9878
J. Phys. Chem. B 2000, 104, 9878-9886
A Comparison between Light Reflectometry and Ellipsometry in the Rayleigh Regime Rene´ C. van Duijvenbode* Leiden Institute of Chemistry, Gorlaeus Laboratories, Leiden UniVersity, P.O. Box 9502, 2300 RA Leiden, The Netherlands
Ger J. M. Koper Laboratory of Physical Chemistry, Delft UniVersity of Technology, Julianalaan 136, 2628 BL Delft, The Netherlands ReceiVed: May 17, 2000; In Final Form: August 7, 2000
The adsorption of Rayleigh particles is analyzed in terms of particle radius and surface coverage with thin island film theory with both scanning angle light reflectometry and ellipsometry around the Brewster angle. A comparison between both techniques shows that an additional uniformity parameter can be extracted out of the experimental reflectivity data. This gives information about the distribution of the adsorbed mass normal to the surface. Fixed angle reflectometry is less sensitive to surface properties than fixed angle ellipsometry. This is closely related to the fact that ellipsometry measurements provide an extra measurable physical quantity, the change in ellipticity at the surface, which has a weaker but different dependence on surface coverage and layer thickness. This enables ellipsometry to distinguish between a broad range of combinations of surface coverage and particle radii that give similar reflectivity. Fixed angle reflectometry can therefore only lead to an interpretation in terms of adsorbed mass. Scanning angle reflectometry measurements, on the contrary, can easily be interpreted in terms of surface concentration and thickness and make further ellipsometry measurements unnecessary.
Introduction Adsorption processes play an important role in various industrial and biological applications, such as chromatography, catalysis, adhesion, stabilization of colloidal particles, biocompatibility of artificial organs in medicine, mineral processing, and so on. Because of the very broad character of these applications, numerous scientific disciplines are involved, with different types of particles varying from colloids to (bio)polymers. To study the kinetics and statics of adsorption different techniques are applied, each with its own advantages and drawbacks. Many adsorption studies involve optical microscopy, that has a very straightforward analysis, albeit that only a limited range of particle sizes can be studied. Electron microscopy makes the Rayleigh regime accessible but requires rather difficult and time-consuming sample preparation, that can also affect the properties of the system of interest. More favorable means to study the adsorption are reflection techniques; they are nondestructive and can be applied in situ. A powerful tool to characterize thin film layers, and in particular to obtain the density profile through the adsorbed film, is neutron reflectivity. The wavelength of the reflecting neutron beam is of the order of the dimensions of the film and thus the technique contains more information about the adsorbed layer than any other technique. Unfortunately, neutron sources are not readily available. Another drawback is related to the small radiation wavelength. Neutron reflectivity does not only give more information about the layer, but it contains also a large contribution from the bulk media. This hampers the extraction * To whom correspondence should be addressed at Leiden University. Fax: +31715274397. Email:
[email protected].
of information from the reflectivity data. To enhance the contrast, selective deuteration is necessary, which can be difficult, and can lead to changes in the properties of the system, especially in the case of biological macromolecules. A good alternative is light reflectivity. Because the wavelength of light is large in comparison to the characteristic thickness of most macromolecular layers, these optical techniques can only be used under special conditions. These conditions can easily be derived from the Fresnel equations. Together with Snell’s law, these relations describe the reflection and transmittance of light at an interface in terms of the bulk refractive indices (n1,n2) and the angle of incidence θ1 and transmittance θ2 going from medium 1 to 2, for the s-wave, with the E-field component perpendicular to the plane of incidence
r12s )
n2 cosθ2 - n1 cosθ1 2n1 cosθ1 and t12s ) n2 cosθ2 + n1 cosθ1 n2 cosθ2 + n1 cosθ1 (1)
and the p-wave, with the E-field component parallel to the plane of incidence
r12p )
n2 cosθ1 - n1 cosθ2 2n1 cosθ1 and t12p ) n2 cosθ1 + n1 cosθ2 n1 cosθ2 + n2 cosθ1 (2)
When we consider a two component system for which the transition region (the region in which the dielectric permittivity changes from the bulk value of the first component to the bulk value of the second component) is very small compared to the wavelength of the light, the reflection and transmittance of the light from the interface are quantified by the Fresnel equations.
10.1021/jp001832m CCC: $19.00 © 2000 American Chemical Society Published on Web 09/28/2000
Light Reflectometry and Ellipsometry From eq 2, we can derive the special condition where r12p ) 0 at tan θ1 ) n2/n1. This angle is called the Brewster angle θB. The fact that at the Brewster angle the Fresnel amplitude for p -polarized incident radiation vanishes for a perfect, abrupt interface, implies that at, and around, this angle, the reflectivity (Rp ) |rp|2) and ellipticity (F ) rp/rs) are extremely sensitive to particle adsorption. The reflection amplitudes rp and rs are then the result of a summation over the reflection and transmittance coefficients of all interfaces present in the system. This enables techniques such as optical reflectometry, which simply measures the light reflectivity, and optical ellipsometry, which measures both the amplitude and the phase of the ratio rp/rs, to study film formation by small particles, despite the large wavelength. The latter technique is especially widely used for these purposes. A disadvantage of both techniques is the less straightforward manner of data interpretation. In the data analysis, quite often the adsorbed material is treated as a homogeneous layer. The, in reality nonhomogeneous layer is effectively smeared out over the surface, and the analysis results in an “optical” thickness and an effective refractive index for this layer. For particles of colloidal dimensions adsorbed in a random manner, it is doubtful whether this gives a reasonable idea of what happens at the interface. The precise physical meaning of the optical thickness of the layer remains unclear. An extensive review regarding these effective medium theories has been written by Landauer.1 Born and Wolf2 and Lekner3 present a method in which the adsorbed material is analyzed as a thin stratified layer, that is a layer in which the dielectric permittivity is allowed to vary in the direction normal to the interface. Bedeaux and Vlieger4,5,6 have developed a more general theory that allows for the description of so-called thin island films. This term is used to stress the two main features of the system, namely that the thickness is small is compared to the wavelength of the light and that the system is discontinuous. The dielectric permittivity is allowed to vary in all directions in the film. They analyze this situation by placing “excess polarization densities” and “excess magnetization densities” on a “dividing surface”, which is usually taken to be the substrate-ambient interface. The restriction is that the fields far away from this surface are invariant to the precise location of the dividing surface. The reflectivity and ellipticity of this system, when it only contains dielectric materials, can then be described in terms of optical invariants, quantities that do not depend on the choice of location of the dividing surface at which the excess polarizabilities and excess magnetizations are calculated. For isotropic spherical particles, the polarizabilities can be assumed to be uniform in all directions, which simplifies the theory again to the stratified layer theory of Born and Wolf and Lekner. The optical invariants can be expressed in terms of particle parameters: size, refractive index, and coverage.7 The approach is further simplified, considering only interactions at the dipolar level between the particles and neglecting the interactions between particles and substrate. This seems reasonable for low coverages and small differences in refractive indices of the films. A relative simple expression for the reflectivity and ellipticity can be derived with an expansion of the reflection up to first order in the phase difference. That way the particles are assumed to be situated directly on top of the Fresnel interface. In practice it is often observed that there is a more gradual change in the refractive index of the substrate close to the surface, and it has to be modeled with an extra layer between the Fresnel interface and the particles. The latter approach implies a full expansion of the phase factor and especially for thicker layers this can become of great importance.
