Langmuir 2002, 18, 2069-2074
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An Alternative Method To Quantify Surface Plasmon Resonance Measurements of Adsorption on Flat Surfaces Sander Haemers,* Ger J. M. Koper, Mieke C. van der Leeden, and Gert Frens Laboratory for Physical Chemistry, Delft University of Technology, Julianalaan 136, 2628 BL Delft, The Netherlands Received April 4, 2001. In Final Form: October 30, 2001 We reformulated the expressions for the interpretation of surface plasmon resonance (SPR) signals presented in the paper of Jung and co-workers (Jung, L. S; Campbell, C. T.; Chinowsky, T. M.; Mar, M. N.; Yee, S. S. Langmuir 1998, 14, 5636.) so that only two experimentally accessible parameters are needed to quantify the SPR signal of a coated sensor disk with unknown optical properties: a specific sensitivity and the specific decay length of the evanescent field. The sensitivity was measured using the refractive index difference between two different ionic strength solutions. The decay length was determined using the rate of adsorption of latex spheres while comparing these with theoretical predictions using the diffusion coefficients of these spheres. In addition, the errors involved in a linearized version of the equation were assessed.
Introduction In a system where light propagates from an optically less dense into a more dense medium, total reflection occurs for incidence angles larger than the critical angle and an evanescent field is generated in the dense medium. When this evanescent wave couples with the electrons in the optically dense layer, the intensity of the reflected light is reduced. This surface plasmon resonance (SPR) effect is especially pronounced for thin gold and silver films. The incidence angle at which the SPR occurs is very sensitive to the optical properties of the medium on top of the metal film. Instead of angular changes, one may likewise consider changes in wavelength at which the SPR effect occurs. Here and in what follows we shall only consider angular changes. A change in refractive index, for example due to the accumulation of macromolecules near the surface, results in a shift of the angle at which the resonance occurs (Figure 1). The widest application of SPR-based sensors can be found in biophysical studies where interactions between a receptor and a biomolecule are measured. In most cases the technique of choice is to link a hydrogel (dextrane) to the metal surface and bind specific receptors to this gel. This enhances the specific surface for receptor binding and hence the sensitivity of the technique.1 SPR-based sensors are less commonly used for measuring the adsorption of molecules on flat surfaces. This despite the fact that SPR is a sensitive technique with which it is easy to realize fast response times2 making it an excellent tool to measure the dynamics of adsorption. Moreover, the gold films normally used in SPR-based instruments are a good substrate for spin coating thin polymer layers, making it easy to prepare surfaces with any desired property. A problem in the interpretation of the signal is that the evanescent field decays exponentially into the material on top of the gold film, which results in a nonlinear response to an increasing layer thickness. A useful approach of this problem was published by Jung and co-
workers.3 In their paper they demonstrated that the equations based on Maxwell’s laws, normally used in describing the properties of an evanescent field, could be well fitted with simple exponentially decaying functions. An important parameter in their models is the decay length (ld) of the evanescent field, and Jung et al. presented an exact equation to calculate the decay length. However, when the gold surface is covered with a coating layer, this might influence the penetration depth of the evanescent field in the solution. In a number of papers1,3,4 methods were presented to calculate the new decay length, but for these methods detailed information about both the thickness and the refractive index of the coating layer is needed. The aim of the present work is to use the mathematical formalism proposed by Jung and co-workers and to reformulate this in terms of adsorbed mass instead of layer thickness. We subsequently develop a method to determine the decay length of the evanescent field as well as the sensitivity of the SPR setup experimentally without the need of detailed knowledge of the optical properties of the coating layer on the sensor disks. This results in a full calibration of the setup without resort to a secondary
(1) Liedberg, B.; Lundstro¨m, I.; Stenberg, E. Sens. Actuators, B 1993, 11, 63. (2) Lenferink, A. T. M.; Kooyman, R. P. H.; Greve, J. Sens. Actuators, B 1991, 3, 261.
(3) Jung, L. S.; Campbell, C. T.; Chinowsky, T. M.; Mar, M. N.; Yee, S. S. Langmuir 1998, 14, 5636. (4) Stenberg, E.; Persson, B.; Roos, H.; Urbaniczky, C. J. Colloid Interface Sci. 1991, 143, 513.
Figure 1. Typical example of an SPR signal: the shift of the minimum, from φ to φ + ∆r, in the intensity of the reflected light due to a change in refractive index above the substrate.
