Langmuir 1997, 13, 3177-3186
3177
Characterization of Thin Protein Films through Scanning Angle Reflectometry L. Heinrich,† E. K. Mann,†,‡,§ J. C. Voegel,‡ and P. Schaaf*,†,|,⊥ Institut Charles Sadron (CNRS-ULP), 6, rue Boussingault 67083, Strasbourg Cedex, France, Fe´ de´ ration de Recherche “Odontologie” U424 INSERM-ULP, Universite´ Louis Pasteur, 11, rue Humann, 67085 Strasbourg, France, and Ecole Europe´ enne de Chimie, Polyme` res et Mate´ riaux de Strasbourg, 1, rue Blaise Pascal, BP 296, 67008 Strasbourg Cedex, France Received October 22, 1996. In Final Form: March 17, 1997X This article has two major purposes. The first is to demonstrate the utility of the invariant method for the analysis of the optical properties of thin films. The use of invariants allows the determination of a maximum of information on the layer, independent of arbitrary optical models. In particular, it is possible to determine the surface concentration and an average thickness of the adsorbed layer, with additional information about the distribution of matter in the layer. The limitations of the invariant method are discussed in detail. Second, the optical properties of an antigen/antibody (both immunoglobulin G proteins) film adsorbed at the water/silica surface are explored, comparing the results of the invariant method of analysis and those of optical models, in particular the uniform isotropic layer and the bilayer. The results consistently indicate a depletion layer near the solid interface.
I. Introduction Films of adsorbed macromolecules occur at interfaces in most practical situations. Proteins adsorb on the surface of almost anything with which they come in contact, including prostheses, teeth, and cooking utensils, and play a controlling role in the proper operation of these implements. Polymer films are used to control adhesion, wettability, biocompatibility, and other properties intrinsically associated with surfaces. As of yet, little is known about even the structure of such films, although theoretical models have been developed for the case of simple polymer chains. Experimentally, the structure of protein films shows intriguing properties, tied to the very slow kinetics of protein desorption; on practical time scales, the adsorption shows signatures of irreversibility, including the dependence of the properties of the film on the adsorption history. Thus, the ease and success of removing the protein depend on how long it has been in contact with the surface. Furthermore, the structure of the layer may depend on the adsorption rate, which controls the time given for conformational changes in one adsorbed macromolecule before the arrival of a neighboring one, which may block such changes. Although the observed history dependence of both total surface concentration and the response to changes in conditions are assumed to be due to conformational changes in the macromolecules, little direct data exists for the structure of the macromolecular film, and in particular for the changes in that structure. The structure of macromolecular films has been characterized by many different techniques, each with its own advantages and drawbacks. In particular, reflection techniques are nondestructive and can be used in situ. Structural information, and in particular the density profile of the adsorbed film, has been obtained for polymer films using neutron reflectivity, which is particularly welladapted since the wavelength of the reflecting beam is †
Institut Charles Sadron. Universite´ Louis Pasteur. § Present address: Federal Institute of Technology (ETH)-Zurich, Grabenstrasse 3, CH-8952 Schlieren, Switzerland. | Ecole Europe ´ enne de Chimie, Polyme`res et Mate´riaux de Strasbourg. ⊥ Institut Universitaire de France. X Abstract published in Advance ACS Abstracts, May 15, 1997. ‡
S0743-7463(96)01031-1 CCC: $14.00
usually of the same order of magnitude as the characteristic thickness of the layer under study.1 The extraction of a maximum of information from the reflectivity is an ongoing problem.2 This technique generally requires deuteration of the macromolecules for good contrast with the bulk media. This may be an advantage in allowing selective deuteration of different parts of the macromolecule or of different species for more precise structural information. However, deuterating biological macromolecules without changing their properties is a difficult, to our knowledge generally unsolved, problem. Furthermore, neutron reflectivity cannot be used routinely in the laboratory, which is a severe disadvantage in a field with many possible systems and many variables with unknown impact. Light reflectivity is thus also used to study such films. Since 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, for example, at total reflection3 or for waves polarized in the plane of incidence (“p” waves) in the vicinity of the Brewster angle.4 At this angle and for this polarization, no energy is reflected from the perfect planar abrupt, or Fresnel, interface; the reflection coefficient is thus very sensitive to small departures from such an interface. Ellipsometry, which measures the amplitude and the phase of the ratio rp/rs of the amplitude reflection coefficients for waves with the two polarizations p and s, is widely used to obtain structural information about adsorbed layers.5,6 Scanning angle reflectometry (SAR), used here, simply measures the reflectivity Rp ) |rp|2 at (1) Lee, L. T.; Guiselin, O.; Farnoux, B.; Lapp, A. Macromolecules 1991, 24, 2518. Henderson, J. A.; Richards, R. W.; Penfold, J.; Thomas, R. K. Macromolecules 1993, 26, 65. Karim, A.; Satija, S. K.; Douglas, J. F.; Ankner, J. F.; Fetters, L. J. Phys. Rev. Lett. 1994, 73, 3407. (2) Crowley, T. L.; Lee, E. M.; Simister, E. A.; Thomas, R. K. Physica B 1991, B173, 143. Sivia, D. S.; Hamilton, W. A.; Smith, G. S. Physica B 1991, 173, 121. Chen, S. H.; Zhou, X. L.; Carvalho, B. L. Prog. Colloid. Polym. Sci. 1993, 93, 85. Bucknall, D. G.; Ferna`ndez, M. L.; Higgens, J. S. Faraday Discuss. 1994, 98, 19. (3) Tien, P. K. Rev. Mod. Phys. 1977, 49, 361. Rondelez, F.; Ausserre´, D.; Hervet, H. Annu. Rev. Phys. Chem. 1987, 38, 317. (4) Meunier, J. In Light Scattering by Liquid Surfaces and Complementary Techniques; Langevin, D., Ed.; Marcel Dekker, Inc.: New York, 1992; p 17. (5) Azzam, R. M. A.; Bashara, N. M. Ellipsometry and Polarized Light; North-Holland: Amsterdam, 1989. (6) Malmsten, M. J. Colloid Interface Sci. 1994, 166, 333.