J. Phys. Chem. B, Vol. 104, No. 42, 2000 9879
Figure 1. Assembly of noninteracting spheres adsorbed on a glasswater interface is placed on a mathematical dividing plane a distance h . a above the interface. The data analysis is split up in a Abeles' matrix method with a glass surface layer between the Fresnel interface and the adsorbed particles (a), and a calculation with optical invariants where the particles adsorb directly on top of the Fresnel interface (b). The mathematical picture corresponding to each situation is shown on the right-hand side.
In the present paper, we will use both light reflectometry and ellipsometry to study the adsorption of particles in the Rayleigh regime in order to get more insight into the experimental differences. We pay special attention to the differences in data interpretation and the advantages and disadvantages of both techniques therein. We will further check the assumption of the experimentally used substrate as a theoretically sharp boundary in order to ease the data interpretation. Therefore, we also perform a data analysis in which we include a gel layer at the glass surface with a different refractive index from the one in the bulk of the glass. Theory To interpret the reflectometry and ellipsometry data in terms of particle properties, we use the thin island film theory proposed by Bedeaux and Vlieger.4,5 In this perturbation theory, a dividing surface is introduced, which is usually taken to be located at the position of the interface. The bulk dielectric permittivities are extrapolated to this mathematically well-defined interface and the optical properties are described by the Fresnel equations. In reality, there is a difference in the real dielectric permittivities and the extrapolated bulk values, taken into account by excess polarization and magnetization densities on the dividing surface. The Maxwell equations, together with these new boundary conditions, lead to a modification of the Fresnel equations stated above. Details about the proper choice of these excess quantities and the constitutive relations for the excess polarizabilities parallel and perpendicular to the surface can be found elsewhere in the literature.4,5,7,8 The theory is applied here to dielectrics, but includes magnetic materials as well.9 Here, it is sufficient to know that the proper choice of the position of the mathematical dividing surface leads to a nonlocal dependence of the polarization densities of the particles on the field. In the description using an equivalent polarization density on a sharply defined dividing surface, the whole distribution of polarization is shifted to a dividing surface at a height h above the substrate-ambient interface (the dashed line in Figure 1a). The shifted polarization densities are, however, coupled to the fields at the original
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position, the glass surface layer(2)-water(3) interface. This way, it is taken into account that if a surface is spatially dispersive, the polarization is no longer related to the fields in a local manner. The nonlocal response found is caused by the heterogeneous nature of the interface. The polarization is distributed over a thin layer at this height h, and this layer may contain discrete islands and may be rough. In view of the fact that the shift h is only over a small distance, and the fact that the bulk fields vary slowly over this distance, it is sufficient to expand the fields to linear order in the gradient. For the case where one considers a film consisting of small spherical particles adsorbed at an interface, expressions for the constitutive coefficients were derived by Haarmans.7 The response of the particles to the field is characterized up to second order in the sphere radius over the wavelength of the incident light. For the particles in the Rayleigh regime, we shall neglect the intrinsic quadrupole contributions, which reduces the number of constitutive coefficients to two. There is a response to the electric field experienced by the particles γ parallel to and β normal to the substrate
( ) ( )
γ ) 4k02 β)
L)
F)
where a is the radius of the particles, φ is the fractional surface coverage, and k0 ) 2π/λ, where λ is the wavelength of the light source in a vacuum. Note that the fractional surface coverage φ itself is also a function of the particle radius, for spherical particles it is the surface concentration times the consumed area πa2. In principle, these constitutive coefficients are affected by the presence of other spheres and substrate, but is shown by Haarmans et al.8 that a truncation at the dipolar level yields an accuracy of a few percent for nonabsorbing materials. The simplified relations presented in eq 3 are therefore sufficient to describe the optical properties of the Rayleigh particles at reasonably low coverages. The modified Fresnel equations for the reflection and transmission amplitudes in terms of radius, coverage, and refractive index of the spheres are used in a model in which reflections from additional layers apart from the Fresnel interface (substrate - ambient) are summed up, assuming these layers are stratified. An expansion is made in the film thickness or, for instance, the size of the discrete islands of adsorbed material divided by the wavelength. A phase factor corrects for the retardation of the reflected beam coming from the extra layers on top of the Fresnel interface. An illustration of this is presented in Figure 1a and 1b, which also shows that there are different approaches to model the reflectometry and ellipsometry data. A. Abeles’ Matrix Method. The most common and convenient method is to use the Abeles' matrix method.10 The method uses two types of complex 2 by 2 matrixes to describe light propagation through the interface. This method is applicable to both the p- and s-wave of the reflected and transmitted light. An interface matrix I12 deals with the transition from the first layer into the next, in terms of the Fresnel amplitude transmission and reflection coefficients
