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The ability to size accurately low concentrations of nanoscale particles in small volumes is useful for a broad range of disciplines. Here, we characterize confocal correlation spectroscopy (CCS), which is capable of measuring sizes of both fluorescent and non-fluorescent particles, such as quantum dots, gold colloids, latex spheres, and fluorescent beads. We measured accurately particles ranging in diameter from 11 nm to 300 nm, a size range that had been difficult to probe, owing to a phenomenon coined biased diffusion that causes diffusion times, or particle size, to deviate as a function of laser power. At low powers, artifacts mimicking biased diffusion are caused by saturation of the detector, which is especially problematic when probing highly fluorescent or highly scattering nanoparticles. At higher powers (>1 mW), however, autocorrelation curves in both resonant and non-resonant conditions show a structure indicative of an increased contribution from longer correlation times coupled with a decrease in shorter correlation times. We propose this change in the autocorrelation curve is due to partial trapping of the particles as they transit the probe volume. Furthermore, we found only a slight difference in the effect of biased diffusion when comparing resonant and non-resonant conditions. Simulations suggest the depth of trapping potential necessary for biased diffusion is >1 kBT. Overcoming artifacts from detector saturation and biased diffusion, confocal correlation spectroscopy is particularly advantageous due to its ability to size particles in small volumes characteristic of microfluidic channels and aqueous microdroplets. We believe the method will find increasing use in a wide range of applications in measuring nanoparticles and macromolecular systems.
Accurate characterization of nano-scale particles has wide application in several cross-disciplinary fields. For example, liposomes and other nano-containers are used effectively as agents for isolation and controlled delivery of drugs and genes to treat specific regions of the body.1,2 In addition, nano-scale investigation of sub-cellular structures offers new insights into roles that size or shape may play in cellular processes, such as the release of synaptic-vesicle contents at synapses.3 Furthermore, synthetic nanoparticles composed of materials such as polymers or metals are finding broad utility in a range of applications such as diagnostic imaging.4,5 In many cases, the unique chemical and physical properties of nanoparticles results from not only material characteristics, but also from the size and shape. Currently, detailed imaging of nano-scale particles is performed using microscopy techniques such as scanning electron microscopy (SEM), atomic force microscopy (AFM), transmission electron microscopy (TEM), and cryogenic electron microscopy (Cryo-EM), which achieve atomic to micron scale resolution for visualizing particle morphologies.6,7 Unfortunately, these approaches are not ideal in some cases because of long preparation times needed to acquire images, expensive equipment and, for SEM/TEM, samples must be dried in ultra-high vacuum and particles must be conductive or coated in gold to image at nanometer resolutions. Furthermore, fixing samples for imaging with cryo-EM or adsorbing the particles on a surface for AFM can result in morphological deformation, thus giving rise to inaccurate or misleading measurements.
For many applications, direct measurement of freely diffusing particles in solution is more convenient, provides a non-invasive way to gather data in real-time, and offers a means to size samples not suited for imaging with static methods. Dynamic light scattering, perhaps the most popular method for sizing nanoparticles, has been successful owing to its ability to size accurately particles ranging from nanometers to microns, requirement of relatively inexpensive equipment and its applicability to many different types of samples such as synthetic nanoparticles, DNA, and liposomes.8 In addition to size, analysis at multiple scattering angles can also reveal shape information. Unfortunately, dynamic light scattering is limited to higher concentrations (μM) and requires rather large sample volume (mL) to avoid artifacts caused by scattering off of surrounding surfaces near the probe volume. These disadvantages are particularly important when considering the increasing number of applications involving nanoparticle synthesis in microfluidic channels,9-13 encapsulation of particles in immiscible fluid droplets,14,15 and for biological samples that after purification are too dilute and must be analyzed in small-volume aliquots.