© 1997 American Chemical Society
3178 Langmuir, Vol. 13, No. 12, 1997
a series of different angles around the Brewster angle. For both these techniques, the usual method of interpreting the optical data is to determine the thickness and refractive index of the uniform layer that corresponds most closely to the optical data. However, although this may define an average layer thickness and a surface concentration may also be deduced, it is unclear a priori how these values depend on the particular, very simple, optical model chosen. In principle, the data should be reanalyzed with a full range of physically reasonable models in order to clarify the meaning of the “optical thickness”, for example. An alternative, model-independent way of analyzing the optical data from reflection experiments on thin films is to characterize the layer by optical invariants. The invariants appear naturally in the analysis of the reflection of light in the long wavelength limit, in an expansion of the reflection coefficients in terms of increasing order in the parameter k0d, where d is the characteristic thickness of the layer and k0 ) 2π/λ the wavenumber of the light in the vacuum.7-9 The reflectivity coefficients rp and rs of a nonabsorbing film are, up to second order in k0d, functions of one first-order invariant Ie and two invariants of second order in the thickness, Iτ and Iδ.10 For stratified media, these invariants are given in terms of integrals over functions of the dielectric constants.7,9 For thin layers, for which higher orders of k0d can be neglected, in principle no more than three parameters should be used to entirely characterize these layers optically. In section II below, we will review the theory of the reflectivity of thin films in terms of invariants. The different definitions of the invariants found in the literature are given, along with a minor variation that allows the determination of three independent invariants from the usual SAR data. For a homogeneous layer, or any other two-parameter model, these three invariants must be related in a particular way. A test parameter, a function of the three invariants that will take on the value zero for a homogeneous layer, was thus defined recently.11 A more intuitive physical meaning, in terms of moments of the optical index distribution, is provided for the three invariants and for this test parameter in the context of the approximation of small refractive index augmentation. We will also discuss the relative sensitivity of different optical methods, in particular ellipsometry and SAR, to the structure of the layer. We are thus able to deduce the total quantity of adsorbed material and an average film thickness, where this average now has a precise physical interpretation. Furthermore, with sufficient precision in the data over a large enough angular range, one can determine not only the surface concentration and the optical thickness but also a higherorder moment. This higher-order moment provides a test for the consistency of any two-parameter model, such as that of the homogeneous layer. In particular, it is sensitive to the presence of either a dense adsorption layer or a depletion layer near the surface. This method of analysis is applied to a particular interfacial layer. In a recent article12 we used SAR to investigate the antigen-antibody reaction when the antigens are preadsorbed on a solid surface. The analysis, using the uniform thin film model, of the structure of the protein layer thus formed demonstrated a long-time (7) Lekner, J. Theory of Reflection; Martinus Nijhoff Publishers: Dordrecht, 1987. (8) Bedeaux, D.; Vlieger, J. Physica 1973, 67, 55. (9) Haarmans, M. T.; Bedeaux D. Thin Solid Films 1995, 258, 213. (10) Notations of Haarmans and Bedeaux.9 (11) Mann, E. K.; Heinrich, L.; Voegel, J. C.; Schaaf, P. J. Chem. Phys. 1996, 105, 6082. (12) Heinrich, L.; Mann, E. K.; Voegel, J. C.; Koper, G. J. M.; Schaaf, P. Langmuir 1996, 12, 4857.
Heinrich et al. Table 1. Relations between the Three Sets of Invariants Found in the Literature ref 14
refs 8 and 9
J1/k0 J2,1/k02 J2,2/k02
Ie Iτ Iδ
ref 7 -J 1
i2/2(2 - 1) j2/2(2 - 1)
evolution of this structure. Naively, one would expect that when a layer formed by preadsorbed antigens is brought into contact with a solution of its antibodies, a layer of the antibodies forms on top of the antigen layer, provided the conformation of the adsorbed antigen is such as to allow the antigen-antibody reaction to take place. We demonstrated that although such reactions occur, the layer that is built up is very different from this expectation, and may reach 3 times the thickness of a single protein layer. Furthermore, the structure as characterized by its optical thickness evolves with time long after the increase in surface concentration stops. Here, we apply the invariant method of analysis in order to obtain a further means of characterizing the thin film. We find clear evidence for a depletion layer: the density within the film increases as one moves away from the wall before it decreases again to the bulk value. II. Theory As discussed above, the reflectivity, to second order in the wavelength, of a nonadsorbing interface can be written in terms of three parameters of the film called optical invariants. Various versions of these invariants exist in the literature,7-9,13,14 but they differ only by trivial factors, as summarized in Table 1. Here, we have chosen to use a nondimensionalized form of the invariants as they are given by Haarmans and Bedeaux. For an isotropic stratified film, these invariants are given by15
[(z) - 1][(z) - 2]
∫-∞∞dz
J1 ) k0 J2,1 ) -k02
(z)
∫-∞∞dz∫-∞∞dz′ ×
{
}
[(z) - °(z - z′)][(z′) - °(z′ - z)] 2(2 - 1)
J2,2 ) k02
{
∫-∞∞dz∫-∞∞dz′ ×
[
(1)
]}
°(z′ - z) 1 1 [(z) - °(z - z′)] 2 - 1 (z′) °(z′ - z)
The invariants are seen to be integrals over the difference between the true dielectric constant (z) and the dielectric constant associated with an ideal Fresnel interface defined by
{
for z < 0 °(z) ) 1 2 for z > 0 where 1 and 2 are defined so that (z) - °(z) ) 0 far outside the interfacial region, that is, for |k0z| .1. The light is assumed to originate in medium 1 (z < 0; see Figure (13) Bedeaux, D.; Koper, G. J. M.; van der Zeeuw, E. A.; Vlieger, J.; Wind, M. Physica A 1995, 207, 213. (14) Mann, E. K.; van der Zeeuw, E. A.; Koper, G. J. M.; Schaaf, P.; Bedeaux, D.; J. Phys. Chem. 1995, 99, 790. (15) Reference 9 provides a more general expression, allowing anisotropy between the dielectric constants parallel and perpendicular to the interface. A trivial error of a factor of 2 appeared in the reference in going from eqs 3.22 and 3.23. Similarly, a factor of 2 should appear in eq 39, p 68 of ref 7, for agreement with his calculated invariants in Table 3-1 (p 75), which are indeed appropriate for his expression for the reflectivity in eq 50.
Thin Protein Films
Langmuir, Vol. 13, No. 12, 1997 3179
(a)
(b) Figure 2. Reflectivity as a function of incidence angle for a bare buffer/silica surface (×) and for the same surface after the successive adsorption of IgG (O) at 0.2 g/L, leading to a surface concentration of 2 mg/m2, and anti-IgG (b) at 0.3 g/L, leading to a surface concentration of 10 mg/m2.
Figure 1. (a) Schematic representation of the reflectometer: L, polarized HeNe laser; P1 and P2, polarizers; Eip and Erp, electric field of the incident and reflected beam polarized in the incident plane (p-waves); D, photomultiplier; S, silica prism; F, fluid. (b) Schematic view of the interfacial region, showing the light path and defining the various dielectric constants. i, r, and t indicate the light incident, reflected, and transmitted at the silica/fluid interface. θ1 and θ2 are the incident and transmitted angles. S indicates the silica prism of dielectric constant 1, F the fluid with dielectric constant 2, and AL the adsorbed layer with dielectric constant (z) ) 2 + ∆(z), which can vary with height z above the fluid/silica interface.
