ARTICLE pubs.acs.org/ac
Fluorescence Correlation Spectroscopy: An Efficient Tool for Measuring Size, Size-Distribution and Polydispersity of Microemulsion Droplets in Solution Nibedita Pal, Sachin Dev Verma, Moirangthem Kiran Singh, and Sobhan Sen* Spectroscopy Laboratory, School of Physical Sciences, Jawaharlal Nehru University, New Delhi 110067, India
bS Supporting Information ABSTRACT: Fluorescence correlation spectroscopy (FCS) is an ideal tool for measuring molecular diffusion and size under extremely dilute conditions. However, the power of FCS has not been utilized to its best to measure diffusion and size parameters of complex chemical systems. Here, we apply FCS to measure the size, and, most importantly, the size distribution and polydispersity of a supramolecular nanostructure (i.e., microemulsion droplets, MEDs) in dilute solution. It is shown how the refractive index mismatch of a solution can be corrected in FCS to obtain accurate size parameters of particles, bypassing the optical matching problem of light scattering techniques that are used often for particle-size measurements. We studied the MEDs of 13 different W0 values from 2 to 50 prepared in a ternary mixture of water, sodium bis(2-ethylhexyl) sulfosuccinate (AOT), and isooctane, with sulforhodamine-B as a fluorescent marker. We find that, near the optical matching point of MEDs, the dynamic light scattering (DLS) measurements underestimate the droplet sizes while FCS estimates the accurate ones. A Gaussian distribution model (GDM) and a maximum-entropy-based FCS data fitting model (MEMFCS) are used to analyze the fluorescence correlation curves that unfold Gaussian-type size distributions of MEDs in solution. We find the droplet size varies linearly with W0 up to ∼20, but beyond this W0 value, the size variation deviates from this linearity. To explain nonlinear variation of droplet size for W0 values beyond ∼20, we invoke a model (the coated-droplet model) that incorporates the size polydispersity of the droplets.
F
luorescence correlation spectroscopy (FCS) is a powerful tool for studying molecular diffusion, chemical kinetics, and molecular interaction in vitro and in vivo.16 Since the first demonstration of FCS by Elson and Webb,7 and subsequent upgradation with confocal microscopy by Rigler,8 FCS, as a technique, has come a long way to evolve tremendously, in terms of its sensitivity and applicability.9 These developments not only provided high sensitivity in FCS to measure molecular diffusion and chemical kinetics at the single molecule level, but also gave rise to several variants of the technique, such as fluorescence crosscorrelation spectroscopy,10 fluorescence lifetime correlation spectroscopy,11,12 inverse-FCS,13 and even ultracompact FCS-kit.14 FCS is used extensively in biological and biomedical research.16 However, its use in the studies of chemical systems is less prominent. FCS is ideal for measuring molecular diffusion and, thereby, the molecular size in dilute solutions.1517 However, its power has not been tested to its best in measuring the size, size distribution, polydispersity, etc. of many important chemical systems. Among them are the nanometer-sized supramolecular nanostructures (droplets) that are formed in a mixture of two immiscible liquids when separated by a surfactant layer. These thermodynamically stable droplets are generally called microemulsion droplets (MEDs).18 They form a core of a polar liquid, stabilized by the polar part of the surfactant, and disperse r 2011 American Chemical Society
in a nonpolar liquid (see Figure 1).18 The importance of these nanostructures lies in the fact that they can encapsulate high amount of liquids and large macromolecules inside their core. Because of this, MEDs have attracted extensive applications that range from encapsulating proteins,19 DNA,20,21 polymers,22 ionic liquids,23 etc. in their core to their use in the purification/ extraction of biomolecules,24 nanoparticle synthesis,25 and drug delivery.26 This fact also led to several experimental22,2731 and simulation32,33 studies, which helped to understand the structure and dynamics of confined molecules inside these systems, and thereby unfolded the unique properties of these nanostructures for numerous applications. Efficient application of MEDs requires detailed knowledge about their basic properties, such as composition, structure, size, size distribution, polydispersity, etc. Although NMR,34 smallangle neutron scattering (SANS),23,35,36 small-angle X-ray scattering37 (SAXS), viscosity measurements,38 and molecular dynamics simulations32,33 have been used to characterize these systems, contrast variation static and dynamic light scattering (SLS, DLS) are used more often, mainly because of their easy Received: May 18, 2011 Accepted: September 7, 2011 Published: September 07, 2011 7736
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Figure 1. Structure of a microemulsion droplet (MED); rw is the radius of the polar core, δ is the thickness of the surfactant layer, Rh is the hydrodynamic radius, and εp, εs, and εnp are the dielectric constants of the polar phase, surfactant layer, and nonpolar phase, respectively.