10.1021/la010506a CCC: $22.00 © 2002 American Chemical Society Published on Web 02/13/2002
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technique for the determination of layer thicknesses or adsorbed masses. In addition, we quantify the possible errors introduced by neglecting the effect of layer thickness in the determination of the adsorbed mass.
for the attenuation of the signal due to the presence of the coating layer. Because this is nothing more than a scaling factor,3 we defined a specific sensitivity m ˜ , which can be determined the same way as m, so that
(
Theory A change in refractive index causes the SPR angle to change. The response of an SPR-based instrument to a bulk layer of homogeneous refractive index is defined as the sensitivity of a surface plasmon setup. This sensitivity depends on the thickness of the metal layer and the properties of the optical system. The sensitivity can be determined by measuring the difference of the SPR angle for two liquids put on top of the metal surface with a different refractive index. The resulting shift in SPR angle will in general not be exactly linear with the change in refractive index but tends to increase with higher refractive index differences. The response r over a wide range of refractive index values N is usually described by a second-order polynomial3,4
∆r ) m1∆N + m2∆N2
(1)
in which m1 and m2 are instrument-dependent constants representing the sensitivity. If the refractive indices show no larger difference than approximately 0.05 units of refraction, the response can be taken to be linear
∆r ) m ∆N
(2)
In this equation m is the sensitivity of the used SPR setup. Because m is the local slope of the full r versus N curve, it has to be determined in the same range of N as the refractive index of the solvent used in the adsorption experiments; see also ref 3. In the case the change in refractive index is not in a bulk layer, the decay of the evanescent field becomes important because then not only the change in refractive index is important but also the layer thickness. To deal with this problem, an effective refractive index Neff was defined3 which takes into account the effect of the decaying intensity of the evanescent field. Liedberg et al.1 proved that the proper weighting factor with height z is exp(-2z/ld), in which ld is a characteristic decay length. For example, consider the system were the gold surface is coated with a layer of thickness dc and has a refractive index Nc. On top of this coating layer particles adsorb, forming a layer with thickness d and refractive index Na. The solvent has a refractive index of Ns. The effective refractive index then becomes
(∫
( ) ( )
-2z Nc exp dz + 0 ld ∞ dc+d -2z -2z Na exp dz + d +d Ns exp dz (3) dc c ld ld
2 Neff ) ld
∫
dc
( ) )
∫
Using this formalism while assuming that one single decay length is sufficient to describe the decay of the field in the adsorbed particle layer and in the bulk solution, Jung et al combined eqs 2 and 3 and derived the following equation for the specific case that the gold substrate is coated with a thin layer with thickness dc on which the adsorbing molecules adhere:
∆r ) m(Na - Ns) exp
(
)
-2dc -2d 1 - exp ld ld
(4)
The first exponential factor in eq 4 is basically correcting
∆r ) m ˜ (Na - Ns) 1 - exp
)
-2d ld
with
( )
m ˜ ) m exp
-2dc ld
(5)
The refractive index of an inhomogeneous layer consisting of a solvent with refractive index Ns and solute at concentration c can be approximated by3,5,6
Na ) Ns + c
dN dc
(6)
The validity of this equation for dense adsorbed protein layers was experimentally verified by de Feijter and coworkers.5 The refractive indices of pure proteins range from 1.4 to 1.9, so this equation will also be valid for the latex spheres (N ) 1.59) to be used in the present adsorption experiments. The latex spheres used here are much larger than protein molecules. However, in our opinion eq 6 still holds because the spheres are still smaller than the probe depth.3 The refractive increment of a particle in solution can be measured using a refractometer or can be calculated for particles of known refractive index, Np, and partial specific volume ϑp to be6
dN/dc ) (Np - Ns)ϑp
(7)
The adsorbed amount, Γ, can be written in terms of the solute concentration c in an adsorbed layer of thickness d
Γ ) cd
(8)
If the refractive index difference between the solution and solvent is negligible, the equations above can be combined into an equation, which relates the adsorbed mass Γ to the angular shift
Γ)
d
∆r -2d dN 1 - exp m ˜ ld dc
(
)
(9)
The first factor on the right-hand side of the equation takes into account the effect of the layer thickness on the adsorbed amount. When d is much smaller than ld, we can approximate this by
Γ)
ld ∆r 2 dN m ˜ dc
(10)
which is a linear equation that is very useful to study the adsorption of small colloids such as proteins. Because of their small size,7 proteins form adsorbed layers with a thickness much smaller than ld. Apart from the specific sensitivity m ˜ , eqs 9 and 10 contain one remaining unknown, namely, the decay length (5) De Feijter, J. A.; Benjamins, J.; Veer, F. A. Biopolymers 1978, 17, 1759. (6) Ball, V.; Ramsden, J. J. Biopolymers 1998, 46, 489. (7) Haynes, C. A.; Norde, W. Colloids Surf., B 1994, 2, 517.