1). The appellation “invariant” for the three integrals refers to the fact that they are invariant with respect to a change in the position (here, z ) 0) assumed for this ideal interface. Since the ideal interface is merely a construct to perform the calculation, the reflectivity must depend only on properties of the interfacial region that are invariant to its position. In our particular case, the solution/silica interface itself behaves experimentally like a Fresnel interface on which the protein adsorbs. The position z ) 0 will thus be taken at this interface with 1 the silica dielectric constant and 2 the dielectric constant of the solution. (The light is incident on the surface through the silica; see Figure 1.) The reflectivity of p-polarized light from a nonadsorbing interface is then given by7,9,13,14
Rp(θ1) ) R°p(θ1) - 2t12t21r12(J2,1 - 1(sin2 θ1)J2,2) + 2 2 (t12 t21/(42)) sin2 θ1 tan θ1(tan θ2)J12 (2)
where R°p(θ1) ) r122 is the Fresnel reflectivity and r12, t12, and t21 are the standard Fresnel reflection and transmission coefficients:16
r12 )
n2 cos θ1 - n1 cos θ2 2ni cos θi , tij ) n2 cos θ1 + n1 cos θ2 nj cos θi + ni cos θj (3)
n1 ) (1)1/2 and n2 ) (2)1/2 are the refractive indices of the incident and transmission media, respectively, and θ1 and θ2 correspond to the angles of incidence and refraction. All terms in Rp leading to a departure from the Fresnel (16) Jackson, J. D. Classical Electrodynamics, 2nd ed.; John Wiley & Sons: New York, 1975.
law are of second order in the layer thickness (higherorder terms will be ignored). This property is exploited in SAR to optically characterize thin films. In this technique,17,14 the reflection coefficient for “p”-waves is measured as a function of the incidence angle, typically, for protein layers, for about 1° around the Brewster angle (see Figure 2). Linear least-squares fits to such data find cross-correlation coefficients18 for J2,1 and J2,2 on the order of 0.9999. The very high value implies that the determination of these two coefficients is unreliable. The reason for this failure is obvious from inspecting eq 2. The variables J2,1 and J2,2 occur only in the combination J2,1 - 1(sin2 θ1)J2,2, where 1 sin2 θ1 changes by less than 5% over the 1° angular range explored. We thus redefine more appropriate parameters for the fit: J1, J2,2, and J2,3, an additional second-order invariant defined by
12 J2,3 ≡ J2,1 - 1(sin2 θB)J2,2 ) J2,1 - J2,2 (4) 1 + 2 where θB represents the Brewster angle. Expression 2 for the reflectivity of p-polarized light now reads
Rp(θ1) ) R°p(θ1) - 2t12t21r12(J2,3 - 1(sin2 θ1 2 2 sin2 θB)J2,2) + (t12 t21/(42)) sin2 θ1 tan θ1(tan θ2)J12 (5)
Both r12 and sin2 θ1 - sin2 θB vary significantly around the Brewster angle so that in eq 5 the angular functions that multiply the three parameters all vary significantly with respect to the others over the angular range explored. As a result, the three invariants are determined independently, with cross-correlation coefficients less than 0.4. The effect of the reparametrization on the precision with which the three parameters can be determined from experimental data will be seen in section IV below. SAR is thus sensitive to J1, which is proportional to the zeroth-order moment of ∆(z) ) (z) - °(z) and to J2,3 and J2,2, which are, in first approximation, proportional to the first order moment in ∆(z). Higher-order moments of ∆(z) with no simple physical picture also appear, as we will see below. (17) Schaaf, P.; De´jardin, Ph.; Schmitt, A. Rev. Phys. Appl. 1986, 21, 741. Schaaf, P.; De´jardin, Ph.; Schmitt, A. Langmuir 1987, 3, 1131. (18) Bevington, P. R. Data Reduction and Error Analysis for the Physical Sciences; McGraw Hill: New York, 1989.
3180 Langmuir, Vol. 13, No. 12, 1997
Heinrich et al.
If one assumes a uniform isotropic layer, with thickness L and dielectric constant , the three invariants are easily calculated from eq 1:
( - 1)( - 2) ( - 1)( - 2) J1 ) Lk0 ; ; J2,1 ) L2k02 2(1 - 2) ( - 1)( - 2) ; J2,2 ) L2k02 (1 - 2) ( - 1)( - 2) 2 12 1(6) J2,3 ) L2k02 1 + 2 2(1 - 2)
[ ( )]
Eliminating the two structural parameters L and , one can easily show that the three invariants corresponding to this model must satisfy the identity F)1-
{
}{
}
1(1 - 2) 2(1 - 2) J2,2 2J2,3 + J 1 + 2 1 + 2 2,2 ) 12 2 2J1 J2,3 + J 1 + 2 2,2 0 (7)
(1 - 2) 2J2,3 -
(
)
The coefficient F thus defined, equal to 0 for a uniform isotropic layer, can be used to investigate the validity of this model. For a physical picture of F, as well as of the three invariants, let us consider the case of a layer of adsorbed macromolecules on a solid substrate with dielectric constant 1 so that the dielectric constant at a position z from the surface is (z) ) 1 for z < 0 while for z . 0, (z) ) 2, the dielectric constant of the solution. In the intermediate region, (z) ) 2 + ∆(z). Expanding the invariants in a power series in ∆(z), one finds
1 - 2 1 + 2 J1/k0 ) -M0 2 2
∫0∞ dz(∆(z))2 + O((∆)3) (8a)
2
J2,2/k0
-2M1 M02 M21 ) + 2 + O((∆)3) 2 2(2 - 1)
(8b)
1 - 2 M02 1M21 M1 + 1 + 2 2(1 + 2) 2(1 + 2) O((∆)3) (8c)
and
F)1-
M21 (1 + O(∆)) M02
(8d)
where M0 and M1 are the first two moments of the distribution of ∆, Mn ) ∫∞0 dz zn∆(z), and M21 is a somewhat peculiar moment defined by
M21 )
∫0∞dz z(∆(z))2 + ∫0∞dz1∫0z
1
L(1) ) k0-1(1 - 2)J2,2/J1 + O(∆)
dz2 ∆(z1) ∆(z2) (8e)
Note that the coefficient F is of order 0 in both the thickness and refractive index augmentation ∆ of the adsorbed layer. These expressions have been obtained assuming that the layer is isotropic and stratified, which should be a good approximation for adsorbed macromolecular layers. They also assume both ∆/2 , 1 and ∆/|2 - 1| , 1.