availability.22,3942 From all these studies, it is found that the size of MEDs can be characterized by a single parameter, R: R ¼
½polar phase ½surfactant
For water-in-oil (w/o) microemulsions, R ¼ W0 ¼
½ H2 O ½surfactant
This parameter suggests that the size of core—and, thus, the size of the entire droplet—varies with the amount of polar liquid present in the system at a constant surfactant concentration. However, it turns out that the constituent phases of this ternary system (i.e., a polar liquid phase (εp), a surfactant layer (εs), and a nonpolar liquid phase (εnp)) have different dielectric constants (see Figure 1).4042 In addition, in most cases the relationship among them follows an order: εp < εnp < εs.4042 As a result, at a particular R-value, the overall dielectric constant of the entire droplet becomes the same as that of the bulk nonpolar phase (εnp).4042 This condition is called the optical matching point (OMP), where the excess polarizability of the droplets (relative to pure nonpolar phase) vanishes. Consequently, the intensity of the scattered light from these droplets reaches zero. This make the droplets invisible in light scattering experiments.4042 In reality, however, the scattered intensity reaches a minimum value (not zero), because of the existence of some size polydispersity.4042 This excess polarizability of the droplets (relative to pure nonpolar phase) is directly related to the refractive index change (Δn) of the droplets.4042 Now, as Δn changes with varying R, the scattered intensity shows a dip near the OMP, which gets reflected as an unusual sigmoidal variation of the droplet size (hydrodynamic radius, Rh) with R.3942 This fact suggests that the accurate size measurements of supramolecular nanostructures such as MEDs are difficult in light scattering experiments, especially near the OMP. Besides, for particles with low scattering cross sections (such as MEDs, in the present case), the scattered signal becomes very difficult to detect in dilute solutions. Eventually, the calculated size distribution of the particles may become biased toward higher values, because larger particles scatter more than the smaller ones. This paper addresses these problems and shows how FCS can be used efficiently to measure the size and, most importantly, the size distribution and polydispersity of MEDs in dilute solution,
bypassing the difficulties of light scattering experiments. Although this paper examines the problems associated with MEDs, the methods described here can be well-adopted for accurate measurement of the size parameters of other particles in FCS. Furthermore, because FCS measures the diffusion time of individual droplets in dilute solution and gives the number-averaged size of the particles, the final results of size distribution do not become biased toward larger-sized particles. In this paper, we describe a detailed methodology to correct the inherent refractive index (n) mismatch of a system (MEDs in the present case) in FCS measurements. It also presents a detailed analysis procedure for a single fluorescence correlation curve with a Gaussian distribution model (GDM) and a maximum-entropy-method-based FCS data fitting model (MEMFCS) to extract the size and size distributions of particles. As a model system, we used well-studied MEDs, viz, w/o AOT reverse micelles prepared in a ternary mixture of water, sodium bis(2-ethylhexyl) sulfosuccinate (AOT), and isooctane, with sulforhodamine-B (SRhB) as a fluorescent marker. To study a broad range of size variation, we prepared 13 AOT reverse micelles of W0 = 250 in a dilute (15 mM) AOT solution. Finally, we calculated the polydispersity of the droplets by fitting the Rh vs W0 plot with an equation derived from the coated-droplet model,42 as well as by analyzing individual size distributions. To the best of our knowledge, only two previous studies—by Eicke and co-workers43 and by Robinson and co-workers44—used FCS to measure diffusion constants and sizes of MEDs, respectively. However, none of these studies focus on either the refractive index correction to extract the accurate droplet size or describe the fitting analysis of correlation curves to obtain the size distribution and polydispersity of droplets. The reader will soon realize the importance of following the procedure described here in order to obtain the accurate size, size distribution, and polydispersity of the nanometer-sized particles, such as MEDs, in solution using FCS.