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ld. In ref 3 an equation was presented to calculate this length
ld ) (λ/2π)/Re(-Neff4/(Neff2 + metal))1/2
(11)
in which metal is the complex dielectric permittivity of the used metal at wavelength λ. A problem in the calculation of ld for a multilayer system is that Neff depends itself on ld (eq 3). In the case of a somewhat thicker spin-coated layer, one would need a secondary technique, for example, ellipsometry, to characterize the coating first. Another problem is that the dielectric permittivity of the thin gold layer is influenced by the manufacturing technique used, e.g., sputtering or evaporation, and the annealing conditions.8 So, instead of calculating the decay length using eq 11, we measured ld on an already coated gold surface, thereby automatically taking into account the effect of the coating layer as well as deviations in the gold dielectric permittivity. This was achieved by measuring the formation rate of latex spheres monolayers; this provides for independent information on the thickness and the refractive index of the adsorbing layer. The thickness of the layer is equal to the diameter of the used spheres, while the refractive index of the layer is related to the surface coverage through eq 6 and hence to the rate of adsorption. A requirement for this to be true is that the rate of adsorption of the used particles can be accurately predicted. In the present situation, where there is no energy barrier for adsorption, e.g., every particle arriving at the surface will adsorb immediately, the sticking probability is equal to 1. Under this condition of diffusion-controlled adsorption, the initial rate of adsorption is9
J ) (dΓ/dt)tf0
(12)
which depends only on the diffusion coefficient of the adsorbing particles and on the hydrodynamic conditions in the measuring cell. Materials and Methods The solutions were made with double distilled water and analytical grade KNO3. The refractive index of the solutions was measured with an ATR-SW refractometer from Schmidt+Haensch, which measures at 589 nm with an accuracy 2 × 10-5 units of refraction. The 0.1 and 1 M KNO3 solutions had refractive indices of 1.33324 and 1.34129, respectively. In the adsorption experiments we used commercially available latex spheres ranging in diameter from 20 to 100 nm thus providing a size range where the effect of the decay of the evanescent field can be probed. These “sulfate white polystyrene latex spheres” were purchased from Interfacial Dynamics Corp., Oregon, U.S.A. The sulfate functional groups on the surface give the spheres a pHindependent negative surface charge. The refractive index of the spheres is 1.591 at 590 nm; the density is 1.055 g/cm3. The refractive index increment of the latex particles was calculated with eq 7 to be 0.24 mL/mg in the 0.1 M KNO3 solution. The surface plasmon resonance instrument is a commercially available Bio Molecular Interaction Sensor from IBIS Instruments BV, Enschede, The Netherlands (Figure 2). In this setup the intensity of the reflected laser light (670 nm) is measured by a diode array over a range of 5°. (8) Aspnes, D. E.; Kinsbron, E.; Bacon, D. D. Phys. Rev. B 1980, 21, 3290. (9) Duijvenbode R. C.; van Koper, G. J. M. J. Colloid Interface Sci. 2001, 239, 581.
Figure 2. The IBIS Technologies surface plasmon setup fitted with the stagnation point flow cell.