(
1 1 r12 t12 r12 1
)
(4)
A layer matrix L describes the propagation of the light through
)
(5)
(
1 1 -rf tf rf t2f - r2f
)
(6)
For p-polarized light the amplitude reflection and transmission amplitudes for the Rayleigh spheres read
rfp ) (3)
(
e-i∆ 0 0 ei∆
with a phase factor ∆ ) 2k0a cosθ, which accounts for the retardation of the light. 2k0a is the ratio of the layer thickness over the wavelength of the beam. Instead of a description of the adsorbed particles in terms of a uniform layer with an optical thickness and an effective refractive index, we use the modified reflection and transmission amplitudes rf and tf.11 In the abovedescribed matrix formalism, this means that the interface matrix for the adsorbed particles is modified into
3 - 2 φa 3 + 22
4k0 3 - 2 φa 2 3 + 22
I12 )
the layer
iX iX + iY + 2XY iY and tfp ) 1 - iX 1 - iY (1 - iX)(1 - iY)
(7)
γ cos θ2 βn32 sin2 θ2 and Y ) 2 cos θ2 2n2
(8)
where
X)
and for s-polarized light
rfs )
X X and tfs ) 1-X 1-X
(9)
iγ 2n2 cos θ2
(10)
with
X)
In our experimental setup we take a glass surface gel layer on top of the glass-water interface, to which the particles are adsorbed (see Figure 1a), and the product A of such matrixes is
A ) I12L2I23F
(11)
The total amplitude reflection coefficient of interest is the ratio of the matrix element A21 to A11. The existence of an additional glass surface layer is well-established in the literature.12,13 Notice that there is absolutely no correlation between the position h of the mathematical dividing surface and the thickness of the surface layer in Figure 1a and that the layer is calculated from the position of the glass surface layer on which the particles are adsorbed. B. Optical Invariants Method. When this surface gel layer is small and the particles themselves are in the Rayleigh regime, the retardation of the reflected beam can be assumed to be small. The model can then be further simplified using an expansion of the reflection and transmission amplitudes to first order in the phase factor, thus thereby taking the phase difference into account up to first order in the thickness. This leads to expressions as introduced by Bedeaux and Vlieger for the reflectivity of p-polarized light and ellipticity F from a non-
Light Reflectometry and Ellipsometry
J. Phys. Chem. B, Vol. 104, No. 42, 2000 9881
TABLE 1: Relationship between the Three Sets of Optical Invariants in Literaturea ref (14, 16, 17, 31)
ref (4, 5, 8)
ref (3)
J1/k0 J2,1/k20 J2,2/k20 J2,3/k20
Ie Iτ Iδ Iτ - (12/(1 + 2)) Iδ
-J1 i2/2(2 - 1) j2/(2 - 1) i2 - (12/(22 - 21)) j2
a
Note the additional factor two in the conversion of the second order optical invariants J2,2 and Iδ into j2 compared to earlier publications (ref 14).
adsorbing interface in terms of parameters of the film called optical invariants (J1, J2,2, J2,3)3,14
Rp(θ) ) R0p(θ) + ((t212p t221p/42) sin2 θ1 tan θ1 tan θ2)J21 2t12pt21pr12p(J2,3 - 1(sin2θ1 - sin2θB)J2,2) (12) and3,15
F(θ) ) F0(θ) - i
t212p sin θ1 tan θ1 J1 2n1r12s
t312p sin2 θ1 tan2 θ1 2 t12pt21p(1 + 2) sin2 θ1 J1 J2,3 (13) 4n1n2r12s 2r12s respectively, with n1, n2 the refractive indices of the substrate and the ambient in our experimental setup, θ1 the angle of incidence, and R0p(θ) ) |r12p|2 and F0(θ) the Fresnel reflectivity and ellipticity. All of the terms leading to a departure from the Fresnel law are up to second order in the layer thickness (higherorder terms are ignored). With this approach, the adsorption is simplified to a layer of adsorbed particles directly on top of the Fresnel interface, as is shown in Figure 1b. These optical invariants J1, J2,2, J2,3 are combinations of the constitutive coefficients chosen such that the analysis is independent of the choice of the position of the dividing surface. It is already stated earlier that the effect of the nonlocal nature of the interfacial coefficients is related to the choice of the position h of the dividing surface, and that the coefficients depend sensitively on this choice. As the optical properties themselves do not depend on this choice, they should be h-independent combinations of these coefficients (see also Figure 1b). The latter two invariants contain extra intrinsic quadrupole contributions of the spheres, which were neglected in the Abeles' matrix method. Various versions of these invariants exist in the literature but they differ only by trivial factors, as summarized in Table 1. Notice the extra factor 2 in the conversion between (J2,2, Iδ) and j2 compared to ref 14. Here, we have chosen to use a dimensionless form of the invariants as they are given by Haarmans and Bedeaux8
J1 ) γ - 12β J2,2 ) -k0aγ +
γ2 2(1 - 2)
1 - 2 γ(γ - 12β) + J2,3 ) k0aγ 1 + 2 2(1 - 2)
(14)
The first invariant is of first order in the thickness of the layer, whereas the other two are of second order in the thickness. The invariants can be related to moments of the dielectric permittivity distribution. As such, the first invariant, J1, is the
zeroeth moment of the dielectric permittivity distribution.3 It is directly related to the total mass adsorbed at the interface
Γ≈
n2
J1
(15)
2k0(2 - 1)dn/dc