(9a)
or from eq 8c,
2(1 + 2) (1 + O(∆)) 2
L(2) ) -k0-1(J2,3/J1)
(9b)
The first definition, L(1), has the advantage of being exact for a thin uniform layer. Unfortunately, experimentally, J2,2 is determined with less precision than J2,3. On the other hand, if the layer is uniform, eq 6 can be used to deduce the thickness directly, leading to
(
)
2(1 + 2) 1 + 2 ∆ 12 1 - 2 2
L(3) ) -k0-1(J2,3/J1)
-1
(9c)
Since generally |1 - 2| , 1 + 2, the correction to eq 9b can be very large even when ∆ , 2. In our case of adsorption at the solution/silica interface, ∆n as small as 0.03, corresponding to ∆/2 ) 0.045, leads to a factor of 2 underestimate in the thickness. The error in using eq 9b to calculate 2M1/M0 from experimental data can be found by estimating ∆, again assuming the uniform layer model. This can be done from the two best-determined parameters alone, in particular by taking the ratio J2,3/ J12, which is independent of L. The resulting quadratic equation, valid for the uniform layer,
(
)
J23 1 - 2(1 - 2) 2 (∆)2 + J1 2
2
J2,3/k02 )
When one can neglect the terms of order (∆)2 in relation 8a, J1 gives directly the zeroth-order moment for the distribution, or the total adsorbed quantity if ∆ can be related to the protein concentration, by eq 14 below, for example. An optical thickness for the film may be defined by 2M1/M0, which is given to lowest order in ∆ by eq 8b as
(
)
22 J23 22(1 - 2) + (1 - 2)2 2 ∆ ) 0 (9d) 1 + 2 1 + 2 J 1
can be solved directly for ∆. Notice that there will be two possible solutions for ∆. Generally, physical reasons will dictate the choice. This estimate of ∆ then serves to calculate L(3). We will compare the values for the thickness found from eqs 9a-c and the best-fit invariants from experimental data, along with the thickness found via a direct fit of the data to the uniform layer model, in section IV below. An optical thickness and surface concentration may thus be determined directly from the three invariants with various levels of accuracy and precision. The third parameter that can be deduced from the data, F, is a more unusual moment of the distribution in dielectric response and is more difficult to interpret intuitively. It may, however, be calculated for several different model films within the approximation ∆/2 , 1 and ∆/|2 - 1| , 1. Some examples are given in Table 2. We see that the coefficient F gives information about the density distribution in the layer, although the information is naturally not complete. Information on F may serve to eliminate various two-parameter models, for example. For models with more parameters, in general F can serve only to provide an additional relation between these parameters, not test the model itself. However, one qualitative characteristic of the layer can be deduced from F by inspection: if ∆ decreases continuously beginning at the surface, F is always positive. This can be seen in a number
Thin Protein Films
Langmuir, Vol. 13, No. 12, 1997 3181
Table 2. Uniformity Coefficient F for Different Nonuniform Distributions of the Dielectric Constant model uniform
)
linear decrease
)
linear increase
)
parabolic decrease
)
{ { {
{
2 + ∆ for 0 < z < L 2 for z > L
0
2 + ∆(1 - z/L) for 0 < z < L 2 for z > L
1/6
2 + z∆/L for 0 < z < L 2 for z > L
-1/2
exponential decrease
2 + ∆(1 - z/L)2 for 0 < z < L 2 for z > L ) 2 + ∆e-z/L for z > 0
bilayer
)
triple layer
2 + ∆1 2 + ∆2 ) 2 + ∆3 2
parabolic (symmetric)
)
{
{ {
Fa
dielectric constant distribution
1/5 1/4
2 + ∆1 for 0 < z < L1 2 + ∆2 for L1 < z < L1 < L2 2 for z > L1 + L2 for for for for
L1L2∆2(∆1 - ∆2) (L1∆1 + L2∆2)2
0 < z < L1 L1 < z < L 1 + L 2 L1 + L2 < z < L1 + L2 + L3 z > L 1 + L2 + L3
2 + ∆(1 - (1 - 2z/L)2) for 0 < z < L 2 for z > L
(L1∆1 + L2∆2 + L3∆3)2
-1/10
sparsely seeded spheres random distribution, with density F, of spheres with radius a, polarizability R, and centers at z ) a b a
L1L2∆2(∆1 - ∆2) + L1L3∆3(∆1 - ∆3) + L2L3∆3(∆2 - ∆3)
1/2
Calculated from expressions 8d and 8e. b Ignoring all multiple scattering, as supported experimentally for low densities in refs 14 and
20.
of different examples given in Table 2. A negative value for F indicates a maximum in ∆ away from the wall, i.e., a depletion layer near the solid, provided that the film is isotropic and the refractive index is a function of z alone (no roughness). The case of a rough surface is harder to treat. One simple model system has been treated analytically, that of small, dielectric spheres of radius a and polarizability R, sparsely seeding a substrate with density F.9,19 If the interactions between spheres and between spheres and the surface can be ignored, in the spirit of refs 14 and 20,
( )
J1 ) k0FR 1 -
1 2
FR(1 + 2) -2a + 2 222(1 - 2)
(
Rp )
)
)
1 - 2 FR J23 ) k02FR a 1 + 2 22
so that F ) 0.5 in the low-density limit. Note that this value is different from that calculated for the stratified layer that corresponds to taking the average optical density at each level within the layer, which would give a parabolic distribution with F ) -1/10 (see Table 2). The value of 0.5 has been verified in a preliminary fashion, for SAR data presented previously20 using a different method of analysis, for polystyrene latex spheres with diameter 93 nm. A more complete test is underway. The reflectivity as calculated by the invariants can be compared with the exact values calculated from various models of the interfacial region. The most common model is that of the uniform isotropic film, for which the (19) Haarmans, M. T.; Bedeaux, D. Thin Solid Films 1993, 224, 117. Haarmans, M. T.; Bedeaux, D Physica A 1994, 207, 340. (20) Mann, E. K.; Bollender, A.; Heinrich, L.; Koper, G. J. M.; Schaaf, P. J. Opt. Soc. Am. A 1996, 13, 1046.