’ MATERIALS AND METHODS Samples. AOT (g99%, SigmaUltra) was purified with activated charcoal in diethyl ether. The mixture was stirred for hours and filtered through a 0.2 μm filter, and then the solvent was evaporated. This method removes fluorescent impurities efficiently. The purity was checked by placing the AOT reverse micellar solution (without SRhB) in FCS setup and measuring the nonexistence of correlation curves (see Figure S1 in the Supporting Information). AOT was dried under vacuum for 24 h. The w/o reverse micelles of W0 = 250 were prepared from 15 mM stock of AOT in isooctane (Spectrochem, UV grade) by mixing and vortexing the requisite amount of water that contains SRhB (Sigma) (see the Supporting Information for molecular structures). The value of W0 was confirmed by the ratio of the added concentration of water and 15 mM AOT (recall that W0 = [H2O]/[surfactant]). (The error in W0 can only come from the minimum graduations of the apparatus used, and, in this case, the maximum error does not exceed 1.5%.) The final concentration of SRhB in MED solution was kept at 0.7 nM in all samples. Taking the surfactant aggregation number as 33 (for reverse micelles of W0 = 2)45 and 1380 (for reverse micelles of W0 = 50),34 the number ratios of dye to droplet are calculated to be 1:7 106 (for W0 = 2) and 1:1.5 103 (for W0 = 50). Thus, in the entire range of W0 values, the number of dyes remained much lower, compared to available droplets. Increasing (or decreasing) the dye concentrations only 7737
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populates the number of droplets more (or less) with the dye. Also, because SRhB is a charged molecule, it is insoluble in nonpolar phases but soluble in polar water. Thus, SRhB preferentially remains near the charged interface of AOT inside the polar core of the droplets. For FCS measurements, the samples were kept inside a closed cell prepared on a glass coverslip (No. 1), which ensured sample stability during the experiments. FCS Setup. We used a highly sensitive home-built FCS setup that is based on an Olympus (IX71) inverted confocal microscope equipped with a 60 water-immersion objective with a correction collar (NA 1.2, WD 0.28 mm, UPlanSApo, Olympus). Samples were excited with a 532-nm CW DPSS laser (3 mW, Shanghai Dream Laser Tech.) that had a (expanded) beam diameter of 11 mm that overfills the back aperture of the objective. The fluorescence bursts from samples were collected with the same objective, and then were passed through a dichroic (Model XF2016, Omega Optical, Inc., USA) and an emission filter (Model 607AF75, Omega Optical, Inc.). The (only) fluorescence signal was then focused through a tube lens onto a multimode fiber patch cord (Model QMMJ-3S3S-UVVIS-25/125-3-1, Oz Optics, Canada) 25 μm in diameter, which acted as the confocal pinhole. Fluorescence signal was then fed into a single-photon avalanche photodiode (SPCM-AQRH-13-FC, PerkinElmer). Autocorrelation of the fluorescence bursts was obtained by hardware correlation, using a FLEX correlator card (FLEX990EM-12D, Correlator.com, USA). Autocorrelation curves were collected in a routine written in LabView. The correlation curves were finally analyzed in IGOR-Pro software (WaveMetrics, USA). MEMFCS analysis was performed using software developed and generously provided by Profs. N. Periasamy and S. Maiti of TIFR, Mumbai, India. Theory of FCS. FCS uses laser and confocal microscopy to produce a very small observation volume (on the order of femoliters (fL)) inside a sample.113 As fluorescent molecules diffuse in and out of that volume, the fluorescence intensity fluctuates. These fluctuations are time-correlated to get a normalized autocorrelation function G(τ):113 GðτÞ ¼
ÆδFðtÞδFðt þ τÞæ ÆFðtÞæ2
ð1Þ
where F(t) is the fluorescence intensity at time t, and τ is the time shift. δF is the fluctuation in fluorescence, relative to the average value ÆF(t)æ. For a three-dimensional (3-D) Gaussian-shaped observation volume with radial (r) and axial length (l), G(τ) is given by113 2 !1=2 1 τ 1 r τ 1 þ GðτÞ ¼ 1 þ ð2Þ N τD l τD where N is the average number of particles in the observation volume, and τD is the time a particle takes to cross the volume. The translational diffusion constant (D) can then be related to τD as113 τD ¼
r2 4D
ð3Þ
Our setup is calibrated with Rhodamine-6G (Rh6G; Aldrich) in water. Correlation curves of Rh6G, analyzed using eq 2, extracted a τD value of 50 ( 1 μs. From this τD and a known D-value of Rh6G in water,46 r and the observation volume (Veff) were determined to be 283 nm and 1.10 fL, respectively (see the Supporting Information). Finally, the Rh value of the
Figure 2. Geometrical ray-diagram of the refractive index mismatch between objective immersion liquid (water) and MED solution in isooctane (sample) of refractive indices n1 and n2, respectively (n1 < n2).
droplet was determined from the StokesEinstein equation (eq 4).46 Rh ¼
kT 6πηD
ð4Þ
where k is the Boltzmann constant, T is the temperature (25 C in the present case), and η is the viscosity of the solvent. In eq 4, the viscosity (η) of AOT solution was calculated from eq 5:47 ηsolution ¼ ηisooctane ð1 þ ½ηCÞ
ð5Þ
where C is the concentration of AOT (in units of g cm3) and [η] is the intrinsic viscosity. For AOT in heptane, [η] = 3.01 cm3 g1 at 25 C. This [η] value was assumed to be the same as that for isooctane and was used to calculate ηsolution for the AOT solutions. The value of ηsolution was calculated to be 0.51 cP for a 15 mM AOT solution (assuming ηisooctane = 0.5 cP). In FCS measurements, the choice of fluorescent molecule is an important aspect, because the correlation data can be severely influenced by photophysical effects of the molecule, such as saturation and photobleaching. We have chosen a molecule (SRhB) and used the laser power such that there was no saturation and photobleaching in the samples (see the Supporting Information for details). We also checked whether the measured correlation data are dependent on the concentration of added SRhB in AOT solution. No signature of such dependence was observed (see the Supporting Information).
