Fluorescence Correlation Spectroscopy with Patterned

herent laser beams creates an excitation fringe pattern from which fluorescence emission is monitored. Spon- taneous concentration fluctuations of flu...
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Anal. Chem. 1998, 70, 1281-1287

Fluorescence Correlation Spectroscopy with Patterned Photoexcitation for Measuring Solution Diffusion Coefficients of Robust Fluorophores Richard L. Hansen, X. Ron Zhu,† and Joel M. Harris*

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

Patterned fluorescence correlation spectroscopy is developed as a new technique for measuring diffusion coefficients of photostable fluorescent probe molecules. In this method, interference between two intersecting, coherent laser beams creates an excitation fringe pattern from which fluorescence emission is monitored. Spontaneous concentration fluctuations of fluorescent molecules within the excitation volume are detected as excess noise on a fluorescence transient; concentration fluctuations are driven primarily by diffusion of these molecules between interference fringes although contributions from photobleaching and diffusion over the entire pattern dimensions can also be observed. Autocorrelation of the fluorescence transient allows analysis of the temporal characteristics of the fluctuations, which were used to determine solution diffusion coefficients; the method was applied to study the diffusion of Rhodamine 6G (R6G) in water/methanol solutions containing added electrolyte and in pure ethanol. The method can be used to characterize the diffusive transport of fluorescently labeled species, which is an important issue in designing smallvolume detection experiments. Recent developments in high-sensitivity fluorescence detection at or near the single-molecule level have generally relied on excitation of fluorescence from very photostable chromophores. Examples include detecting molecules in a flowing stream,1-6 molecules diffusing in free solution,7-12 molecules being electro† Radiation Oncology, St. Luke’s Medical Center, P.O. Box 2901, Milwaukee, WI 53201-2901. (1) Nguyen, D. C.; Keller, R. A.; Jett, J. H.; Martin, J. C. Anal. Chem. 1987, 59, 2158-2161. (2) Castro, A.; Fairfield, F. R.; Shera, E. B. Anal. Chem. 1993, 65, 849-852. (3) Shera, E. B.; Seitzinger, N. K.; Davis, L. M.; Keller, R. A.; Soper, S. A. Chem. Phys. Lett. 1990, 174, 553-557. (4) Soper, S. A.; Shera, E. B.; Martin, J. C.; Jett, J. H.; Hahn, J. H.; Nutter, H. L.; Keller, R. A. Anal. Chem. 1991, 63, 432-437. (5) Soper, S. A.; Davis, L. M.; Shera, E. B. J. Opt. Soc. Am. B 1992, 9, 17611769. (6) Soper, S. A.; Mattingly, Q. L.; Vegunta, P. Anal. Chem. 1993, 65, 740-747. (7) Nie, S.; Chiu, D. T.; Zare, R. N. Science 1994, 266, 1018-1021. (8) Nie, S.; Chiu, D. T.; Zare, R. N. Anal. Chem. 1995, 67, 2849-2857. (9) Zander, C.; Sauer, M.; Drexhage, K. H.; Ko, D.-S.; Schulz, A.; Wolfrum, J.; Brand, L.; Eggeling, C.; Seidel, C. A. M. Appl. Phys. B 1996, 63, 517-523. (10) Xu, X.-H.; Yeung, E. S. Science 1997, 275, 1106-1109. (11) Rigler, R.; Mets, U ¨ . SPIE Proc.-Soc. Int. Opt. Eng. 1992, 1921, 239-248.

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© 1998 American Chemical Society