with dn/dc the bulk refractive index increment. Although the analysis of reflectometry measurements gives next to J1 two other optical invariants to second order in the thickness (J22 and J23 in eq 12), only J23 can be obtained from ellipsometry (eq 13). Thus, reflectometry gives the possibility to obtain additional information about the adsorbed layer next to radius and coverage. These second-order invariants J22 and J23 are related to first moments of the dielectric permittivity distribution3 and, in principle, yield information on the distribution of the adsorbed mass. Mann et al.16 have studied this dependence and defined yet another invariant F that they termed the optical uniformity parameter. In terms of the previously defined invariants, it is written as F
F)1-
(
2J2,3 -
1(1 - 2) J 1 + 2 2,2
(
)(
2J21 J2,3 +
2J2,3 +
2(1 - 2) J 1 + 2 2,2
12 J 1 + 2 2,2
)
)
(16)
The coefficient F, thus defined, is equal to 0 for a uniform isotropic layer, and can be used to investigate the validity of this model. It depends more sensitively than any of the original invariants on the uniformity of the thin films. The most important property of the function F is that for stratified films it is independent of both the thickness and the average refractive index and only depends on the distribution of the material within the film. Consider a film on top of a substrate. The parameter F is positive when the layer is denser near the substrate and negative when the layer is denser near the ambient.17 Indeed, it vanishes for a uniform film. Also, the uniformity parameter may be proven to depend on the lateral distribution of material in a film.16 Experimental Section A schematic diagram of the used reflectometer/ellipsometer is presented in Figure 2. The light source is a stabilized 1 mW HeNe laser (λ ) 632.8 nm). The laser beam passes through two Glan-Thompson polarizers and a quarter wave plate (all Melles Griot). The intensity of the reflected beam is measured by means of a photomultiplier. The angle of incidence is selected by simultaneously rotating the laser and the detector supports. The two supports, and the optical elements mounted on them, are fully automated and computer-controlled with an accuracy of 1/1000°. With reflectometry, the fast axis of the compensator and the polarization axes of the polarizers are aligned in the plane of incidence (p-wave), and the intensity is measured as a function of the angle of incidence θ1. When the polarization axis of P and the fast axis of quarter wave plate C are aligned, the laser beam hitting the prism will be linearly polarized. The measured intensities are directly related to the reflectivity Rp(θ1) by
I(θ1) ) I0 + ARp(θ1 + δ)
(17)
where I0 is the residual intensity at the Brewster angle θB, A is an instrument-dependent constant, and δ allows for a small
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Figure 2. Schematic diagram of the scanning angle reflectometer/ ellipsometer: (L) light source; (P), (A) polarizers; (C) quarter wave plate or compensator; (S) substrate; (D) photodetector. For reflectometry measurements, the polarization and fast axes of the optical elements P, C, and A are aligned in the plane of incidence (p-wave), in case of null ellipsometry the compensator C is fixed at an angle of 45° with the plane of incidence, and the polarizers are rotated systematically in a search for minimal intensity.
systematic indeterminacy of the incidence angle.18 The amplification, residual intensity, and offset are determined prior to the adsorption experiments analyzing a chemically etched glassNaOH (0.1 M) interface in terms of a Fresnel interface.19,20 The error in the reflectivity with the present setup is of the order of 10-7. We compare this relatively simple technique to the most classical variant of ellipsometry, i.e., null ellipsometry. The transmission axis of P is now rotated in such a way that the elliptically polarized light caused by the compensator or quarter wave plate C is changed after reflection into linearly polarized light and can be extinguished by polarizer A. With the setup shown in Figure 2, which is sometimes addressed as the PCSA arrangement,10 the ellipticity F(θ1) of the interface is determined by
rp tan C - i tan(P - C) F(θ1 + δ) ) | | ei∆ ) -tan A rs 1 + i tan C tan(P - C)
(18)
where A, P, and C symbolize the angle of the transmission and fast axes of the different optical elements with respect to the plane of incidence. To achieve “null” intensity in these experiments, the compensator is fixed at 45°, and both polarizers are rotated according to a simplex algorithm in the intensity minimization process. In data analysis, the ellipticity is often split up in the ratio of the amplitude |rp/rs| or |F|, and the phase difference ∆ between the p- and s-wave. The error with which these parameters are determined with this setup are of the order 10-4 and 1°, respectively. The adsorption experiments have been performed on polystyrene latex particles of five different sizes, a ) 10, 20, 30, 42, and 50 nm, as determined with electron microscopy by the manufacturer (Interfacial Dynamics Corporation, Portland, OR). The latex particles are negatively charged because they are stabilized by sulfonate groups. The refractive index of the spheres is given as 1.591 (at 590 nm) by the manufacturer. The original particle concentrations (5 % wt/wt) were diluted 1000 times with Millipore water (used throughout the experiment).
Figure 3. Examples of experimentally obtained scanning angle reflectivity curves corresponding to particle radii 10, 30 and 50 nm at the glass-water interface. Drawn lines are fits of the thin island film theory to the data; the results of the fit are presented in Table 2.