’ RESULTS AND DISCUSSION Refractive Index Mismatch Correction. Diffusion time measurements of MEDs using FCS are not straightforward, because of the existing inherent refractive index (n) mismatch among the droplets of different sizes.4143 Furthermore, the objective immersion liquid (water in this case) and the microemulsion solution have different n-values. In light scattering techniques, the signal depends on the (excess) polarizability of the droplets. However, in FCS, the fluorescence signal is generated from a fluorophore embedded inside the droplet (SRhB remains inside the polar core of the droplet). Moreover, the emitted fluorescence only senses an average dielectric medium (of average n) when it passes through the microemulsion solution. Figure 2 7738
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Figure 4. Normalized fluorescence correlation curves for AOT MEDs of different W0. As W0 increases, the curves shift to the longer lag times, indicating the increase in size of the droplets. The correlation curve for SRhB in water (dotted line) is also included for comparison.
Figure 3. Effect of refractive index mismatch: (A) dependence of the τD value of Rh6G in water with objective collar positions in the presence of different concentrations of GdnHCl. (In pure water, the collar setting of 0.16 only corrects the refractive index introduced by the glass coverslip.) (B) Dependence of τD on depth (d) for different GdnHCl concentrations. Lines through points are present as a guide to the eye. Errors in τD values are also included.
shows the aberration effect when the average n of the microemulsion solution (sample) is considered to be n2 and that of immersion liquid (water) is considered to be n1 (n1 < n2). In that case, it can be shown that48 tan sin1 ðNA=n2 Þ 0 d d ¼ ð6Þ tanðsin1 ðNA=n1 ÞÞ where d and d0 are the depths of nonaberrated and aberrated focal positions, respectively (see Figure 2). In the present case, however, n2 can change as the overall refractive index of the entire droplet varies with W0. In fact, the final (average) n2 value of the solution gets modified, based on the number of droplets present in the solution. With increases or decreases in droplet size, the amount of water and the aggregation number of the surfactant changes, which, in turn, changes the overall refractive index of the entire droplet and modifies the average n2 value of the solution. In such a condition, the focusing property of the objective gets distorted with changing W0, giving rise to different d values and distorted Veff values, which leads to increased τD values. We used an aqueous solution of guanidine hydrochloride (GdnHCl) and the correction collar of objective to standardize our FCS setup and, hence, determine the correct collar positions for matched n-values in AOT microemulsions.49 (The n-value of the GdnHCl solution is a linear function of the GdnHCl concentration.50) We measured the correlation curves (and τD) for 5 nM Rh6G in water at different GdnHCl concentrations. Figure 3A shows the τD variation with collar position in different GdnHCl concentrations measured at an arbitrary d ≈ 40 μm. Figure 3A shows that increasing the collar position initially compensates the aberration; however, beyond an optimum position, it overcorrects. (See Figure S5 in the Supporting Information for the collar setting dependence on n.) Note that, in Figure 3, the minimum τD at high GdnHCl concentrations does not reach the same
value as that in water (50 μs), because of the increase in solution viscosity. In a second set of experiment, the collar was kept at the minimum τD position, as obtained for a particular GdnHCl concentration (from Figure 3A). The height of the objective was then changed to get the minimum τD position again. Figure 3B shows the dependence of τD on depth (d). This figure clearly shows that, for a given collar position (minimum τD) and [GdnHCl] e 9 M, the optimum depth can be safely considered to be d ≈ 30 μm. Thus, all FCS measurements on AOT reverse micelles were carried out at d = 30 μm. Under these conditions, it is expected that the effect of aberration is compensated in AOT solutions, and the r and Veff values become similar to those in water, where no aberration is present. This r and Veff values in AOT solution under aberration-compensated conditions were determined experimentally which give similar values (r = 282 nm, Veff = 1.25 fL) to those in water (see the Supporting Information). FCS Data for AOT MEDs. The present study uses MED solutions prepared in water (np= 1.3349)/isooctane (nnp= 1.3914)/ AOT (ns= 1.4850), where isooctane remains the dominant phase at low AOT concentrations. However, in the present case, the Δn value of the droplet (relative to pure isooctane) may vary from approximately +0.08 (at W0 = 0) to approximately 0.02 (at W0 = 50).41 Thus, the average n2 value of the AOT solution can range from ∼1.4714 (at W0 = 0) to ∼1.3714 (at W0 = 50), which corresponds to [GdnHCl] ≈ 8.53 M. Under these conditions, Figure 3A shows that the optimum collar position should be within the range of ∼0.170.20. However, note that the average n2 value of the AOT solution depends on the AOT concentration (i.e., on the droplet numbers).51 Thus, at low AOT concentration (15 mM in this case), the contribution from Δn of droplets remains low, which leads to a situation where the effective n2 value is dominated by the bulk nonpolar phase. In such a case, the optimum collar positions may remain near ∼0.190.20. Nonetheless, in the actual FCS experiments with MEDs, the collar was varied systematically to determine the optimum position where τD was minimum. As expected, the minimum τD value was obtained at a collar position of ∼0.20 for low W0, and this position got shifted systematically toward ∼0.19 at high W0. Figure 4 shows the correlation curves for AOT MEDs (W0 = 250), measured at the optimum collar positions, and d = 30 μm. Gaussian Distribution Model (GDM) Analysis. To start with, we fitted our FCS data of AOT MEDs with eq 2, assuming no size heterogeneity in the solution. The fit for W0 = 50 is shown in Figure 5, where the residual, which reflects the goodness of fit, is not satisfactory (see also below). For a better fit, eq 2 could be 7739
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Figure 5. Fits to the correlation curve and residuals of the fits for AOT MED of W0 = 50: Single component fit is shown as a solid blue line, GDM analysis is shown as a solid red line, and MEMFCS analysis is shown as a dashed green line. (See Figure S11 in the Supporting Information for fits for all 13 W0 values.)