phoretically moved through an observation volume,13-16 or molecules trapped in levitated microdroplets.17,18 A general strategy of these experiments has been to lower probed volumes to the femtoliter regime or below to reduce extraneous background from solvent Raman scattering or fluorescence emission from solution impurities. Small probe volumes also facilitate sample handling and preparation because dye concentrations as high as several nanomolar can yield on average one molecule within these probe volumes. Molecular diffusion plays an important role in the design of small-volume detection experiments. Small probe volumes are traversed rapidly by molecules diffusing freely in solution; a small molecule in a low-viscosity solvent diffuses 1 µm in ∼1 ms, assuming a diffusion coefficient of 3 × 10-6 cm2 s-1. The number of photons detected per molecule depends on the residence time of the fluorophore in the observation region,19 which can be limited by diffusion out of the experimental detection volume; molecules can also diffuse into and out of the probe volume during an experiment, creating short dark periods bounded by bursts of photons.8 Thus, knowledge of diffusive transport of analyte molecules is important in the design of small-volume detection experiments. Measurements of diffusion of fluorescently labeled biomolecules can also be useful in understanding their effective size in solution and changes in tertiary structure. Our own interest in the diffusion of robust fluorescent probes is in modeling the solution transport contribution to adsorption/desorption kinetics at liquid/solid interfaces. Solution-phase diffusion coefficients can be measured by several classes of experiments. For example, diffusion coefficients can be measured electrochemically if the species is electroactive20 or by observing diffusive transport over macroscopic distances (12) Mets, U ¨ .; Rigler, R. J. Fluoresc. 1994, 4, 259-264. (13) Lee, Y.-H.; Maus, R. G.; Smith, B. W.; Wineforder, J. D. Anal. Chem. 1994, 66, 4142-4149. (14) Castro, A.; Shera, E. B. Anal. Chem. 1995, 67, 3181-3186. (15) Chen, D.; Dovichi, N. J. Anal. Chem. 1996, 68, 690-696. (16) Fister, J. C., III.; Jacobson, S. C.; Davis, L. M.; Ramsey, M. J. Anal. Chem. 1998, 70, 431-437. (17) Ng, K. C.; Whitten, W. B.; Arnold, S.; Ramsey, J. M. Anal. Chem. 1992, 64, 2914-2919. (18) Barnes, M. D.; Ng, K. C.; Whitten, W. B.; Ramsey, J. M. Anal. Chem. 1993, 65, 2360-2365. (19) Barnes, M. D.; Whitten, W. B.; Ramsey, J. M. Anal. Chem. 1995, 67, 418A423A. (20) Bard, A. J.; Faulkner, L. R. Electrochemical Methods; Wiley: New York, 1980.

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or through a permeable barrier.21 Optically based methods can also be used to measure fluorophore diffusion. A successful class of optical experiments utilize patterned illumination profiles. These experiments are efficient because transport within the small dimensions of the pattern can be rapid, and many pattern elements can be observed simultaneously. For example, fluorescence recovery after patterned photobleaching22 and transient holography experiments involving diffraction of a laser beam by an optically induced volume grating23,24 have both been used to measure diffusion coefficients. These optical methods exploit photochemical bleaching as a means of rapidly perturbing a population of molecules to create a periodic concentration gradient; the rate at which the gradient relaxes due to diffusion is detected and used to calculate a diffusion coefficient. Patternbased experiments provide a convenient and accurate means of changing the time scale of the diffusive recovery by changing the characteristic spacing of the pattern. In this way, the observed recoveries can be checked experimentally to ensure that they are entirely based on diffusion and not reversible photobleaching. Since high-sensitivity fluorescence detection experiments generally utilize robust chromophores, rapid photobleaching perturbations can be difficult, requiring very energetic laser pulses or long pulses of lower energy. If the latter approach is taken, diffusion can randomize the induced concentration profile during the perturbation step making the recovery difficult to detect and the data more difficult to interpret. A nonperturbative optical method of measuring solution-phase diffusion coefficients relies on detecting random fluctuations in the population of solute molecules within a defined observation region. The fluctuations depend on processes that can alter the molecular population within the region, such as diffusion, convection, or chemical reaction. Fluorescence correlation spectroscopy (FCS)25,26 is an example of such an experiment in which the population of fluorescent molecules within an observation volume is monitored as a function of time. The time-dependent fluorescence signal contains information about the population fluctuations, which appear as excess noise on the fluorescence transient. Analysis of the fluctuations by autocorrelation allows the magnitude and temporal characteristics of the fluctuations to be related to the size of the population observed and rates of diffusion and chemical reaction, respectively.25 FCS experiments require that population fluctuations be detected when superimposed on the mean population signal, thus establishing an upper bound to the size of the population that is experimentally observable. It is therefore important to collect as many photons per molecule as possible, requiring efficient light collection and often aggressive excitation of fluorescence. Slow photobleaching at high laser fluences can be problematic, establishing a lower bound to experimentally measurable diffusion or reaction rates. One approach to solving this problem is to flow (21) Bidstrup, D. E.; Geankoplis, C. J. J. Chem. Eng. Data 1963, 8, 170-173. (22) Davoust, J.; Devaux, P. F.; Leger, L. EMBO J. 1982, 1, 1233-1238. (23) Zulli, S. L.; Kovaleski, J. M.; Zhu, X. R.; Harris, J. M.; Wirth, M. J. Anal. Chem. 1994, 66, 1708-1712. (24) Terazima, M.; Okazaki, T.; Hirota, N. J. Photochem. Photobiol. 1995, 92, 7-12. (25) (a) Elson, E. L.; Magde, D. Biopolymers 1974, 13, 1-27. (b) Magde, D.; Elson, E. L.; Webb, W. W. Biopolymers 1974, 13, 29-61. (26) Ehrenberg, M.; Rigler, R. Chem. Phys. 1974, 4, 390-401.