As a complementary approach to dynamic light scattering, fluorescence correlation spectroscopy (FCS) can be used to determine rates of movement for very small species at the molecular or large-protein size scale.16-19 Owing to impressive signal-to-noise ratios characteristic of confocal systems, the method is particularly suited for measuring single fluorescent molecules at low concentrations (pM-nM), and because the probe volume is typically on the order of 0.3 fL very little sample volume is needed for analysis. When analyzing larger particles ranging in diameter from tens to hundreds of nanometers, obtaining accurate diffusion coefficients or sizes can be problematic because diffusion times appear to be power dependent. This observation has been coined biased diffusion and is believed to originate from optical trapping forces imparted onto particles as they enter into the laser beam thus slowing the trajectory of the particle in the beam vicinity.20-23 Not only does the effect appear to slow diffusion, the probability of the particle entering the probe volume, which is governed by Poisson statistics, deviates as laser power is increased causing particles to look as if they are locally concentrated near the laser beam waist.
Here, we characterize confocal correlation spectroscopy (CCS), a method built on the foundation of FCS and capable of accurately sizing particles that are either fluorescent or non-fluorescent.24 We provide insights into the effects of true and apparent biased diffusion over a wide range of laser powers and present new findings as to how to overcome such artifacts. We found for bright particles with peak photon count rates in the MHz range under low excitation powers (nW-μW), size dependence on power was due only to detector saturation. The dead time of the detection system has long been known to be a potential source of problems for photon counting experiments including FCS experiments.17,25,26 Expressions for the effect of the detector dead time on the FCS correlation function have been derived to first order in the ratio of the detector dead time to the sample interval.17,26 The count rates, however, for some of our laser powers were so large that it was not clear if these expressions would still be valid, so the effect of the detector dead time was studied by Monte Carlo simulations, which fit well to the data.
At powers >1 mW, we observed, under resonant and non-resonant conditions, correlation curves with an increase in longer duration bursts, or particle transit times, coupled with a corresponding decrease in shorter bursts, thus suggesting the occurrence of biased diffusion. Moreover, simulations suggest that biased diffusion did not occur until potential well depths of >1 kBT were reached. Awareness of the power ranges that result in biased diffusion should provide a convenient means to recognize particle interactions with the laser beam. Compared to other sizing techniques, such as dynamic light scattering, the advantage of CCS lies in its ability to measure dilute samples (down to a single particle or molecule) and in small volumes (femtoliters). With advances in non-fluorescent modes of single-particle and single-molecule detection, we believe correlation spectroscopy will find increasingly broad use in measuring the dimensions of nanoparticles and macromolecular systems.
Fluorescent (60-, 110-, 180-, 290-nm) and non-fluorescent (110-, 180- and 300-nm) polystyrene beads were purchased from Molecular Probes (Eugene, OR), Duke Scientific Corporation (Palo Alto, CA), and Bangs Laboratories (Fishers, IN). Gold nanoparticles (40-, 60-, 110-nm) were from Ted Pella, Inc. (Redding, CA). Quantum dots (QDs), which emitted at 565 nm and were composed of a CdSe core capped with carboxyl-terminated polymer, were purchased from Invitrogen Corporation (Carlsbad, CA).
Experiments were conducted on a Nikon TE2000U microscope equipped with a home-built add-on for confocal microscopy. For excitation, 488 nm light from a solid-state diode pumped laser (Coherent Sapphire, Santa Clara, CA) was collimated and directed into a Nikon 100× objective (NA 1.45). Laser power was measured immediately prior to the beam entering the objective. Fluorescence from the object plane was collected with the objective, passed through a dichroic mirror (z488rdc, Chroma, Rockingham, VT), imaged onto a 50μm pinhole (Thorlabs, Inc., Newton, NJ) placed at the image plane, and filtered with a bandpass filter (HQ550/100M, Chroma, Rockingham, VT) before being focused onto the avalanche photodiode (APD) (SPCM-AQR-16, Perkin Elmer, Fremont, CA). Scattering bursts were collected through the same beam path, but without a bandpass filter. For non-resonant conditions, we used a 632.8 nm HeNe laser (Coherent, Santa Clara, CA), which was overlaid with the beam focus of the 488 nm laser. Autocorrelation was recorded using a Flex02−12D multiple-tau autocorrelator from correlator.com (Bridgewater, NJ) and fitted with data analysis software. A power meter (PD 300-UV, Ophir Optronics, Wilmington, MA) was used to record laser power before entrance into the objective.
