Measuring Reversible Adsorption Kinetics of Small Molecules at Solid

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Anal. Chem. 1998, 70, 4247-4256

Measuring Reversible Adsorption Kinetics of Small Molecules at Solid/Liquid Interfaces by Total Internal Reflection Fluorescence Correlation Spectroscopy Richard L. Hansen and Joel M. Harris*

Department of Chemistry, University of Utah, Salt Lake City, Utah 84112

Reversible adsorption kinetics of a cationic dye molecule at a C-18 modified silica surface are measured in a fluorescence correlation spectroscopy experiment. An interfacial observation volume is defined by total internal reflection of a laser beam which excites fluorescence emission from Rhodamine 6G (R6G+) in adsorption equilibrium with the surface. Spontaneous fluctuations in the population of R6G+ molecules within the interfacial volume are observed as excess low-frequency noise superimposed on a fluorescence transient. Autocorrelation of the transient allows the temporal characteristics of the fluctuations to be examined. The rate at which fluctuations occur depends on transport of molecules to the interface and their adsorption kinetics. Adsorption and desorption rates of R6G+ in water/methanol solutions at a C-18 modified silica/solution interface were measured over a 3 order of magnitude range of the retention equilibrium; both rates were found to depend strongly on the composition of the overlying solution phase. Most of our understanding of chemical interactions at the solid/liquid interface has been inferred from solute retention equilibria measured by chromatography, equilibrium adsorption isotherms, or other steady-state experiments. The subtlety of chemical interactions that play a role in interfacial phenomena cannot generally be identified by equilibrium measurements alone. Interfacial kinetic measurements afford an opportunity for understanding interfacial transport and reaction mechanisms responsible for adsorption equilibria and allow examination of the kinetic barriers that govern the rates of adsorption and desorption. Despite the importance of kinetic measurements in understanding the solid/liquid interface, few techniques are available to probe chemical reaction kinetics at dielectric (nonconductive) interfaces. Generally, a rapid perturbation of concentration or activity is needed to observe a rate of reaction. Flow methods, which can alter chemical composition at an interface, are limited by molecular diffusion through the stagnant solvent layer adjacent to a solid surface. The slow transport of molecules to the interface limits rates of adsorption that can be measured. Relaxation kinetic methods1 avoid the need for fast concentration changes by instead perturbing the activity of species through (1) Eigen, M. Discuss. Faraday Soc. 1954, 17, 194-205. S0003-2700(98)00925-1 CCC: $15.00 Published on Web 09/18/1998

© 1998 American Chemical Society

a rapid change of conditions such as temperature, pressure, or electric field strength. The equilibrium constant of a reversible reaction is thus shifted to a new value. The “relaxation” of the system to the new equilibrium position is followed by monitoring changes in concentrations of species that govern the equilibrium. Two approaches to perturbing adsorption/desorption equilibria have recently been developed for studying the kinetics of reversedphase chromatographic systems: a pressure jump can produce a shift in an equilibrium constant that depends on the change in molar volume.2,3 Alternatively, a temperature jump provided by Joule discharge4-6 or laser7 heating of the sample can be used to perturb an adsorption/desorption equilibrium based on the temperature dependence of the equilibrium constant; the magnitude of the perturbation scales with the enthalpy of adsorption. These two relaxation methods have successfully revealed details of adsorption kinetics at reversed-phase chromatographic interfaces,8 but they are limited in their application to systems that exhibit a large change in molar volume or enthalpy upon adsorption. In the present study, total internal reflection fluorescence correlation spectroscopy (TIRFCS) is examined as a method to study adsorption and desorption kinetics of small molecules at C-18 modified silica/solution interfaces. In TIRFCS, no perturbation is required to measure adsorption kinetics; random fluctuations in the number of molecules in equilibrium with the interface are instead monitored by fluorescence emission from an interfacial observation region. The rate at which fluctuations evolve is determined by an autocorrelation of the fluorescence transients, which can be used to determine the adsorption and desorption rates that govern the fluctuation relaxation. TIRFCS was first described by Thompson, Burghardt, and Axelrod9,10 and has been used to examine the surface binding (2) Marshall, D. B.; Burns, J. W.; Connolly, D. E. J. Chromatogr. 1986, 360, 13-24. (3) Marshall, D. B.; Burns, J. W.; Connolly, D. E. J. Am. Chem. Soc. 1986, 108, 1087-1088. (4) Waite, S. W.; Marshall, D. B.; Harris, J. M. Anal. Chem. 1994, 66, 20522061. (5) Ren, F. Y.; Waite, S. W.; Harris, J. M. Anal. Chem. 1995, 67, 3441-3447. (6) Ren, F. Y.; Harris, J. M. Anal. Chem. 1996, 68, 1651-1657. (7) Waite, S. W.; Holzwarth, J. F.; Harris, J. M. Anal. Chem. 1995, 67, 13901399. (8) Harris, J. M.; Marshall, D. B. J. Microcolumn Sep. 1997, 9, 185-191. (9) Thompson, N. L.; Ph.D. Thesis, University of Michigan, Ann Arbor, MI, 1982.

