Diffusion Coefficients of CdSe/CdS Quantum Rods in Water Measured Using Polarized Fluorescence Correlation Spectroscopy
 Author: Lee Jaeran, Pack ChanGi, Kim Soo Yong, Kim Sok Won
 Organization: Lee Jaeran; Pack ChanGi; Kim Soo Yong; Kim Sok Won
 Publish: Current Optics and Photonics Volume 18, Issue5, p598~604, 25 Oct 2014

ABSTRACT
A polarization fluorescence correlation spectroscopy system based on a confocal microscope was built to study the rotational and translational diffusion of CdSe/CdS quantum rods (Qrods), with the same and different polarization states between the polarizer and the analyzer (i.e. the
XXX andXYY states). The rotational diffusion amplitude showed the dependences on polarization of 0.75±0.05 in theXXX state and 0.26±0.03 in theXYY state, when the translational diffusion amplitude was 1. The diffusion coefficients of the Qrods were found based on their translational and rotational diffusion times in the two polarization states, in solutions with viscosity ranging from 0.9 to 6.9 cP. The translational and rotational diffusion coefficients ranged from 1.5×10^{11} to 2.6×10^{12} m^{2} s^{1} and from 2.9×10^{5} to 5.6×10^{4} s^{1}, respectively.

KEYWORD
Fluorescence correlation spectroscopy , Quantum rod , Photon counting

I. INTRODUCTION
Fluorescence correlation spectroscopy (FCS) was developed to study molecular diffusion dynamics and the photochemical properties of various fluorescent probes. FCS is based on the analysis of timedependent fluorescence fluctuations from diffusing fluorescent probes. Since the late 1990s FCS has progressed due to advances in confocal systems and correlators, and due to the development of detectors with high quantum efficiency. It is well known that the diffusivity of a small particle in solution depends on the size of the particle and the viscosity of the solution. In most studies employing FCS, the relative size of the particle and viscosity of the sample solution have been determined using the StokesEinstein equation, based on the diffusion coefficient of a wellknown probe in water [15]. These studies were unable to confirm particle shape, but the translational diffusion of fluorescent probes has often been investigated assuming that particles are spherical [6, 7].
The timescale of rotational diffusion for nanometersized probes introduces technical challenges, because the detectable time ranges of the correlator become expanded. However, photon antibunching and triplet formation occur on a timescale similar to that of rotational diffusion [8]. Fortunately, larger fluorescent probes (nanometer or submicrometersized fluorescent particles such as quantum dots (Qdots) and Qrods) have different time scales for antibunching and lifetime, and they have no triplet state to complicate the analysis. Quantum particles are useful as probes due to their broad absorption bands, narrow emission bands, and high quantum yields [9, 10]. In particular, Qrods of welldefined size and shape can be used as a standard to verify experimentally the diffusion properties of various rodlike molecules and particles.
In this study we built a simple confocalmicroscopebased polarization FCS system using CWlaser excitation to investigate the rotational and translational diffusion of CdSe/CdS quantum rods. By autocorrelation analyses of various combinations of polarization and viscosity, we characterized the dependence of the rotational and translational diffusion of Qrods on polarization by using the amplitude and characteristic time of the correlation function [11].
II. THEORY
In a confocal microscope system, the movements of fluorescent particles into and out of the focal volume cause the detected fluorescence to vary in intensity, and this intensity variation inherently contains information about the size, number, shape, and speed of the particles. It is difficult to confirm a large number of totaltime series data varying over less than a few milliseconds, so instead these data may be analyzed via the following correlation function:
where
δF _{1}(t ) is the fluctuation deviation in fluorescence intensity on one detector at timet ,δF _{2}(t +τ ) is the fluctuation deviation in fluorescence intensity on the other detector at time (t +τ ), and <F _{1}(t )> and <F _{2}(t )> are the average fluorescence intensities of the time series signals at each detector. In general the correlation function obtained from experimental measurements shows several exponential decays, including those due to antibunching, triplet state, and translational and rotational diffusions [8]. The wellknown correlation model equation includes three terms representing translational diffusion (G_{D} (τ )), triplet state (G_{T} (τ )), and rotational diffusion (G_{R} (τ )):This equation can be expanded as follows:
where
R is the rotational fraction,τ_{R} is the rotational diffusion time,F is the fraction of the triplet state,τ_{T} is the triplet state lifetime,G (0) is the initial amplitude of translational diffusion, andτ_{D} is the translational diffusion time. The termsτ_{R} ,τ_{T} , andτ_{D} are characteristic times corresponding to the FWHM of each region. The beam waistw and axial lengthz are related to the form of the confocal volume [12].The third term of Eq. (3) has been well established by various experiments showing that the parameters of the translational diffusion term carry quantitative information about the average particle number contained in the focal volume during
τ_{D} , the diffusion coefficient, the hydrodynamic radius of the particles, and the viscosity of the solvent as determined by the StokesEinstein equation. The average particle number is simply the reciprocal ofG (0). The second term indicating the tripletstate differs according to the presence or absence of each fluorophore. For fluorescent molecules such as rhodamine, this term varies according to background conditions [13], and this term is not present for semiconductors such as Qdots. The first term shows a rotational diffusion; in the rotational diffusion, the rotational fractionR varies according to the polarization state of the FCS system [14,15]. In FCS the rotational diffusion coefficientD_{R} and translational diffusion coefficientD_{T} are often used to compare the dynamical properties of particles, given by Eq. (4) [16].III. EXPERIMENT
3.1. Optical Setup
The FCS setup is illustrated schematically in Fig. 1. A 532nm DPSS laser (Cobolt, Samba; 25 mW) and a singlephoton counting module (SPCM; Id Quantique, id10050) were used as the excitation source and detector respectively. Alignment of the optical components was accomplished using a commercial optical microscope (Olympus IX71). The optical components in the FCS setup were a polarization beam splitter (PBS), nonpolarized beam splitter (NPBS), excitation filter (ExF), dichroic mirror (DM), objective lens (OB), cover glass (CG), emission filter (EmF), lens (L), mirror (M), and pinhole (PH).
In this setup, the laser beam from the light source passes through the PBS and enters the optical microscope. The beam then passes through the ExF (Semrock, FF01531/4025) and reflects off the DM (Semrock, FF545/650Di02). The reflected beam passes through the OB (Olympus, 60×, 1.2 NA), the immersion water, and the CG, and then focuses on a very small volume of a few fL inside the solution. The fluorescent particles in the focal volume are excited by the laser beam and then emit fluorescent light; a portion of this light enters the OB. After the light passes through the DM, EmF, and L, it escapes the microscope by the inner mirror. The light emerging from the microscope passes the PH, which is 30 μm in diameter, and the PBS. The PH and PBS are used to make a confocal conjugate point for the confocal volume, and to pass only the polarized fluorescence light, respectively. The light is split into two beams by the NPBS, and these beams are detected by two SPCMs. The fluorescent light detected at the SPCMs is converted into TTL signals, which are then converted into a correlation function by using a timecorrelated singlephoton counting system (PicoQuant, Picoharp300). The correlation function is monitored as the crosscorrelation function of the responses of the two SPCSMs, using the FluoFit program (PicoQuant) [16], the delay time of the correlation being limited to the range from 0.15 μs to 0.1 s.
3.2. Preparation of Probe Sample
To investigate the rotational diffusion of rodlike particles, we used Qrods (Qrod591) of length 35 nm and width 5 nm (aspect ratio of 7) as the fluorescent particles. SEM analysis showed that the Qrods were of uniform high quality, and TEM analysis showed that they were of uniform size (Fig. 2). The Qrods were of a core/shell configuration, with CdSe cores and CdS shells. Qrod growth is described in detail in Ref. [17]. The Qrods had a maximum emission peak at 591 nm and could be sufficiently excited by the 532nm laser beam.