rewritten as sum of two or three diffusing components. However, for highly heterogeneous systems such as MEDs, a smaller number of diffusing components might not provide the accurate description of the system. Earlier DLS studies revealed Gaussian type size-distributions for AOT MEDs that range over broad sizes.22 If we assume that there are m number of species diffusing through the observation volume with times τDi, then eq 2 can be modified as 2 #1=2 " m τ 1 r τ ai 1 þ 1 þ ð7Þ GðτÞ ¼ τ l τ Di Di i¼1
∑
where ai represents the relative amplitudes of the species that relate to their average number in the observation volume and brightness. Microemulsion systems are highly heterogeneous for which the lowest and highest τD values are expected to differ by orders of magnitude. In such a case, the τD-distribution should be considered on a logarithmic time-scale. This τD-distribution can thus be obtained through eq 7 by assuming a Gaussian distribution in ai on a logarithmic time-scale, as 2 #1=2 " m τ 1 r τ ai ðτDi Þ 1 þ 1 þ GðτÞ ¼ τ l τ Di Di i¼1
∑
ð8Þ where
2
lnðτDi Þ lnðτp Þ ai ðτDi Þ ¼ Ai exp4 b
!2 3 5
ð9Þ
with relative amplitudes of components (Ai), peak diffusion time (τp), and b, which is related to the width of the distribution. We fitted the FCS data using eq 8. A total of 150 fixed τDi time components, logarithmically spaced within 20106 μs, were used in these fits. (Fitting was performed from 20 μs to neglect the triplet contribution.) In the least-squares fitting analysis, the Ai and b values were varied to get the best distribution of ai(τDi) that fit the experimental correlation data correctly. Figure 5 includes the fitted results for GDM analysis for W0 = 50. The residual in GDM analysis can be readily observed to be much better (sum-of-squares deviation is given as SSD = 0.00148, and reduced χ2 = 1.063), compared to the single component fit (SSD = 0.00336 and
reduced χ2 = 3.72). (Also see Figure S10 and Table S2 in the Supporting Information to check that GDM analysis improves the fitted results, compared to single-component fit, and one needs a large size distribution to fit the correlation data for MEDs.) Although visual improvement of the residual and the SSD value gives first-hand proof of the goodness-of-fit in any least-squares fitting analysis, calculating the reduced χ2 value is always preferred. However, such a calculation for FCS data is not straightforward, because the hardware modules generally do not provide quantitative information about noise.52,53 We calculated the reduced χ2 value by repeating (15 times) the measurements of correlation curves and obtaining the standard error for each data point.52,53 For checking, the reduced χ2 values for four MEDs of W0 = 2, 10, 25, and 50 were calculated. It is found that the reduced χ2 values improve in GDM analysis, compared to single-component fit (see the Supporting Information for details). At this juncture, it should also be noted that eq 9 can be modified by adding a greater number of Gaussian distributions around other τp values to capture multimodal size distributions. We checked this issue for AOT MEDs: Although least-squares fitting was started with largely separated bimodal Gaussian distributions, the fitting ultimately converged to a unimodal one after several iterations in all cases. Although MEDs are expected to have some size distributions in solution, it is important to check whether the improvement of fitting in GDM analysis is real, compared to the single-component fit, and not an instrumental artifact. This was checked by analyzing the correlation data for pure dye diffusing in pure solvents of different viscosities and for a dyeprotein complex in water where no size heterogeneity is expected. For these systems, no improvement in GDM analysis was found, compared to the single-component model (see the Supporting Information). Thus, we can safely conclude that the present size-distribution of MEDs is real and not an instrumental artifact. MEMFCS Analysis. Maximum entropy method analysis is based on an algorithm proposed by Skilling and Bryan.54 Previously, this method has been successfully used to obtain the distribution of fluorescence lifetimes and diffusion coefficients.55,56 Recently, this method is implemented as MEMFCS to analyze FCS data.53 MEMFCS fitting analysis of the correlation data seeks the best distribution of ai(τDi) in eq 9, where not only is the χ2 value at a minimum, but the entropy (S) is also at a maximum. Here, entropy is given as53 S¼