To adsorb the latex particles onto the also negatively charged optically flat hypothenuse of a rectangular prism made of glass (Schott BK7, refractive index n1 ) 1.515) a positive layer of a fifth generation polypropyleneimine dendrimer (DSM Geleen, The Netherlands) was attached to the glass prior to the latex adsorption. More detailed information about the adsorption of the dendrimers is given elsewhere in the literature.20,21 With bulk concentrations of the order of 0.1% wt/wt the dendrimer adsorption was typically of the order of 0.2 mg/m2. The molecular size of the dendrimer is of the order of 3 to 4 nm in the bulk,22,23 but it is shown for similarly built dendrimers that they flatten while being adsorbed at the interface.24 The particle adsorption ran, depending on the experiment, for 2 to 3 h, with a flow rate of 1.3 mL/min. A peristaltic pump (LKB 12000 VarioPerpex, Sweden) was used for this purpose. At the end of an experiment, pure water was flushed through the measurement cell, ensuring all nonadsorbed dendrimers and latex were removed. Because the curves did not change significantly after that, the conclusion that only adsorbed particles contribute to the evolution in reflectivity and ellipticity is justified. Examples of typical experimental scanning angle reflectivity and ellipticity (in terms of |F| and ∆) curves are shown in Figures 3 and 4 for particle sizes a ) 10, 30, and 50 nm. Data Analysis To interpret the experimental reflectivity and ellipticity curves with the thin island film theory, equations 12 and 13 and the Abeles' matrix method (eq 11) were implemented in a nonlinear least-squares fitting routine. The two fit parameters a and φ were systematically varied in a search for the least sum of squares χ2min. The sum of squares χ2 is related to the difference (either Rp, |F|, or ∆) and the between the experimental data fexp i theoretically calculated values f defined by the two fit parameters a and φ for N angles of incidence θ1
χ ) 2
1
∑
N i)1,N
(
)
fexp - f(θ1,i; a, φ) i σi
2
(19)
where σi is a weighting factor that corresponds to the maximum
Light Reflectometry and Ellipsometry
J. Phys. Chem. B, Vol. 104, No. 42, 2000 9883
Figure 4. Examples of experimentally obtained scanning angle ellipsometry measurements corresponding to particle radii 10, 30, and 50 nm at the glass-water interface. The ellipticity is plotted as the modulus of the ratio of rp over rs (|F|) and the phase difference ∆ between rp and rs. Drawn lines are fits of the thin island film theory to the data; the results of the fit are presented in Table 2. exp
accuracy with which f i can be measured (see the Experimental Section). The least sum of squares χ2min corresponds to the situation where the reflectivity and ellipticity data is modeled with radius a0 and coverage φ0, apart from an experimental noise contribution τi
f
exp i
) f(θ1,i; a0, φ0) + τi
(20)
The evolution of the sum of squares around (a0,φ0) gives more insight in the accuracy with which the data can be fitted with these two fit parameters. We therefore model the average experimental reflectivity and ellipticity 〈fexp i 〉 with f(θ1,i; a0, φ0) (the average contribution of the experimental noise 〈τi〉 is zero, and 〈τ2i 〉 ) 〈σ2i 〉) and calculate the change in the average sum of squares 〈χ2〉 for f(θ1,i; a, φ), varying a and φ. The eigenvalues of the Hessian matrix with the second derivatives of 〈χ2〉 with respect to a and φ give the largest and smallest dependency of the sum of squares on these fit parameters. For this type of function, the eigenvalues in the minimum given by (a0,φ0) are generally given by
λ1 ) 0 and λ2 ) 2
(
) (
∂2 f(θ1,i; a0, φ0) ∂a2
2
+2
)
∂2 f(θ1,i; a0, φ0) ∂φ2
(21) The fact that the sum of squares gives a zero eigenvalue in the minimum, implies that there always exists one axis along which there is no variation in 〈χ2〉, and another axis with maximum curvature in 〈χ2〉 (up to second order in deviations). To fit the parameters independently, it is important that the eigenvectors
Figure 5. Evolution of the average sum of squares 〈χ2〉 with particle radius a and surface coverage φ for a fixed angle measurement is plotted for both reflectivity Rp (top), and ellipticity F (bottom), using the thin island film theory. The experimental situation is modeled with the 2 function f (a0 ) 30 nm, φ0 ) 15%). 〈χ2F〉 is defined as the sum 〈χ|F| 〉+ 2 2 〈χ∆〉. The dependence of 〈χ|F|〉 on the fit parameters is similar to the observations in the top figure. 〈χ2∆〉 (middle) shows a different, albeit much weaker, dependence.
do not coincide with the axes of the fit parameters: a dominant fit parameter decreases the accuracy with which the other parameter can be determined. In Figure 5, the evolution of 〈χ2〉 with a and φ is plotted for Rp (top), ∆ (middle), and a summation of 〈χ2〉 for both |F| and ∆ (bottom), simulating the reflectivity and ellipticity data for a system where Rayleigh particles with size a0 ) 30 nm are adsorbed on a glass substrate with surface coverage φ0 ) 15%. Figure 5 is the result of calculations at a fixed angle of incidence θ1 close to the Brewster angle for this particular system, which simplifies eq 19 to just the summation over one angle of
9884 J. Phys. Chem. B, Vol. 104, No. 42, 2000 incidence. The experimentally obtained ellipsometry data are 2 〉 + 〈χ2∆〉, identical to the approach plotted also fitted using 〈χ|F| in Figure 5. The influence of a change in particle properties is especially noticeable close to the Brewster angle for the p-wave, the change in the reflection coefficient for the s-wave upon adsorption is negligible. The evolution of the average sum of squares 〈χ2〉 for |F| is therefore similar to the observations for Rp. Both 〈χR2 p〉 2 〉 do not show a sharp minimum but a series of and 〈χ|F| combinations of sphere radius and coverage which give values close to the reflectivity Rp and modulus F for a0 ) 30 nm and φ0 ) 15%. In the literature, an alternative way of measuring the reflectivity, the so-called "poor man’s" ellipsometer is proposed. In addition to the reflectivity of the p-wave, the reflectivity of the s-wave is measured simultaneously. The ratio of both reflectivities Rp and Rs, |F|2, is related to the adsorption via the squared modulus of eq 13. The sensitivity of this method to surface coverage and particle radius is again identical to the trend in Figure 5 (top) because the relative change in