Analytical Chemistry, Vol. 70, No. 20, October 15, 1998 4247

kinetics of immunoglobulin G biomolecules labeled with a fluorescent dye at a protein-coated silica/water interface;11 variations in binding kinetics between insulin and immunoglobulin G were observed. Other forms of TIRFCS have been used to characterize virus particles traversing through an evanescent wave excitation volume.12 Theory has also been developed to measure adsorption kinetics of a nonfluorescent species competing with a fluorophore for binding sites.13 A chemical system analogous to that used in reversed-phase chromatographic separations is an ideal system to examine by TIRFCS; a wide range of adsorption equilibrium constants can be explored by varying the solvent compositions. Challenging chemical systems from a separations standpoint, such as cationic or basic solutes adsorbing at a C-18 modified silica surface,14-16 can be explored. Silica surfaces, derivatized to form a reversedphase chromatographic packing material by reaction with monomeric alklysilane reagents, typically retain greater than half the original silanol groups on the surface.17 Silanol groups are acidic, leading to strong association with basic solutes including ionexchange interactions.18,19 In contact with water, the silanols deprotonate to produce a negatively charged surface14 that causes problems of peak broadening in the separation of cationic species on silica-based phases; this has prompted studies to understand the chemical environment that a cationic probe experiences at a silica/C-18 surface20 and underlying energetic contributions involved in adsorption.14 Adsorption of cationic molecules is important not only in standard reversed-phase separations but also in ion-pair separations in which a cationic ion-pairing reagent assists in retention of anionic species.21-23 In the present experiment, the adsorption/desorption kinetics of the cationic dye Rhodamine 6G (R6G+) at a C-18 derivatized silica surface are examined using the TIRFCS technique. To the authors’ knowledge, this is the first application of TIRFCS to examine adsorption/desorption kinetics of small molecules. The method was tested over a range of solvent compositions, and kinetics were measured over nearly a 3 order of magnitude range in the adsorption equilibrium constant, where Keq was varied from ∼3 to 2000 M-1. THEORY The theoretical basis of TIRFCS was pioneered and developed by Thompson et al.9,10 This comprehensive work describes in detail how transport and kinetic processes influence the observed correlation signal. To discuss the interpretation of our results, a (10) Thompson, N. L.; Burghardt, T. P.; Axelrod, D. Biophys. J. 1981, 33, 435454. (11) Thompson, N. L.; Axelrod, T. P.; Biophys. J. 1983, 43, 103-114. (12) Hirshfeld, T.; Block, M. J.; Mueller, W. J. Histochem. Cytochem. 1977, 25, 719-723. (13) Thompson, N. L. Biophys. J. 1982, 38, 327-329. (14) Huang, X.; Kovaleski, J. M.; Wirth, M. J. Anal. Chem. 1996, 68, 41194123. (15) Wirth, M. J.; Fairbank, R. W. P.; Fatunmbi, H. O. Science 1997, 275, 4448. (16) Cox, G. B. J. Chromatogr., A 1993, 656, 353-367. (17) Roumeliotis, P.; Unger, K. K. J. Chromatogr. 1978, 149, 211-224. (18) Cox, G. B. J. Chromatogr. 1987, 384, 315-336. (19) Bildlingmeyer, B. A.; Del Rios, J. K.; Korpi, J. Anal. Chem. 1982, 54, 442447. (20) Shaksher, Z. M.; Seitz, W. R. Anal. Chem. 1989, 61, 590-593. (21) Liu, H.; Cantwell, F. F. Anal. Chem. 1991, 63, 993-1000. (22) Liu, H.; Cantwell, F. F. Anal. Chem. 1991, 63, 2032-2037. (23) Del Rey, M. E.; Vera-Avila, L. E. J. Liq. Chromatogr. 1987, 10, 2911-2929.

4248 Analytical Chemistry, Vol. 70, No. 20, October 15, 1998

brief summary of their theory is repeated here. The TIRFCS technique is based on observing a population of molecules that are in equilibrium with a surface to detect spontaneous concentration fluctuations within a defined region at the interface. An interfacial probe volume, from which the population is monitored by fluorescence emission, is created by total internal reflection of an excitation laser beam.24 The dimensions of the volume are determined by the depth of penetration of the resulting evanescent wave into the solution and a limiting aperture in the image plane of the collection lens, comparable to the projected spot size of the beam at the point of internal reflection. If the number of fluorophores within this probe volume is kept small, spontaneous fluctuations in the population are observable as excess lowfrequency noise superimposed on a fluorescence transient F(t′).

∫

F(t′) ) QIo I(rj)C(rj,t′) d2r

(1)

where I(r) is the unitless illumination intensity profile, which when multiplied by the maximum intensity, Io, collection and excitation efficiencies, Q, and concentration of surface species, C(r,t′), predicts the fluorescence observed at the detector. The rate at which population fluctuations occur is measured by autocorrelating the fluorescence transient.

G(t) )

〈F(t′)F(t′+t)〉 - 〈F(t′)〉2 〈F(t′)〉2

(2)

where

∫

1 〈F(t′)F(t′ + t)〉 ) lim Tf∞T

T/2

F(t′)F(t′ + t) dt′

-T/2

(3)

Fluctuations are controlled by kinetic factors that determine the residence time of molecules within the observation volume, including diffusion, adsorption and desorption at the surface, and possibly photochemical bleaching of the probe. Analysis of the temporal characteristics of the fluctuations by autocorrelation can, in principle, be used to determine the rates at which these processes occur. The magnitude of the fluctuations is determined by the size of the fluorophore population observed. This is due to the fact that the molecules in the observation region are drawn from a large reservoir in bulk solution and thus follow a Poisson distribution. Because the variance of a Poisson distribution scales with its mean,25 the relative fluctuation magnitude drops with the inverse square root of the population mean. The upper bound to the number of molecules observable is determined by the amplitude resolution of the detection system;26 the number of molecules observed is typically less than 105. The theory describing fluctuations of molecules adsorbed to a surface is based on the following chemical system: a fluorescent molecule, A, is in solution and can bind to an available site on the (24) Thompson, N. L.; Pearce, K. H.; Hsieh, H. V. Eur. Biophys. J. 1993, 22, 367-378. (25) Barlow, R. In Statistics; Mandl, F., Ellison, R. J., Sandiford, D. J., Eds.; John Wiley & Sons: Chichester, U.K., 1989. (26) Elson, E. L.; Webb, W. W. Annu. Rev. Biophys. Bioeng. 1975, 4, 311-334.

surface, B, to become a bound species C. kads

}C A + B {\ k

(4)

des

where the surface species, C, and available sites, B, are defined in units of number per area making the adsorption rate constant, kads, and desorption rate constant, kdes, defined in units of M-1 s-1 and s-1, respectively. A binding or adsorption equilibrium constant can also be defined when the rates of adsorption and desorption are equal:

Keq )

kads [C] ) kdes [A][B]

(5)

Spontaneous fluctuations of solution species A and surface species C are described by relating the instantaneous concentration at a given position and time to their average concentrations:

DA and DC are the solution and surface diffusion coefficients of species A and C, respectively. The symbol ∇2 denotes the Laplacian operator. Equation 9 describes time-dependent changes in the concentration of the solution species through diffusive transport. Equation 10 relates changes in the bound species to both surface diffusion and adsorption and desorption events. Only the concentration of A directly at the interface (zf0) is considered for adsorption events. Equation 11 is a boundary condition equating the rate of flux at the interface with the difference between adsorption and desorption rates. Lateral diffusion of adsorbed species across the surface10 can generally be neglected because the observation area is typically large and surface diffusion relatively slow.27,28 The rate of surface transport in to and out of the observation region is, therefore, negligibly small, which leads to the following solution to eq 8:10

G(t) )

G(0) [ν-1/2w(-ixν+RRt) ν-1/2 - ν+1/2 ν+1/2w(-ixν-RRt)] (12)

δA ) A(rj,z,t) - 〈A〉 δC ) C(rj,t) - 〈C〉

(6)

The fluctuations of the solution species can depend on both a radial surface dimension, r, and a distance normal to the interface, z. Concentration correlation functions are constructed from the spontaneous fluctuations; fluctuations in the A and C are correlated with fluctuations in C at different times and radial positons.