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or sweep the sample through the excitation beam.27-29 Fresh fluorophores can be moved into the excitation region faster than can be accomplished by diffusion. Alternatively, the excitation region can be swept through a stationary sample.30 The later approach has recently been demonstrated by utilizing a moving interference pattern, created by the intersection of two coherent laser beams.31 This experiment measured diffusion of polystyrene spheres containing a fluorescent label suspended in water/glycerol solutions. In the present work, a simpler patterned excitation FCS experiment is described in which a stationary interference pattern is used for fluorescence excitation. Although photobleaching can still contribute to measured population fluctuations, patterned excitation allows the time scale of the experiment to be determined by pattern spacing, which provides an experimentally straightforward means to separate effects of diffusion from photobleaching or chemical reaction. THEORY Several processes driving population fluctuations, such as photobleaching and diffusion, may simultaneously occur in a patterned FCS experiment and contribute to the observed autocorrelation. If the processes occur at different rates, their contributions can be evaluated if one of the rates is independently variable. Population fluctuations arising from diffusion are dependent on the geometry of the excitation volume; the time required for diffusion over a characteristic distance scales with the distance squared. Excitation with an interference pattern has several potential advantages over traditional FCS experiments, which use a focused laser beam to define the excitation volume. Interference patterns created by the intersection of two coherent laser beams have easily characterized fringe spacings, d, which depend on the wavelength of the laser line, λo, and the angle at which the beams intersect, θ, as described by eq 1.33 It is

d ) λo/(2 sinθ/2)

(1)

experimentally simple to change the intersection angle, thus changing the fringe spacing to provide a convenient internal check that the observed autocorrelation relaxation is dominated by diffusion within the fringes. Fluctuations are monitored by collecting a fluorescence transient, F(t), from fluorophores diffusing in the excitation pattern; population fluctuations are observable as excess noise superimposed on the fluorescence transient. The transient will contain both fluorescence emission from the solution fluorophores and a contribution from background Raman scatter. In the present experiment, the number of solvent molecules that can Raman scatter are ∼1012 times more concentrated than the dye molecules of interest, but fortunately, Raman scattering is ∼1014 times less (27) Peterson, N. O.; Johnson, D. C.; Schlesinger, M. J. Biophys. J. 1986, 49, 817-820. (28) Palmer, A. G., III.; Thompson, N. L. Biophys. J. 1987, 51, 339-343. (29) Magde, D.; Webb, W. W.; Elson, E. L. Biopolymers 1978, 17, 361-376. (30) Meyer, T.; Shindler, H. Biophys. J. 1988, 54, 983-993. (31) Hattori, M.; Shimizu, H.; Yokoyama, H. Rev. Sci. Instrum. 1996, 67, 40644071. (32) Aragon, S. R.; Pecora, R. J. Chem. Phys. 1976, 64, 1791-1803. (33) Zhu, X. R.; McGraw, D. J.; Harris, J. M. Anal. Chem. 1992, 64, 710A719A.

efficient than fluorescence from a good fluorophore. The transient is autocorrelated so that the temporal dependence of the fluctuations can be examined.

