J. Phys. Chem. C 2009, 113, 16121–16127
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Detailed Analysis of Quartz Crystal Microbalance and Surface Plasmon Resonance Spectroscopy in Probing Molecular Adsorption onto Solid-Liquid Interfaces Jiajie Fang,†,‡ Ping Wang,†,‡ Xianbin Du,† and Da-Ming Zhu*,†,‡ Department of Modern Physics, UniVersity of Science and Technology of China, Hefei, People’s Republic of China 230027, and Department of Physics, UniVersity of Missouri-Kansas City, Kansas City, Missouri 64110 ReceiVed: May 1, 2009; ReVised Manuscript ReceiVed: July 22, 2009
The responses of quartz crystal microbalance (QCM) and surface plasmon resonance (SPR) spectroscopy in probing molecular adsorption on solid-liquid interfaces have been analyzed in detail. The two techniques respond differently to variation in the concentration profile of an adsorbed film. Quartz crystal microbalance measures the acoustic contrast between the changes caused by the adsorbed molecules in the medium next to the interface. Thus, an extended concentration profile on the film-solution interface could result in a large resonant frequency shift. Surface plasmon resonance, however, measures accumulative changes in the refractive index of the adsorbed film to the solution. Because of the exponential decay of the evanescence wave, the contribution to the resonance angle shift from adsorbed molecules decreases with the distance of the molecules to the solid-liquid interface. Thus, an extended interface would reduce the shift of the resonance angle, but such an effect is very small unless the film thickness and film-solution interface width are close to or beyond the optical decay length of the evanescence waves. 1. Introduction Molecular adsorptions onto interfaces play important roles in many physical, chemical, and biological processes involving interfaces and have continuously drawn considerable interest from researchers in various fields.1-10 However, molecular adsorptions onto solid-liquid interfaces are difficult to study quantitatively, mainly because adsorption occurs onto an interface sandwiched between two condensed media. Consequently, adsorption is not only affected by interactions between adsorbed molecules and solid substrate but also by interaction of adsorbed molecules and solutions. Adding to the difficulty is that many techniques developed for probing adsorption of molecules onto a solid-vapor interface are difficult to use in probing adsorption onto a solid-liquid interface due to strong attenuation of the probing signals by the liquid phase that covers the interface. Several novel techniques that have been developed for the study of adsorption onto solid-liquid interfaces in recent years probe directly the characteristic features of adsorbed molecules onto the interfaces. Among them, quartz crystal microbalance (QCM)11-21 and surface plasmon resonance (SPR) spectroscopy22-28 have been increasingly used for investigating adsorption behaviors and properties of the adsorbed macromolecules onto solid surfaces from solutions, and sometimes they are combined to yield more accurate information.29-35 Quartz crystal microbalance (QCM) replies on the inverse piezoelectric effect of a quartz crystal.11-21,36-38 By applying an alternating current (ac) electric field across a quartz crystal, a shear deformation (strain) oscillation can be excited in the crystal. A small amount of mass deposited on the surface of the quartz crystal can change the resonant frequency of the oscillation. Thus, by measuring the resonant frequency shift of a quartz crystal oscillating in a thickness shear mode, a mass deposited on the order of a subnanogram can be detected. Such * To whom correspondence should be addressed. E-mail:
[email protected]. † University of Science and Technology of China. ‡ University of Missouri-Kansas City.