Most commonly, confocal correlation spectroscopy has been performed in the fluorescence mode to determine diffusion coefficients of fluorescent particles and molecules undergoing Brownian motion. Here, accurate information was also recorded in a confocal geometry from back-scattered bursts off non-fluorescent nanoparticles. Photon bursts corresponding to particles passing through the laser probe volume were autocorrelated and the resulting correlation curve was fit with the following equation:16
where Go is the amplitude of the correlation function, which for an ideal system is inversely proportional to the average number of particles in the probe volume, G∞ equals one for an ideal system, K is the shape factor, or the ratio of height (z0) to width (w0) of the probe volume, and D is the diffusion coefficient for the fluorescent or non-fluorescent species. Dimensions of the probe volume were determined by calibration with particles of three known sizes (diameter; 110-, 180-, 300-nm). Dimensions for the back-scattering mode (wo; 230 nm, zo: 1 μm) were smaller than for fluorescence measurements (wo: 280 nm, zo: 1.25 μm), which may have been due to a directional dependence of scattering. The measured diffusion time, τD, was used to calculate the diffusion coefficient (D) and the particles' radius (R) using Stokes-Einstein equation (D=kBT/6πηR), where kB is the Boltzmann constant, T is temperature, and η is viscosity of the surrounding medium.
Simulations similar to those used previously were performed to test the effect of the detector dead time.19 The sample was modeled as fluorescent beads with a radius of 55 nm in a container, which was a sphere of radius R=3000 nm. There were 40 beads in the container, which corresponded to a concentration of 0.6 nanomolar. The simulation temperature was 20 °C. A trial configuration was created for each time step in the simulation by applying a small random displacement to each of the particles. Three random deviates, uniformly distributed between -ε0 and +ε0 , were chosen for each particle to give random displacements of the particle along the x, y and z directions. The random number generator RANLUX was used to generate the random deviates.27 The relationship between the time step size, Δt , and the translation diffusion coefficient, D, is the following:28
where is the mean squared displacement along one axis for a single step. For displacements that are uniformly distributed between -ε0 and +ε0 , . For Δt = 25 nsec and ε0 = 7.64482×10−8 cm, equation 2 gives D =3.896×10−8 cm2/sec, which is the translational diffusion coefficient for a 110nm diameter sphere at 20 °C in water. Due to the small 25 nsec time step, it was not considered necessary to draw the individual displacements from a Gaussian distribution. If two or more of the particles overlapped, then the configuration was rejected and a new one was created. Otherwise, the trial configuration was accepted for the simulation. If a particle reached the surface of the spherical sample volume, it was displaced by a distance 4ε0 back towards the center of the sample volume. If that resulted in an excluded volume overlap with another particle, then the particle was placed at random at a point R −4ε0 away from the center of the sample volume. Other than excluded volume, there were no interactions between the particles.
For each time step of the simulation, the intensity was calculated using a Gaussian beam profile which is similar to that used by Aragon and Pecora,29
For each simulation, a count rate I0 was chosen for a bead at the center of the container. Then for each time step, the intensity was calculated from
where ri is the position of the ith bead at time step t, and the sum goes over all of the N beads in the container. Poisson distributed noise was added to I'(t) to obtain the simulated total intensity, I(t), which can then be autocorrelated in the same manner as the experimental signal. I0 was set to 1 MHz for a simulated laser intensity of 10 nW. This number was chosen to match approximately the experimentally observed peak count rates for the 110 nm diameter beads, when the count rate was averaged over intervals of 1 ms (1.35 MHz). For larger simulated laser intensities, I0 was assumed to scale linearly with the power. The same values of wo and K that were determined from our experimental measurements were used both in the simulation and fitting of the simulated correlation functions.