Rs with adsorption is negligible. Figure 5 shows that under fixed angle conditions, the extra parameter ∆, the phase difference between the p- and s-wave, leads to a more clearly defined combination of the fit parameters a and φ in case of ellipsometry. The largest eigenvalue of the Hessian matrix of 〈χ2〉 in (a0,φ0) at fixed angle of incidence is for the present conditions about 603 for Rp, 161 for |F| and only 11 for ∆. The change in 〈χ2〉 is largest for the case where Rp, ∆ does not show such a strong dependency of 〈χ2〉 on the fit parameters. The corresponding eigenvector, which gives the axis of maximum curvature in 〈χ2〉, makes an angle with the a-axis of 35° for both Rp and |F|, and 20° for the case of ∆, clearly showing a different dependency on radius and coverage. Despite the relatively low curvature in 〈χ2〉 in the minimum for the latter case, it is still of importance because of the different dependencies. It results in a better defined well in the bottom 2 figure, where 〈χ2〉 is the sum of 〈χ|F| 〉 and 〈χ2∆〉. The results presented in Figure 5 are based on calculations with optical invariants, but identical trends are observed in the case of the Abeles' matrix method. Even for the case of the widely used uniform film model, there is a broad range of combinations of surface concentration and layer thickness, which give comparable reflectivity signals. In either case, there is a very tedious relation between the experimentally accessible quantities and the properties of interest, which could easily lead to erroneous combinations of thickness and surface concentration. Note also that the 〈χ2〉 shown in this figure is the result of averaging out the experimental noise τi, which in real experiments complicates the analysis. In Figure 6 〈χ2〉 is a summation over various angles within a domain of ( 1° around the Brewster angle, similar to a typical experimental scanning angle curve (see Figures 3 and 4). The extra information on the change of the Brewster angle and the shape of the curve lead to a more reliable fit of the layer thickness and coverage. In Figure 6, both the optical invariants method and the homogeneous slab model are tested on their applicability in fitting the experimental data. Because we consider here Rayleigh particles directly attached to the Fresnel interface, the phase factor is small, and thus, the difference between the optical invariants method and the Rayleigh model, which differ in the expansion of the phase factor, is negligible. Therefore, the minimum shown in Figure 6 is identical for the optical invariants method and the Rayleigh model. It should be taken into account that this is no longer the case when a system
van Duijvenbode and Koper
Figure 6. 〈χ2〉 is plotted for both scanning angle reflectometry (top) and ellipsometry (bottom), identically to Figure 5, but now summed over various angles of incidence with a domain of ( 1° around the Brewster angle (see also Figures 3 and 4). The optical invariants method is also compared to the uniform film model. In the film model, the adsorbed particles are smeared out as a slab with an effective refractive index (via the Lorenz-Lorentz equation2 related to the surface coverage) and an optical thickness. The color scale is the same as in Figure 5.
is considered where particles are adsorbed on thicker layers of, for instance, polymers.11 A fit of the scanning angle reflectometry curve in terms of coverage and thickness gives then only one possible combination of these parameters, with a deeper well than that which is observed for ellipsometry. Figure 5 already demonstrated that the change in 〈χ2〉 for Rp was much more pronounced than for |F| and ∆ together, and the summation over various angles in Figure 6 will enhance that difference even more. In contrast to the fixed angle measurements, the minimum is easier to locate for scanning angle reflectometry data than for ellipsometry data. Results The thin island film theory developed by Bedeaux and Vlieger is fitted to the experimental data in Figures 3 and 4. The outcome in the optical invariants |J1|, J2,2, and J2,3 in case of reflectometry (eq 12), and J1, J2,3 for ellipsometry (eq 13) are nonlinear combinations of γ and a (eq 14). Mann et al.14 suggested different direct routes to calculate the size of the particles out of combinations of the invariants with various levels of accuracy and precision. We chose to extract these parameters with a nonlinear fit procedure involving all optical invariants together. This gives results for the layer thickness that were systematically 10% higher than calculated with equations 9a and 9b in ref 14, where terms larger than first order in ∆ are neglected. The results in terms of radius and coverage (between brackets) are plotted together in Table 2 for all combinations of the two models and two techniques. In case of an analysis with the Abeles' matrix method the reflectivity and ellipticity of the background glass-water interface are interpreted as coming from a Fresnel interface with on top a uniform glass layer with an optical thickness of 4.5 nm and an effective refractive index of 1.45 (see Figure 1a). The particle properties come directly out of the fit parameters γ and a. Table 2 does not show any
Light Reflectometry and Ellipsometry
J. Phys. Chem. B, Vol. 104, No. 42, 2000 9885
TABLE 2: Comparison between Radii, as Obtained from Fitting the Ellipsometry and Reflectometry Data in Terms of Excess Polarization Densities and Values from the Manufacturer (Obtained with Electron Microscopy)a Abeles' matrix method
optical invariants
radius
ellipsometry
reflectometry
ellipsometry, eq 13
reflectometry, eq 12
10 ( 2 nm 20 ( 3 nm 30 ( 3 nm 42 ( 3 nm 50 ( 2 nm
8 nm (18%) 22 nm (17%) 37 nm (13%) 35 nm (20%) 58 nm (13%)
9 nm (20%) 19 nm (20%) 36 nm (13%) 36 nm (20%) 58 nm (13%)
10 nm (14%) 22 nm (17%) 37 nm (14%) 40 nm (17%) 53 nm (14%)
12 nm (19%; F ) 0.2) 22 nm (19%; F ) 0.5) 35 nm (15%; F ) 0.3) 37 nm (20%; F ) 0.5) 50 nm (16%; F ) 0.4)
a A distinction is made between the Abeles' matrix method with a glass surface layer between substrate and spheres (Figure 1a), and a direct analysis with optical invariants using equations 12 and 13 (Figure 1b). The coverages also determined with the analysis are indicated between brackets. With the equations 12 and 16 the uniformity coefficient F is also determined from the reflectometry data.