|
r1f + rf2 eiδ
|
2
(10)
1 + r1frf2eiδ
where δ ) 4πLn cos θf/λ, the subscript f refers to values within the film, θf the angle of refraction in the film is given by the Snell law, n sin θf ) n1 sin θ1, and the reflectivity coefficients rij are given by eq 3. Similarly, the reflectivity of a bilayer is given by
Rp )
(
J22 ) k02FR
reflectivity is
|
(r1f + rfg eiδf)(1 + rg2 eiδg)
(1 + r1frfg eiδf)(rfg + r1f eiδf)rg2 eiδg
|
2
(11)
where the indices f and g refer to the values for the first and second layers, respectively. This discussion has been framed entirely within the context of reflectivity measurements. It applies equally to any other type of optical measurement, including the very widely used ellipsometry. The ratio rp/rs determined in ellipsometry is a linear function of Ie, Ie2, and a particular angle-independent combination of Iτ and Iδ, the prefactors depending only upon the incidence angle and the dielectric constants of the two bulk media.7 In terms of the version of the invariants used above, the invariants involved are J0, J02, and J23. This result implies that even scanning angle ellipsometry is able to provide at most two structural parameters for a thin film. For more information, spectroscopic ellipsometry is required.21,22 Moreover, the linear term in the first-order invariant, related to the surface concentration, dominates so that ellipsometry is quite insensitive to higher-order terms. On the other hand, in the intensity reflection coefficient Rp of p waves, the three invariants all intervene independently to second order in the layer thickness. In principle, three quantities (21) See, for example, the journal volume dedicated to the subject. Thin Solid Films 1993, 233. (22) Tonova, D.; Konova, A. Surf. Sci. 1996, 349, 221.
3182 Langmuir, Vol. 13, No. 12, 1997
Figure 3. Expected sensitivity of optical techniques to the structure of the layer: the percent difference between the theoretical values for a bilayer (model defined in Table 2) and the best-fit monolayer as a function of incident angle. Solid lines refer to the reflectivity, long dashes to the ellipsometric coefficient ∆, and short dashes to the ellipsometric coefficient ψ. In this example, the differential refractive index of the layer is ∆n ) 0.02 in all cases. The two layers are of equal thickness: (light lines) L1 ) L2 ) 10 nm and (heavy lines) L1 ) L2 ) 25 nm.
can be determined: the surface concentration, average thickness, and the uniformity coefficient F. The well-known simple bilayer and single layer models can also be used to compare the sensitivity of different optical techniques to the homogeneity of the adsorbed layer. In particular, it is possible to compare the reflectivity, as well as the ellipsometric coefficients, for a simple double layer model with the nearest single layer model (i.e., the single layer that gives the best fits either to the reflectivity curve as found in SAR or to the two ellipsometric coefficients as can be determined by scanning angle ellipsometry). Such a comparison can be seen in Figure 3 for a double layer in which the ∆n within the second layer is twice that within the layer near the wall, and the two layers have equal thickness L1 ()10 and 25 nm). The single layer thickness and optical densities deduced from the ellipsometric and the reflectivity data were virtually identical, implying that the “average optical thickness” as deduced by the two methods should be identical in this thickness range. We see that the reflectivity difference between the single layer and double layers is too small to be visible experimentally with ellipsometry for all these layer thicknesses, but it is significant in reflectometry for the thicker layers. This is consistent with the expressions for the invariant expansion to second order given by Lekner as just discussed:7 although the intensity reflectivity depends on all three invariants, to second order in k0d, the ratio of the reflective coefficients for the two different polarizations, which is measured in ellipsometry, depends only on J1 and J23. III. Material and Methods The materials and methods used in these experiments have been presented in considerable detail in a previous article.12 The details that are important for understanding the present results will be summarized here. A. Proteins. Both proteins belong to the immunoglobulin G (IgG) family of antibodies. The first is a human IgG, which acts as the antigen and will be referred to as such in what follows. The second is a rabbit IgG (which will be called “antibody” here) produced in an immune reaction against injected human IgG molecules. The proteins are very similar in size and shape, but different
Heinrich et al.
in where they bind to each other. IgG antibodies are roughly Y-shaped;23 at the top of each of the two arms of the Y there is an active domain, which is able to fix an antigen. A given active region reacts with one particular molecular pattern on the antigen. However, our antibody sample is polyclonal; different molecules in the sample fix on different parts of the antigen. We thus see that, in spite of the considerable similarity of the two molecules, the antigen/antibody interaction is asymmetric; only two regions on the antibody molecules play a role in binding, while many sites on the antigen are available to be bound to some antibody. Proteins were received in powder form from the Centre Re´gional de Transfusions Sanguines of Strasbourg and were dissolved in a phosphate buffer (PBS) at pH ) 7.5. Protein solutions were kept frozen in small 1 mL aliquots at concentrations of about 3 g/L for the antigen samples and 5 g/L for the antibody samples. The concentrations were determined by absorbence with a UV spectrophotometer (Beckman Model DL 640, wavelength of 280 nm). For all these experiments, the final protein solution was prepared less than 1 h before introduction into the measurement cell. The aliquots were immersed in a bath between 30 and 40 °C for a few minutes, just until the protein solution was nearly melted. The solution was then added to more PBS to achieve the desired protein concentration. Both buffer and protein solutions were filtered with a Millex GV filter with a pore size of 0.22 µm. Before each experiment, the sample cells were cleaned over a period of about 2 days in a laboratory detergent, Hellmanex II (Helma GMBA, D-79379 Mullheim), at 3% followed by copious rinsing in pure water and several minutes in a sulfuric acid bath (Prolabo; quality “for analysis”, diluted to 1%). Finally, everything was rinsed several more times with pure water. Deionized SuperQMillipore water was used throughout. B. Scanning Angle Reflectometry. The technique has been extensively described in previous articles12,24 and we present it only briefly here. In SAR one measures the reflection coefficient of an electromagnetic wave polarized in the plane of incidence (p-wave) for various incidence angles around the Brewster angle. Not only does the reflected intensity at the Brewster angle differ from zero in the presence of an adsorbed layer but the whole reflectivity curve (reflected intensity as a function of the incidence angle) changes shape and position (see Figure 2). It is these variations in the reflectivity curve that are analyzed to extract information about the adsorbed layer. The fact that for a Fresnel interface the reflection coefficient vanishes at the Brewster angle leads to a high sensitivity to adsorbed layers. The light source of the reflectometer is a 5 mW HeNe laser (λ ) 632.8 nm). The beam is polarized before entering almost perpendicularly through one face of a prism. It reflects from the hypotenuse of the prism, which is in contact with a buffer solution. This face constitutes the adsorption surface and is optically flat to λ/4, where λ corresponds to the wavelength of light in the vacuum. The reflected beam then leaves through the third face of the prism, again nearly perpendicularly, before it passes through a second polarizer. This second polarization is required because we will measure reflection coefficients on the order of 10-7, so s-polarization, with its high reflectivity, must be eliminated to this degree. The intensity is then measured by means of a photomultiplier. The angles of incidence are selected by rotating the cell (and thus the interface) by means of a high-precision (23) Bagchi, P.; Birnbaum, S. M. J. Colloid Interface Sci. 1981, 83, 460. (24) Schaaf, P.; De´jardin, Ph.; Schmitt, A. Rev. Phys. Appl. 1986, 21, 741.