The manufacturer's specification for the APD claims a dead time of 50 nsec, whose effect on the simulated fluorescence intensity was approximated by limiting the maximum number of counts that could be observed in any 25 ns time interval to 1, and by requiring that if a count was observed in the ith bin, then zero counts would be recorded for the (i+1)th bin. This limits the simulated observed count rates to an absolute maximum of 20 MHz. Simulations were also performed in which neither of these restrictions were included, which corresponds to a detector with zero dead time. The autocorrelation function of the fluorescence intensity was calculated in a manner analogous to a hardware autocorrelator as described by Kojro, et al.30 For each simulated power, five simulations of 2.4×109 moves each were performed as described above. Each simulation represented 1 minute of real time. The simulated correlation functions were then analyzed in the same manner as the measured autocorrelation functions to obtain five values of G0, τD and G∞ for each intensity, which were then averaged. For the simulations at the lowest simulated power, where the effects of dead time were negligible, the best-fit diffusion coefficient was D = 3.9±0.2×10−8 cm2/sec, in good agreement with the theoretical value. Simulations with a range of particles for a detector with zero dead time also yielded best-fit diffusion coefficients in very good agreement to the expected results.
The effect of an external potential on the measured FCS correlation function was studied in simulations with an external potential. The simulation procedure, which is a Brownian dynamics simulation,31 was similar to that for the dead time simulations with the following changes. Instead of drawing random particle displacements from a uniform distribution for each time step, the displacement of a particle along a given coordinate axis is given by
where D is the diffusion coefficient of the particle and Δt is the time for each move in the simulation, (0) and (Δt) are the positions of the particle before and after the move, is the force due to the external potential acting on the particle at (0) , and (Δt) is a vector of random deviates drawn from a Gaussian distribution with width
Equations 5 and 6 are a much simplified version of equation (15) of Ermak and McCammon31 and apply only for the case where hydrodynamic interactions between the spheres are ignored as they are in these simulations. For the biased diffusion simulations, D = 1.54 ×10−8 cm2/sec, the diffusion coefficient of a 290 nm diameter sphere, and Δt = 200 ns. The size of physically meaningful time steps is restricted to
where m is the mass of the particle. Assuming the density of the polystyrene spheres is approximately the same as the density of water, for these simulations equation 7 becomes the condition Δt >> 5 nsec, which is satisfied here. As before, if the excluded volumes of two particles overlapped that configuration was rejected and a new one was generated. No other interaction between the particles was included in the simulation.
The simulated trapping force imparted onto a polystyrene particle was calculated for a 633 nm laser focused into water with an oil immersion lens. The force, F, is dominated by Fgrad, the force due to the gradient of electric field:32
where E() is the electric field near the focus, a is the radius of particle, nparticle is refractive index of particle, nwater is the refractive index of water, and ε0 is the permittivity of vacuum. Following Richards and Wolf, E() equals the Debye integral of a highly focused linearly polarized Gaussian beam:33,34
where P(θ ,) is the electric field right after the objective lens, is the unit vector in the direction of propagation of the light, and k = 2π / λ . Radiation pressure caused by the scattering or absorption of light by the particle is negligible compared to Fgrad, due to the low laser power and the small size of particle.35 In this calculation, the light is propagating along the z axis and is polarized along the x direction. The polarization of the light breaks the symmetry between the x and y axes, and thus the resulting forces along the x and y axes, while similar, are not identical. Equations 6 and 7 are used to provide a force field that approximates the one experienced by a particle in the FCS experiment. As will be discussed below, the simulation uses only the relative magnitudes and directions of the calculated force field, and the value of any constants that are independent of position are not relevant.