1 Tf∞2T

G(τ) ) lim

∫

〈δF(t) δF(t + τ)〉 ) A2π3

T

F(t)F(t + τ) dt ) 〈F(t)F(t + τ)〉 ) -T 〈F(t)〉2 + 〈δF(t) δF(t + τ)〉 (2)

The brackets in eq 2 denote an ensemble average. The autocorrelation describes the time dependence of fluctuations around a mean fluorescence intensity and is the sum of two components: 〈F(t)〉2 is the mean fluorescence intensity squared, and 〈δF(t) δF(t + τ)〉, in which δF(t) ) F(t) - 〈F(t)〉, describes the time-dependent deviations from the mean. The illumination profile in solution consists of two interfering laser beams of much smaller spot size than their propagation distance through the sample cuvette. Contributions from diffusion along the beam axis are therefore much slower than across the beam, normal to the propagation axis. Two dimensions are initially considered here, but with the approximations made below, only one dimension (across the interference pattern) is critical to short-term diffusional fluctuations. The fluorescence intensity is dependent on the local concentration of fluorophores in solution, C(x,y,t), the excitation intensity profile, p(x,y), and a term, A, which incorporates excitation and collection efficiencies.

F(t) ) A

∫ p(x,y) C(x,y,t) dx dy

variables, solving the time-dependent integrals, and then inverse Fourier transforming the solution:31

(3)

(

)

1 4Dτ 1+ 2 2 ω

-1

((

)

Po2 2 4Dτ ω [C] 1 + 2 4 ω

(

exp -4π2D

(

-1

+

) ))

τ 4Dτ 1+ 2 d2 ω

-1

This solution is valid only if the number of fringes in the excitation profile is large (>10). Equation 7 contains terms describing diffusion within the overall Gaussian intensity profile25 and a singleexponential relaxation describing diffusion across the interference fringes. The above solution neglects contributions to the autocorrelation from slow photobleaching of the fluorophores. Although diffusion over the entire spot and slow photobleaching may contribute to the observed autocorrelation decay over long times, diffusion within the fringe pattern will dominate the decay at short times when the pattern dimensions are much smaller than the entire excitation volume and when moderate laser intensities are used to control excess photobleaching. These slower rates can be predicted by knowing the spot size and power densities of laser beams used to form the excitation interference pattern, and the photobleaching yield of the dye. The time constant for diffusion over the entire excitation region, τω, is dependent on the e-2 spot size of the interfering laser beams, ω, and the solution diffusion coefficient, D.25,34

tω ) ω2/4D Because the measured fluorescence fluctuations derive from local concentration fluctuations, it is necessary to relate the fluorescence autocorrelation to solution concentration deviations.

〈δF(t) δF(t + τ)〉 ) A2

∫ ∫ p(x,y)p(x′,y′) 〈δC(x,y,t) δC(x′,y′,t + τ)〉 dx dy dx′ dy′

(7)

(8)

The photobleaching lifetime, τPB, is dependent on the intensity of the laser radiation, I, the absorption cross section of the fluorophore, σ, and the photodestruction quantum yield, φPB:

τPB ) 1/[IσφPB]

(9)

(4) where δC(x,y,t) ) C(x,y,t) - 〈C(x,y,t)〉. To solve eq 4, the excitation profile must be known, along with the time dependence of local deviations in the solution concentration which depend on diffusion. The excitation profile is a stationary interference fringe superimposed onto a Gaussian excitation profile:

p(x,y) )

(

) (

)

Po 2πx 2 1 + cos exp - 2 (x2 + y2) 2 d ω

(5)

The intensity profile is dependent on the fringe spacing, d (eq 1) and the e-2 size of the Gaussian laser beams, ω, interfering to create the fringe pattern of peak intensity Po. Concentraiton fluctuations occur through diffusive transport and are described by Fick’s second law.