A mirror scans the angular range with a frequency of 44 Hz, which results in a sample time of about 20 ms. The incidence angle offset can be varied manually with a spindle in the range of 68-78°, corresponding with a solvent refractive index range of 1.33-1.43. A semicylinder is used to focus the beam in one single measuring spot of about 2 mm2. For the adsorption kinetics experiments the original IBIS cuvette/syringe system was replaced by a stagnation point flow cell10 (see Figure 2). The sensor disks were 2 mm glass slides coated with a 1.8 nm Ti layer. On top of the Ti layer, a 47.8 nm layer of gold was sputtered. The SPR sensor disks are also supplied by IBIS Technologies. Optical contact between the sensor disks and the prism was assured with index matching oil (Cargille series A, refractive index 1.518). It is our experience that coating the gold surfaces of the sensor disks with a thin polymer layer (especially polar polymers) reduces baseline instabilities significantly compared to the bare gold surfaces. Because of this, all experiments where done with sensor disks coated with a thin layer of poly(vinyl alcohol) (PVA) of molecular mass 105 Da (Aldrich). These were prepared by spin coating the IBIS sensor disks with a 0.2% PVA solution in Millipore water. Prior to spin coating, the gold surface was cleaned by exposing the surface in air to 185/255 nm UV light for 30 min. Directly after being cleaned, the gold surface was totally covered with the PVA solution. After 8 min of incubation, the disk was spun at 6000 rpm for 15 s. After the disks were spun, the surfaces were dried overnight in a vacuum exsiccator. Ellipsometry showed that the PVA coatings had a thickness of approximately 5 nm (with a refractive index of 1.55). The contact angle of water on these surfaces was between 8 and 11°, indicating their hydrophilic character. This simple method for spin coating turned out to give stable surfaces in the pH range of 4.57.5, although they show some swelling in contact with water. The sensitivity m ˜ (eq 5) was determined by a two-point calibration. Instead of pure liquids with tabulated refractive index, we chose to use two solutions of a different (10) Dabros, T.; van de Ven, T. G. M. Colloid Polym. Sci. 1983, 261, 694.
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Figure 3. Example of SPR shift versus time for an adsorption measurement. The first jump in the baseline is due to the switch from 0.1 to 1.0 M KNO3 solution. This gives the sensitivity m ˜ of the sensor surface. The slope ∆r/∆t is due to the adsorption of 85 nm latex spheres.
KNO3 concentration. Using this method, we were able to measure the sensitivity prior to each adsorption experiment without contaminating the surface. Calibration was done by measuring the baseline while flowing a 0.1 M KNO3 solution and subsequently switching to a 1.0 M KNO3 solution. The jump in the baseline divided by the refractive index difference directly yields the sensitivity of the coated sensor disk (Figure 3). This procedure turned out to be a good check of the surface smoothness as well. Rough surfaces, for example, strong acid-treated gold surfaces, show a tail after switching from 1.0 M KNO3 to 0.1 M KNO3 solution because the exchange of the solution in the pores is a slow process. In the stagnation point flow cell we used (Figure 2), the surface can be regarded as a perfect sink for the adsorbing particles and the surface concentration, cs, is small compared to the bulk concentration, cb. The flux J toward the surface is given by11
J ) 0.776(R j Reν)1/3(Df)2/3R-1(cb - cs)
(13)
where ν is the kinematic viscosity, Df is the diffusion coefficient of the particles, R is the radius of the inlet tube, Re is the Reynolds number of the flow at the end of the inlet tube, and R j is the flow intensity parameter which depends on the Reynolds number and on the cell dimensions. In this cell the distance h from the end of the inlet tube to the surface is 1.0 mm and the radius of the inlet tube R is 0.60 mm. For this h/R ratio of 1.7, Dabros and van de Ven numerically calculated10 the flow intensity parameter R j for Reynolds numbers ranging from 2 to 50. To calculate the Reynolds number, we use the definition of the above authors Re ) vj R/ν, with vj the average velocity at the end of the outlet tube. With the kinematic viscosity of water12 ν ) 0.891 × 10-6 m-1 s-1, the Reynolds number for the used flow cell is equal to the flow rate (expressed in ml min-1) multiplied by 10; this relation was used throughout. We used Figure 5 of ref 10 to determine R j for the thus calculated Reynolds number. The diffusion coefficient of the latex particles has been calculated with the Stokes-Einstein equation for rigid spheres
Df ) kT/3πηdp
(14)
(11) Lide, D. R. Handbook of Chemistry and Physics, 72nd ed.; CRC Press: Boston, MA, 1991. (12) Dijt, J. C.; Cohen Stuart, M. A.; Hofman, J. E.; Fleer, G. J. Colloids Surf. 1990, 51, 141.