Equations 8 and 9 were used to calculate a force field for a cubic lattice of points with a spacing of (0.04)λ , where λ=633 nm is the wavelength of the trapping laser. The force field was calculated for points extending from −2λ to +2λ in the x and y directions and from −4λ to +4λ in the z direction; the origin of the system was placed at the focal point of the trapping laser beam. For those points in between the cubic lattice of points, the force was obtained by linear interpolation of the force vectors of the eight lattice points that surround it. The force on any particle outside the cubic lattice was set to zero. The spacing of the points in the lattice limits the spatial resolution of our force field to approximately (0.02)λ ≈ 13 nm, which was judged to be sufficient for these simulations.
The force field was scaled by a multiplicative factor to produce a desired well depth, U, so only the direction and relative magnitude of the force from the above calculation were needed for the biased diffusion simulations. The reversible work necessary to remove a particle from the force field by moving it along one of the coordinate axes is
where s is x, y or z, Fs(s) is the s component of the force for points on the s axis, and L is a positive constant large enough such that Fs(s) = 0 for both s > L and s < -L . The well depth, W, for the calculated force field was taken to be the average of the six reversible works calculated by integrating equation 10 along each of the three axes in both the positive and negative directions. The six numerical results agreed to within ±0.25%. The forces were then scaled by (U / W ) , where U is the desired value of the potential well depth for a given simulation. For U = 1 kBT, the resulting forces are shown in Figure 1A, and the potential energy is shown in Figure 1B. The corresponding plots for any other value of the well depth can be obtained by multiplying the numbers on the ordinate of Figures 1A and 1B by the desired value of U expressed in units of kBT.
The intensities for each configuration of beads were calculated using equations 3 and 4 and the results autocorrelated and fit as before. Simulations were performed for three different values of the well depth (U= 0, 1, and 3kBT, where kB is Boltzmann's constant and T is the temperature). For each value of U, simulations were run with N = 10 particles in the sample volume. Each simulation consisted of 1.2×109 moves and represented 4 minutes of real time. Ten simulations were performed for each value of U. The temperature for these simulations was set at 21.5 °C, which was judged to be close to the average room temperature of the laboratory where the FCS measurements were made. The viscosity of water was 0.966 cP. As before, G(τ) was calculated and fit for each simulation in the same manner as the experimental data.
The simulations with U=0 provide a check on the simulation protocol and program. Fits of the correlation function for the case U=0 yielded a best-fit value for the diffusion coefficient of D = 1.52 ± 0.01×10−8 cm2/sec, in good agreement with the theoretical value. As an additional check, simulations were also run using a single 2 nm diameter particle with U=25 kBT. For these simulations, the particle was started at the bottom of the potential well, and the well depth was large enough that the particle did not escape during the simulation. The choice of a smaller particle size in this instance was simply to speed up the equilibration of the simulation by giving the particle a larger diffusion coefficient, and so from equation 6, the simulation would use larger step sizes. So long as the particle spends most of its time near the bottom of the potential well, the equilibrium distribution should be approximately Gaussian with width
where ks is the force constant along the s axis near the origin, and equation 11 is just the expected result for a particle trapped in a harmonic potential. The x, y and z force constants for our force field equal the negative of the slope of the force versus position curves in Figure 1A near the origin. After scaling to obtain a well depth of 25 kBT, the force constants are kx = 4.93×10−3, ky = 7.53×10−3 , and kz = 7.16 ×10−4 dyne/cm. Using equation 11 we get σx = 2.86 ×10−6 , σy = 2.32 ×10−6 , and σz = 7.52 ×10−6 cm. The largest of these widths, σz , corresponds to a distance of 0.12λ, which as can be seen from Figure 1B, indicates that the particle spends most of its time near the bottom of the potential well. These predicted results can be compared with the results of the five single-particle simulations of 1.2×109 moves each, where the position of the particle was recorded every 5000 moves to produce an equilibrium distribution of particle positions within the potential well. These distributions were fit to a Gaussian, to yield best-fit widths of , and , in good agreement with the values above.