A ˇ (rj,rj′,z,t) ) 〈δA(rj,z,t′+t) δC(rj′,t′)〉/〈C〉2

(7)

C ˇ (rj,rj′,t) ) 〈δC(rj,t′+t) δC(rj′,t′)〉/〈C〉2 If most of the signal arises from surface-adsorbed species, as opposed to solution species within the evanescent wave, the autocorrelation observed experimentally can be described by eq 8. To solve eq 8, it is necessary to relate the concentration

∫∫I(rj)I(rj′)Cˇ (rj,rj′,t) d r d r′ G(t) ) [∫I(rj) d r] 2

2

2

2

(8)

correlation function to physical processes that govern the transport of species in to and out of the observation region. It is assumed that fluctuating surface concentrations may arise from diffusion of species A through solution, surface diffusion of the bound species C into the observation region, and adsorption and desorption.10

∂A ˇ ˇ ) DA∇2r,zA ∂t ∂C ˇ ˇ + kadsA ˇ zf0〈B〉 - (kads〈A〉 + kdes)C ˇ ) DC∇2r C ∂t ∂A ˇ ∂z

( )

DA

zf0

) kadsAzf0〈B〉 - (kads〈A〉 + kdes)C ˇ

(9)

(10)

(11)

where

∫

I2(rj) d2r β G(0) ) (1 - β) [〈B〉 + 〈C〉][ I(rj) d2r]2 β)

ν1/2 ( )

∫

〈B〉 (〈B〉 + 〈C〉)

x

1 2

RR [-1 ( x1-4RBND/RR] RBND

w(iη) ) eη erfc(η) (η complex) 2

(13)

(14)

(15)

(16)

The value of the function at zero offset, G(0), is the inverse of the number of molecules contributing to the observed fluctuations within the observation volume, 1/N. When the terms containing β in eq 13 are expanded and multiplied by 1/(〈B〉 + 〈C〉), the result is the inverse of the mean surface concentration, 1/〈C〉. The integral quotient is equal to the inverse of the intensity weighted illumination area and when multiplied by 1/〈C〉 is equal to 1/N. Two characteristic rates determine the shape of the autocorrelation described by eq 12. The observed rate that relates to surface reactions, RR,

RR ) kads[A] + kdes

(17)

is the sum of the rates of adsorption and desorption, which both work to relax fluctuations in the adsorbed population; this is analogous to the relaxation rate observed in a perturbation/ relaxation experiment.4 The second characteristic rate is due to (27) Zulli, S. L.; Kovaleski, J. M.; Zhu, X. R.; Harris, J. M.; Wirth, M. J. Anal. Chem. 1994, 66, 1708-1712. (28) Hansen, R. L.; Harris, J. M. Anal. Chem. 1995, 67, 492-498.

Analytical Chemistry, Vol. 70, No. 20, October 15, 1998

4249

bulk normal diffusion, RBND in eq 15 where

RBND )

At zero offset, eq 20 is equal to

DA

(18)

(β[C]/[A])2

is the rate at which a change in concentration on the surface is balanced by a change in concentration in solution. The denominator of RBND contains the ratio of the surface to solution concentrations, [C]/[A]; this term is the distance into solution that contains the same number of molecules as are adsorbed to the surface. This distance is related to a relaxation rate by dividing its square into the solution diffusion coefficient; it represents the upper bound to measurable adsorption kinetics where transport to the interface is rate limiting. Equation 12 requires that most of the observable fluorescence emission arises from molecules adsorbed to the surface. In cases of low surface retention, one may observe a significant fraction of fluorescence from molecules in the solution volume of the evanescent wave. This case requires the addition of another characteristic rate, REW, the evanescent wave relaxation rate.

REW ) DA/dp2

(19)

This rate is simply the time required for diffusional relaxation across the evanescent wave which penetrates into solution a distance dp. Equation 20 describes the autocorrelation with contribution from solution molecules in the evanescent wave.9

G(t) )

[

βF { ν R w(-) - xν-RRw(+)} + ∆ x + R

2βFRR

{xR-w(+) - xR+w(-) + ∆w(EW)} ∆xR+R-

βFRR {R ν R w(+) - Rxν-RRw(-) + 2∆[RR + ∆R+R- -x + R

xREW(xν+RR + xν-RR)]w(EW)} + βFRR

}

xR+R-

{

(1 - F) +

]

{4xREWt/π + (1 - 8REWt)w(EW)} /[(〈C〉 +

∫

〈A〉dp/2)( I(rj) d2r)] (20)

where

〈C〉 (〈C〉 + 〈A〉(dp/2))

(21)

w(() ) w(-ixν(RRt)

(22)

w(EW) ) w(2ixREWt)

(23)

xR( ) 2xREW + xν(RR

(24)

F)

4250 Analytical Chemistry, Vol. 70, No. 20, October 15, 1998

G(0) )

βF + (1 - F)

∫

(〈C〉 + 〈A〉(dp/2))( I(rj) d2r)