a technique has been widely used for monitoring thin film depositions from vapor over the past several decades. In recent years, the QCM technique has been extended to the investigation of adsorption of molecules and molecular interactions onto solid-liquid interfaces.11-21,29-35 In these cases, adsorption of molecules not only causes a shift of the resonant frequency but also broadens the resonant peak. The broadening of the resonant peak is associated with the energy dissipation among the adsorbed molecules and the interaction between the molecules and solutions.36-38 Thus, by monitoring the frequency shift and broadening of the resonant peak of a quartz crystal, which is commonly termed as quartz crystal microbalance with dissipation monitoring (QCM-D), one can assess the viscoelastic properties of the adsorbed layers.12,16,18-20,30,32 Surface plasmon resonance (SPR) spectroscopy is a surface sensitive analytical method that relies on coupling of optical evanescent waves and surface plasma excitations on the surface of a noble metal.22,23,39,40 In a system where light propagates from an optically less dense into a more dense medium, total reflection occurs as the incident angles are larger than the critical angle. In this case, an evanescent field is generated in the denser medium. When this evanescent wave couples with the electron in the optically dense layer, the intensity of the reflected light is reduced. The incident angle at which the SPR occurs is very sensitive to the optical properties of the medium on top of the metal surface layer. Thus, the technique is sensitive in detecting adsorption of molecules from solutions on top of the metal surfaces. This technique can detect refractive index changes smaller than 10-7 with a time resolution of a few seconds or a resonant index as small as 0.1 RU.41,42 Although QCM and SPR techniques are capable of monitoring the kinetics of adsorption, they are sensitive to different aspects of the adsorbed layers.29-35 QCM probes the acoustic response of the adsorbed layers or the acoustic contrast between the adsorbed layers and solution, while SPR probes the refractive index contrast between the adsorbed layer and solution. Currently available experimental results indicate that the results on
10.1021/jp9040666 CCC: $40.75 2009 American Chemical Society Published on Web 08/14/2009
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(
β 2πFqhq
)
(1)
(
)
(2)
∆f ) Im
∆D ) Re
β πfFqhq
where
β ) κξ
A) Figure 1. (a) Illustrations of quartz crystal microbalance (QCM) and (b) surface plasmon resonance (SPR) spectroscopy in probing molecular adsorptions onto solid-liquid interfaces.
the adsorbed masses obtained on the same adsorbed molecular layers using the two techniques can be quite different.29-35 For adsorption of macromolecules or biological entities onto a solid surface, differences have been attributed to water trapping by molecular layers that are mechanically coupled to the shear mode of the QCM surface, while such a coupling does not exist for SPR.29-35 Thus, there is a general belief that the results from SPR measurements account accurately for the mass of adsorbed layers onto a surface. An alternative but equivalent explanation, which is suitable for more general cases, considers the concentration variation in adsorbates in the direction normal to the interface a diffusive interface between the adsorbed layers and solution.5,43-46 However, a detailed analysis of the response of QCM-D and SPR to the concentration variation in adsorbed molecules is needed in order to understand the results from these interface sensitive techniques in characterizing the adsorbed molecular layers onto solid-liquid interfaces. In this work, we numerically calculated the acoustic and optical responses of adsorbed molecular films with a concentration profile in the normal direction deposited onto a solid-liquid interface. The numerically calculated results show that the shift of resonant frequency of QCM and that of the resonance angle in SPR change approximately proportionally to the thickness of the adsorbed film until the film thickness is close to either the acoustic or optical decay length of the system. However, the slopes of the changes in terms of the properties of the adsorbed molecules are different for the two techniques. The resonant frequency shift and peak broadening of QCM are affected strongly by the changes in viscosity profiles of the adsorbed layers onto the solid-liquid interface. However, the shift of the resonance angle in SPR is not sensitive to the concentration file of the adsorbed molecules until the width of the film-solution interface is close to or beyond the evanescence decay length. 2. Method and Model Let us first consider the case of quartz crystal microbalance. A metal electrode surface on a quartz crystal is in contact with a solution overlayer as illustrated in Figure 1a. Assuming that the overlayer is uniform and can be described as a viscoelastic element that consists of a spring, a dashpot in parallel, and a “no-slip” boundary condition between the overlayer and electrode surface, a well-known derivation of the resonant frequency shift ∆f and dissipation factor change ∆D of the quartz crystal gives37,38