While confocal correlation spectroscopy can size particles in batch volumes like dynamic light scattering, a unique advantage lies in its ability to size both non-fluorescent and fluorescent particles in real time and in small-volume (pL-nL) samples. Accurate sizing of nanoparticles with ranging compositions and diameters depended on several factors, such as particle composition, signal intensity, background level, and laser power. Figure 2A shows burst intensities at 10 nW (488 nm) for slightly different concentrations of 60 nm diameter gold colloids, 110 nm non-fluorescent latex beads, and 110 nm fluorescent beads. In the backscattering mode, we observed strong scattering from the 60 nm gold colloids, which peaked at ~1300 counts/msec. In comparison, 110nm latex beads peaked at ~350 back-scattered photon/msec, a reflection that polystyrene beads are weaker scatterers than gold particles. Because the backscattering mode was recorded without a bandpass filter, excess Rayleigh background scattering from solution limited the sensitivity of the backscattering mode in comparison to the fluorescence mode, which had low backgrounds from Raman scattering from water and was capable of detecting single dye molecules. Also, backscattering off of the glass coverslip plagued sampling when the laser focus was placed closer than ~5 microns from the substrate. As a result, for each run, the probe volume was positioned ~25 microns above the glass surface. In the fluorescence mode, QDs of ~11 nm in diameter composing both the core and shell were readily measured. In both the backscattering and fluorescence modes, reproducibility and error were primarily caused by the quality of the sample as aggregates caused unwanted inaccuracies in the measurements. In addition, more concentrated solutions allowed for faster acquisition and more accurate measurements due to the larger number of passes that were averaged over the duration of the experiment.
With proper care in acquiring measurements, confocal correlation spectroscopy was versatile for sizing a wide range of nanoparticles (see Figure 2B). The method can accurately size particles composed of gold and polystyrene, as well as CdSe QDs and dye-doped fluorescent polystyrene nanospheres.
For many particles, measurements with low excitation powers sufficed to provide an accurate size; however, for smaller, less bright particles, low powers can pose a problem due to poor signal-to-noise ratios. Unfortunately, measured particle sizes appeared to be power dependent and were inaccurate at higher powers. First suggested by Chiu and Zare, this power dependence has been coined biased diffusion and is believed to be due to optical trapping forces imparted onto particles as they enter into the laser beam. These forces slow the trajectory of the particle through the beam and also cause a deviation in the probability that a particle will enter the beam.21 These effects have been characterized for single dye molecules; Klenerman and coworkers reported that the measured diffusion time of individual rhodamine 6G molecules increased linearly with power ranging from 300−900 μW and the probability of observing a particle deviated from Poisson statistics.22 In addition, Chirico and coworkers reported a similar linear increase in diffusion time as a function of power, but found that two-photon microscopy does not render a power dependence and could be used to overcome such artifacts.23 Several possibilities have been suggested to explain the effect of power on particle diffusion. Most notably, the gradient force induced by an optical trap can disrupt the path of a freely diffusing particle enough to slow its movement across the probe volume. At constant powers, the magnitude of this interaction depends primarily upon the polarizability of the particle,20 which is dependent upon particle volume, composition, and for fluorescent particles the number of dye molecules doped inside. In addition, the deviation has also been attributed to resonant versus non-resonant forces, which for single rhodamine 6G molecules biased diffusion is claimed to be enhanced due to excitation with a laser on resonance with the dye absorption.22 Lastly, another potential artifact results from molecular saturation of the molecules as they transit the probe volume and can indirectly affect the probe volume measurements used to determine the diffusion time.36-38
In contrast to previous reports on single molecules, for larger nanoscale particles, we found that the power dependence of the diffusion times was non-linear and different for each particle type and diameter. As shown in Figure 3, QD intensity increased linearly with power, but we observed no deviation in the measured particle with laser power (from a minimum power of 250 nW needed for detection out to 10 μW). In contrast, the 110 nm non-fluorescent latex beads, and even more so the 290 nm fluorescent beads, showed problematic deviations from their true size as power was increased from 10 nW to 10 μW. For fluorescent particles, brighter beads had a greater deviation of the diffusion time from their true values as measured at low powers. Moreover, intensity was related to volume as well as the number of dyes in the particle. Interestingly, the gold colloids showed an opposite effect with increasing power: the diffusion times decreased rather than increased. We are currently working to understand the result for the gold colloids; however, this effect may be due to possible heating of the nanoparticle due to absorption or a scattering force that pushes the colloids out of the probe volume more quickly as power is increased. Nevertheless, for accurate sizing, low laser powers can be used where the diffusion times of gold colloids do not show deviations.