(25)

and contains contributions from both solution and surfaceadsorbed molecules. G(0) is again equal to the inverse of the number of molecules in the observation region contributing to fluctuations. Although eq 20 appears unwieldy, it is in effect governed simply by the three characteristic rates, RR, RBND, and REW. The factor F weights the contribution to the autocorrelation amplitude from adsorbed species, increasing its influence at higher adsorption equilibrium constants. The first two terms in eq 20 are the same as eq 12, which is used when the contribution from solution-phase species in the evanescent wave volume can be neglected. The next line in eq 20 contains cross-correlation terms between solution species, A, and adsorbed species, C. The final three lines of the numerator relate autocorrelations of solution species A, which are modified by adsorption and desorption at the interface.9 EXPERIMENTAL SECTION Chemical Preparation. Fused-silica substrates (Esco) were prepared by first cleaning in an oxidizing acid mixture to remove any residual organic material; the substrates were placed in a stirred 1:1 mixture of concentrated nitric and sulfuric acid for ∼30 min. The substrates were then rinsed thoroughly with distilled and deionized 18 MΩ cm water. Underivatized substrates were stored under methanol for future use. Substrates to be derivatized with monomeric C-18 ligands were dried in an oven at 130 °C for several hours. The substrates were then refluxed under an atmosphere of dry nitrogen for 12 h with a large excess of octadecyldimethylchlorosilane in ∼250 mL of toluene with 5 mL of pyridine added as a catalyst. The octadecyldimethylchlorosilane reacts with silanol functional groups on the substrate surface to covalently bind the alkyl chains through a siloxane linkage. Following the derivatization reaction, the slides were thoroughly rinsed in toluene and methanol to remove unreacted reagent from the surface and stored under methanol for future use. Adsorption/desorption kinetics of R6G+ (Exciton) at a C-18 derivatized silica surface were measured at five R6G+ concentrations in several water/methanol compositions. Solutions were made by mixing water and methanol in ratios (by volume) of 5:95, 20:80, 40:60, and 50:50 water/methanol. Solutions containing R6G+ concentrations of 5.1 × 10-11, 1.2 × 10-10, 2.5 × 10-10, 5.1 × 10-10, and 1.2 × 10-9 M were prepared by adding appropriate aliquots of 2.54 × 10-7 M R6G+ from a stock solution in methanol; all solutions also contained 10 mM added NaCl. The NaCl was added to reduce electrostatic interactions between the cationic R6G+ with the negatively charged silica surface. The 10 mM concentration of added electrolyte was determined from its improvement of the peak shape and reduction of chromatographic retention of R6G+ on a C-18 column. At higher concentrations of NaCl, only modest decreases in retention of R6G+ were observed. The C-18 (end-capped) chromatographic stationary phase was bound to a large-pore (300 Å) silica to mimic as closely as possible the planar C-18 surface used in the kinetic experiments. The observed adsorption kinetics in the presence of 10 mM NaCl

appeared to be homogeneous and reversible and were well fit by theory (see below); without the addition of electrolyte, the adsorption kinetics were not simple as would be expected from the chromatographic results. Adsorption equilibrium constants at the planar surface measured by quantifying the fluorescence fluctuations29 were similar but not identical to those predicted from retention measurements on this column. Data Acquisition. TIRFCS experiments were carried out using a laser-based instrument previously described.29 Briefly, the 514.5-nm line from an argon ion laser focused to the point of total internal reflection in p-polarization at the derivatized silica slide/solution interface within a flow cell. Note that polarized excitation could give rise to fluctuations due to rotational motions of the adsorbate; for small molecules, however, these motions have been shown to occur on a nanosecond time scale,30 which is much faster than is accessed in these experiments. The laser light was coupled to the slide through a fused-silica dove prism, indexmatched with glycerol. A long working distance (1.44 mm), high numerical aperture (0.7 NA) microscope objective collected the fluorescence emission through the solution in the flow cell. Sample solutions flowed through the cell at rate of 0.75 mL/min; the effect of flow on the concentration profile near the surface is negligible. At our slow linear velocities corresponding to a Reynold’s number, Re ≈ 0.2,31 the flow profile should be laminar or parabolic. At the largest distance from which adsorbate must diffuse to relax concentration changes at the interface, [C]/[A] ) 2 µm (see above), the solution velocity is 65 µm/s, giving a residence time in the illumination area, for molecules at this distance from the interface, of more than 3 s; this is 15× longer than the longest time-scale data reported. Light collected from the illumination area of the flow cell was passed through a holographic notch filter to remove Raleigh scatter. The internal reflection spot was reimaged onto a limiting aperture the size of which was comparable to the projected excitation laser beam spot size, to maximize light collection from the interface while reducing stray radiation from the laser beam propagating through the dove prism, index matching fluid, and substrate. Radiation passed by the aperture was refocused through a second long-pass filter onto a photomultiplier tube. The data were collected in photon-counting mode by amplifying photoelectrons with a 10×, 300-MHz amplifier and counting pulses that exceeded a preset threshold with a stand-alone discriminator. A current pulse was created by the discriminator for each photoelectron event detected. The pulses were recorded by a multichannel analyzer (EG&G Turbo MCS). The data were then transferred to a Pentium PC for analysis. Files of finite length containing discrete time bins were processed off line by a program using an autocorrelation algorithm compiled in FORTRAN.32 Data were Fourier-transformed, multiplied by their complex conjugate to form a power spectrum, and then inverse Fourier-transformed to form the autocorrelation. Typically, 100 files each containing 16 384 time bins were collected for a given experimental condition. (29) Hansen, R. L.; Harris, J. M. Anal. Chem. 1998, 70, 2565-2575. (30) Burbage, J. D.; Wirth, M. J. J. Phys. Chem. 1992, 96, 5943-5948. (31) Giddings, J. C. Dynamics of Chromatography; Marcel Dekker: New York, 1965; Chapter 5. (32) Press: W. H.; Flannery, B. P.; Teukolsky, S. A.; Vetterling, S. A. Numerical Recipes; Cambridge: London/New York, 1986.