1 - A exp(2ξh) 1 - A exp(2ξh)
(3)
κξ + κbξb tanh(ξbhb) κξ - κbξb tanh(ξbhb)
µ κ)η-i , ω
-
ξ)
κb ) ηb,
ξb )
(4)
Fω2 µ + iωη
(5)
iFbω ηb
(6)
where h, F, η, and µ are thickness, density, viscosity, and shear modulus of the overlayer, respectively; and hb, Fb, and ηb are the thickness, density, and viscosity of the medium above the adsorbed overlayer, respectively. The shear modulus of the medium is assumed to be zero. It should be noted that ∆f and ∆D depend strongly on h, F, η, and µ. Figure 2 shows a typical dependence of -∆f and ∆D on the thickness h of the overlayer, assuming that the rest of the viscoelastic parameters of the overlayer as well as the medium above it remain constant. Both -∆f and ∆D increase approximately linearly with h until the thickness becomes close to δ ) (2η/Fω)1/2, the viscous penetration depth of the overlayer. After reaching maxima, -∆f and ∆D decrease and then saturate at values that are derived directly from eqs 1 and 2 by assuming h f ∞
∆f ≈ -
1 2πFqhq
(
F ηω 2
√µ2 + η2ω2 + µ µ2 + η2ω2
µ
∆D ≈
1 πfFqhq
( F ηω 2
√µ2 + η2ω2 - µ µ2 + η2ω2
√µ2 + η2ω2 - µ µ2 + η2ω2
µ
-
)
(7)
+
√µ2 + η2ω2 + µ µ2 + η2ω2
)
(8)
Thus, for a thick overlayer, ∆f and ∆D can be used to determine its viscosity and shear modulus if the overlayer can be treated as uniform. If the overlayer is purely viscous (µ = 0), the above equations (eqs 7 and 8) are reduced to the wellknow Kawazana-Golden equations.47 So quartz crystal microbalance can be used to determine either the thickness of an overlayer, which is thinner than the viscous penetration depth δ, or the viscoelastic properties of the overlayer, when it is much thicker than δ. The penetration depth δ increases with the viscosity of the overlayer. For overlayers with larger viscosities,
Molecular Adsorption onto Solid-Liquid Interfaces
J. Phys. Chem. C, Vol. 113, No. 36, 2009 16123
kz )
ω c
εaεm 2π - εa ) εa + εm λ
-εa2 εa + εm
εm and εa are the dielectric constants of the metal film and dielectric media, respectively. The optical decay length δp of the evanescence wave is the inverse of kz. This part of the wave, so-called the evanescence wave, allows probing the dielectric media next to the metal-dielectric interface, provided the metal film is thin enough. Consider that p-polarized light is incident upon the metal film through a glass prism or hemisphere (Figure 1b). Its reflection is affected by the metal film and overlayer on top of the film and can be described using the Fresnel equation.50 By deriving the mean-square electric fields induced by plane electromagnetic radiation on the interfaces between a glass prism, metal film, adsorbed layer, and semi-infinite solution, the reflectance R of this four-phase stratified medium is given as follows51
R ) |r| 2 )
|
(M11 + M12qa)qp - (M21 + M22qa) (M11 + M12qa)qp + (M21 + M22qa)
|
2
(9)
where
[
] [
]
M11 M12 cos(βm) -i sin(βm)/qm ) × M21 M22 -iqm sin(βm) cos(βm) cos(βs) -i sin(βs)/qs -iqs sin(βs) cos(βs)
[
qk ) Figure 2. Numerically calculated shift of the resonant frequency and dissipation change in a quartz crystal microbalance as a function of the thickness of adsorbed molecular films on a solid-liquid interface, with viscosity η chosen to be 0.01 Ns/m2 (dashed lines), 0.05 Ns/m2 (solid lines), and 0.1 Ns/m2 (dotted lines), respectively. Shear modulus µ is 104 N/m2.
the linear increasing regimes in -∆f extends to higher h values, but the slope of the increase remains the same. Thus, for uniform overlayers (η and µ uniform throughout the overlayer), -∆f is proportional to the layer thickness or the mass per unit area of the overlayer as long as h < δ. The surface plasmon resonance technique relies on a surface charge density wave propagating along the interface between a metal and dielectric medium.22,23,39,40,48 In an attenuated total reflection setting, where a metal film is deposited directly onto a high-index prism or hemisphere (Figure 1b), the light is reflected at angles greater than the critical angle for total reflection; most of the light will be total reflected. Part of the electromagnetic waves associated with the light, however, penetrate outside the metal film into the dielectric media and decay exponentially, following approximately39,49
Ez(z) ≈ Ez(0) exp[-kzz]
where
βk ) hk
√εk - εp sin2 θ εk
2π ε - εp sin2 θ, λ0 √ k
k ) p, m, s, a