∂δ C(x,y,t) ) ∇2 δ C(x,y,t)D ∂t

(6)

where ∇2 is the two-dimension Laplacian. Equation 4 can be solved by Fourier transforming with respect to the spatial

As discussed in more detail below, both of these factors are significantly slower than diffusion within the fringes. On the short time scale in which diffusion between the interference fringes dominates the autocorrelation decay, photobleaching and wholespot diffusion are very slow processes, and their effect on the measured autocorrelation is well described by a sloping baseline that represents the linear term of a Taylor series expansion of their time dependence. Under this approximation, the autocorrelation data can be fit to eq 10, which consists of a singleexponential relaxation describing the diffusional relaxation within the fringes, an offset term corresponding to the squared mean fluorescence signal, 〈F(t)〉 2, and a sloping baseline that captures any photobleaching or whole-spot diffusion:

G(τ) ) R exp(-τ/τd) + mτ + β

(10)

where τd ) d2/4π2D. To confirm that linearization of the small contribution from photobleaching and whole-spot diffusion is justified, data can be (34) Thompson, N. L. Topics in Fluorescence Spectroscopy, Volume 1: Techniques; Lakowicz, J. R., Ed.; Plenum Press: New York, 1991.

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Figure 1. Patterned fluorescence correlation spectroscopy instrument used to measure diffusion coefficients of Rhodamine 6G. See text for description.

taken at several fringe spacings and examined based on the time constant of the exponential in eq 7. The plot of inverse time constants (1/τd) from the exponential decay versus inverse fringe spacings squared (1/d2) should be linear, since (1/τd) ) 4π2D/ d2 as shown in eq 7. The intercept of this plot estimates the autocorrelation decay rate at infinite fringe spacings. If other processes contribute significantly to the observed autocorrelation decay, there will be a nonzero intercept indicating that a baseline correction for photobleaching and whole-spot diffusion is not justified. EXPERIMENTAL SECTION Rhodamine 6G (R6G) solutions were made by diluting a stock solution of R6G (Exciton Rhodamine 590) dissolved in methanol into an appropriate water/methanol solution; 5 µL of 2.54 × 10-7 M stock were diluted into 50 mL, creating solutions that were ∼2.5 × 10-11 M in R6G. Water/methanol solutions were prepared by volume percent and consisted of 5:95, 20:80, 40:60, 50:50, and 60:40 water/methanol. The solutions also contained 10 mM NaCl, with the exception of two 60% water solutions containing 1 and 100 mM NaCl. The viscosity of the water/methanol solutions was measured with a no. 25 Ostwald viscometer calibrated with water and methanol at 23 °C. R6G was also added to 50 mL of ethanol containing no added electrolyte via a 5-µL aliquot of the stock R6G in methanol. Following equilibration with the laboratory air temperature of 23 °C, the samples were then placed in a 1-cm cuvette for experiments. A holographic grating was created by interfering two coherent laser beams within the sample as shown in Figure 1. A single beam from an argon ion laser operating at 514.5 nm was weakly focused with 3-m focal length lens L1 and passed through PellinBrocca prisms P1 and P2 to remove any incoherent plasma lines. The resulting beam was propagated through Ronchi Ruling R (Diffraction International) of period 25, 40, or 67 µm. The ( one diffraction orders were allowed to pass through aperture A1 but 1284 Analytical Chemistry, Vol. 70, No. 7, April 1, 1998