Thin Protein Films
Langmuir, Vol. 13, No. 12, 1997 3183
(0.001°) goniometer (Microcontrole, Evry, France). The position of the photodetector is adjusted to find the position of greatest intensity with the same precision. The experiment was conducted as follows. The reflected intensity was first measured for a pure buffer/silica interface, with a Brewster angle of approximately 42.5°, at discrete incidence angles 0.02° apart, typically over the range 41.5°-43.5°. This curve was compared to the Fresnel reflection law in eq 3, supposing that the measured signal was described by
I ) I0 + AR°p(θ; n1, n2)
(12)
where I0 is a residual signal and A a calibration factor that depends on the apparatus. The incidence angle θ is determined with a precision of 0.001° with respect to a reference that is known to 0.01°. The index of fused silica at the operating temperature (22° ( 2°) is n1 ) 1.4570 ( 0.0001, and the index of pure water is n2 ) 1.3328 ( 0.0002, increasing by 0.003 for the buffer solutions used with the proteins. The reflectivity curve was fit to eq 12 with parameters I0, A, and one other choice among the refractive indices and the error in the reference angle. We chose to use this liberty to determine the refractive index of the buffer solution from the Fresnel curves but verified that varying either the reference angle or n1 within their error limits left the results unchanged within the error bars presented with the results below. In principle, I0 as defined in eq 12 may depend on angle θ. In particular, some part of I0 is due to s-polarized light, with reflectivity Rs, which is not eliminated by the two polarizers. However, Rs varies by only 10% for the angular range explored. Since I0 is small (typically corresponding to a reflectivity of 10-8), even if it were entirely due to this Rs, the error introduced by ignoring its angular dependence would be everywhere more than 2 orders of magnitude less than the purely statistical noise level. Taking I0 as constant had no effect on the results of the curve-fitting procedure. After this calibration of the apparatus, the antigen solution was rapidly injected into the cell. The adsorption of the proteins on the silica surface was followed by measuring reflectivity curves at repeated time intervals. The determination of each reflectivity curve took approximately 7 min, with at least an additional 7 min between curves. During the adsorption process, the solution was at rest in the cell so that only the diffusion process of the molecules in the solution and the interaction mechanism between the proteins and the surface influenced the adsorption kinetics. In order to stop the adsorption process and to replace the antigen solution in the cell by an antibody solution, the cell was first flushed with pure buffer, which also removed any loosely adsorbed antigen molecules. This flushing of the cell was very extensive, with about 100 times the volume of the cell over a 12 h period, in order to minimize the antigen concentration in the cell before antibody molecules were injected; otherwise, these two kinds of molecules would have interacted one with each other in the solution and no longer only on the surface. After this flushing step, an antibody solution was injected into the cell and the reaction of the antibody molecules with the adsorbed antigen molecules was again followed by determining reflectivity curves as a function of time. Parameters defining the layer were determined by fitting the experimental data to the theoretical expressions, either in terms of invariants (eq 5) using a linear least-squares fit or in terms of the parameters defining one (eq 10) or two (eq 11) uniform layers using a more general least-squares fitting procedure. The results of these fits can be compared using the quality function,
which is the standard least-squares parameter 2
χ )
∑i
(Rexp(θi) - Rth(θi, ∆n, L))2 σ2(θ)
(13)
where the expected error σ(θ) is estimated to be 0.01Rexp(θ), plus additional base line noise corresponding to reflectivities of ∼10-9. From the structural parameters, it is also possible to determine the protein surface concentration, assuming that
Γ)
∫ dz ∆n(dn dc )
Γ)
d ∫ dz ∆(dc )
-1
(14a)
or alternatively -1
(14b)
where dn/dc is the refractive index augmentation on the order of 0.18 g-1 cm3 for a large variety of protein solutions,25 which leads to d/dc ≈ 2n dn/dc ) 0.48. We will assume these values in what follows. However, the values for dn/dc are measured for dilute bulk protein solutions, and one may ask whether they are appropriate within the relatively dense protein layer. From inspection of the Lorenz-Lorenz relation,16 or the various effective medium approximations26 that are designed to hold at higher densities, we see that deviations from strict linearity are probably not more than 1%. C. Additional Characterization Techniques. We observed surprisingly high values for the thickness of the antigen/antibody layer. It is thus particularly important to verify the purity of the antibody solutions and in particular possible contamination by large molecules (most likely IgM molecules, also involved in the immune response, which correspond roughly to a cyclic IgG pentamer) or aggregates in the antibody solution. Three independent tests were performed: electrophoresis, lightscattering, and immunoprecipitation. These are described in detail elsewhere.12 No large proteins or aggregates were detected by any of these methods. IV. Results The values for the invariants J1, J2,1, and J2,2 determined by LLSQ fits to eq 1 for a typical experiment are given in Figure 4. We can see that although there is very little noise in the values for J1, there is much more in J2,1 and J2,2; the variation in these may be as great as 30% for the single antigen layer, with a corresponding uncertainty in the deduced parameters. Furthermore, although crosscorrelation coefficients between J1 and the other two parameters are generally less than 0.4 in absolute value, the cross-correlation coefficients between J2,1 and J2,2 are on the order of 0.9999, which is not acceptable. As discussed in section II, more appropriate parameters for the fit are J1, J2,3 ) J2,1 - 1(sin2 θB)J2,2 and J2,2. These three are determined independently, with correlation coefficients less than 0.4. Since the equation for R is linear, the results for J1 and J2,2 are unchanged. The values found for J2,3 are also given in Figure 4. With this parametrization, both J1 and J2,3 are well-determined with variations 10 times less than for J2,1, with as before considerably more noise in the third parameter J2,2. These three invariants can now be used to explore the properties of the adsorbed layer, as discussed in the theory (25) De Feijter, J. A.; Benjamins, J.; Veer, F. A. Biopolymers 1978, 17, 1759. (26) Frey, W.; Schief, W. R., Jr.; Vogel, V. Langmuir 1996, 12, 1312.
3184 Langmuir, Vol. 13, No. 12, 1997
Heinrich et al.