systematic difference in the fit parameters when comparing reflectometry and ellipsometry with one another. The different methods in data analysis do not seem to have a systematic effect on the outcome either. Only for 50 nm particles the direct approach using optical invariants gives a value for the particle radius that is much closer to the manufacturer’s value than the results from an interpretation in which the spheres are situated on a glass surface layer. These observations are in line with the expectations. Mann et al. suggested that the particles are embedded in a gel layer,26 but so far, there is no way to test their hypothesis. For small interfacial layers such as the 3 to 4 nm for the glass layer in this study there should be no significant difference between the two approaches. But in case of a larger interfacial layer or a multilayered system, it should be the more extended approach with additional layers that gives a better picture of the physical reality because it takes the full phase factor into account. The fact is that in the case of the extra glass surface layer, an additional set of parameters is needed, which reduces the overall accuracy in the fit parameters. The extra optical invariant J2,2 in eq 12 gives the possibility of extracting additional information about the layer of adsorbed particles by means of a uniformity coefficient F (eq 16) in the analysis of reflectometry measurements, next to the particle radius and the surface coverage. The results are also plotted in Table 2. This uniformity parameter F gives information about the distribution of the particles through the layer. Possible values for F and its corresponding distribution of the dielectric constant through the layer are presented in Table 2 in ref 14. For all particle sizes, we found F ≈ 0.4-0.5, which agrees with a random distribution of spheres with the centers at z ) a. It also shows that an analysis in terms of a uniform layer with an overall effective refractive index and an optical thickness is not a proper way of interpreting the results. Such a homogeneous, uniform distribution of the dielectric constant would imply F ) 0. The results of an analysis as a uniform layer are left out of the present paper, it is sufficient to state that an interpretation of the experimental data with a uniform film model still gives reasonable values for the thickness and surface concentration. However, the coverages in Table 2 are reasonably large, and the film islands can be interpreted as a uniform layer smeared out over the substrate. For smaller surface coverages, we do indeed see that the film model deviates more from the thin island film theory. The particle radii in Table 2 are the results of an interpretation of the adsorbed particles as monodisperse spheres. In reality, there is a size distribution to be taken into account, which is of the order of 10% according to the manufacturer, see also Table 2. Although in principle polydispersity can be introduced in the equations, fitting calculated curves corresponding to a sample with a polydispersity of several percent does not yield different results for the particle radius.18,27 An analysis of a polydisperse
system assuming no size distribution leads to an underestimation of the surface coverage, whereas the fitted radius is still very close to the mean radius of the particles. Especially in the case of a ) 10 and 20 nm the effect of size dispersion on the reflectivity/ellipticity of the system plays a role. From the analysis of the reflectivity curves, we observe a gradual increase in the mean particle radius in time. This is completely in line with what in the literature is called the Vroman effect for (bio)polymers.28,29 Although the initial adsorption is dictated by the smaller and thus faster diffusing particles, the final thermodynamically more favored situation is the one where the larger particles exchange with these initially adsorbed small particles. The radii for the particles under consideration in Table 2 are the outcome of the latter, thermodynamically favored case. Discussion We now elaborate on the advantages and drawbacks of reflectometry and ellipsometry, from both theoretical as well as experimental perspective. One of the experimental advantages of ellipsometry over reflectometry is that for the former case the positions of the optical elements directly relate to the ellipticity, there are no apparatus constants to be determined. The only calibration parameter is the systematic indeterminacy of the incidence angle, δ. This, in contrast to reflectometry measurements, where next to this angle δ, the residual intensity I0 and amplification A are also to be determined prior to the experiments. A more important difference between the two techniques is the time needed to obtain the shape of the reflectivity/ellipticity curve around the Brewster angle. The main drawback of null ellipsometry is that it is rather time-consuming. With the present setup it takes two minutes to determine F at a certain angle of incidence, where for reflectometry this is a matter of seconds, including the movement of the supports and alignment of the reflected beam into the photomultiplier pinhole. The poor man’s ellipsometer, measuring Rp and Rs simultaneously, has the advantages over normal p-wave reflectometry and null ellipsometry that the influence of laser beam undulations are ruled out, and that the ellipticity in terms of |F|2 is measured at much faster time scales. However, to relate the intensities to the theoretical reflectivities, a calibration method is still needed to subtract the background intensity and find the so-called sensitivity factor.30 Furthermore, and that shifts the discussion to the theoretical point of view, the poor man’s ellipsometer and the null ellipsometer relate the adsorption to the optical invariants J1 and J23, whereas the reflectometer gives an additional optical invariant J22. Therefore, p-wave reflectometry gives, next to the particle radius and coverage, access to additional information about the adsorbed layer, the gradient of the dielectric permittivity normal to the substrate.