∑pi ln pi
ð10Þ
where pi = ai(τDi)/∑ ai(τDi). The main difference between GDM and MEMFCS analysis is that GDM uses a Gaussian function to model the distribution, whereas MEMFCS does not use any model function. Furthermore, GDM analysis seeks a distribution for which only χ2 is minimized, but, in MEMFCS analysis, both χ2 is minimized and S is maximized. We analyzed our correlation data for AOT MEDs using MEMFCS to determine if it extracts similar results as GDM analysis or not. Figure 5 includes the fitted data for W0 = 50 from MEMFCS analysis. It shows the residual obtained in MEMFCS analysis is identical to that of the GDM analysis. This similarity applies to all the MEDs of other W0 values (see Figure S11 in the Supporting Information). Thus, GDM and MEMFCS model FCS data in the same fashion. Size and Size Distribution of MEDs. The distributions of hydrodynamic radii (Rh) of MEDs were calculated, using eqs 3, 4, and 5, from ai(τDi) vs τD distributions obtained in GDM and 7740
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MEMFCS analyses. Results are shown in Figure 6. GDM analysis provides size distributions that nicely match the MEMFCS results. Hence, both models support the fact that AOT MEDs possess unimodal Gaussian-type size distributions, similar to those seen in previous DLS measurements.22 Both models provide very similar peak Rh and full width at half maximum (fwhm) values for a given W0 (see Table 1). The Rh values are in good agreement with previous NMR,34 IR,45 and simulation results,33 and partially agree with the DLS results (see below).39 Interestingly, Figure 6 (and Table 1) show that the fwhm increases monotonically with W0. This suggests that the heterogeneity of the AOT MED system increases with increasing W0. It is now essential to check how refractive index mismatch of the AOT solution affects the Rh distributions and peak Rh values, if it is not corrected properly. This is easily checked by analyzing the correlation data taken at different collar positions for a particular W0 value. Performing such analysis, we find that the Rh-distribution gets systematically broadened and the peak Rh value gets shifted toward the higher side as the collar position moves out of the optimum one (see Figure S12 in the Supporting Information). This clearly suggests that, without correcting the refractive index mismatch in FCS, it is impossible to obtain the accurate size and
size distribution of any system that is comprised of media of different dielectric constants. Figure 7 plots the variation of peak Rh with W0. It has been proposed earlier that, for low W0 (up to 20), Rh is a linear function of W0.31,38,47,57 We find that our Rh vs W0 plot can be modeled nicely with a linear equation (eq 11) up to W0 = 20, but beyond this W0 value, the data deviates from this linearity (see Figure 7). Rh ¼ 0:175W0 þ 1:5ðnmÞ
ð11Þ
Equation 11 matches with previously reported equation that relate Rh and W0.57 The intercept of 1.5 nm on the Rh-axis obtained in eq 11 gives the thickness (δ) of surfactant layer of the droplet (see Figure 1). However, the data beyond W0 = 20 can only be explained if one incorporates the effect of size polydispersity in a nonlinear equation (see discussion below). Coated-Droplet Model and Polydispersity. To explain the nonlinear dependence of Rh on W0 at higher values, we derived an equation (eq 20) from the coated-droplet model. This model was proposed earlier to explain the light scattering data.42 Fits to the Rh vs W0 plot with eq 20 directly provides the polydispersity
Figure 7. Dependence of droplet size (Rh) on W0. The droplet sizes from W0 = 2 to W0 ≈ 20 fit eq 11 (black line) nicely. Beyond W0 ≈ 20, the size increment deviates from the linear relationship depicted by eq 11 and is better modeled by the nonlinear expression given as eq 20 (red line). Equation 20 is not valid for low W0 values, as can be seen by the dotted part of the red line.
Figure 6. Distributions of hydrodynamic radii of AOT MEDs for different W0 values, plotted on a logarithmic size-scale. Size distributions show Gaussian-type features for all MEDs. Solid and dotted lines represent data for GDM and MEMFCS analysis, respectively. See Table 1 for values.