To characterize the rotational diffusion of the Qrod591 particles, we diluted them to a concentration of a few nM. At this concentration the number of Qrods in the effective focal volume of our system would be less than 10. Therefore, we prepared samples of two different concentrations such that the average particle number in the focal volume would be approximately 4 (the “highconcentration” solution) or 0.5 (the “lowconcentration” solution). In addition, Qrod solutions with various viscosities were prepared, to study the dependence of rotational diffusion on viscosity, and to allow a comparison of that result to the effect of translational diffusion on viscosity. Glycerin was used to adjust the viscosity of the solutions with volume percentages of 0, 10, 20, 30, 40, and 50%; the corresponding viscosities of these solutions at 23 ℃ were 0.9, 1.2, 1.7, 2.6, 4.0, and 6.9 cP, as calculated using a formula for the viscosity of a glycerinwater mixture [18].
3.3. Combination of Polarization States
To examine the dependence of excitation and fluorescence on polarization states, two linear polarizers were used, one near the laser and one between the PH and the NPBS. The fluorescent light was split into two beams for cross correlation. The fluorescent light split by the NPBS was polarized into two states,
XXX andXYY , in which the first letterX refers to the excitation light, and the last two letters denote the polarization states parallel to the excitation light (XX ) and perpendicular to the light (YY ), respectively.3.4. Effective Focal Volume of FCS System
Before analyzing the diffusion of Qrods, we confirmed the effective focal volume of the FCS system based on polarization states using a rhodamine 6G (R6G) solution of about 10 nM concentration. At our laser intensity the correlation curve of the R6G solution revealed a triplet state and a translational diffusion state. Thus, the curve was fitted by Eq. (3) assuming the triplet state and linear translational diffusion (
a =1). The fitting parameters was obtained via Eq. (3); for all polarization states the structure parameters =z/w , triplet state timeτ_{T} , and initial amplitude of translational diffusionG (0) were found to be 4.0±0.2, 1.4±0.22 μs, and 0.57±0.03 respectively, and the translational diffusion timeτ_{D} was 43±4.7 μs for the XXX state and 43±5.0 μs for the XYY state. This result indicates that the effective focal volume (V _{eff}) of our FCS system does not depend on the polarization state. The beam waist was calculated to be 0.266 μm by Eq. (4), using the translational diffusion coefficient of R6G [19].V _{eff} was calculated from this result, using the equationV _{eff} = π^{3/2}w ^{2}z , yielding a value of 0.42 fL [1]. Additionally, to verify the performance of our FCS system we calculated the concentration of the R6G solution used in the measurements. The calculated concentration was 6 nM, which agreed fairly well with the expected value of 10 nM, with a difference of 4 nM; this indicated that the performance of the system was suitable to determine the concentration of the Qrod solution. Therefore we determined the concentrations of both the lowconcentration and highconcentration samples used in this study to be 2 and 16 nM respectively.IV. RESULTS & DISCUSSION
To investigate the polarization dependence of the rotational diffusion of Qrods, we built the polarization FCS system and measured correlation functions of the Qrods’ fluorescence responses. Measured functions were fitted by Eq. (3) to determine parameters related to the rotational and translational diffusion of the Qrods. In Eq. (3) we neglected the triplet state because it does not appear in semiconductors such as Qdots and Qrods [19]. However, because the diffusion of Qrods cannot be expected to be linear like that of Rhodamine 6G, due to the small differences in particle sizes and the irregular blinking in the full duration of the experiments, the anomalous diffusion exponent α was fitted by Eq. (3) using other parameters.
Before analyzing the influence of polarization, we examined the change in correlation function caused by varying the Qrod concentration. Figure 3 shows the correlation functions obtained from Qrod solutions of two different concentrations and the two solvent viscosities of 2.6 and 6.9 cP, in the