The files were processed sequentially and then combined by averaging the calculated autocorrelations. Data were acquired on two different time scales depending on retention of R6G+ at the interface. For the 5, 20, and 40% water in methanol conditions, data were acquired at 10 µs/bin. At 50% water, data were acquired at 100 µs/bin. The time scale of data acquisition was chosen such that sufficient bandwidth was available to fully describe the shape of the autocorrelation at early offsets. Because of the coupling to terms in eq 20, adsorption and desorption kinetics can affect the shape of the experimental autocorrelation at all parts of the curve. This is important approaching conditions of high retention, where the bulk normal diffusion rate decreases significantly. A characteristic of TIRFCS is that the slowest fluctuating component has the greatest influence on the shape of the autocorrelation at long times; data must be taken at sufficiently long time delays to find the fluorescence intensity level where no further correlation in the data is observable. Unfortunately, long time scale autocorrelations are the most susceptible to any type of low-frequency process or noise, such as photostability of the fluorophore or drift of the laser power. Because of this, data were parsed into two point densities; a point density was selected that was high enough to describe the curvature of the fast, initial portion of the curve, and a lower point density describing the tail of the curve was also selected. In this way, the initial portion of the curve could be searched for the best-fit reaction rate without undue influence from the tail of the autocorrelation, while the lower point density data could be used to establish the mean fluorescence level for normalizing the data (see below). Normalization of the data consistent with eq 2 required removal of the contribution from background scatter to the averaged autocorrelations. Background intensities were determined from a blank sample of identical solvent composition run before acquisition of data with R6G+-containing solutions. The autocorrelation of the background intensity exhibited no delay-time dependence beyond the first time increment,29 so that its power spectrum is white on the time scale of these experiments and contributes no shape to the measured fluorescence correlation. The mean background signal, 〈B〉, was estimated from the square root of the blank autocorrelation amplitude; this level was subtracted from the mean total signal from the sample, 〈S〉, to determine the mean fluorescence, 〈F〉, used to normalized the data as in eq 2. Data Fitting. In the present work, experimental conditions varied over a large range of adsorption equilibrium constants, Keq, (see Table 1) necessitating use of eq 20 to include the contribution from solution species in the evanescent wave at low surface retention. Equation 20 is, however, the general solution of the autocorrelation function for all conditions. Once a suitable fitting algorithm was developed, eq 20 was used exclusively to fit the data regardless of surface retention. The characteristic rates RR, RBND, and REW that describe the autocorrelation in eq 20 produce effects on the shape of the relaxation that are not independent. Mutual compensation between rates proved to be problematic when fitting the data to all three rates. This is not surprising since there is real correlation between the physical processes described by these rates; for Analytical Chemistry, Vol. 70, No. 20, October 15, 1998

4251

Table 1. Equilibrium and Rate Constants for R6G+ Adsorption at a C-18 Silica/Solution Interface % water in MeOH

Keqa (M-1)

RBNDb (s-1)

REWb (s-1)

kdesc (s-1)

kadsd (M-1 s-1)

5 20 40 50

3.4 ( 5 22 ( 5 298 ( 14 1940 ( 90

4(( 8) × 107 6(( 3) × 105 2.6((0.2) × 103 55(( 5)

4.1((0.4) × 104 2.7((0.4) × 104 2.1((0.15) × 104 1.9((0.1) × 104

4200 ( 1100 3400 ( 710 655 ( 140 44 ( 9

1.4(( 2) × 104 7.5(( 2) × 104 2.0((0.4) × 105 8.5((1.8) × 104

a These values are taken from surface populations estimated from the amplitude of the fluorescence fluctuations.29 The diffusion distances, [C]/[A], that govern RBND are proportional to the equilibrium constant; these vary from 4 × 10-7 cm at 5% water to 2.1 × 10-4 cm at 50% water. b Values of the solution diffusion coefficient, D , used in calculation of these rates were determined by patterned fluorescence correlation spectroscopy A in solution;33 the depth of penetration of the evanescent wave varies slightly with solvent composition due to changes in refractive index, dp ) 1.05 -5 × 10 cm in 5% water compared to 1.16 × 10-5 cm in 50% water. c Error bounds express the 90% confidence limits. d The 90% confidence limits include uncertainties in Keq.

example, a desorption event from the surface changes the local concentration at the interface, affecting the concentration gradient and therefore net flux away from the interface. Because of the high degree of correlation between these kinetic processes, it was necessary to fix the rates that can be determined a priori. Both the evanescent wave relaxation rate, REW, and the bulk normal diffusion rate, RBND, can be estimated from independent experiments and fixed, leaving only RR to float when the fit of the transients is optimized to theory. To calculate REW (eq 19), knowledge of the evanescent wave depth of is required along with the solution diffusion coefficient of species A. The depth of penetration can be predicted from the angle of incidence, θ, the refractive indices of the denser and rarer media at the interface, n1 and n2, respectively, and the vacuum wavelength of the laser line, λo.24

dp )

λ0 2

4π(n1 sin2θ - n22)1/2

(26)

Solution-phase diffusion coefficients for R6G+ molecules were measured for each solvent condition by a patterned fluorescence correlation spectroscopy experiment33 and used in calculation of both REW and RBND. RBND (eq 18) can be calculated if the adsorption equilibrium constant, Keq, is known from separate experiments; for this work, Keq was measured by using the magnitude of the fluorescence fluctuations to determine the number of molecules at the interface.29 To calculate equilibrium values of A and C, the value for the available surface site density, B, was required. This was obtained by measuring the adsorption isotherm of R6G+ in a surface-dosing experiment.34 The maximum coverage obtainable at the C-18 interface was 1.1 ( 0.1 × 10-6 mol/m2, corresponding to 6.6 × 1013 sites/cm2. At the low solution concentrations of R6G+ used in this experiment, adsorbate coverages ranged from 1.5 × 10-9 to 2 × 10-6 of a monolayer. Because coverages were so low, the value of B was set at its maximum value for all calculations. Data fitting required calculation of the function in eq 16, in which a complementary error function containing a complex argument must be evaluated. This can be accomplished by evaluating the following summation in which the entire w function (33) Hansen, R. L.; Zhu, X.-R.; Harris, J. M. Anal. Chem. 1998, 70, 1281-1287. (34) McCain, K. S.; Hansen, R. L.; Ladaa, T.; Harris, J. M., to be published.