in which dp is the diameter of the spheres and η is the viscosity of the solution. Although the stagnation point flow is a good method for a quantitative control of the flux,11 care has to be taken of the systematic errors introduced because of the finite size of the detection area. Bo¨hmer and co-workers13 evaluated the systematic error made in calculation of the flux J. They studied scanning electron microscopy pictures taken at different distances from the stagnation point and determined the surface coverage. A picture made at 2 min after adsorption showed a rather sharp maximum in adsorbed amount at the stagnation point. The area outside the stagnation point is detected as well and has a lower adsorbed amount. From this work we conclude that the average flux impinging on the detected area is 16% less than the value of the flux J calculated for the stagnation point with eq 13. PTFE chromatographic tubing (Ismatec, Germany) of 0.3 mm i.d. was used together with a turbulence poor PTFE microvalve without dead volume (Ismatec), so that switching between solutions caused no disturbance of the flow. After the sensitivity was determined (see above), the 1.0 M KNO3 solution was replaced by the latex sphere solution. The feeding tube of the microvalve was flushed and filled with the same latex solution. This was done to prevent premixing in the relatively wide feeding tubes (see Figure 2). During the actions described above, the surface was prevented from running dry by blocking the outlet tube of the flow cell. Immediately after switching to the latex sphere solution, the flow rate was determined by sampling the mass flowing through the cell during approximately 90 s. During the first minutes the slope ∆r/∆t was usually constant and therefore taken equal to the initial rate of adsorption. The procedure to determine ld is to measure the initial rate of adsorption of latex spheres and to compare this with the calculated flux toward the surface. We did this by constructing a dΓ/dt versus J plot and by adjusting ld until the slope of the linear least-squares fit becomes equal to 1.0 (see eq 9). Results Determination of m ˜ from nine different experiments, see Table 1, resulted in a sensitivity of 121 ( 2.6°/unit of refractive index. According to the manufacturer of the SPR instrument, the sensitivity m of the used type of sensor disks is 120°/unit of refractive index. From the fact that the determined value for m ˜ is almost the same as the sensitivity m of the bare sensor, we conclude that the effect of the 5 nm thick PVA layer on the sensitivity is negligible. In Figure 3 a typical adsorption experiment is shown which demonstrates the linearity of the initial adsorption versus time. Figure 4 shows the experimental dΓ/dt values plotted versus the flux J. The flux J was calculated, from eq 13, using the diffusion coefficient, from eq 14, and the Reynolds number. The dΓ/dt values were obtained using the diameter of the latex spheres as the layer thickness d (see eq 9). The decay length was obtained by adjusting the dΓ/dt vs J plot until a slope equal to 1 was obtained. The scatter in the experimental data results in a variation in the slope of 12% for a 95% confidence interval. The thus determined decay length is 283 ( 34 nm. The fact that we are able to fit all experiments with one single decay length and the absence of a systematic deviation with particle (13) Bo¨hmer, M. R.; Zeeuw, E. A.v.; Koper, G. J. M. J. Colloid Interface Sci. 1998, 197, 242.
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Table 1. Parameters Used in the Calculation of the Flux J and the Rate of Adsorption dΓ/dt with Their Variances diameter of latex spheres (nm)
concn (104 mg m3) ((4.0%)
flow rate (mL min-1) ((1%)
flow intensity parameter (R j)
∆r/∆t (10-3 deg s-1)
diffusion coefficient Df (10-12 m2 s-1)
flux J (mg s-1 m-2)
rate of adsorptiona (mg s-1 m-2)
21 ( 14.8% 21 ( 14.8% 41 ( 13.8% 41 ( 13.8% 80 ( 8.3% 80 ( 8.3% 80 ( 8.3% 85 ( 6.0% 100 ( 4.1%
4.05 4.05 4.25 8.50 2.05 4.10 8.20 8.08 8.10
3.34 3.25 2.75 3.51 3.29 3.18 3.29 2.94 3.40
18.0 ( 0.2 17.6 ( 0.2 15.6 ( 0.2 18.4 ( 0.2 18.0 ( 0.2 17.2 ( 0.2 18.0 ( 0.2 16.4 ( 0.2 18.4 ( 0.2
58.4 ( 5% 54.8 ( 5% 29.5 ( 4% 68.1 ( 5% 10.0 ( 2% 21.0 ( 2% 41.0 ( 2% 33.2 ( 2% 30.0 ( 2%
23 ( 14.8% 23 ( 14.8% 12 ( 13.8% 12 ( 13.8% 6.1 ( 8.3% 6.1 ( 8.3% 6.1 ( 8.3% 5.7 ( 6.0% 4.9 ( 4.1%
0.29 ( 11% 0.29 ( 11% 0.18 ( 10% 0.40 ( 10% 0.060 ( 6.9% 0.12 ( 6.9% 0.24 ( 6.9% 0.21 ( 5.8% 0.21 ( 5.0%
0.304 ( 6% 0.285 ( 5% 0.164 ( 5% 0.379 ( 5% 0.063 ( 3% 0.133 ( 3% 0.259 ( 3% 0.213 ( 3% 0.202 ( 3%
a
Decay length adjusted to match the values of the flux.