XXX polarization state. The correlation functions varied greatly with viscosity and sample concentration (Fig. 3(a)). In samples of viscosity 2.6 cP, the amplitude of the correlation functionG (τ ) increased as the sample concentration decreased. This result was also observed in samples of viscosity 6.9 cP, indicating thatG (τ ) is inversely proportional to the concentration of the solution. In general, the initial value of the translational diffusionG (0) is proportional to the reciprocal of the number of particles in the effective focal volume. In the 2nM samples,G (τ ) increased as the viscosity of the sample was increased from 2.6 to 6.9 Qrods’ fluorescence response, but the variation inG (τ ) was less than the concentration variation. A similar result was also obtained for the 16nM samples, but the increase inG (τ ) with viscosity was very small compared to that for the 2 nM samples. This means thatG (τ ) depends more strongly on concentration than on viscosity. AlthoughG (τ ) showed a weaker dependence on viscosity than on concentration, the variation ofG (τ ) with viscosity cannot be neglected at the 2nM concentration level.As shown in Fig. 3(a), in samples of concentration 2 nM the
G (0) values of the correlation functions for viscosities of 2.6 and 6.9 cP were 1.63 and 1.89 respectively. This result indicates that the effective focal volume was reduced as the viscosity of the solvent was increased. In both 2nM samples the concentration of Qrods was the same; therefore, the difference in viscosity was responsible for the variation inG (0). The samples of viscosity 6.9 cP used in this study included more glycerin than those of viscosity 2.6 cP; this increment of glycerin concentration increases the refractive index of the solution [20], thereby reducing the focal volume, and consequently decreasing the number of Qrods passing through the reduced focal volume. This can be regarded as a reduction of the concentration of the solution.According to the results in Fig. 3(a), the correlation functions obtained based on measurements of the Qrod solutions were affected by the solutions’ concentration and viscosity. As mentioned previously, the goal in this study is to investigate the rotational diffusion of Qrods in combination with their polarization states. To carry this out, the influences of the concentration and the viscosity in the translational diffusion of the correlation function were eliminated by using a normalized correlation function. The normalization was achieved by dividing
G (τ) byG (0), which yielded the same normalized results for the two concentrations; the normalized functions differed only between the 2.6 and 6.9cP viscosity groups (Fig. 3(b)). The function for the 6.9cP group was timedelayed relative to that for the 2.6cP group, but the normalized functions had the same amplitude in the rotational and translational diffusion regions, respectively. The functions were separated into regions corresponding to rotational and translational diffusion, described by two decay functions before and after approximately 10^{5} s. For translational diffusion the amplitudeG (0) was 1, and the amplitude of rotational diffusion was 1.75±0.05. The difference of the amplitudes of rotational diffusion and translational diffusion is the rotational fraction; in this case, the rotational fraction was 0.75±0.05. This fraction has same value as the parameterR obtained by fitting. At a viscosity of 2.6 cP, the initial value of the function was lower than the rotational fraction. This means that the rotational region begins before 0.15 μs.To analyze the rotational diffusion of Qrods in combination with the polarization state, we measured and normalized the correlation function using viscosities of 0.9, 1.2, 1.73, 2.57, 4.06, and 6.9 cP and the polarization states
XXX andXYY , and the normalized functions were fitted by Eq. (3) to obtain the parametersR ,τ_{R} ,τ_{D} , andα . For all polarization states the translational and rotational diffusion times increased as the viscosity increased. For theXXX state, the rotational diffusion time increased from 0.58±0.055 μs to 5.17±0.50 μs as the viscosity of the solvent increased from 0.9 to 6.9 cP; the diffusion time for theXYY state showed a similar trend, with the rotational diffusion time increasing from 0.42±0.074 μs to 3.27±0.22 μs as the viscosity increased in the same range. For the lowest viscosity of 0.9 cP, the diffusion times of theXXX andXYY states differed slightly, by 0.16 μs, but for the viscosity of 6.9 cP the difference was 1.90 μs. This result for the 6.9 cP viscosity may be due to nonlinear diffusion caused by irregular blinking and the small differences in size of the Qrods in the highviscosity solvent. Furthermore, the anomalous diffusion exponent α, quantifying the nonlinear diffusion of the Qrods, decreases from 0.75 to 0.61 with increasing solvent viscosity [21]. This result also may indicate that the difference in diffusion times under highviscosity conditions is due not to the polarization state but to nonlinear diffusion [22].Figure 4 illustrates the normalized