A possible source of the power dependence of the apparent size and local concentration of particles is the dead time of the detector. The detector dead time, which is the time after the detection of a photon during which the detector is blind to the arrival of additional photons, has long been known to affect the observed distribution of photon counts,17,25,26 and expressions for the effect of the detector dead time on the FCS correlation function have been derived to first order as a ratio of the detector dead time to the sampling interval.17,26 Furthermore, these effects have also been investigated for FCS measurements on single dye molecules.39
The presence of a finite dead time in the detector or electronics will cause the lost of some photons. While the average count rate (Figure 3) seems modest enough given the dead time of our detector (~50 ns), when the fluorescent particle is near the center of the probe volume, the count rate, averaged over 1 ms, can be as high as 1.35 MHz with a 10 nW of excitation power. Even at the lowest power used (10 nW), the number of photons counted is reduced by approximately 7% due to the detector dead time. As the power is increased, the fraction of photons lost can become quite large. In addition, peak shapes for each burst event will increasingly deviate from those defined by the fixed excitation profile of the probe volume. Notably, the maximum intensity flattens in relation to the width of the burst, thereby resulting in the appearance that diffusion time increases as a function of power. Figure 4 shows three curves of normalized diffusion times (τD) for 110 nm diameter fluorescent beads freely diffusing and measured at varied laser powers. The uncorrected diffusion time (squares) increased non-linearly with power and fit well to simulations of 110 nm beads (circles) diffusing into and out of a probe volume of the same dimensions and with a simulated detector dead time of 50 nsec. In practice, placement of a neutral density filter in front of the detector resulted in peak intensities that did not register over ~1.5 MHz, thus causing no deviation within error for the measured particle size out to 10 μW (triangles).
Another effect that has been attributed to biased diffusion is the increase in local particle concentration around the beam focus. For brighter particles, we also found the number of particles per unit volume, or the reciprocal of the y-intercept in the autocorrelation curve, increased non-linearly with power. Figure 5A shows the effective increase in the frequency and duration of photon bursts over a given period of time. At low power (10 nW), we only saw ~30 burst events over 60 seconds. In contrast, at high power (100 μW) we recorded ~65 burst events and the bursts were significantly longer in duration. Similar to the deviation in diffusion time, dead-time artifacts from the detector also caused an apparent increase in concentration or the average number of particles contained within the probe volume. As shown in Figure 5B, the normalized number of particles per unit volume was calculated from the intercept of the correlation curve from experiments and compared to simulations that factored in the dead-time of the detector. For both diffusion times and particle concentration, without proper care and awareness this effect can be incorrectly interpreted as a biasing of diffusion of particles in solution.
Palmer and Thompson derived an expression for the FCS correlation function to first order in τdt/T , where τdt is the dead time of the detection system, and T is the sampling interval.26 Some of these correction terms to F(t) (the correlation function in the absence of the detector dead time) are proportional to τdt/T , and have the same time dependence as F(t). As a result, their inclusion will not affect the determination of the D, though they might affect the determination of G(0). Other terms, however, are proportional to τdtq/T , where q is the average number of photons per time interval, and have a different time dependence from F(t). For large count rates, τdtq can be comparable to T, and so these terms can affect the determination of both D and G(0), and our results here are in agreement with their results. However, for smaller count rates, such as for our measurements taken with 10 nW excitation power, the effect of these additional terms is expected to be negligible, as was demonstrated by our simulations that showed best-fit diffusion coefficients in good agreement with the theoretical diffusion coefficients for the lowest excitation power.