Table 1. Peak Rh Values and Full Width at Half Maxima (FWHM) of Size Distributions for AOT MEDs of Different W0 Values Obtained in GDM and MEMFCS Analysis (see also Figure 6) GDM
a
MEMFCS
W0
peak Rh (nm)
full width at half maximum, fwhm (nm)
peak Rh (nm)
full width at half maximum, fwhm (nm)
RhAvg (nm)a
2
1.96
1.71
1.99
1.74
1.98
4
2.34
2.03
2.42
2.20
2.38
6
2.67
2.53
2.84
2.54
2.76
8 10
3.06 3.27
2.70 3.60
3.22 3.30
2.78 4.58
3.14 3.29
15
4.17
4.50
4.50
4.66
4.34
20
5.11
4.80
5.35
4.29
5.23
25
5.85
6.25
6.15
6.54
6.00
30
7.07
7.27
7.22
7.79
7.15
35
8.20
8.37
8.11
8.95
8.16
40
9.51
8.85
9.10
10.83
9.31
45 50
10.45 11.80
10.61 14.54
10.94 12.42
11.00 14.35
10.70 12.11
RAvg is the average of the peak Rh values obtained in GDM and MEMFCS. h 7741
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of the droplets. Methods for determining polydispersity from a single size distribution had also been proposed earlier.58 We tested both methods to check if the coated-droplet model can explain the Rh vs W0 data and provide the polydispersity of the AOT MEDs, along with other structural parameters. The coated-droplet model assumes a water core of volume Vw and radius rw (in the case of w/o microemulsion), surrounded by a surfactant layer of volume Vs, which is defined as42 vs Vs ¼ 4πrw 2 δ; δ ¼ ð12Þ as
Table 2. Comparison of FCS Results with Previous DLS Resultsa
(see Figure 1). Here, vs is the volume of a single surfactant molecule and as is the area occupied by the headgroup of the surfactant inside water core; δ is the thickness of the surfactant layer (see Figure 1). The total volume of a MED is then given by V ¼ Vw þ Vs
or
4π 3 4π 3 Rh ¼ ðrw þ 3rw 2 δÞ 3 3
ð13Þ
Hence,
RAvg (FCS) (nm) h
Rh (DLS)b (nm)
2
1.98
2.15
4 6
2.38 2.76
2.50 2.80
γ
8
3.14
3.20
10
3.29
3.40
15
4.34
4.11
20
5.23
4.40
0.056
25
6.00
5.20
0.059
30
7.15
6.20
0.046
35 40
8.16 9.31
7.60 9.80
0.039 0.029
45
10.70
11.16
0.031
50
12.11
11.80
0.041
a
Rh ¼ ðrw 3 þ 3δrw 2 Þ1=3
ð14Þ
MEDs are polydisperse in size. Thus, one writes eq 13 as 4π 3 Ærw æ þ 3δÆrw 2 æ ÆV æ ¼ ÆVw æ þ ÆVs æ ¼ ð15Þ 3 Here, the volume of water is proportional to rw3, whereas that of the surfactant is proportional to rw2. Hence, W0 can be written as42 W0 ¼
y Ærw 3 æ ; 3δ Ærw 2 æ
y¼
vs vw
ð16Þ
where vw is the volume of a water molecule. The size polydispersity (γ) can then be defined as γ¼
W0
Ærw 2 æ 1 Ærw æ2
Here, Ærw æ ¼
ð17Þ
Z f ðrw Þrw drw
ð18Þ
where f(rw) is the distribution function. For distributions with low size polydispersity (γ , 1), one can use a general recursion relation (with integer q) for any distribution, as42 Ærw q þ 1 æ ¼ ð1 þ qγÞÆrw q æÆrw æ
ð19Þ
In eq 17, rw can be replaced by Rh for constant δ to obtain the size polydispersity of the entire droplet. In that case, the error in the numerator and denominator are of same order of magnitude, and hence will be compensated. Assuming γ is independent of W0, and using eqs 14, 16, and 19, Rh can then be written as
" # 3δW0 yð1 þ 2γÞ y2 ð1 þ 2γÞ2 5y3 ð1 þ 2γÞ3 1 þ Rh ¼ þ 3W0 yð1 þ 2γÞ 9W0 2 81W0 3
ð20Þ In this equation, we have only three adjustable parameters to model the Rh vs W0 plot: y, δ, and γ. Note that eq 20 is only valid when δ/rw ,1, i.e., for large W0. As seen in Figure 7, the data for W0 > 20 can be modeled reasonably well with eq 20 for parameters δ = 1.5 nm, γ = 0.043 (polydispersity index of γ1/2 = 0.21),
Polydispersity (γ) at each W0 is calculated from the size distribution, using eqs 17 and 18. γ values for W0 < 20 are not calculated, because the size varies linearly with W0 in this region. Bold numbers show the region where OMP affects the measured sizes in DLS (see text). b Data taken from refs 34 and 39.