G (τ ) for viscosities of 0.9 and 2.6 cP and the two polarization states studied; as shown in the figure, the rotational fractions were 0.75±0.05 in theXXX state and 0.26±0.03 in theXYY state. The fractions in theXYY state were smaller than those in theXXX state, which might an artifact of our FCS system. The system used in this study may be regarded as having three polarizers: a linear polarizer, a rotating linear polarizer, and an analyzer. The linear polarizer is in front of the light source, the rotating linear polarizer is the Qrods in solution, and the linear analyzer is in front of the detector. The angle between the first polarizer and the last analyzer is 0° in theXXX state and 90° in theXYY state. The intensity of light passing through the three polarizers can be described by Malus’ law [23]: At 0° the maximum intensity ratio of input to output is 1, and at 90° the ratio is 0.25. According to this assumption the fraction for theXXX state would be 4 times as large as the value for theXYY state, but the observed difference was about 3 times as large, indicating that the Qrods do not act as an ideal linear polarizer. If the Qrods were rotating linear polarizers then the rotational diffusion time would also be expected to change with polarization state, but the rotational axis in Brownian rotation of the rod like particles is random at the full duration of the experiments.The rotational and translational diffusion coefficients were calculated by Eq. (4) using the diffusion time; the results are shown in Fig. 5(a) and Table 1. The results agree with the trend in diffusion time. As shown in Fig. 5a, the translational and rotational diffusion coefficients in 0.9cP samples were (1.6±0.2)×10^{11} m^{2} s^{1} and (2.9±0.4)×10^{5} s^{1} for the
XXX state, and (1.5±0.2)×10^{11} m^{2} s^{1} and (3.9±0.6)×10^{5} s^{1} for theXYY state, respectively.The relationship between
D_{R} andD_{T} was fairly linear in all polarization states (Fig. 5(b)); the linearfit values of the slopeD_{R}/D_{T} were 2.0×10^{16} m^{2} for theXXX state and 2.2×10^{16} m^{2} for theXYY state. According to previously published theories on the diffusion coefficient [22],D_{R}/D_{T} has a constant value determined by the particle’s aspect ratio and the square of the particle’s length. In the results herein, the linear fit withD_{R} =0 andD_{T} =0 gave a better result for theXXX state than for theXYY state.V. CONCLUSIONS
We built a polarized fluorescence correlation spectroscopy system based on a confocal microscope, and measured and analyzed the correlation functions of CdSe/CdS Qrods in solution for various concentrations and viscosities, and for two polarization states. In our analysis of the correlation functions with respect to polarization states, translational diffusion did not depend on the polarization state, whereas rotational diffusion depended strongly on polarization. The rotational fraction was 0.75±0.05 in the
XXX state and 0.26±0.03 in theXYY state, and the rotational diffusion time in theXXX state was similar to that in theXYY state. We determined the diffusion coefficients based on the relation between the rotational and translational diffusion coefficients. The translational and rotational diffusion coefficients respectively ranged from 1.5×10^{11} m^{2} s^{1} to 2.6×10^{12} m^{2} s^{1} and from 2.9×10^{5} s^{1} to 5.6×10^{4} s^{1}. Our results can be used to determine the size of rodlike particles, or to determine the rotational diffusion coefficients of particles of other sizes [25].

17. Deka S., Quarta A., Lupo M. G., Falqui A., Boninelli S., Giannini C., Morello G., Giorgi M. D., Lanzni G., Spinella C., Cinglani R., Pellegrino T., Manna L. 2009 “CdSe/CdS/ZnS double shell nanorods with high photoluminescence efficiency and their exploitation as biolabeling probes,” [J. Am. Chem. Soc.] Vol.131 P.29482958

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[FIG. 1.] The polarization FCS setup, based on an optical microscope.

[FIG. 2.] (a) SEM and (b) TEM micrographs of 5 nm × 35 nm (aspect ratio: 7) CdSe/CdS Qrods.

[FIG. 3.] (a) Fluorescence correlation functions for two viscosities and two particle concentrations; (b) normalized function of (a).

[FIG. 4.] Normalized fluorescence correlation functions for two viscosities and both the XXX and XYY states. Symbols and solid lines respectively denote measured and fitted data.

[TABLE 1.] Translational and rotational diffusion coefficients for various viscosities

[FIG. 5.] (a) Effects of viscosity on translational and rotational diffusion coefficients. (b) Rotational diffusion coefficient versus translational diffusion coefficient for the XXX and XYY states.