the zero and higher orders were blocked. A focusing lens L3 was used to reimage (cross) and focus the beams to a spot of ∼67 µm e-2 in radius within a cuvette C containing the R6G solution; the lens combination L2 and L3 was chosen such that the diffracted laser beams were brought to a waist at the point of intersection in the sample cuvette. The interference fringes created were 3.02 ( 0.06, 4.86 ( 0.04, and 8.05 ( 0.29 µm for the 25-, 40-, and 67-µm rulings, respectively. The fringe spacing was calculated by determining the angle at which the beams crossed (see eq 1). Although the angle of propagation changes according to Snell’s law at the air/cuvette and cuvette/solvent interface due to differences in refractive index, this is exactly canceled in eq 1 by the change in the wavelength of the light propagating in the new media;35 therefore, if the beams cross the interface at the same incidence angle, eq 1 remains applicable. The spacing of the fringes in air was confirmed by translating a knife edge mounted to a precision translation stage (Newport) through the interference pattern. The light diffracting around the knife edge produced a modulation pattern in the far field as it was translated through the fringes. Translating the knife edge through many fringes within the laser spot allowed the spacing to be calculated and was used to establish error bounds on the reported spacings. Fluorescence from the interference pattern was collected orthogonal to the plane of intersection of the laser beams and therefore in the plane of the interference fringes. Microscope objective MO (Figure 1, inset) of numerical aperture 0.7 with a working distance of 1.44 mm and 60× magnification (Universe Kogaku) was used to collect the fluorescence emission. Variable aperture A2 located in the image plane limited the field of view of the detector; the aperture was set at a 6-mm diameter corresponding to 100 µm in the sample plane, which is slightly smaller than the full laser spot with e-2 radius ω ) 67 µm. A minimum of 12.4 fringes for the largest spacing was observable at the detector. Following the image plane aperture, the light was loosely focused with lens L4 through a long-pass colloid filter F (Schott) onto a photomulitiplier tube PM (Hamamatsu R928). The PMT was kept in a Pelltier cooled housing. The collected emission was passed through a 300-MHz bipolar amplifier followed by a discriminator (Phillips Scientific); photocurrent pulses exceeding a preset threshold were counted by a multichannel scaler (EG&G Turbo MCS). Data were transferred and processed on a Pentium PC. Data were collected in photon counting mode into discrete time bins. An autocorrelation was calculated by Fourier transforming a fluorescence transient, multiplying by its complex conjugate to form a power spectrum, and then inverse Fourier transforming to generate the autocorrelation. Typically, autocorrelations from 100 files of 16 384 bins were averaged together for a single experiment. When fitting the autocorrelations, the first point, which is dominated by photon shot noise, was removed. Shot noise has a bandwidth much greater than that of the molecular fluctuations and had no correlation past the first time bin. The data were fit to eq 10 by a nonlinear least-squares routine using a Marquardt algorithm36 compiled in FORTRAN. (35) Smith, B. A. In New Characterization Techniques for Thin Polymer Films; Tong, H.-M., Nguyen, L. T., Eds.; John Wiley: New York, 1990. (36) Press: W. H.; Flannery, B. P.; Teukolsky, S. A.; Vetterling, W. T. Numerical Recipes; Cambridge: London/New York, 1986.

Figure 2. Autocorrelation of Rhodamine 6G in ethanol diffusing in 4.86-µm fringes. The data are fit to a single-exponential function with a time constant 2.4 ms, which describes diffusion within the fringes. Additionally, a linear baseline is included in the fit which describes slow photobleaching or diffusion over the entire excitation volume.

RESULTS AND DISCUSSION An example autocorrelation of 2.5 × 10-11 M R6G diffusing between 4.86-µm fringes in ethanol is shown in Figure 2 along with the best fit to eq 10 determined by nonlinear optimization. The predicted volume of the probed interference pattern includes the area passed by the image-plane aperture and the field of view. This volume is expected to be ∼1.1 × 10-6 cm3. Given the concentration of dye solution used, the number of molecules producing the autocorrelation in Figure 2 is expected to be 1.6 × 104. FCS experiments have the property that the autocorrelation at zero offset, not including shot noise, is equal to the inverse of the number of molecules in the probe volume25 if the autocorrelation is normalized by subtracting and dividing by the squaredmean fluorescence intensity. Based on the data in Figure 2, the number of molecules observed, predicted by the magnitude of the population fluctuations, is ∼1.9 × 104. The data in Figure 2 are dominated by fluctuations associated with diffusion within the fringes described by a single-exponential decay, but they also contain a component due to photobleaching or transport within the laser spot; photobleaching and diffusion over the laser spot have time constants much longer than diffusion between fringes. The time scale of these contributions can be estimated through eqs 7-9 knowing the excitation intensity, beam intersection angle, observation area, molar absorptivity, and photodestruction quantum yield. In this experiment, all data were taken within the interference pattern created by intersecting two 67-µm e-2 radius coherent laser beams of 25-mW intensity. Predicted contributions to the autocorrelation from diffusion through the entire Gaussian spot and relaxation between the fringes can be compared by calculating the respective recovery time constants (eqs 7 and 8). The ratio of time constants for diffusional recovery through the entire spot and diffusional recovery between the fringes is equal to π2ω2/d2 and describes how much longer whole-spot diffusional recovery takes relative to diffusion in the fringes. In the present experiment, this ratio ranges from 4.9 × 103 with a 67-µm spot size and 3.02-µm fringes to 680 with 8.05-µm fringes. The spot size of the interfering beams produced a peak photon flux of 3.7 × 1021 photons cm-2 s-1 at the central fringe.33 However, during the time course of the experiment, the molecules