]
(10)
(11)
(12)
where the subscripts p, m, s, and a are the abbreviation of glass prism, metallic film, overlayer, and ambient medium (water or air), respectively; εk and hk are the dielectric constants; and thickness, λ0, is the wavelength of incident light. As the incident angle θ is increased, the shift of reflectance R remains nearly a constant until the incident angle reaches the resonance angle, and then the reflection decreases sharply to a minimum, corresponding to the resonance of surface plasmon in the film. The resonance angle, θSPR, is related to the thickness and dielectric constant of the overlayer. Figures 3a and 3b display numerically calculated shifts of the resonant angle versus the thickness of the overlayer, assuming a uniform density and typical dielectric constants of polymer films for the overlayer. The response of SPR shows a similar behavior as that for QCM; it increases with film thickness until the film thcikness reaches about a few hundred nanometers when the changes in the resonant reflection angle saturates and reaches a plateau. The saturated values in the resonant frequency of QCM and angle of SPR depend on the viscosity and refractive index of the adsorbed films. For the overlayer with a larger viscosity, the linear regime of the resonant frequency change would extend to larger thickness of the films (Figure 3). However, a higher refractive index of the adsorbed film would result in a larger resonant angle shift, but the saturation always occurs at about
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Fang et al. the adsorbed mass measured using a quartz microbalance can be several times larger than the actual mass of the layer.29,30,32,34,35 The discrepancy, together with the consideration that strong interactions between adsorbates and solvent molecules tend to produce a diffuse interface between the adsorbed molecular layer and solution, prompted several studies that consider the effects of a diffused interface to the measured results.52 It is realized that the density of an adsorbed molecular film on a solid-liquid interface is more appropriately described in terms of a continuous concentration profile that gradually diminishes over a distance, which can be substantially larger than the average film thickness. The exact form of the profile probably is system specific. Attempts have been made to describe the solid-melt interface profile in terms of a Gaussian, simple exponential or hyperbolic functions, and error function.5,43-46 For polymers films chemically grafted onto a solid substrate, an exponential form has been used in describing the concentration variation. For polymers adsorbed in a diluted solution where the interface is saturated, the variation in polymer density may follow a power law profile.53 For physisorbed multilayer films, the density in the bottom layers might be relatively uniform, and thus, the profile might be appropriately described by a hyperbolic-tangent function44
c(z) ) cb + (c0 - cb){1 - tanh[(z - b)/a]}/2
(13)
Figure 3. Numerically calculated reflectance as a function of the incident angle with a thickness of adsorbed molecular films onto solid-liquid interfaces of 100 nm and the shift of the resonant angle due to surface plasmon resonance as a function of the thickness of adsorbed films. Changes in refractive index ∆n due to adsorbed layers were 0.03 (dashed lines), 0.06 (solid lines), and 0.09 (dot lines).
a few hundred nanometers in thickness. This is a manifestion that the viscous penetration depth varies sensitively with changes in the sensing medium, while the decay length of the evanescence wave is not sensitive to the change in the medium due to the adsorption on the metal-solution interface.39,49 Therefore, SPR always probes a depth of only a few hundred nanometers in the sensing layer, while the probing depth of QCM varies with the penetration depth of the acoustic wave in the adsorbed medium. 3. Probing an Adsorbed Overlayer with a Concentration Profile The above consideration assumes that the adsorbed overlayer has a uniform boxed concentration. For adsorption of molecules in a solution, the strong interaction between adsorbates and solvent molecules may result in a diffused interface profile between the adsorbates and solution. In the case of chemisorption of macromolecules, this interaction could result in the coil structures of macromolecules extending into the solution, forming a density variation in the direction normal to the interface.45,46 Currently available experimental results show that