9886 J. Phys. Chem. B, Vol. 104, No. 42, 2000 The main drawback of the reflectometer, although the same holds also for the poor man’s ellipsometer, is the sensitivity of the reflectivity curves to the surface concentration and thickness of the layer. Both techniques give only one parameter. Rp, |F| and |F|2 are in a very tedious way related to the properties of the adsorbed layer at fixed angle measurements close to the Brewster angle (see Figure 5), and therefore extra attention is needed to extract this information from the experimental curves. Null ellipsometry gives extra experimental information in the form of ∆. Because of the different dependence of this phase factor on the surface coverage and layer thickness, the total sensitivity of ellipsometry is enhanced dramatically, and this makes fitting with optical invariants, but the same is also true for fitting the adsorbed layer as a uniform film layer, more reliable. Although fixed angle ellipsometry measurements give a well-defined minimum, multiple minima are obtained from reflectometry data (Figure 5). However, as becomes clear from Figure 6, the situation is completely reversed when performing scanning angle measurements around the Brewster angle. The curve, containing extra information about the change in shape of the curve and the shift of the Brewster angle upon adsorption, makes it possible to distinguish between the concentration of particles adsorbed on the surface and the thickness of that layer, even in case of reflectometry. In fact, the minimum is then even more pronounced than for scanning angle ellipsometry measurements; the sum of the relative small changes in 〈χ2〉 for ∆ over various angles is overwhelmed by the stronger dependence for |F|, and does no longer give additional information. Conclusion In summary, we have shown that both light ellipsometry and reflectometry can be useful in situ tools to study the adsorption of particles in the Rayleigh regime. Although ellipsometry is more frequently used, reflectometry has the advantage over ellipsometry that, in addition to the fact that the time scales to perform reflectometry are much shorter, extra information can be extracted out of the experimental data. The adsorbed layer can be analyzed in terms of surface concentration and particle radius, and a uniformity parameter that gives information about the distribution of the adsorbed mass normal to the surface. An important side-note is that for fixed angle reflectivity measurements there is no clear dependence on the surface properties, but a broad range of combinations of surface coverage and particle radii which give similar values for the reflectivity. Fixed angle measurements can only lead to an interpretation of the data in terms of adsorbed mass. With fixed angle ellipsometry an additional experimental quantity ∆ is measured, and this makes it possible to distinguish between (a0,φ0) and other pairs of (a,φ). For scanning angle reflectivity and ellipticity measurements, the situation is completely different, the fit parameters can easily be determined from reflectometry, and the additional ∆ determined with ellipsometry does not give extra information. A comparison of different fitting methods to extract the surface properties out of the experimental data showed that a
van Duijvenbode and Koper more straightforward method with optical invariants and a linearized phase factor is just as efficient as analyzing the system in terms of an Abeles' layered structure in case of a small layer between the Fresnel interface and the adsorbed particles, with a full expansion of the phase factor. The advantage of both models over the homogeneous slab model is that the interpretation of the layer thickness with the thin island film theory has a more physical meaning than the “optical thickness” defined in the homogeneous slab model. Acknowledgment. The authors would like to thank Jacques van der Ploeg for sharing his expertise in software programming and Henk Verpoorten for his expert help with the instrument. References and Notes (1) Garland, Ed. J. C.; Tanner, D. B. Proc. First Conf. Electron. Opt. Prop. Inhomogeneous Media 1978, 40, 2-42. (2) Born, M.; Wolf, E. Principles of Optics; Pergamon Press: Oxford, 1959. (3) Lekner, J. Theory of Reflection; Martinus Nijhoff Publishers, 1987. (4) Bedeaux, D.; Vlieger, J. Physica 1973, 67, 55-73. (5) Vlieger, J.; Bedeaux, D. Thin Solid Films 1980, 69, 107. (6) Bedeaux, D.; Koper, G. J. M.; van der Zeeuw, E. A.; Vlieger, J.; Wind, M. M. Physica A 1994, 207, 285-292. (7) Haarmans, M. T. Ph.D. Thesis Leiden, Leiden University, 1995. (8) Haarmans, M. T.; Bedeaux, D. Thin Solid Films 1995, 258, 213223. (9) Lazzari, R.; Jupille, J.; Borensztein, Y. Appl. Surf. Sci. 1999, 142, 451-454. (10) Azzam, R. M. A.; Bashara, N. M. Ellipsometry and Polarized Light; North-Holland Publ. Co.: Amsterdam, 1989. (11) Bo¨hmer, M. R.; van der Zeeuw, E. A.; Koper, G. J. M. J. Colloid Interface Sci. 1998, 197, 242-250. (12) Iler, R. K. The Chemistry of Silica; Wiley: New York, 1979. (13) Doremus, R. H. Glass Science, Wiley; New York, 1994. (14) Heinrich, L.; Mann, E. K.; Voegel, J. C.; Schaaf, P. Langmuir 1997, 13, 3177-3186. (15) Bedeaux, D.; Vlieger, J. Optical Properties of Surfaces; work in progress. (16) Mann, E. K.; Heinrich, L.; Semmler, M.; Voegel, J. C.; Schaaf, P. J. Chem. Phys. 1998, 108, 7416-7425. (17) Mann, E. K.; Heinrich, L.; Semmler, M.; Voegel, J. C.; Schaaf, P. J. Chem. Phys. 1996, 105, 6082-6085. (18) van der Zeeuw, E. A.; Sagis, L. M. C.; Koper, G. J. M. J. Chem. Phys. 1996, 105, 1646-1653. (19) Fu, Z.; Santore, M. M. Colloids Surfaces A 1998, 135, 63-75. (20) van Duijvenbode, R. C.; Koper, G. J. M.; Bo¨hmer, M. R. Langmuir, accepted for publication. (21) van Duijvenbode, R. C.; Rietveld, I. B.; Koper, G. J. M. Langmuir, accepted for publication. (22) Scherrenberg, R.; Coussens, B.; van Vliet, P.; Edouard, G.; Brackman, J.; de Brabander, E.; Mortensen, K. Macromolecules 1998, 31, 456-461. (23) Rietveld, I. B.; Smit, J. A. M. Macromolecules 1999, 32, 46084614. (24) Tsukruk, V. V.; Rinderspacher, F.; Bliznyuk, V. N. Langmuir 1997, 13, 2171-2176. (25) AKZO, Eur. Patent Application No. 88200230.6. (26) Mann, E. K.; Bollander, A.; Heinrich, L.; Koper, G. J. M.; Schaaf, P. J. Opt. Soc. Am. A 1996, 13, 1046-1056. (27) Mann, E. K.; van der Zeeuw, E. A.; Koper, G. J. M.; Schaaf, P.; Bedeaux, D. J. Phys. Chem. 1995, 99, 790-797. (28) Ball, V.; Bentaleb, A.; Hemmerle, J.; Voegel, J. C.; Schaaf, P. Langmuir 1996, 12, 1614-1621, and references therein. (29) Hoogeveen, N. G.; Cohen Stuart, M. A.; Fleer, G. J. J. Colloid Interface Sci. 1996, 182, 146-157. (30) Dijt, J. C.; Cohen Stuart, M. A.; Fleer, G. J. AdV. Colloid Interface Sci. 1994, 50, 79-101. (31) Koper, G. J. M. Colloids Surfaces A 2000, 165, 39-57.
