angle reflectometry geometries in Figure 2b, 1 ) s. The dielectric constant within the film is || parallel to the surface and ⊥ perpendicular to the surface. Exact expressions for the optical properties of uniaxial layers are given elsewhere;28 here we wish to treat the layers in terms of invariants. Exact expressions for the invariants of layered thin films are given in the appendix, but here, for simplicity, we consider films that are uniform through their depth as well as laterally. Define the increment ∆ of the dielectric constant and the optical anisotropy R as (28) Lekner, J. Pure Appl. Opt. 1994, 3, 821.
Optical Techniques for Determining Film Structure
∆)
| - w w
R)
and
Langmuir, Vol. 17, No. 19, 2001 5877
⊥ - | ⊥ - | ) | - w w∆
(11)
For proteins, it is reasonable to treat ∆ and R as being of the same order: Scanning angle reflectometry on IgG protein layers found ∆ < 0.04,20 while a Maxwell/Garnettstyle estimation would suggest R < 0.03 for most proteins.25 To second order in these two parameters, the invariants, as given in equations in ref 1429 and in eqs A2-A4 of Appendix A, are then given by
γ˜ e ) wk0L∆
[(
J1 ) - (1 - w) 1 -
(12a)
)/
w∆ - (1 - w∆)R 1 - w
]
(1 + ∆(1 + R)) k0L∆
(
1 (∆ - R) k0L∆ 1 - w
J21 ) -
w w∆ 1k 2L2∆ 2 1 - w 0
≈ - (1 - w) 1 -
[( (
J22 ) - 1 ≈- 1-
(12c)
]
)
1∆ R + k 2L2∆ 1 - w 2 0
(12d)
1w 1 + w
w(1 - w)
(
≈
)
2(1 + w)(1 + ∆(1 + R)) 1-
)
(12b)
w∆(1 + R) R + /(1 + ∆(1 + R)) k02L2∆ 1 - w 2
J23 ≡ J21 - J22 )
(
)
×
(1 + 2w)∆ + 1R ∆R(1 + w) 1 - w 1 - w
(
w(1 - w) 2(1 + w)
)
∆2(1 + R)w(1 + w)
1-
(1 - w)
2
)
k02L2∆
(21 + w)∆ 1R + k 2L2∆ 1 - w 1 - w 0 (12e)
For the final, approximate forms of these equations, it was necessary to assume not only ∆,R , 1 but also ∆/ (1 - w), R/(1 - w) , 1. Note that only two of the three second-order invariants are independent. J21 and J22 appear naturally in the theoretical calculations.12,14 However, elllipsometry depends only on the single second-order invariant J23. Further, J21 and J22 are highly correlated when determined from fitting scanning angle reflectivity data, while J23 is uncorrelated with either of the other two second-order invariants.20 J23 and either of the other two second-order invariants are more appropriate for describing optical data in the region of the Brewster angle. The Brewster angle is irrelevant for OWLS data, and expressions for J21 and J22 are slightly simpler than for J23. All three secondorder invariants are provided for convenience. (29) Note a factor of 2 missing in the denominator of Iτ in eq 3.23 of ref [14].
Even in approximate form, only the equation for γ˜ e is direct and intuitive: as the excess polarizability density parallel to the interface, it is just the excess parallel dielectric constant Ew∆ multiplied by the film thickness (nondimensionalized by the wavenumber). The invariants are, by nature, unintuitive combinations of the different film parameters. J21 is the simplest, in that it depends only on the multipole polarization densities parallel to the interface: R does not appear. In the other invariants, ∆ and R appear together as correction terms to the dominant k0L∆ dependence of the first-order invariant J1 and the dominant k02L2∆ dependence of the second-order invariants. The implications of this entanglement of ∆ and R on the extraction of structural information from optical data will be discussed in detail in the next section. It can already be seen that quantities both first and second order in k0L will be necessary to deduce L and ∆ separately, to even moderate accuracy; at least three such values would in principle give R as well. However, the possibility of nonuniform films should also be considered. Appendix B discusses the treatment of data in which both R and the nonuniformity of the film may be factors. IV. Comparison of the Effect of Anisotropy on Film Parameters Deduced Using the Isotropic Film Model on Data from OWLS, Ellipsometry, and Scanning Angle Reflectometry If L/λ is sufficiently small (