Figure 3. Inverse time constants vs inverse fringe spacings squared for diffusion of Rhodamine 6G in ethanol. The slope of the weighted linear fit is equal to 4π2D, and the negligible intercept indicates that no other relaxation is present in the autocorrelation (eq 7). The error bars indicate ( one standard deviation.

can diffuse between the fringes in the pattern, sampling both light and dark regions. A better estimate of the photon flux that a molecule experiences, when it is within the field of view of the detector, is determined by averaging the intensity of the Gaussian laser beams over the area viewed by the detector. An average photon flux of 1.33 × 1021 photons cm-2 s-1 is estimated over this region. The photodestruction yield of R6G ranges from 5.0 × 10-7 in ethanol to 1.9 × 10-5 in water;37 approximately 2 × 106 and 5.3 × 104 photons per molecule on average can be emitted in ethanol and water, respectively. Given the absorption cross section of R6G at 514.5-nm excitation (2.2 × 10-16 cm2),37 the lower limit to the lifetime of the dye in the excitation profile is expected to range from 0.2 s in water to 6.9 s in ethanol, with lifetimes in the mixed water/methanol solutions expected to lie somewhere between these values. These lifetimes range from approximately 33 to 103 times longer than the slowest relaxations measured for diffusion within the interference fringes. Both photobleaching and whole-spot diffusion contributions are therefore not expected to contribute significantly to the observed autocorrelations at short times. A single check that the data are dominated by diffusion within the interference fringes is accomplished by noting that the autocorrelation observed when one of the beams creating the interference pattern is blocked lacks the fast exponential component. A more rigorous check is completed by changing the fringe spacing to alter the time scale of the experiment (see eq 7). A plot of inverse exponential time constants against inverse fringe spacings squared is expected to be linear with a slope of 4π2D. In Figure 3, inverse time constants for diffusion of R6G in ethanol at three fringe spacings are plotted; three autocorrelation time constants were averaged at each fringe spacing and used to estimate dispersion in the rates. The weighted linear fit shown allows examination of the inverse time constant at the intercept, corresponding to an infinite fringe spacing. The best-fit line has an intercept that is indistinguishable from zero at 95% confidence, indicating that the data are adequately described by eq 10. A similar plot for each experimental condition found no significant intercepts, verifying that the autocorrelations were dominated by diffusion within the fringes. (37) Soper, S. A.; Nutter, H. L.; Keller, R. A.; Davis, L. M.; Shera, E. B. Photochem. Photobiol. 1993, 57, 972-977.

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Table 1. Measured Rhodamine 6G Diffusion Coefficients and Solution Viscosities water/methanola

D × 106 (cm2/s)b

η (cP)c

5:95 20:80 40:60 50:50 60:40

4.5 ( 0.6 3.4 ( 0.4 2.8 ( 0.3 2.5 ( 0.2 2.5 ( 0.3

0.771 1.18 1.63 1.73 1.78

a Solution composition by volume fraction water/methanol; each contains 10 mM NaCl added electrolyte. b Error bounds indicate the 95% confidence limits. c Standard deviations for viscosity measurements are estimated to be (5%.

Since the uncertainties in fitted time constants are proportional to their magnitude, the inverse time constants measured at different fringe spacing provide estimates of the diffusion coefficient of equivalent precision (since D ) (d2/4π2τd) and τd is proportional to d2). Each time constant was thus used to calculate a diffusion coefficient; the results were then averaged together, and the variance in diffusion coefficient could be estimated. To test the validity of this approach, diffusion of R6G in ethanol was measured and compared with a value from the literature. A diffusion coefficient of (2.9 ( 0.3) × 10-6 cm2/s was measured for R6G in ethanol. This value is indistinguishable from a result previously reported in the literature, D ) 3 × 10-6.34 Diffusion coefficients of R6G in various water/methanol solutions were measured and are presented in Table 1. Diffusion coefficients decreased with increasing water content until ∼40% water. This trend agrees qualitatively with measured bulk viscosities also reported in Table 1. The data were fit to the Stokes-Einstein equation under “stick” limit conditions:38