Figure 4. Invariants as determined by LLSQ fits to reflectivity curves measured during the successive adsorption of IgG proteins at 0.2 g/L (to the left of the dashed line) and anti-IgG proteins at 0.3 g/L (to the right of the dashed line). The invariants are (O) J1 × 103, (3) J21 × 103, (b) J22 × 103, and (1) J23 × 2 × 104.
section. The most usual way of characterizing such a layer is by the overall surface concentration of the adsorbed molecules and an average layer thickness. However, since we have three quantities available, these two values are overdetermined; there are many different ways of deducing them from the measured parameters. Four different methods of extracting the layer thickness are discussed in section II: L(1), L(2), and L(3) as given by the invariants in eq 9 or bypassing the invariants altogether, and fitting the experimental reflectivity curves directly to the reflectivity given by the uniform layer model in eq 10. There are three reasonable methods of extracting the surface concentration Γ using relations 14 between the refractive index or dielectric constant augmentations and the bulk protein concentration within the layer. (1) The first is directly from J1, using eq 8a to order 1 in ∆, which implies Γ ) M0(d/dc)-1 ) -(2/(1 - 2))k0-1(d/dc)-1J1. This expression is expected to hold to first order in ∆. The result will be called Γ(1). (2) The second is from J2,3 and J1, using the expressions for these two invariants for the uniform layer, as discussed for L(3) above. This approximation will hold for any ∆, and the result will be called Γ(2). (3) The third bypasses the invariants and directly fits the experimental reflectivity curves to the uniform layer model. Both L and Γ can thus be determined in several different ways, given by these different expressions. The results for one typical data set, with an antibody bulk concentration of 0.3 g/L, are shown in Figure 5. We see there is good agreement between L(3) and the uniform layer model. L(1), based on the less well-determined J2,2, naturally gives much noisier results. Furthermore, varying the instrumental calibration parameters and the bulk refractive indices within the experimental uncertainties may change the value of L(1) at low Γ (here, this corresponds to the antigen alone on the surface) by a factor of 2 or more. These uncertainties have little effect on the parameters deduced by the other two methods. L(2) consistently underestimates the thickness of a uniform layer by a factor of 2, which is what is expected for the values of ∆ found here. This can be seen from L(3), which corrects for this underestimation. There are also deviations of up to 10% in the values of L and Γ found using the invariants assuming a uniform layer and those found using the exact uniform layer expression (eq 10) for thicknesses above 50 nm. This is because the invariant method assumes k0L , 1, while for L ) 50 nm and λ ) 632.8 nm, k0L ≈ 0.5.
Figure 5. (a) Thickness and (b) surface concentration of the layer formed by successive adsorption of IgG proteins at 0.2 g/L (to the left of the dashed line) and anti-IgG proteins at 0.3 g/L (to the right of the dashed line) as determined by the different methods discussed in the text: (O) single layer model; (+) double layer model, supposing that L1 ) L2 (×) double layer model, supposing that L1 ) 16 nm. The following is determined from the invariants: (b) L(1), Γ(1); (9) L(2), Γ(2); (2) L(3).
The approximation approaches the limit of its validity at these thicknesses. The deviations of the parameters determined using the invariant method with the assumption of a uniform layer to analyze the uniform layer reflectivity calculated from the exact expression [eq 10] are given in Figure 6. Each of the different methods of calculating L and Γ has its advantages and drawbacks, corresponding to different levels of approximation. L(1) - L(3) correspond to an approximation in which Lk0 , 1, while the uniform layer model gives a result which is good for any L but makes an assumption about the distribution of mass in the layer. L(1) and L(2) bypass any assumption about the mass distribution within the layer but have other drawbacks: L(1) because it is very sensitive to small experimental inaccuracies, and L(2) because it does not give the exact value for the thickness in the case of a uniform layer unless ∆/|1 - 2| , 1. L(3) may be useful for checking this approximation. However, it assumes a particular model of the mass distribution, the uniform layer. Which of these determinations of a characteristic layer thickness is the most appropriate depends on the experimental situation. If in fact ∆/|1 - 2| , 1, L(2) would be the best choice, being both independent of the model and precise. Unfortunately, in our case we find ∆/|1 - 2| ≈ 0.3, which is somewhat large for the lowest
Thin Protein Films
Figure 6. Limit of validity for the invariant expressions for the reflectivity of a thin uniform layer. The deviation with layer thickness of characteristics was calculated (eq 6) from the values of the invariants determined by LLSQ best fits to the reflectivity calculated using the exact expression (eq 10) over an angular range θ ) θB ( 1°. Fits are everywhere good within 0.01% Solid line represents the thickness L. Long dashes represent the differential refractive index ∆n. Short dashes represent the surface concentration Γ.
Figure 7. Coefficient F indicating the uniformity of the layer formed by successive adsorption of IgG proteins at 0.2 g/L (to the left of the dashed line; note that error bars are larger than the field) and anti-IgG proteins at 0.3 g/L (to the right of the dashed line). The nearly horizontal solid line corresponds to the value of F found for the uniform layer that is the best fit to the experimental reflectivity curves. Error bars correspond to the range found using different parameters for the basic Fresnel reflectivity over the limits of experimental fluctuations during a comparable time period.
order approximation in eqs 8a and 8b, so that the values for L(2) are somewhat misleading physically. However, with a little care, the thicknesses and surface concentrations determined from the invariant method and those deduced assuming a uniform layer are in agreement for layer thicknesses less than about 80 nm (or more generally, ∼λ/8). The usefulness of the invariant method is, however, that it can be independent of the model as long as the layer is sufficiently thin. It can be used to characterize the layer and offers the further possibility of testing models for the distribution of mass in the layer. This can be done through the parameter F, defined in eq 7 above, which is shown in Figure 7 for the same data set as displayed in Figures 4 and 5. We see that for the single layer of IgG protein molecules, there is a very large uncertainty in F, due mainly to the uncertainty in the
Langmuir, Vol. 13, No. 12, 1997 3185
Figure 8. Deviation between the experimental reflectivity values and the best fits: (×) bare surface and the Fresnel expression (eq 3) for data from the IgG/anti-IgG layer with surface concentration of 10 mg/m2; (b) the exact monolayer expression, eq 10; (O) the invariant expression 6, which implies a nonuniform layer with F ) -0.33. The expected error σ is taken as 1% Rexp plus a background noise of ∼10-9.