and fixed y = 20.4 (vs = 0.61 nm3, vw = 0.03 nm3).42 The thickness δ = 1.5 nm obtained from this fit is in excellent agreement with that obtained from the linear fit, using eq 11 at low W0 range (see above). This fact suggests that the variation of Rh with W0 for W0 > 20 can be explained reasonably well with the coated-droplet model by just incorporating the size polydispersity and keeping the thickness (δ = 1.5 nm) the same. This also gives a physically meaningful picture of the MEDs, which suggests that polydispersity only arises from the size variation of the water core, and not the surfactant layer thickness, which is expected to be constant in droplets of any W0. Table 2 includes the RAvg h values (from Table 1), as well as the sizes obtained in previous DLS experiments.39 The most important point to note here: near the OMP of the AOTwater isooctane system (W0 ≈ 25),40,41 DLS experiments fail to reproduce the accurate size of the droplets, giving smaller values than expected (see Table 2). This is because, as discussed above, near the OMP, the low scattering intensity from the droplets directly affects the measured Rh value. This problem prevails in the W0 range from ∼15 to 35 (see Table 2 and Figure S13 in the Supporting Information). Below and above these W0 values, FCS and DLS provide very similar droplet sizes. It is of high importance to note here that the OMP of MEDs can change if any of the three constituent phases is replaced with another phase of different dielectric constant,41,42 or a new phase is added to make a quaternary system (for example, a macromolecule is embedded inside the polar core).40 Finding the OMP for such new microemulsion systems may become complicated. In addition, with unknown OMPs, it might then be disadvantageous to use light scattering techniques to obtain accurate size parameters of MEDs, especially near the OMP. FCS is then certainly preferred over DLS in such a scenario. Table 2 also includes the polydispersity (γ) of the MEDs for W0 g 20, calculated numerically (using eqs 17 and 18) from the individual size distribution obtained in GDM analysis. It is quite interesting to see that the size polydispersity (γ = 0.043) obtained from the fit using eq 20 is in excellent agreement with the (average) γ-value calculated numerically from the individual size 7742
dx.doi.org/10.1021/ac2012637 |Anal. Chem. 2011, 83, 7736–7744
Analytical Chemistry distribution. Overall, this suggests that the coated-droplet model is capable of describing the important size parameters of microemulsion systems.
’ CONCLUSION This paper has described how fluorescence correlation spectroscopy (FCS) can be used efficiently to extract accurate size and, most importantly, the size distribution and polydispersity of nanometer-sized microemulsion droplets in solution, bypassing the difficulties of light scattering techniques. Detailed methodologies for refractive index mismatch correction and analysis of fluorescence correlation data have been discussed. We show the importance of compensating the refractive index mismatch to extract the accurate size parameters of particles that possess media with different dielectric constants. We found that both the Gaussian distribution model (GDM) and the maximum entropy-based FCS (MEMFCS) model provide almost-identical results, suggesting that a Gaussian-type distribution is sufficient to describe the size distributions and polydispersity of pure microemulsion droplets in solution. We also found the variation of hydrodynamic radius (Rh) with core size (i.e., W0) follow a linear relationship up to W0 ≈ 20. However, beyond this W0 value, the data can be explained only by a model (the coated-droplet model) that incorporates the size-polydispersity parameter (γ). Most importantly, we determined that, near the optical matching point, dynamic light scattering (DLS) measurements underestimate the size of the droplets, whereas FCS provides the accurate size measurements. Note that, although this paper deals with the problems associated with MEDs, the methods described here can be adopted for measurement of the size parameters of other particle systems. Nonetheless, the reader should not assume that DLS must be replaced by FCS whenever measuring the size parameters of any particle system. This is because FCS also encounters problems in providing accurate sizes when the particle size become comparable to the size of the diffraction-limited spot. In such a scenario, DLS is certainly preferred over FCS for accurate measurements of the particle size on the order of hundreds to thousands of nanometers. Thus, rather than replacing one technique with another, FCS should be used as complementary of DLS. In addition, both techniques should serve in tandem to provide accurate size parameters of any particle system over a broad range of sizes. ’ ASSOCIATED CONTENT
bS
Supporting Information. Additional figures, text, and data analysis. (PDF) This material is available free of charge via the Internet at http://pubs.acs.org.
’ AUTHOR INFORMATION Corresponding Author
*E-mail:
[email protected]. Tel: +91-11-26738803. Fax: +91-1126717537.
’ ACKNOWLEDGMENT This work is supported by the project “Construction and multisite commissioning of multiple fluorescence correlation spectrometers (FCS, a single molecule biophotonic tool)” (No. 12(4)/2007-PDD of Department of Information Technology (DIT), Govt. of India). This work is also supported by Department of Science and Technology (DST), Govt. of India (SR/FTP/PS-16/2007),
ARTICLE
and University Grants Commission (UGC), Govt. of India. We thank Prof. S. Maiti and his students for many helpful technical discussions on FCS setup as well as for providing the data collection software developed by them in LabView. We also thank Profs. N. Pariasamy and S. Maiti for providing MEMFCS analysis software. N.P. and M.K.S. thank CSIR, and S.D.V. thanks UGC for their fellowships.
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