where z is the coordinate in the direction perpendicular to the solid-liquid interface, c0 is the concentration of a saturated layer that corresponds to the closest packed structure of the adsorbed molecules onto the surface, cb is the concentration of the molecules in solution, a is a parameter that describes the width of the film-solution interface, and b is a parameter that together with a, gives the effective thickness of the adsorbed film. Assuming that the viscosity, shear modulus, and refractive index of adsorbed layers depend only on the concentration of the adsorbed molecules in the layer following a linear relationship, these properties take similar forms as those of concentration profiles
µ ) µ0{1 - tanh[(z - b)/a]}/2
(14)
η ) ηb + (η0 - ηb){1 - tanh[(z - b)/a]}/2
(15)
n ) n0 + c0(∂n/∂c){1 - tanh[(z - b)/a]}/2
(16)
where η0 and µ0 are the viscosity and shear modulus of the adsorbed layer with saturated concentration, respectively; ηb the viscosity of the solution, and n0 is the refractive index of solvent. Here, we assume that the shear modulus of the solution is zero. Following the analysis by Voinova et al.,37,38 the adsorbed film is divided into multiple layers with each layer, assuming that each layer can be treated as a single Voigt element, having a uniform physical property, and the property variation from layer to layer follows the concentration profile. If the thickness of the adsorbed film is much less than the viscous penetration depth δ (i.e, ξk∆h , 1 and h , δ), the exponential terms in eqs 1 and 2 can be approximated by their leading terms in the expansion. Assuming no slippage between adjacent layers, ∆f and ∆D can be expressed as
Molecular Adsorption onto Solid-Liquid Interfaces
∆f ≈ -
{ ∑[ () { ∑
ηb 1 + 2πFqhq δb n
Fω - 2
j)1
∆D ≈
]}
ηb 2 η(zj)ω2 ∆h δb µ2(z ) + η2(z )ω2 j j n
ηb 2 + ωFqhq δb
J. Phys. Chem. C, Vol. 113, No. 36, 2009 16125
Fbω
j)1
ηbµ(zj)ω µ (zj) + η2(zj)ω2 2
(17)
}
∆h
(18)
The first terms in ∆f and ∆D are the contributions from the solution, which is proportional to ηb1/2. If the dependence of the solution viscosity on the concentration is known, the contribution from the solution can be subtracted from the measured ∆f and ∆D to obtain adsorption isotherms. Given the concentration and viscoelastic profiles of an adsorbed film, ∆f and ∆D can be calculated using eqs 17 and 18. Figure 4a displays a calculated ∆f as a function of the adsorbed layer-solution interface width for a fixed adsorbed mass per unit area (b ) 100 nm). The results show that ∆f is proportional to the width of the film-solution interface, indicating that the shift of frequency is determined by the region where the adsorbed molecules extend to the solution, when the viscosity of this extended region is several times larger than the solvent. A similar approach can be applied to surface plasmon resonance in probing an adsorbed film with a density profile at a solid-solution interface. Consider the film on the prism-solution interface is divided into n multiple layers with the dielectric constant of each layer being εk, which varies following the refractive index profile (eq 16). When p-polarized radiation impinges at the prism-multilayer interface with an angle of incidence θ, the reflectance R is given by eq 9, with the characteristic matrixes of the system being the product of the matrix of the individually bounded plane layer51 n
Mij ) (
∏ Mk)ij
i, j ) 1, 2
(19)
k)0
and
Mk )
[
cos(βk) -i sin(βk)/qk -iqk sin(βk) cos(βk)
]
(20)
where the definition of βk and qk are the same as that in eqs 11 and 12, subscripts p, 0, 1, 2, ..., k, ..., n + 1 denote the prism, metal film, first layer, second layer, ..., kth layer, ..., and solution, respectively. Figure 4b shows the calculated reflectance as a function of the incident angle for adsorbed films with a refractive index profile described by eq 16 with b ) 100 nm and a varying from 0 to 100 nm. The mass per unit area for both profiles are kept the same. For profiles with effective thickness b and interface width a, both considerably less than the optical decay length, changes to the shift of the resonance angle are small. Only when the effective film thickness and interface width become comparable to the optical decay length does the change in the resonant angle become appreciable. The difference in ∆f and ∆θ in characterizing an adsorbed film with a concentration profile is quite dramatic; ∆f increases with the interface width of profile a following an approximately linear relationship, while ∆θ is insensitive to the interface width when the effective thickness is much less than the optical decay
Figure 4. Calculated shift of resonant frequency of (a) QCM and that of resonance angle of (b) SPR as a function of the interfacial width a of an adsorbed molecular film with a fixed amount of mass per unit area. Viscosity and refractive index can be described by eqs 15 and 16, with η0 and ∆n chosen to be 0.1 Ns/m2 and 0.08, respectively. Effective thickness of the film is 100 nm.