conversion of the signal to a reflectivity, because of fluctuations in laser and detector efficiency for example. No additional information, beyond the thickness and concentration of the layer, can be determined. However, for the thicker layers that result from the addition of the antibodies, the effect of these uncertainties is much less. Typical differences between the experimental reflectivities and both those expected for a uniform layer and those calculated from the best-fit invariants are given in Figure 8. The invariants, which correspond to a value of F (-0.33; see Figure 7) inconsistent with the uniform layer model, yield reflectivities clearly much closer to the data than the uniform layer model, outside the experimental uncertainties. Since the thicker layers are at the limit of the validity of the invariant approximation, which here keeps terms to only second order in d/λ, it is advisable to check for any possible breakdown in the expression for F for a homogeneous layer. This is also indicated in Figure 7, where the solid line represents the F calculated from the bestfits of the invariant expression for the reflectivity (eq 5) to the uniform-layer reflectivity (eq 10) curves that correspond best to the experimental data and thus to the approximate layer thickness. F will be zero for a uniform film, if k0d , 1 as assumed for the analysis in terms of invariants, but may deviate for larger d. However, we see that there is no significant effect of the layer thickness on F in the thickness range explored here. For the antigen/ antibody layer on the other hand, the parameter F is clearly negative. This implies that the layer is certainly not uniform and isotropic. It is, however, almost certainly optically isotropic. If the layer can be additionally taken as uniformly stratified, the negative value implies that the density layer increases toward a maximum away from the surface, or in other words, that there is a depletion layer of the protein near the solid surface. This does not exclude an additional tailing of the density into the solution, as can be seen from the negative value of F for the centered parabolic profile or from using the threelayer model for example (see Table 2). The existence of a depletion layer can also be tested by fitting the data to a bilayer model (see Table 2). There are, however, four independent parameters to this model, while the data is sensitive to only three. We thus consider two models with different constraints: in the first, that the layer near the surface keep the same thickness L1 as
3186 Langmuir, Vol. 13, No. 12, 1997
Figure 9. Goodness-of-fit parameter χ2 found for the experimental reflectivity curves on the IgG/anti-IgG layer, as compared to different bilayer models, as a function of the ratio between the optical density of the two layers: (solid lines) L1 ) L2; (dashed lines) L1 ) 15 nm; (a) Γ) 2 mg/m2; (b) Γ) 6 mg/m2; (c) Γ ) 10 mg/m2.
the initial antigen layer, and in the second, the two layers be of equal thickness: L1 ) L2. Fits to the data were then performed for a series of different ratios f ) ∆n2/∆n1 between the refractive index augmentation of the two layers, letting vary only two parameters, the overall thickness of the layer, and the refractive index of one of the layers. This two-step fitting procedure was used, because the data were in some cases at the limit of allowing the determination of three independent parameters. The results of the best fits for the overall thickness L1 + L2 and for the surface concentration are shown in Figure 5, with that given by the previous analysis in terms of invariants or the uniform layer. The parameters L1 + L2 and Γ for the two bilayer models differ little and are in agreement with the other results for other models. Figure 9 presents the result of the quality of these fits as a function of the ratio f ) ∆n2/∆n1 for several points during adsorption. We see that for the antigen alone on the surface, for an overall Γ ) 2 mg/m2, the quality function is virtually flat. As before, we are unable to test the uniform layer approximation. However, for the data with Γ ) 6 mg/m2 and Γ ) 10 mg/m2, there are clear minima in the quality function for ratios f clearly superior to 1. Furthermore, the f corresponding to the two different models are of similar order. Again, we see that the data clearly indicate that the layer is not uniform and homogeneous. The stratified layer model (here, a double layer) again indicates a depletion layer at the surface. In fact, the two results, from the double layer and from the invariant analysis, are also in quantitative agreement: f ) 2, as observed, corresponds to F ) -0.2. Conclusions We have seen that the invariant method of analyzing reflectivity data allows the determination of an overall layer thickness and surface concentration essentially independent of any optical model. One major limitation is in terms of the overall thickness of the layer, since k0d is assumed small. In practice, the results are in agreement with both single and double layer models even for our thickest layers for which k0d ≈ 0.6. The other limitation is that the simple interpretation of the invariants, in terms of the first two moments of the optical (or mass) density distribution, depends on assuming that the optical density
Heinrich et al.
of the adsorbed layer is very close to the optical density of the diluting media 2, with | - 2| , 2 but also | 2| , |2 - 1|. This latter approximation was not entirely valid in our case, for which - 2 ≈ 0.08. In such cases, some care should be taken in interpreting the results for the thickness of the adsorption layer. Here, we can confirm that the average thickness for our antigen/antibody layers is 60 nm, nearly independent of the method of analysis. This confirms the result of an earlier article,12 which discussed these surprisingly thick layers equivalent to the length of three protein molecules. In addition to giving a result that is independent of the model for the layer, the invariant method of data analysis also has the potential of giving information about the distribution of mass in the layer through a rather complicated higher-order moment. In particular, any twoparameter model for the distribution of mass can be tested. In our case, the layer was clearly not isotropic and uniform; the higher-order moment suggests a depletion layer at the surface. Note that this does not imply that the layer does not slowly decrease in density into the solution after reaching a maximum. From Table 2, we see that even a symmetric parabolic distribution can give a negative value for the homogeneity coefficient, indicating a depletion layer. This information on the mass distribution was, however, limited to relatively thick layers. No significant information was available for thicknesses of about 20 nm, while clear deviations from the uniform layer were observed for thicknesses above 50 nm. This makes it difficult to conclude about any evolution of the structure in the layer as the antibodies interact with the initial antigen layer. Modification of the experimental apparatus, for greater precision and in particular stability, may make such information available. The structure of the layer may also be explored, with greater precision, using neutron reflectivity. Good contrast generally requires deuteration of the macromolecules, and varying the contrast can be particularly useful for lifting ambiguities in the determined structure for the adsorbed layer. Unfortunately, to our knowledge, deuteration, without denaturation, of protein molecules is not yet possible in general. A very interesting recent study27 does use neutron reflectometry to study a single protein layer, with only the buffer solution deuterated. Information on the structure of the layer, up to the level of a double layer model, was deduced. It would obviously be extremely interesting to apply such a technique to our antigen/antibody layers. However, the possibility of routine laboratory use gives optical techniques an important role in the exploration of the structure of macromolecular films; the invariant method of analysis of such data offers clear advantages to the more usual modelbased analysis. Unfortunately, although it would be interesting to use the invariant method of analysis for ellipsometric data on adsorbed layers, two invariants alone, J1 and J23, are sufficient to describe data in the limit k0d , 1. Model calculations demonstrate that nonuniformity in dielectric films would cause deviations of less than 0.1% in the ellipsometric parameters, even for d ≈ 50 nm. The major advantage from the invariant analysis in this case is simply to give a model-independent estimation of the first two moments of the mass distribution (i.e., the surface concentration and the average thickness.) This is in itself a worthwhile goal. LA9610314 (27) Liebmann-Vinson, A.; Lander, L. M.; Foster, M. D.; Brittain, W. J.; Vogler, E. A.; Majkrzak, C. F.; Satija, S. Langmuir 1996, 12, 2256.