Figure 4. Initial adsorption rate dΓ/dt versus impinging flux J. See the text for further details. The solid line is the result of a linear least-squares fit. The decay length was adjusted to yield a slope of 1.0 ((0.12).
size is in itself a strong indication that the sticking probability is indeed close to 1.9 After having determined the decay length, we can check the validity of the linearization by calculating dΓ/dt using eq 10. If we divide the thus calculated dΓ/dt by the flux J, we obtain the resulting underestimation of the adsorbed amount (see Figure 5). The solid line in Figure 5 is equal to the quotient of eqs 10 and 9 which gives the theoretical underestimation. Discussion The determination of m ˜ is straightforward. The advantage of using the described method is that variations in coating layers are easily corrected for. There is no need to determine the optical properties of each new coating layer; one simply has to measure the new m ˜. The difference in refractive index between 589 nm (used in the refractometer) and 670 nm (the SPR setup) is acceptable as long as all the optical properties of solvent and adsorbed layers are expressed at/or measured with one particular wavelength. The reason is that the dN/dλ values for different organic liquids, proteins, and water are nearly the same, so that systematic errors made in the determination of the individual refractive indices will be equal and therefore cancel. Often it is more convenient to use the value obtained from 589 nm because the majority of refractive index data in the literature is given for that wavelength.
Figure 5. Comparison of the full equation for the adsorbed mass (eq 9) with its linearized form (eq 10) as a function of layer thickness (drawn line). The data points are the results of the adsorption kinetics experiments. For clarity, for each particle diameter the lower data point has been shifted to the right.
When measuring the adsorption of particles of high refractive index, it is important to investigate whether the basic assumption behind eq 2 still holds, i.e., that m is constant over the range of refractive indices used. For example, this assumption does not hold when the surface coverage with latex spheres exceeds 20%. Because in this system the difference between the refractive index of the adsorbed layer and the solvent exceeds 0.05 units of refractive index, the sensitivity m (or likewise m ˜ ) is changing while the adsorption layer is building up. The adsorption curve depicted in Figure 3 shows this (small) effect. An important assumption behind the method presented in this paper is that the sticking probability (i.e., (dΓ/ dt)/J) is indeed equal to 1; any deviation from this value would immediately show up in the determined decay length. We do not expect deviations, because of the following reasons: (1) PVA has no dissociative groups and hence a barrier for adsorption due to electrostatic repulsion is not likely to occur. Moreover, (2) possible Coulomb interactions are effectively screened in the 0.1 M KNO3 solution used. Also, (3) the PVA surface is very homogeneous and hence it is unlikely that the latex particles can only adsorb at specific sites.14 Finally, (4) to our experience the adsorption of the latex spheres on PVA is highly (14) Lyklema, J. Fundamentals of Interface and Colloid Science, Volume II: Solid-Liquid Interfaces; Academic Press Limited: London, 1995; Chapter 2.
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irreversible so that there will be no effect of desorption on the initial rate of adsorption.14 Because the initial rates of adsorption were used, the surface coverage of the latex spheres is zero and therefore the effective refractive index Neff is equal to that of the solution together with the PVA coating. When the experimentally determined decay length of 283 ( 34 nm is used to calculate the effective refractive index Neff, using eq 3, we find a value that is less than 1% larger due to the presence of a 5 nm thick coating layer having a refractive index of 1.55. Therefore, we can compare our experimental value directly with the theoretical prediction, eq 11, for a bare gold sensor surface. Using an effective refractive index of 1.333 and a dielectric permittivity for gold15 of metal ) (-12.01 + 1.131i), eq 11 yields a decay length of 194 nm, which is at variance with our experimental value. However, in ref 3 a similar deviation was reported. Differences may be attributed to different sensor systems, e.g., wavelength (15) Palik, E. D., Ed. Handbook of Optical Constants of Solids; Academic Press: Orlando, FL, 1985.
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shift compared to angular shift, or by differences in sensor disks, i.e., a 50 nm evaporated gold layer on a 2 nm chromium layer and a 48 nm sputtered gold layer on a 1.8 nm titanium layer. Conclusions We have demonstrated that it is possible to use the rate of adsorption to determine the decay length of the evanescent field without knowing all the optical parameters of sensor and coating layers. This results in an in situ calibration of the instrument without resort to external techniques. Also we have experimentally determined the errors made by using a simple linear equation (eq 10) that neglects the effect of the decay of the evanescent field. Acknowledgment. We thank the Dutch Technology Foundation STW for funding this research. LA010506A