4252 Analytical Chemistry, Vol. 70, No. 20, October 15, 1998

is estimated:

w(z) ) e-z erfc(-iz) ≈ 2

1

∞

∑

πin)-∞

he-tn

1 - (-1)ne-πi(R-z)/h

2

tn - z

(27)

Note that z in eq 27 is a complex number. Equation 27 derives from an application of the sampling theorem in which the sampling grid is defined by tn ) R + nh, where R is an arbitrary offset to the sampling period h ranging over integers n.32 Application of eq 20 requires normalization of the autocorrelated data to the form of eq 2, which in turn requires accurate determination of both the blank contribution and the mean fluorescence intensity squared, which establishes the baseline of the observed autocorrelation (see above). To account for the amplitude of the autocorrelation, the value of ∫I(r) d2r in the denominator of eq 20 must be known. This is the intensity weighted area of the elliptical-Gaussian, internal-reflection excitation spot, which can be calculated from the e-2 intensity bounds of the ellipse.29 To account for uncertainty in these variables, the data were fit by allowing both the denominator of eq 20 (correlation amplitude) and the baseline to float as the error space was searched for the optimum reaction rate. Estimated values of the denominator from the measured spot size of the laser beam were compared with the optimum fitted values of the amplitude and found to agree typically within a factor of 2. The quality of the fit achieved when the amplitude was floated during the fit was significantly improved compared to when an a priori estimate of the denominator was used. Because both the offset and amplitude are linear terms in eq 20, they can be optimized by linear least squares. A series of reaction rates, RR, were fed to a program compiled in FORTRAN to calculate G(t); the program held the other rates, RBND and REW, at fixed a priori values. While carrying out a nonlinear least-squares fit of the shape of the kinetic curve by varying RR, the best values of the amplitude and offset were calculated by a matrix linear least-squares technique.35 Optimum reaction rates were determined by plotting χ2 achieved by each fit against the reaction rate used in the fit. The relative standard deviations for rates measured under all solvent conditions averaged ∼30%. All results were combined to calculate a pooled standard deviation for the reproducibly of a measured reaction (35) Draper, N. R.; Smith, H. Applied Regression Analysis, 2nd ed.; Wiley: New York, 1981; Chapter 2.

rate; 90% confidence bounds were then calculated based on the number of observations at each solvent condition (typically 5). RESULTS Autocorrelation data for R6G+ encountering a C-18 modified silica/solution interface was acquired for multiple concentrations of R6G+ in four different water/methanol compositions which also contained 10 mM NaCl. Data for solutions of 5:95, 20:80, and 40:60 water/methanol (by volume) were taken using 30 mW of laser energy into an internally reflected area of 1.14 × 10-4 cm2 with p-polarization. This produced an average power density of ∼735 W/cm2.29 The data for 50:50 methanol/water solutions were taken at lower power densities to avoid complications from photobleaching due to the longer residence time of the probe at the interface; the incident power was 2.5 mW into an internally reflected area of 9.03 × 10-5 cm2, which produced an average power density of 77.5 W/cm2. The photodestruction quantum yield of R6G+ ranges from 5.7 × 10-7 in ethanol to a value that is 30 times greater in water, 1.9 × 10-5;36 in water/methanol solutions, the dye presumably exhibits a photodestruction yield intermediate between the values in water and ethanol. From the molar absorptivity of R6G+36 at 514.5 nm and using the greater photodestruction quantum yield in water, an upper limit to the photobleaching rate of the dye at the interface is estimated to be 10 and 1.0 s-1 for power densities of 735 and 77.5 W/cm2, respectively. These rates are slower by 1 order of magnitude than any of the rates determined from the data, under their respective excitation conditions. Figure 1 shows example data and resulting fits to eq 20 for each of the solvent conditions. The data were fit by varying the reaction rate while optimizing the amplitude and offset by a linear least-squares step; both the bulk normal diffusion rate and evanescent wave relaxation rate were fixed at their a priori values (see above). Table 1 summarizes the parameters used in fitting the data and the resulting reaction rates, RR. The time scale of the data plotted in Figure 1 varies over a >2 order of magnitude range, from conditions of low Keq at 5% water to much more highly retained conditions at 50% water. Reaction rates of 4200 ( 1100, 3400 ( 710, 655 ( 140, and 44 ( 9 s-1 were determined for the 5:95, 20:80, 40:60, and 50:50 water/methanol solutions, respectively; error bounds represent the 90% confidence limits estimated from replicates. For all solvent conditions, there was no detectable variation in the reaction rate with changes in the R6G+ concentration. The reaction rate is dependent on both the adsorption and desorption rates and the concentration of R6G+ in solution as described by eq 17. Because the concentrations of R6G+ used were very small, the reaction rate is dominated by kdes. It is, however, possible to calculate the adsorption rate constant if the adsorption equilibrium constant, Keq, is known. For this work, Keq was measured by using the magnitude of the fluorescence fluctuations to determine the number of molecules at the interface from the Poisson statistics that govern the fluctuations.29 Based on eq 5, the adsorption rate constant, kads, is simply the product of Keq and kdes. Adsorption rate constants of 1.4(( 2) × 104, 7.5(( 2) × 104, 2.0(( 0.4) × 105, and 8.5(( 2) × 104 M-1 s-1 were (36) Soper, S. A.; Nutter, H. L.; Keller, R. A.; Davis, L. M.; Shera, E. B. Photochem. Photobiol. 1993, 57, 972-977.

Figure 1. Experimental data and resulting fit to eq 20 for R6G+ population fluctuations at a C-18 modified silica surface. From top to bottom, 5:95, 20:80, 40:60, and 50:50 water/methanol solutions which also contained 10 mM NaCl. The fitted values of the reaction rate for these curves, RR are 5300, 3600, 685, and 40 s-1, respectively.

thus estimated for 5:95, 20:80, 40:60, and 50:50 water/methanol ratios, respectively. The 90% confidence limits reported with the adsorption rates reflect errors associated with both the desorption rate and adsorption equilibrium constant. In Figure 2, both the adsorption and desorption rates are plotted as a function of percentage water in the methanol/water solutions. The data in Figure 2 indicate that as the percentage of water in the solutions increased, the desorption rate decreased corresponding to longer residence times on the surface and greater surface retention (Keq values). It can be seen that changes in the desorption rate constants alone, however, are not able to account for the variation in Keq with solvent composition; a significant increase in the adsorption rate with increasing fraction of water in the solution is also observed, up to a volume fraction of 40%. This trend is reversed slightly with 50% water solutions, where a rate constant that is comparable to the 20% water condition was measured. DISCUSSION Range of TIRFCS Applications. The ideal situation for measurement of adsorption and desorption rates by internalreflection fluorescence correlations would have fast diffusional Analytical Chemistry, Vol. 70, No. 20, October 15, 1998