length but decreases noticeably as the interface width is close to the optical decay length. These differences can be understood quite easily. For a liquid-like adsorbed film, its shear modulus, µ = 0 and F ∼ Fb, eq 17 becomes
∆f ≈ -
(
)
n ηb 1 ωFb 1 ∆hj 2πFqhq j)1 η(zj)
∑
(21)
It can be seen from this equation that the resonant frequency shift is determined by the profile of η(z). For those molecular layers in which the concentration of adsorbates is low but the viscosity of the layer due to the adsorbates is appreciably higher than that of the solution, they contribute significantly to the frequency shift. Because the viscosity of an aqueous solution usually is small, the adsorbed film with a spread interfacial density profile can satisfy the condition ηb/η < 1 in a region with a thickness much larger than the average film thickness. In other words, the resonant frequency shift of QCM in probing the adsorption in solution actually measures the change in the viscosity profile caused by the adsorption of the molecules, i.e., the thickness in the solution that QCM probes is the length scale
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normal to the interface over which the viscosity has been significantly increased due to adsorption. This mechanism explains why the masses of adsorbed molecular films in solution determined using the resonant frequency shift of QCM are often found to be much larger than that actually adsorbed.29-35 The shift of the resonance angle in SPR is due to the change in the reflective index to the evanescence wave in the adsorbed film. In the case where the decay length of the electromagnetic wave is much longer than the thickness of the adsorbed film, the effects from different molecular layers to changes in the reflective index of the adsorbed film is accumulative. The effect can be illustrated following a derivation by Pockrand to account for the resonant angle change in continuous refractive index profiles54
an extended density profile with a large film-solution interface would result in a large resonant frequency shift. Surface plasmon resonance, however, measures accumulative changes in the refractive index of the adsorbed film to the solution. Unless the film thickness and film-solution interface is close to the optical decay length of the evanescence waves, the resonant angle shift is roughly proportional to the amount of molecules adsorbed onto the solid-liquid interface.
)√
(1) Fleer, G. J.; Cohen Stuart, M. A.; Scheutjens, J.M. H. M.; Cosgrove, T.; Vincent, B. Polymer at Interfaces, 1st ed.; Cambridge University Press: Cambridge, U.K., 1993, and references therein. (2) Taub, H.; Torzo, G.; Lauter, H. J.; Fain, S. F., Jr. Phase Transitions in Surface Films II, In NATO AdVanced Study Institute, Series B: Physics: Plenum Press: New York, 1991; Vol. 267. (3) de Gennes, P. G. Macromolecules 1980, 13, 1069. (4) Ma, J.; Kingsbury, D. L.; Liu, F.; Viches, O. E. Phys. ReV. Lett. 1988, 61, 2348. (5) Milner, S. T.; Witten, T. A.; Cates, M. E. Macromolecules 1988, 21, 2610. (6) Youn, H. S.; Hess, G. B. Phys. ReV. Lett. 1990, 64, 443. (7) Milner, S. T. Science 1991, 251, 905. (8) Oya, T.; Enoki, T.; Grosberg, A. Y.; Masamune, S.; Sakiyama, T.; Takeoka, Y.; Tanaka, K.; Wang, G.; Yilmaz, Y.; Field, M. S.; Dasari, R.; Tanaka, T. Science 1999, 286, 1543. (9) Zhu, D. M.; Zambano, A.; Migone, A.; Harrington, S. Phys. ReV. E 2001, 63, 011404. (10) Talapatra, S.; Krugleviciute, V.; Migone, A. D. Phys. ReV. Lett. 2002, 89, 246106. (11) Ebersole, R. C.; Foss, R. P.; Ward, M. D. Nat. Biotechnol. 1991, 9, 450. (12) Rodahl, M.; Hook, F.; Fredriksson, C.; Keller, C. A.; Krozer, A.; Brzezinski, P.; Voinova, M.; Kasemo, B. Faraday Discuss. 