D ) RT/6πηaN

(11)

yielding a Stokes radius, a ) 5.2 ( 0.1 Å. When this average radius is used to calculate the diffusion coefficients predicted by eq 11 and the solution viscosities, it is apparent that the change in solvent viscosity does not fully account for the variation in the diffusion coefficient of the dye as seen in Figure 4. This observation is consistent with that of Ibuki and Nakahara for cations diffusing in ethanol/water solutions.39 These authors predicted and measured the friction coefficients for various cations in binary alcohol/water solutions using the Hubbard-Onsager dielectric friction theory.40,41 It was found that the contribution of the dielectric friction to the total friction coefficient (dielectric and viscous) increased with increasing ethanol percentages; this is in part due to the decrease in static dielectric constant of the solutions at high ethanol percentages, which affects screening of the charge on the molecule. As the molecule moves through the solvent, the electric field it produces affects the surrounding solvent molecules, and they must reorient their dipoles accordingly. The larger contribution to the friction coefficient from the (38) Rice, S. A. In Diffusion-Limited Reactions; Bramford, C. H., Tipper, C. F. H., Compton, R. G., Eds.; Comprehensive Chemical Kinetics 25; Elsevier: New York, 1985. (39) Ibuki, K.; Nakahara, M. J. Chem. Phys. 1986, 84, 2776-2782. (40) Hubbard, J.; Onsager, L. J. Chem. Phys. 1977, 67, 4850-4857. (41) Hubbard, J. B. J. Chem. Phys. 1978, 68, 1649-1664.

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Figure 4. Diffusion coefficients for Rhodamine 6G vs volume percent water in water/methanol solutions each containing 10 mM NaCl. The error bounds indicate the 95% confidence intervals. The data were used to calculate an average Stokes radius of 5.2 ( 0.1 Å, which was used to recalculate expected diffusion coefficients for measured viscosities (solid line). It is apparent that a single Stokes radius cannot entirely explain the dependence of the diffusion coefficient on the solvent composition.

dielectric friction term would increase the apparent Stokes radius (slower diffusion) at higher alcohol percentages. Rhodamine 6G diffusion was also measured for several NaCl concentrations in 60:40 water/methanol solution. It was found that the diffusion coefficient was unchanged within the 95% confidence intervals from 1 to 100 mM NaCl. Diffusion coefficients of (2.4 ( 0.1) × 10-6, (2.5 ( 0.3) × 10-6, and (2.3 ( 0.1) × 10-6 cm2/s were measured for 1, 10, and 100 mM NaCl in 60: 40 water: methanol/solution. CONCLUSION A new method to measure solution diffusion coefficients of photochemically stable fluorophores has been developed. The method utilizes fluorescence correlation spectroscopy with photoexcitation provided by an interference pattern. Autocorrelations were calculated from fluorescence transients and fit to a singleexponential relaxation on a sloping baseline. Fluctuations arising from diffusion within the interference fringes are described by the exponential decay, and the sloping baseline accounts for slow photobleaching during the experiment and diffusion over the beam volume. The decay rate of the exponential relaxation was determined for autocorrelations taken at several fringe spacings; plotting this decay rate against the inverse square of the fringe spacing allows a test of the diffusive-transport model used to fit the data. The linearity of these plots and their zero intercepts confirm that diffusion between fringes accounts for the measured decay rate of the correlations, and no other kinetics contribute to the exponential relaxation. The diffusion coefficient measured for R6G in ethanol by this experiment agreed with a previously published result. Diffusion coefficients of R6G in water/methanol solutions decreased with increasing water content until ∼40% water by volume. This trend follows changes in viscosity for water/methanol solutions; a small but significant change in the apparent Stokes radius of the dye was observed as the solvent composition was varied.

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 and by Pfizer, Inc. is gratefully acknowledged.

Received for review September 9, 1997. January 13, 1998.

Accepted

AC9709918

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