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Figure 2. Adsorption (triangles) and desorption (circles) rate constants plotted as a function of water percentage in methanol solutions (by volume). Changes in both adsorption and desorption rate constants with changing solvent composition are detected.

transport to the surface, slow reaction kinetics, and few solution molecules within the evanescent wave compared with those adsorbed to the surface. These conditions are unfortunately mutually exclusive. At the lower retention conditions, the rate of surface molecule exchange by transport from solution is fast because the distance into solution that must be integrated to find the same number of molecules as on the surface is small. For example, at 5% water in methanol RBND is 4 × 107 s-1. Because RBND is fast, this would appear to be the ideal situation for measuring adsorption and desorption kinetics. However, these conditions lead to significant interference from the exchange of molecules in the evanescent wave volume. In the present experiment using a fused-silica/aqueous solution interface, the depth of penetration of the evanescent wave at a 72° incident angle averaged ∼1100 Å. At 5% water where the adsorption equilibrium is weak, 97% of the molecules at the interface are in the solution volume defined by the evanescent wave. The small (3%) fraction of sorbed molecules implies that the correlation amplitude is dominated by the exchange of solution-phase molecules and that the reaction rate influences only a small fraction the total correlation signal. Although these conditions are a good test of the theory describing the evanescent wave relaxation (which had no fitted parameters), the error in determining the reaction rate, RR, was significant, and occasionally the fit of a correlation transient would be insensitive to the reaction rate parameter under these conditions. As the water percentage increased, the fraction of surfaceadsorbed species observed at the interface increased significantly. At 50% water conditions, the fraction of adsorbed molecules exceeds 90%, which means that their exchange dominates the autocorrelation transient and evanescent wave relaxation is not a problem. Unfortunately, at this high surface retention, transport of molecules from solution to exchange with those on the surface is very slow. This slow transport is reflected in the small RBND ) 55 s-1. This rate of bulk normal diffusion places a rather stringent upper bound on detectable adsorption/desorption kinetics; however, for R6G+ at a C-18 interface, the reaction rate (dominated by the desorption rate) of 44 ( 9 s-1 was still measurable. At higher fractions of water in the solution phase, which produced even greater retention, the desorption kinetics become obscured by very slow transport to the surface. While slower 4254 Analytical Chemistry, Vol. 70, No. 20, October 15, 1998

desorption rates would be observed with increasing retention, the bulk normal diffusion rate scales with the inverse square of the adsorption equilibrium constant (see eq 18) at low coverages where competition for surface sites is not a problem. This higherorder dependence on the adsorption equilibrium constant accounts for the rapid falloff in RBND with increasing Keq as listed in Table 1, compared to the relatively slow decrease in the desorption rate; some of the increase in Keq is also carried by increases in the adsorption rate which reduces the rate of change of the desorption rate. Thus, the TIRFCS method has a limited range of equilibrium constants at which adsorption/desorption kinetics can be reliably measured. At low surface retention, evanescent wave relaxation over submicrometer distances dominates the correlation transient. At large values of the adsorption equilibrium constant, the adsorbate concentration on the surface is large compared to its concentration in solution, which leads to large diffusion distances (>2 µm) a slow rate of exchange between the two populations limited by diffusion through a large distance into solution. Adsorption/Desorption Kinetics at C-18 Modified Silica Surfaces. The results of this TIRFCS experiment can be compared with those of related experiments, in which adsorption and desorption kinetics at alkyl chain modified silica surfaces were measured by perturbation/relaxation kinetic techniques. Adsorption and desorption kinetics were measured at a C-1 modified surface5,7 and C-4 surface6 by a temperature jump method, while only adsorption kinetics were measured at a C-18 surface by pressure2,3 and temperature jump4 experiments. The probes used in these previous studies were anionic and neutral molecules smaller than R6G+, 1-anilino-8-naphthalenesulfonate (ANS-) and its neutral analogue N-phenyl-1-naphthylamine (1-NPN). The temperature jump experiments were of two kinds: a Joule-heating experiment, where silica was heated by passing an electrical current through the solution surrounding the silica,4-6 and a laser temperature jump experiment used with a silica colloidal dispersion.7 Because Joule heating required electrical current to be conducted through the sample, electrolyte concentrations were high so that the rate of heating was fast. Typically, 0.4 M NaCl was added as supporting electrolyte, which was shown to reduce contributions to the adsorption and desorption rates from kinetically slow adsorption sites at a C-1 surface.7 This agreed with chromatographic evidence where peaks were distorted and eluted more slowly when no electrolyte was added to the mobile phase; this was attributed to slower adsorption and desorption kinetics under the influence of ionic interactions at the surface. Desorption rate constants for the C-1 surface in the presence of electrolyte were slightly larger than those presented here for a comparable chromatographic capacity factor (1.1(( 0.1) × 104 s-1 at k′ ) 2.95). However, similar to the data in Figure 2, increasing the capacity factor for ANS- on the C-1 surface decreased the observed desorption rate constant. Desorption data on a C-1 surface in the absence of electrolyte was measured by the laser temperature jump experiment and found to contain two characteristic rates. Desorption rate constants for the fast step were similar to those reported in the Joule-heating experiment. However, a second slower relaxation was also measured (2.3(( 0.3) × 103 s-1 for an estimated chromatographic capacity factor, k′ ) 82).7 This value is larger than reported here but nevertheless