At lower powers (10 nW-10 μW), the increases in diffusion times measured with confocal correlation spectroscopy were a result of detector dead-time artifacts. At higher powers (>1 mW), however, we observed that biased diffusion does appear to occur and can be detected under both resonant and non-resonant conditions. Theoretical predictions have suggested that trapping enhancements can be achieved when using a beam that is resonant with the absorption maximum of the particles being analyzed.40 Furthermore, this resonance enhancement has been suggested as a potential way to sort selectively particles based upon the resonance wavelength.41 Yet, we found that trapping enhancement for resonant versus non-resonant forces was only slightly different for heavily dye-doped, fluorescent beads absorbing at 488 nm. For resonant conditions (Figure 6), we increased laser power at 488 nm from 10 nW to 3 mW and added neutral density filters to ensure that signal from fluorescent, 290 nm diameter beads did not peak above 1.5 MHz. For non-resonant conditions (Figure 7A), the focus of the 488 nm beam and a 632.8 nm HeNe beam were positioned in the same location. In this case, the 488 nm beam was held constant at 10 nW of power to excite fluorescence from freely diffusing beads. Power of the HeNe was increased incrementally from 10 nW to 3 mW. Within error, no change in the diffusion times of the beads was recognizable until ~1 mW for both conditions. As shown in Figure 7A, the relative contribution of longer correlation times increased while shorter times decreased. We believe the increase in longer diffusion times can be attributed to passes where the bead entered fully into the probe volume and was partially trapped along its trajectory. Because of these changes in the autocorrelation function, the 3D FCS equation used to calculate the diffusion time no longer fit the data. Interestingly, both the resonant and non-resonant condition showed a deviation in correlation times at similar powers; however, for the resonant condition longer time correlations appear to span past one second. Unfortunately, we observed saturation of fluorescence from the beads at ~100 μW, at which point background from the laser and sample decreased the signal-to-background ratio (SBR) for burst events from the 290 nm beads. As a result, the correlation curves taken in the mW range were noisy in comparison to ones obtained at lower powers, such as the 500 μW data shown in Figure 6A. Nevertheless, the observations suggest that biased diffusion does occur at high powers for both conditions.
In addition to experimental observations, we simulated the non-resonant-condition experiments (Figure 7B) to define the range of potential well depths, which were expressed in relation to kBT, experienced by the beads as they traversed the probe volume. The simulations suggest that the interaction of the laser with the beads is not noticeable until >1 kBT. At 3 kBT, the deviation observed in the simulation was similar in magnitude to the experimentally measured deviation shown for 3 mW. However, the experimental correlation function is bimodal in the presence of the trapping laser. One component exhibits a smaller decay time than the single decay time for the correlation function in the absence of the trapping laser, and one exhibits a larger decay time. These combine to produce the pronounced bulge in the correlation function in Figure 7A. In contrast, the simulated correlation functions exhibit only the larger decay time. These differences, we believe, were caused by a small misalignment between the red and blue focal spots. Experimentally, the perfect alignment of these two diffraction-limited focal volumes is challenging.
This paper characterizes confocal correlation spectroscopy for the accurate sizing of nanoparticles. At low laser powers (10 nW to 10 μW), we found high-frequency (>1.5 MHz) burst events from nanoparticles showed detector dead-time artifacts, which resulted in the appearance of a non-linear increase in diffusion times (or particle size) and particle concentrations with laser power. At higher powers (>1 mW), resonant and non-resonant trapping forces caused an increasing contribution of longer particle transit times. Here, the change in the shape of the autocorrelation curve provides a convenient way to assess whether biased diffusion is occurring. With the increasing interest in both nanotechnology and small-volume analysis and control, we believe proper use of confocal correlation spectroscopy will offer a useful way to accurately measure the sizes of nano-scale particles and macromolecular complexes.
We gratefully acknowledge support from the NIH and the Keck Foundation for support of this work, and the NSF (NSF CHE 0342956) for funding the purchase of the computer cluster used for our theoretical simulations.