1997, 107, 229. (13) Hook, F.; Rodahl, M.; Kasemo, B.; Brzezinski, P. Proc. Natl. Acad. Sci. U.S.A. 1998, 95, 12271. (14) Hook, F.; Rodahl, M.; Brzezinski, P.; Kasemo, B. Langmuir 1998, 14, 729. (15) Keller, C. A.; Kasemo, B. Biophys. J. 1998, 75, 1397. (16) Johannsmann, D. Macromol. Chem. Phys. 1999, 200, 501. (17) Zhang, G. Z. Macromolecules 2004, 37, 6553. (18) Iruthayaraj, J.; Olanya, G.; Claesson, P. M. J. Phys. Chem. C 2008, 112, 15028. (19) Hemmersam, A. G.; Rechendorff, K.; Besenbacher, F.; Kasemo, B.; Sutherland, D. S. J. Phys. Chem. C 2008, 112, 4180. (20) Johannsmann, D.; Reviakine, I.; Rojas, E.; Gallego, M. Anal. Chem. 2008, 80, 8891. (21) Edvardsson, M.; Svedhem, S.; Wang, G. L.; Richter, R.; Rodahl, M.; Kasemo, B. Anal. Chem. 2009, 81, 349. (22) Liedberg, B.; Nylander, C.; Lunstrom, I. Sens. Actuators 1983, 4, 299. (23) Johnsson, B.; Lofas, S.; Lindquist, G. Anal. Biochem. 1991, 198, 268. (24) Nelson, B. P.; Grimsrud, T. E.; Liles, M. R.; Goodman, R. M.; Corn, R. M. Anal. Chem. 2001, 73, 1. (25) Ostuni, E.; Chapman, R. G.; Holmlin, R. E.; Takayama, S.; Whitesides, G. M. Langmuir 2001, 17, 5605. (26) Li, X.; Husson, S. M. Biosens. Bioelectron. 2006, 22, 336. (27) Li, L. Y.; Chen, S. F.; Zheng, J.; Ratner, B. D.; Jiang, S. Y. J. Phys. Chem. B 2005, 109, 29342005. (28) Hnilova, M.; Oren, E. E.; Seker, U. O. S.; Wilson, B. R.; Collino, S.; Evans, J. S.; Tamerler, C.; Sarikaya, M. Langmuir 2008, 24, 12440. (29) Keller, C. A.; Glasmastar, K.; Zhdanov, V. P.; Kasemo, B. Phys. ReV. Lett. 2000, 84, 5443. (30) Hook, F.; Kasemo, B.; Nylander, T.; Fant, C.; Sott, K.; Elwing, H. Anal. Chem. 2001, 73, 5796. (31) Plunkett, M. A.; Wang, Z. H.; Rutland, M. W.; Johannsmann, D. Langmuir 2003, 19, 6837. (32) Reimhult, E.; Larsson, C.; Kasemo, B.; Hook, F. Anal. Chem. 2004, 76, 72112004. (33) Tamerler, C.; Oren, E. E.; Duman, M.; Venkatasubramanian, E.; Sarikaya, M. Langmuir 2006, 22, 7712. (34) Jonsson, M. P.; Jonsson, P.; Hook, F. Anal. Chem. 2008, 80, 7988.
∆sin(θc) ≈
(
2π εmεa nλ εm + εa
2
∫0
∞
1
-εmεa
(εf(z) - εa) dz εf(z)
(22)
Because usually na . ∆n,
εf(z) ) nf2(z) ) (na + ∆n)2 ≈ εa + 2√εa∆n thus, ∆sin(θc) ≈
(
4π εmεa nλ εm + εa
)√ 2
1
-εmεa
∫0∞ ∆ndz (23)
So the resonant angle shift is proportional to the adsorbed layer mass and independent of the density profile. It is shown in Figure 4b that the shift of the resonant angle changes very little as the spread of the profile is less than about a few tens of nanometers. However, as the thickness of an adsorbed film is close to or beyond the decay length of the evanescence wave, the shift caused by those layers that are away from the metal-film interface is less than those that are adjacent to the interface. In this case, an adsorbed film with a spread concentration profile (or a larger film-solution interface width) would result in a smaller change in the resonant angle in SPR compared to that of a film with a boxed concentration profile. It should be noted, however, that adsorbed film with a film-solution interface width on the order of the decay length of an evanescence wave in a typical solution probably can only be realized in certain macromolecular or biological systems. In summary, although quartz crystal microbalance and surface plasmon resonator spectrometry are the tools for studying adsorbed molecules on solid-liquid interfaces, their detailed characteristic responses to film thickness and density profiles of adsorbed films are quite different. The shift of resonant frequency of a quartz microbalance is proportional to the thickness of the adsorbed film until the film thickness is close to the viscous penetration depth; the slope of the linear dependence is independent of the viscoelastic property of the adsorbed film. The shift of the resonance angle in surface plasmon resonance also varies linearly with the film thickness until it reaches the optical decay length of the evanescence wave in the medium, but the slope of the variation increases with the refractive index changes due to the adsorbed film. The two techniques also respond differently to the variation in the profile of an adsorbed film. Quartz crystal microbalance measures the acoustic contrast between the changes caused by the adsorbed molecules in the medium next to the solid-liquid interface. Thus
Acknowledgment. This work is supported in part by a grant from the Research Corporation and by the Kansas City Area Life Sciences, Inc. References and Notes
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