shows that several types of adsorption sites may occur at the interface. The desorption rate constant for ANS- at a C-4 modified silica surface (1.1 × 106 s-1 at k′ ) 20.5) was 2 orders of magnitude greater than the C-1 case and the C-18 data reported here under conditions of similar retention. Adsorption and desorption kinetics at a C-4 interface appear to be unique, but the results also carry considerable uncertainty due to the biexponential form of the relaxation and the similarity of the two relaxation rates. The kinetics on a C-4 surface should differ from C-1 surfaces since solutes can partially partition into the alkyl chains, but the chain length is typically not long enough to fully solvate the molecule.6 C-18 desorption rates were not calculated from the temperature jump data but would appear to be slightly faster than those reported here for corresponding capacity factors. Adsorption rate constants of ANS- in the temperature and pressure jump experiments are typically faster by several orders of magnitude than those observed in the present experiment. Two notable exceptions, however, should be pointed out. In the pressure jump experiment, an adsorption rate constant of 2.2(( 0.7) × 106 M-1 s-1 was measured on a C-18 phase of low carbon content.3 This phase was shown to be inefficient because of its heterogeneous nature. Second, the slow rate constant for adsorption onto a C-1 surface measured in the laser temperature jump experiment was 5(( 2) × 105 M-1 s-1.7 The apparent discrepancy in the adsorption rate constants may arise from differences in the experimental approach and/or chemical systems studied. In the present experiment, the measured reaction rate, RR, was not sensitive to the adsorption rate. While the reaction rate expression contains both adsorption and desorption rate constants, the experiment was run under conditions of very low solution R6G+ concentrations and the relaxations observed as fluorescence fluctuations were dominated by the desorption step, as discussed above. As a check of this assumption, it was noted that the measured desorption rate constants did not have any detectable R6G+ concentration dependence. The adsorption rate constant was determined from the measured desorption rate and the equilibrium constants29 using eq 5. In contrast, much higher solution concentrations of probe were used in the temperature and pressure jump experiments, which made the relaxation dominated by the adsorption step. To calculate adsorption rate constants in the temperature and pressure jump experiments, a plot of inverse relaxation rates against solution concentrations of dye was linear where the slope equals the adsorption rate constant; available surface sites were kept in large excess of the dye concentration and were assumed to be pseudoconstant. Unfortunately, the kinetically accessible concentration of surface sites is an uncertain quantity for a porous packed bed where diffusive transport through the irregular geometry of pores can influence the number of surface sites that are available to react on a microsecond time scale. The large difference in the apparent adsorption rates between these two experiments raises the question of whether the same chemical process is being observed. It is possible that the “fast adsorption” step measured in the temperature and pressure jump experiments is not necessarily a rate-limiting step responsible for the adsorption equilibrium constant. A fast initial adsorption event

followed by a significantly slower (and therefore equilibriumdetermining) step that was not distinguished spectroscopically could account for such an apparent discrepancy in rates. Because the autocorrelation experiment measured only desorption rates directly and inferred adsorption rates from the equilibrium constant, only a slower equilibrium-determining step would be reported. The molecular origins of two steps is speculative at this point; the initial event could be a weak adsorption to the C-18/ solution interface (which was observed on the microsecond T-jump time scale). This could be followed by a slow reorganization of alkyl ligands on a millisecond time scale, which stabilizes the adsorbed state and determines the equilibrium constant. A second contribution to the adsorption rate discrepancy may arise from differences in the chemical systems being studied. The fact that the adsorption rate constants were slower in the present experiment may be indicative of selective interactions between R6G+ and a small population of stronger adsorption sites on the substrate related to residual silanols (recall, only a small (e2 × 10-6) fraction of available surface sites are occupied in the present experiments, compared to T-jump experiments4 where a greater fraction (10-3) were occupied). This idea is further supported by the fact that slow adsorption was reported for both a heterogeneous C-18 phase3 and adsorption onto a C-1 surface in the absence of electrolyte.7 In a separate study, electrostatic and chromatographic contributions to the free energy of adsorption have been examined.29 It was found that, up to 40% water in methanol with 10 mM NaCl, the predicted electrostatic contribution to adsorption was greater than or of similar magnitude as nonpolar interactions. This supports the notion that adsorption and desorption kinetics measured in this experiment may have a large component from interactions with residual silanols or silanolates on the surface. Other authors have shown that interactions of cations at a modified silica interface arise from multiple contributions including electrostatics and ion exchange.14,21,22 An attempt was made to measure adsorption/ desorption kinetics at a C-18 interface under conditions of lower NaCl concentrations and at a bare silica surface. The kinetic data acquired under these conditions do not fit theory well, which may be due to excessive dispersion in adsorption site energies. Analogous to the C-1 data previously published,5 as the desorption rate constants decreased, there was a corresponding increase in the adsorption rate constant up to 40% water, corresponding to a Keq value of 298. This implies that changes in the desorption rate alone are not able to account for changes in the retention equilibrium. Barriers to adsorption, therefore, influence the equilibrium for retention of these species on the surface. Summary. Adsorption and desorption rates of R6G+ at a C-18 modified silica surface were measured with fluorescence correlation spectroscopy technique based on fluctuations in populations of molecules at the interface. To account for relaxation kinetics measured over a wide range of the adsorption equilibrium constant, the full equation describing the observed autocorrelation was applied, which included both desorption kinetics and the exchange of solution-phase molecules in the evanescent wave. It was found that desorption rates decreased with increasing retention at the interface while the adsorption rates increased with increasing retention up to 40% water in methanol. This indicates that changes in the desorption rate alone are not able to account Analytical Chemistry, Vol. 70, No. 20, October 15, 1998

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for the observed changes in retention of the dye at the interface; a detectable barrier to adsorption is also present. The observed desorption rates in this experiment were slightly slower than those observed in a previous work. Adsorption rates were significantly slower than previously reported with the exception of cases of low surface carbon content or low ionic strength. This may indicate that a second slow adsorption step controls the equilibrium or that the adsorption/desorption kinetics are influenced by interactions at a small population of kinetically slower sites. Desorption kinetics of R6G+ from bare silica surfaces or C-18 surfaces with solution electrolyte concentrations lower than 10 mM could not be fit to a homogeneous desorption kinetics model and may indicate that a dispersion in rates exists in these cases due to inhomogeneous surface interactions.

4256 Analytical Chemistry, Vol. 70, No. 20, October 15, 1998

ACKNOWLEDGMENT This work was supported in part by the National Science Foundation under Grant CHE95-10312. Fellowship support for R.L.H., provided by the American Chemical Society Division of Analytical Chemistry (sponsored by the Society of Analytical Chemists of Pittsburgh) and by Pfizer, Inc. is gratefully acknowledged. The authors thank Nancy Thompson for helpful correspondence.

Received for review August 17, 1998. Accepted August 18, 1998. AC980925L